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Why Stock Markets Crash
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Why Stock Markets Crash
Critical Events in Complex
Financial Systems
D i d i e r S o r n e t t e
Princeton and Oxford
Copyright ? 2003 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 3 Market
Place, Woodstock, Oxfordshire OX20 1SY
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Sornette, D.
Why stock markets crash: critical events in complex
financial systems/Didier Sornette.
p. cm.
Includes bibliographical references and index.
ISBN 0-691-09630-9 (alk. paper)

  1. Financial crises—History. 2. Stocks—Prices—History.
  2. Financial crises—United States—History.
  3. Stock exchanges—United States—History.
  4. Critical phenomena (Physics). 6. Complexity (Philosophy).
    I. Title.
    HB3722.S66 2002
    332.63�222–dc21 2002024336
    British Library Cataloging-in-Publication Data is available
    This book has been composed in Times
    Printed on acid-free paper. �
    Printed in the United States of America
    10 9 8 7 6 5 4 3 2 1
    xiii Preface
    Chapter 1
    financial crashes:
    what, how, why,
    and when?
    3 What Are Crashes, and Why
    Do We Care?
    5 The Crash of October 1987
    7 Historical Crashes
    7 The Tulip Mania
    9 The South Sea Bubble
    12 The Great Crash of October 1929
    15 Extreme Events in Complex Systems
    20 Is Prediction Possible?
    A Working Hypothesis
    Chapter 2
    fundamentals of
    financial markets
    27 The Basics
    27 Price Trajectories
    30 Return Trajectories
    33 Return Distributions and
    Return Correlation
    38 The Efficient Market Hypothesis and
    the Random Walk
    38 The Random Walk
    vi contents
    42 A Parable: How Information Is
    Incorporated in Prices, Thus
    Destroying Potential “Free
    45 Prices Are Unpredictable,
    or Are They?
    47 Risk–Return Trade-Off
    Chapter 3
    financial crashes
    are “outliers”
    49 What Are “Abnormal” Returns?
    51 Drawdowns (Runs)
    51 Definition of Drawdowns
    54 Drawdowns and the Detection of
    56 Expected Distribution of “Normal”
    60 Drawdown Distributions of Stock
    Market Indices
    60 The Dow Jones Industrial Average
    62 The Nasdaq Composite Index
    65 Further Tests
    69 The Presence of Outliers Is a
    General Phenomenon
    70 Main Stock Market Indices,
    Currencies, and Gold
    73 Largest U.S. Companies
    75 Synthesis
    76 Symmetry-Breaking on Crash
    and Rally Days
    77 Implications for Safety Regulations of
    Stock Markets
    Chapter 4
    positive feedbacks
    82 Feedbacks and Self-Organization
    in Economics
    89 Hedging Derivatives, Insurance
    Portfolios, and Rational Panics
    91 “Herd” Behavior and “Crowd” Effect
    91 Behavioral Economics
    contents vii
    94 Herding
    96 Empirical Evidence of Financial
    Analysts’ Herding
    99 Forces of Imitation
    99 It Is Optimal to Imitate When
    Lacking Information
    104 Mimetic Contagion and the
    Urn Models
    106 Imitation from Evolutionary
    108 Rumors
    111 The Survival of the Fittest Idea
    112 Gambling Spirits
    114 “Anti-Imitation” and Self-Organization
    114 Why It May Pay to Be
    in the Minority
    115 El-Farol’s Bar Problem
    117 Minority Games
    118 Imitation versus Contrarian Behavior
    121 Cooperative Behaviors Resulting
    from Imitation
    122 The Ising Model of Cooperative
    130 Complex Evolutionary Adaptive
    Systems of Boundedly
    Rational Agents
    Chapter 5
    modeling financial
    bubbles and
    market crashes
    134 What Is a Model?
    135 Strategy for Model Construction
    in Finance
    135 Basic Principles
    136 The Principle of Absence of
    Arbitrage Opportunity
    137 Existence of Rational Agents
    139 “Rational Bubbles” and Goldstone
    Modes of the Price “Parity
    Symmetry” Breaking
    140 Price Parity Symmetry
    viii contents
    144 Speculation as Spontaneous
    Symmetry Breaking
    148 Basic Ingredients of the Two Models
    150 The Risk-Driven Model
    150 Summary of the Main Properties of
    the Model
    152 The Crash Hazard Rate Drives the
    Market Price
    155 Imitation and Herding Drive the
    Crash Hazard Rate
    162 The Price-Driven Model
    162 Imitation and Herding Drive the
    Market Price
    164 The Price Return Drives the Crash
    Hazard Rate
    168 Risk-Driven versus Price-Driven
    Chapter 6
    complex fractal
    dimensions, and
    173 Critical Phenomena by Imitation on
    Hierarchical Networks
    173 The Underlying Hierarchical
    Structure of Social Networks
    177 Critical Behavior in Hierarchical
    181 A Hierarchical Model of
    Financial Bubbles
    186 Origin of Log-Periodicity in
    Hierarchical Systems
    186 Discrete Scale Invariance
    188 Fractal Dimensions
    192 Organization Scale by Scale: The
    Renormalization Group
    192 Principle and Illustration of the
    Renormalization Group
    195 The Fractal Weierstrass Function:
    A Singular Time-Dependent
    Solution of the Renormalization
    contents ix
    198 Complex Fractal Dimensions and
    208 Importance and Usefulness of
    Discrete Scale Invariance
    208 Existence of Relevant
    Length Scales
    209 Prediction
    210 Scenarios Leading to Discrete Scale
    Invariance and Log-Periodicity
    211 Newcomb–Benford Law of First
    Digits and the Arithmetic System
    213 The Log-Periodic Law of the
    Evolution of Life?
    217 Nonlinear Trend-Following
    versus Nonlinear Fundamental
    Analysis Dynamics
    218 Trend Following: Positive Nonlinear
    Feedback and Finite-Time
    220 Reversal to the Fundamental Value:
    Negative Nonlinear Feedback
    223 Some Characteristics of the Price
    Dynamics of the Nonlinear
    Dynamical Model
    Chapter 7
    autopsy of major
    crashes: universal
    exponents and logperiodicity
    228 The Crash of October 1987
    231 Precursory Pattern
    236 Aftershock Patterns
    239 The Crash of October 1929
    242 The Three Hong Kong Crashes of 1987,
    1994, and 1997
    242 The Hong Kong Crashes
    246 The Crash of October 1997 and Its
    Resonance on the U.S. Market
    254 Currency Crashes
    259 The Crash of August 1998
    263 Nonparametric Test of Log-Periodicity
    x contents
    266 The Slow Crash of 1962 Ending the
    “Tronics” Boom
    269 The Nasdaq Crash of April 2000
    275 “Antibubbles”
    276 The “Bearish” Regime on the Nikkei
    Starting from January 1, 1990
    278 The Gold Deflation Price Starting in
    279 Synthesis: “Emergent” Behavior of the
    Stock Market
    Chapter 8
    bubbles, crises,
    and crashes in
    emergent markets
    281 Speculative Bubbles in
    Emerging Markets
    285 Methodology
    286 Latin-American Markets
    295 Asian Markets
    304 The Russian Stock Market
    309 Correlations across Markets: Economic
    Contagion and Synchronization of
    Bubble Collapse
    314 Implications for Mitigations of Crises
    Chapter 9
    prediction of
    bubbles, crashes,
    and antibubbles
    320 The Nature of Predictions
    325 How to Develop and Interpret
    Statistical Tests of Log-Periodicity
    329 First Guidelines for Prediction
    329 What Is the Predictive Power of
    Equation (15)?
    330 How Long Prior to a Crash Can
    One Identify the Log-Periodic
    334 A Hierarchy of Prediction Schemes
    334 The Simple Power Law
    335 The “Linear” Log-Periodic Formula
    contents xi
    336 The “Nonlinear” Log-Periodic
    336 The Shank’s Transformation on a
    Hierarchy of Characteristic Times
    337 Application to the October 1929
    338 Application to the October 1987
    338 Forward Predictions
    339 Successful Prediction of the Nikkei
    1999 Antibubble
    342 Successful Prediction of the Nasdaq
    Crash of April 2000
    342 The U.S. Market, December 1997
    False Alarm
    346 The U.S. Market, October 1999
    False Alarm
    346 Present Status of Forward Predictions
    346 The Finite Probability That No
    Crash Will Occur during a Bubble
    347 Estimation of the Statistical
    Significance of the Forward
    347 Statistical Confidence of the
    Crash “Roulette”
    349 Statistical Significance of a Single
    Successful Prediction via
    Bayes’s Theorem
    351 The Error Diagram and the
    Decision Process
    352 Practical Implications on Different
    Trading Strategies
    Chapter 10
    2050: the end of
    the growth era?
    355 Stock Markets, Economics,
    and Population
    357 The Pessimistic Viewpoint of
    “Natural” Scientists
    359 The Optimistic Viewpoint of
    “Social” Scientists
    xii contents
    361 Analysis of the Faster-Than-
    Exponential Growth of Population,
    GDP, and Financial Indices
    369 Refinements of the Analysis
    369 Complex Power Law Singularities
    371 Prediction for the Coming Decade
    377 The Aging “Baby Boomers”
    378 Related Works and Evidence
    383 Scenarios for the “Singularity”
    384 Collapse
    389 Transition to Sustainability
    393 Resuming Accelerating Growth by
    Overpassing Fundamental
    395 The Increasing Propensity to Emulate
    the Stock Market Approach
    397 References
    419 Index
    Like many other people, I find the stock market
    fascinating. The market’s potential for lavish gains and its playful
    character, made more attractive with the recent advent of the Internet,
    resonates with the gambler in us. Its punishing power and unpredictable
    temper make fearful investors look at it sometimes with awe, particularly
    at times of crashes. Stories of panic and suicides following such
    events have become part of market folklore. The richness of the patterns
    the stock market displays may lure investors into hoping to “beat the
    market” by using or extracting some bits of informative hedge.
    However, the stock market is not a “casino” of playful or foolish
    gamblers. It is, primarily, the vehicle of fluid exchanges allowing the
    efficient function of capitalistic, competitive free markets.
    As shown in Figure 0.1 and Table 0.1, the total world market capitalization
    rose from $3.38 trillion (thousand billions) in 1983 to $26.5
    trillion in 1998 and to $38.7 trillion in 1999. To put these numbers in
    perspective, the 1999 U.S. budget was $1.7 trillion, while its 1983 budget
    was $800 billion. The 2002 U.S. budget is projected to be $1.9 trillion.
    Market capitalization and trading volumes tripled during the 1990s. The
    volume of securities issued was multiplied by 6. Privatization has played
    a key role in the stock market growth [51]. Stock market investment is
    clearly the biggest game in town.
    A market crash occurring simultaneously on most of the stock markets
    of the world as witnessed in October 1987 would amount to the
    quasi-instantaneous evaporation of trillions of dollars. In values of
    xiv preface
    $US Trillion
    Other developed
    United Kingdom
    1986 1989 1992 1995
    1999 2000
    Fig. 0.1. Gross value of the world market capitalization from 1983 to 2000. From
    top to bottom, the developing countries are shown as the top strip, other developed
    countries (excluding the United States, Japan, and the United Kingdom), the
    United Kingdom, Japan, and the United States as the bottom strip. One trillion is
    equal by definition to one thousand billion or one million million. Reproduced with
    authorization from Boutchkova and Megginson [51].
    October 2001, after almost two dismal years for stocks, the total world
    market capitalization has shrunk to a mere $25.1 trillion. A stock market
    crash of 30% would still correspond to an absolute loss of about $7.5
    trillion dollars. Market crashes can thus swallow years of pensions and
    savings in an instant. Could they make us suffer even more by being
    the precursors or triggering factors of major recessions, as in 1929–33
    after the great crash of October 1929? Or could they lead to a general
    Table 0.1
    The growth of world stock market trading volumes (1983–1998) (value traded in billions
    of U.S. dollars)
    Countries 1983 1989 1995 1998 1999
    Developed countries 1203 6297 9170 20917 35188
    United States 797 2016 5109 13148 19993
    Japan 231 2801 1232 949 1892
    United Kingdom 43 320 510 1167 3399
    Developing countries 25 1171 1047 1957 2321
    Total world 1228 7468 10216 22874 37509
    Note the Japan bubble that culminated at the end of 1990: around this time, the trading volume
    on Japanese stock markets topped that of the U.S. market! The bubble started to deflate beginning
    in 1990 and has lost more than 60% of its value. Also remarkable is the fact that the market trading
    volume of the United States is now more than half the world trading volume, while it was less than
    a third of it in 1989.
    Reproduced with authorization from Boutchkova and Megginson [51].
    preface xv
    collapse of the financial and banking system, as seems to have been
    barely avoided several times in the not-so-distant past?
    Stock market crashes are also fascinating because they personify the
    class of phenomena known as “extreme events.” Extreme events are
    characteristic of many natural and social systems, often refered to by
    scientists as “complex systems.”
    This book is a story, a scientific tale of how financial crashes can
    be understood by invoking the latest and most sophisticated concepts in
    modern science, that is, the theory of complex systems and of critical
    phenomena. It is written first for the curious and intelligent layperson
    as well as for the interested investor who would like to exercise more
    control over his or her investments. The book will also be stimulating for
    scientists and researchers who are interested in or working on the theory
    of complex systems. The task is ambitious. My aim is to cover a territory
    that brings us all the way from the description of how the wonderful
    organization around us arises to the holy grail of crash predictions. This
    is daunting, especially as I have attempted to avoid the technical, if
    convenient, language of mathematics.
    At one level, stock market crashes provide an excuse for exploring the
    wonderful world of self-organizing systems. Market crashes exemplify
    in a dramatic way the spontaneous emergence of extreme events in selforganizing
    systems. Stock market crashes are indeed perfect vehicles for
    important ideas needed to deal and cope with our risky world. Here,
    “world” is taken with several meanings, as it can be the physical world,
    the natural world, the biological, and even the inner intellectual and
    psychological worlds. Uncertainties and variabilities are the key words
    to describe the ever-changing environments around us. Stasis and equilibrium
    are illusions, whereas dynamics and out-of-equilibrium are the
    rule. The quest for balance and constancy will always be unsuccessful.
    The message here goes further and proclaims the essential importance of
    recognizing the organizing/disorganizing role of extreme events, such as
    momentous financial crashes. In addition to the obvious societal impacts,
    the guideline underlying this book recognizes that sudden transitions
    from a quiescent state to a crisis or catastrophic event provide the most
    dramatic fingerprints of the system dynamics. We live on a planet and in
    a society with intermittent dynamics rather than at rest (or “equilibrium”
    in the jargon of scientists), and so there is a growing and urgent need
    to sensitize citizens to the importance and impacts of ruptures in their
    multiple forms. Financial crashes provide an exceptionally good example
    for introducing these concepts in a way that transcends the disciplinary
    community of scholars.
    xvi preface
    At another level, market crashes constitute beautiful examples of
    events that we would all like to forecast. The arrow of time is inexorably
    projecting us toward the undetermined future. Predicting the future
    captures the imagination of all and is perhaps the greatest challenge.
    Prophets have historically terrified or inspired the masses by their visions
    of the future. Science has mostly avoided this question by focusing on
    another kind of prediction, that of novel phenomena (rather than that
    of the future) such as the prediction by Einstein of the existence of
    the deviation of light by the sun’s gravitation field. Here, I do not shy
    away from this extraordinary challenge, with the aim of showing how a
    scientific approach to this question provides remarkable insights.
    The book is organized in 10 chapters. The first six chapters provide
    the background for understanding why and how large financial crashes
    Chapter 1 introduces the fundamental questions: What are crashes?
    How do they happen? Why do they occur? When do they occur?
    Chapter 1 outlines the answers I propose, taking as examples some
    famous, or shall I say infamous, historical crashes.
    Chapter 2 presents the key basic descriptions and properties of stock
    markets and of the way prices vary from one instant to the next. This
    frames the landscape in which the main characters of my story, the great
    crashes, are acting.
    Chapter 3 discusses first the limitation of standard analyses for characterizing
    how crashes are special. It then presents the study of the
    frequency distribution of drawdowns, or runs of successive losses, and
    shows that large financial crashes are “outliers”: they form a class of
    their own that can be seen from their statistical signatures. This rather
    academic discussion is justified by the result: If large financial crashes
    are “outliers,” they are special and thus require a special explanation, a
    specific model, a theory of their own. In addition, their special properties
    may perhaps be used for their prediction.
    Chapter 4 exposes the main mechanisms leading to positive feedbacks,
    that is, self-reinforcement, such as imitative behavior and herding
    between investors. Positive feedbacks provide the fuel for the development
    of speculative bubbles, preparing the instability for a major crash.
    Chapter 5 presents two versions of a rational model of speculative
    bubbles and crashes. The first version posits that the crash hazard drives
    the market price. The crash hazard may skyrocket sometimes due to the
    collective behavior of “noise traders,” those who act on little information,
    even if they think they “know.” The second version inverts the logic and
    preface xvii
    posits that prices drive the crash hazard. Prices may skyrocket sometimes,
    again due to the speculative or imitative behavior of investors.
    According to the rational expectation model, this outcome automatically
    entails a corresponding increase of the probability for a crash. The most
    important message is the discovery of robust and universal signatures
    of the approach to crashes. These precursory patterns have been documented
    for essentially all crashes on developed as well as emergent stock
    markets, on currency markets, on company stocks, and so on.
    Chapter 6 takes a step back and presents the general concept of fractals,
    of self-similarity, and of fractals with complex dimensions and their
    associated discrete self-similarity. Chapter 6 shows how these remarkable
    geometric and mathematical objects enable one to codify the information
    contained in the precursory patterns before large crashes.
    The last four chapters document this discovery at great length and
    demonstrate how to use this insight and the detailled predictions obtained
    for these models to forecast crashes.
    Chapter 7 analyzes the major crashes that have occurred on the major
    stock markets of the world. It describes the empirical evidence of the universal
    nature of the critical log-periodic precursory signature of crashes.
    Chapter 8 generalizes this analysis to emergent markets, including six
    Latin-American stock market indices (Argentina, Brazil, Chile, Mexico,
    Peru, and Venezuela) and six Asian stock market indices (Hong Kong,
    Indonesia, Korea, Malaysia, Philippines, and Thailand). It also discusses
    the existence of intermittent and strong correlation between markets following
    major international events.
    Chapter 9 explains how to predict crashes as well as other large market
    events and examines in detail forecasting skills and their limitations,
    in particular in terms of the horizon of visibility and expected precision.
    Several case studies are presented in detail, with a careful count
    of successes and failures. Chapter 9 also presents the concept of an
    “antibubble,” with the Japanese collapse from the beginning of 1990 to
    the present taken as a prominent example. A prediction issued and advertised
    in January 1999 has been until now borne out with remarkable
    precision, correctly predicting several changes of trends, a feat notoriously
    difficult using standard techniques of economic forecasting.
    Finally, chapter 10 performs a major leap by extending the analysis to
    time scales covering centuries to millenia. It analyzes the whole of U.S.
    financial history as well as the world economy and population dynamics
    over the last two millenia to demonstrate the existence of strong positive
    feedbacks that suggest the existence of an underlying finite-time
    singularity around 2050, signaling a fundamental change of regime of
    xviii preface
    the world economy and population around 2050 (a super crash?). We are
    probably starting to see signatures of this change of regime. I offer three
    leading scenarios: collapse, transition to sustainability, and superhumans.
    The text is complemented by technical inserts that sometimes use a
    little mathematics and can be skipped on first or fast reading. They are
    offered as supplements that go deeper into an argument or as useful
    additional information. Many figures accompany the text, in keeping with
    the proverb that a picture is worth a thousand words.
    The story told in this book has an unusual origin. Its roots go all
    the way back, starting in the sixties, to the pioneering scientists, such
    as Ben Widom (professor at Cornell University), Leo Kadanoff (now
    professor at the University of Chicago), Michael Fisher (now professor
    at the University of Maryland), Kenneth Wilson (now professor at Ohio
    State University and the 1982 Nobel prize winner in physics), and many
    others who explored and established the theory of critical phenomena in
    natural sciences. I am indebted to Pierre-Gilles de Gennes (College de
    France and the 1991 Nobel prize winner in physics) and Bernard Souillard
    (then a director of research of the Ecole Polytechnique in Palaiseau,
    at the French CNRS-National Center of Scientific Research), for a most
    stimulating year (1985–86) in Paris as their postdoctoral fellow, where
    I started to learn to polish the art of thinking about critical phenomena
    and to apply this field to the most complex situations. I also cherish the
    remarkable opportunity of broadening my vision of scientific applications
    offered by the collaboration with Michel Lagier of Thomson-Sintra
    Inc. (now Thomson-Marconi-Sonars, Inc.), which began in 1983 during
    my military duty and continues to this day. His unfailing friendship and
    kind support over the last two decades have meant a lot to me.
    In 1991, while working on the exciting challenge of predicting the
    failure of pressure tanks made of Kevlar-matrix and carbon-matrix
    composites constituting essential elements of the European Ariane 4
    and 5 rockets and also used in satellites for propulsion, I realized that
    the rupture of complex material structures could be understood as a
    cooperative phenomenon leading to specific detectable critical behaviors
    (see chapters 4 and 5 for the applications of these concepts to
    financial crashes). The power laws and associated complex exponents
    and log-periodic patterns that I shall discuss in this book, in particular
    in chapter 6, were discovered in this context and found to perform
    remarkably well. A prediction algorithm has been patented and is now
    been used routinely with success in Europe on these pressure tanks
    going into space as a standard qualifying procedure. I am indebted to
    Jean-Charles Anifrani (now with Eurocopter, Inc.) and Christian Le
    preface xix
    Floc’h of the company Aerospatiale-Matra (now EADS) in Bordeaux,
    France (the leader contractor for the European Ariane rocket) for a
    stimulating collaboration and for providing this fantastic opportunity.
    A few years later, Anders Johansen, Jean-Philippe Bouchaud, and I
    realized that financial crashes can be viewed as analogous to “ruptures”
    of the market. Anders Johansen and I started to explore systematically
    the application of these ideas and methods in this context. What followed
    is described in this book. In this adventure, Johansen, now at the Niels
    Bohr Institute in Copenhagen, has played a very special role, as he has
    accompanied me first as my student in Nice, France for two years and
    then as my postdoc for two years at the University of California, Los
    Angeles. A significant portion of this work owes much to him, as he
    has implemented a large part of the data analysis of our joint work. I
    am very pleased for having shared these exciting times with him, when
    we seemed alone against all, trying to document and demonstrate this
    discovery. The situation has now evolved, as the subject is attracting an
    increasing number of scholars and even more professionals and practitioners,
    and there is a healthy debate characteristic of a lively subject,
    associated in particular with the delicate and touchy question of the predictability
    of crashes (more in chapters 9 and 10). I hope that this book
    will help in this respect.
    I also acknowledge the fruitful and inspiring discussions and collaborations
    with Jorgen V. Andersen, now jointly at University of Nanterre,
    Paris and University of Nice, France, who is now working with me
    on an extension of the models of bubbles and crashes described in
    chapter 5. I should also mention Olivier Ledoit, then at the Anderson
    School of Management at UCLA. The first model of rational bubbles
    and crashes described in chapter 5 owes a lot to our discussions and
    work together. Other close collaborators, such as Simon Gluzman, Kayo
    Ide, and Wei-Xing Zhou at UCLA, are joining in the research with me
    on the modeling of financial markets and crashes. I must also single out
    for mention Dietrich Stauffer of Cologne University, Germany, who has
    played a key role as editor of several international scholarly journals in
    helping our iconoclastic papers to be reviewed and published. Witty, concise
    to the extreme, straightforward, and with a strong sense of humor,
    Stauffer has been very supportive and helpful. He has also been an independent
    witness to the prediction on the Japanese Nikkei stock market
    described in chapter 9.
    I am also grateful to Yueqiang Huang at the University of Southern
    California, Per J?gi and Matt W. Lee at UCLA, Laurent Nottale of
    the Observatoire Paris-Meudon, Guy Ouillon at the University of Nice,
    xx preface
    and Hubert Saleur and Charlie Sammis at the University of Southern
    California for stimulating interactions and discussions on the theory and
    practice of log-periodicity. I am indebted to Vladilen Pisarenko of the
    International Institute of Earthquake Prediction Theory and Mathematical
    Geophysics in Moscow, who provided much advice and numerous
    insights on the science and art of statistical testing. I am grateful to Bill
    Megginson at the University of Oklahoma for help in getting access to
    data on the world market capitalization. Cars Hommes, at the Center
    for Nonlinear Dynamics in Economics and Finance at the University of
    Amsterdam, and Neil Johnson at Oxford University, U.K., acted as referees
    on a preliminary version of the book. I thank them warmly for their
    kind and constructive advice. I thank Jorgen Andersen and Paul O’Brien
    for a critical reading of the manuscript. I met Joseph Wisnovsky, the
    executive editor of Princeton University Press, at a conference of the
    American Geophyical Union in San Francisco in December 2000. From
    the start, his enthusiasm and support has been an essential help in crystallizing
    this project. Wei-Xing Zhou helped a lot in preparing the fractal
    spiral picture on the cover, and Beth Gallagher performed a very careful
    and much appreciated job in correcting the manuscript.
    I gratefully acknowledge the 2000 award from the program of the
    James S. McDonnell Foundation entitled “Studying Complex Systems.”
    Last but not least, I am grateful for the support of the French National
    Center for Scientific Research (CNRS) since 1981, which has ensured
    complete freedom for my research in France and abroad. Since 1996,
    the Institute of Geophysics and Planetary Physics and the Department of
    Earth and Space Sciences at UCLA has provided new scientific opportunities
    and collaborations as well as support.
    I hope that at least some of the joy, excitement, and wonder I have
    enjoyed during this research will be shared by readers.
    Didier Sornette
    Los Angeles and Nice
    December 2001
    Why Stock Markets Crash
    This page intentionally left blank
    chapter 1
    financial crashes: what,
    how, why, and when?
    Stock market crashes are momentous financial
    events that are fascinating to academics and practitioners alike. According
    to the academic world view that markets are efficient, only the revelation
    of a dramatic piece of information can cause a crash, yet in reality
    even the most thorough post-mortem analyses are typically inconclusive
    as to what this piece of information might have been. For traders and
    investors, the fear of a crash is a perpetual source of stress, and the onset
    of the event itself always ruins the lives of some of them.
    Most approaches to explaining crashes search for possible mechanisms
    or effects that operate at very short time scales (hours, days, or
    weeks at most). This book proposes a radically different view: the underlying
    cause of the crash will be found in the preceding months and
    years, in the progressively increasing build-up of market cooperativity, or
    effective interactions between investors, often translated into accelerating
    ascent of the market price (the bubble). According to this “critical” point
    of view, the specific manner by which prices collapsed is not the most
    important problem: a crash occurs because the market has entered an
    unstable phase and any small disturbance or process may have triggered
    4 chapter 1
    the instability. Think of a ruler held up vertically on your finger: this
    very unstable position will lead eventually to its collapse, as a result
    of a small (or an absence of adequate) motion of your hand or due to
    any tiny whiff of air. The collapse is fundamentally due to the unstable
    position; the instantaneous cause of the collapse is secondary. In the
    same vein, the growth of the sensitivity and the growing instability of
    the market close to such a critical point might explain why attempts to
    unravel the local origin of the crash have been so diverse. Essentially,
    anything would work once the system is ripe. This book explores the
    concept that a crash has fundamentally an endogenous, or internal, origin
    and that exogenous, or external, shocks only serve as triggering factors.
    As a consequence, the origin of crashes is much more subtle than often
    thought, as it is constructed progressively by the market as a whole, as
    a self-organizing process. In this sense, the true cause of a crash could
    be termed a systemic instability.
    Systemic instabilities are of great concern to governments, central
    banks, and regulatory agencies [103]. The question that often arose in
    the 1990s was whether the new, globalized, information technology–
    driven economy had advanced to the point of outgrowing the set of rules
    dating from the 1950s, in effect creating the need for a new rule set for
    the “New Economy.” Those who make this call basically point to the
    systemic instabilities since 1997 (or even back to Mexico’s peso crisis
    of 1994) as evidence that the old post–World War II rule set is now
    antiquated, thus condemning this second great period of globalization
    to the same fate as the first. With the global economy appearing so
    fragile sometimes, how big a disruption would be needed to throw a
    wrench into the world’s financial machinery? One of the leading moral
    authorities, the Basle Committee on Banking Supervision, advised [32]
    that, “in handling systemic issues, it will be necessary to address, on the
    one hand, risks to confidence in the financial system and contagion to
    otherwise sound institutions, and, on the other hand, the need to minimise
    the distortion to market signals and discipline.”
    The dynamics of confidence and of contagion and decision making
    based on imperfect information are indeed at the core of the book and
    will lead us to examine the following questions. What are the mechanisms
    underlying crashes? Can we forecast crashes? Could we control
    them? Or, at least, could we have some influence on them? Do
    crashes point to the existence of a fundamental instability in the world
    financial structure? What could be changed to modify or suppress these
    financial crashes: what, why, and when? 5
    From the market opening on October 14, 1987 through the market close
    on October 19, major indexes of market valuation in the United States
    declined by 30% or more. Furthermore, all major world markets declined
    substantially that month, which is itself an exceptional fact that contrasts
    with the usual modest correlations of returns across countries and the
    fact that stock markets around the world are amazingly diverse in their
    organization [30].
    In local currency units, the minimum decline was in Austria �?11�4%�
    and the maximum was in Hong Kong �?45�8%�. Out of 23 major
    industrial countries (Autralia, Austria, Belgium, Canada, Denmark,
    France, Germany, Hong Kong, Ireland, Italy, Japan, Malaysia, Mexico,
    the Netherlands, New Zealand, Norway, Singapore, South Africa, Spain,
    Sweden, Switzerland, United Kingdom, United States), 19 had a decline
    greater than 20%. Contrary to common belief, the United States was not
    the first to decline sharply. Non-Japanese Asian markets began a severe
    decline on October 19, 1987, their time, and this decline was echoed
    first on a number of European markets, then in North American, and
    finally in Japan. However, most of the same markets had experienced
    significant but less severe declines in the latter part of the previous week.
    With the exception of the United States and Canada, other markets
    continued downward through the end of October, and some of these
    declines were as large as the great crash on October 19.
    A lot of work has been carried out to unravel the origin(s) of the crash,
    notably in the properties of trading and the structure of markets; however,
    no clear cause has been singled out. It is noteworthy that the strong
    market decline during October 1987 followed what for many countries
    had been an unprecedented market increase during the first nine months
    of the year and even before. In the U.S. market, for instance, stock prices
    advanced 31.4% over those nine months. Some commentators have suggested
    that the real cause of October’s decline was that overinflated
    prices generated a speculative bubble during the earlier period.
    The main explanations people have come up with are the following.
  5. Computer trading. In computer trading, also known as program trading,
    computers were programmed to automatically order large stock
    trades when certain market trends prevailed, in particular sell orders
    after losses. However, during the 1987 U.S. crash, other stock markets
    6 chapter 1
    that did not use program trading also crashed, some with losses even
    more severe than the U.S. market.
  6. Derivative securities. Index futures and derivative securities have been
    claimed to increase the variability, risk, and uncertainty of the U.S.
    stock markets. Nevertheless, none of these techniques or practices
    existed in previous large, sudden market declines in 1914, 1929, and
  7. Illiquidity. During the crash, the large flow of sell orders could not be
    digested by the trading mechanisms of existing financial markets. Many
    common stocks in the New York Stock Exchange were not traded until
    late in the morning of October 19 because the specialists could not find
    enough buyers to purchase the amount of stocks that sellers wanted
    to get rid of at certain prices. This insufficient liquidity may have had
    a significant effect on the size of the price drop, since investors had
    overestimated the amount of liquidity. However, negative news about
    the liquidity of stock markets cannot explain why so many people
    decided to sell stock at the same time.
  8. Trade and budget deficits. The third quarter of 1987 had the largest
    U.S. trade deficit since 1960, which together with the budget deficit, led
    investors into thinking that these deficits would cause a fall of the U.S.
    stocks compared with foreign securities. However, if the large U.S.
    budget deficit was the cause, why did stock markets in other countries
    crash as well? Presumably, if unexpected changes in the trade deficit
    are bad news for one country, they should be good news for its trading
  9. Overvaluation. Many analysts agree that stock prices were overvalued
    in September 1987. While the price/earning ratio and
    the price/dividend ratio were at historically high levels, similar
    price/earning and price/dividends values had been seen for most of the
    1960–72 period over which no crash occurred. Overvaluation does not
    seem to trigger crashes every time.
    Other cited potential causes involve the auction system itself, the
    presence or absence of limits on price movements, regulated margin
    requirements, off-market and off-hours trading (continuous auction and
    automated quotations), the presence or absence of floor brokers who
    conduct trades but are not permitted to invest on their own account,
    the extent of trading in the cash market versus the forward market, the
    identity of traders (i.e., institutions such as banks or specialized trading
    firms), the significance of transaction taxes, and other factors.
    financial crashes: what, why, and when? 7
    More rigorous and systematic analyses on univariate associations
    and multiple regressions of these various factors conclude that it is not
    at all clear what caused the crash [30]. The most precise statement,
    albeit somewhat self-referencial, is that the most statistically significant
    explanatory variable in the October crash can be ascribed to the normal
    response of each country’s stock market to a worldwide market
    motion. A world market index was thus constructed [30] by equally
    weighting the local currency indexes of the 23 major industrial countries
    mentioned above and normalized to 100 on September 30. It fell to
    73.6 by October 30. The important result is that it was found to be
    statistically related to monthly returns in every country during the period
    from the beginning of 1981 until the month before the crash, albeit
    with a wildly varying magnitude of the responses across countries [30].
    This correlation was found to swamp the influence of the institutional
    market characteristics. This signals the possible existence of a subtle
    but nonetheless influential worldwide cooperativity at times preceding
    In the financial world, risk, reward, and catastrophe come in irregular
    cycles witnessed by every generation. Greed, hubris, and systemic fluctuations
    have given us the tulip mania, the South Sea bubble, the land
    booms in the 1920s and 1980s, the U.S. stock market and great crash in
    1929, and the October 1987 crash, to name just a few of the hundreds
    of ready examples [454].
    The Tulip Mania
    The years of tulip speculation fell within a period of great prosperity
    in the republic of the Netherlands. Between 1585 and 1650, Amsterdam
    became the chief commercial emporium, the center of the trade of the
    northwestern part of Europe, owing to the growing commercial activity in
    newly discovered America. The tulip as a cultivated flower was imported
    into western Europe from Turkey and it is first mentioned around 1554.
    The scarcity of tulips and their beautiful colors made them a must for
    members of the upper classes of society (see Figure 1.1).
    During the build-up of the tulip market, the participants were not
    making money through the actual process of production. Tulips acted
    8 chapter 1
    Fig. 1.1. A variety of tulip (the Viceroy) whose bulb was one of the most expensive
    at the time of the tulip mania in Amsterdam, from The Tulip Book of P. Cos, including
    weights and prices from the years of speculative tulip mania (1637); Wageningen
    UR Library, Special Collections.
    financial crashes: what, why, and when? 9
    as the medium of speculation and their price determined the wealth of
    participants in the tulip business. It is not clear whether the build-up
    attracted new investment or new investment fueled the build-up, or both.
    What is known is that as the build-up continued, more and more people
    were roped into investing their hard-won earnings. The price of the tulip
    lost all correlation to its comparative value with other goods or services.
    What we now call the “tulip mania” of the seventeenth century was
    the “sure thing” investment during the period from the mid-1500s to
  10. Before its devastating end in 1637, those who bought tulips rarely
    lost money. People became too confident that this “sure thing” would
    always make them money and, at the period’s peak, the participants
    mortgaged their houses and businesses to trade tulips. The craze was
    so overwhelming that some tulip bulbs of a rare variety sold for the
    equivalent of a few tens of thousands of dollars. Before the crash, any
    suggestion that the price of tulips was irrational was dismissed by all the
    The conditions now generally associated with the first period of a
    boom were all present: an increasing currency, a new economy with
    novel colonial possibilities, and an increasingly prosperous country
    together had created the optimistic atmosphere in which booms are said
    to grow.
    The crisis came unexpectedly. On February 4, 1637, the possibility
    of the tulips becoming definitely unsalable was mentioned for the first
    time. From then until the end of May 1637, all attempts at coordination
    among florists, bulbgrowers, and the Netherlands were met with failure.
    Bulbs worth tens of thousands of U.S. dollars (in present value) in early
    1637 became valueless a few months later. This remarkable event is often
    discussed by present-day commentators, and parallels are drawn with
    modern speculation mania. The question is asked, Do the tulip market’s
    build-up and its subsequent crash have any relevance for today’s markets?
    The South Sea Bubble
    The South Sea bubble is the name given to the enthusiatic speculative
    fervor that ended in the first great stock market crash in England, in
    1720 [454]. The South Sea bubble is a fascinating story of mass hysteria,
    political corruption, and public upheaval. (See Figure 1.2.) It is
    really a collection of thousands of stories, tracing the personal fortunes
    of countless individuals who rode the wave of stock speculation for a
    furious six months in 1720. The “bubble year,” as it is called, actually
    10 chapter 1
    involves several individual bubbles, as all kinds of fraudulent joint-stock
    companies sought to take advantage of the mania for speculation. The
    following account borrows from “The Bubble Project” [60].
    In 1711, the South Sea Company was given a monopoly of all trade to
    the South Sea ports. The real prize was the anticipated trade that would
    open up with the rich Spanish colonies in South America. In return for
    this monopoly, the South Sea Company would assume a portion of the
    national debt that England had incurred during the War of the Spanish
    Succession. When Britain and Spain officially went to war again in 1718,
    the immediate prospects for any benefits from trade to South America
    Fig. 1.2. An emblematical print of the South Sea scene (etching and engraving), by
    the artist William Hogarth in 1722 (now located at The Charles Deering McCormick
    Library of Special Collections, Northwestern University). With this scene, Hogarth
    satirizes crowds consumed by political speculation on the verge of the stock market
    collapse of 1720. The “merry-go-round” was set in motion by the South Sea Company,
    who held a monopoly on trade between South America, the Pacific Islands,
    and England. The Company tempted vast numbers of middle-class investors to make
    quick money through absurd speculations. The wheel of fortune in the center of
    the print is broken, symbolizing the abandonment of values for quick money, while
    “Trade” lies starving to death. On the right, the original inscription on the London
    Fire Monument—erected in memory of the destruction of the City by the Great Fire
    in 1666—has been altered to read: “This monument was erected in memory of the
    destruction of the city by the South Sea in 1720.” Reproduced by permission from
    McCormick Library of Special Collections, Northwestern University Library.
    financial crashes: what, why, and when? 11
    were nil. What mattered to speculators, however, were future prospects,
    and here it could always be argued that incredible prosperity lay ahead
    and would be realized when open hostilities came to an end.
    The early 1700s was also a time of international finance. By 1719
    the South Sea directors wished, in a sense, to imitate the manipulation
    of public credit that John Law had achieved in France with the
    Mississippi Company, which was given a monopoly of French trade to
    North America. Law had connived to drive the price of its stock up, and
    the South Sea directors hoped to do the same. In 1719 the South Sea
    directors made a proposal to assume the entire public debt of the British
    government. On April 12, 1720 this offer was accepted. The company
    immediately started to drive the price of the stock up through artificial
    means; these largely took the form of new subscriptions combined
    with the circulation of pro-trade-with-Spain stories designed to give the
    impression that the stock could only go higher. Not only did capital
    stay in England, but many Dutch investors bought South Sea stock, thus
    increasing the inflationary pressure.
    South Sea stock rose steadily from January through the spring. As
    every apparent success would soon attract its imitators, all kinds of jointstock
    companies suddenly appeared, hoping to cash in on the speculation
    mania. Some of these companies were legitimate, but the bulk were
    bogus schemes designed to take advantage of the credulity of the people.
    Several of the bubbles, both large and small, had some overseas trade
    or “New World” aspect. In addition to the South Sea and Mississippi
    ventures, there was a project for improving the Greenland fishery and
    another for importing walnut trees from Virginia. Raising capital by selling
    stock in these enterprises was apparently easy work. The projects
    mentioned so far all have a tangible specificity at least on paper, if not
    in practice; others were rather vague on details but big on promise. The
    most remarkable was “a company for carrying on an undertaking of
    great advantage, but nobody to know what it is.” The prospectus stated
    that “the required capital was half a million, in five thousand shares of
    100 pounds each, deposit 2 pounds per share. Each subscriber, paying
    his [or her] desposit, was entitled to 100 pounds per annum per share.
    How this immense profit was to be obtained, [the proposer] did not
    condescend to inform [the buyers] at that time” [60]. As T. J. Dunning
    [114] wrote:
    Capital eschews no profit, or very small profit � � � . With adequate profit,
    capital is very bold. A certain 1 percent will ensure its employment
    anywhere; 20 percent certain will produce eagerness; 50 percent, positive
    12 chapter 1
    audacity; 100 percent will make it ready to trample on all human laws;
    300 percent and there is not a crime at which it will scruple, nor a risk it
    will not run, even to the chance of its owner being hanged.
    Next morning, at nine o’clock, this great man opened an office in
    Cornhill. Crowds of people beset his door, and when he shut up at three
    o’clock, he found that no less than one thousand shares had been subscribed
    for, and the deposits paid. He was thus, in five hours, the winner
    of £2,000. He was philosophical enough to be contented with his venture,
    and set off the same evening for the Continent. He was never heard
    of again.
    Such scams were bad for the speculation business and so, largely
    through the pressure of the South Sea directors, the so-called “Bubble
    Act” was passed on June 11, 1720 requiring all joint-stock companies
    to have a royal charter. For a moment, the confidence of the people was
    given an extra boost, and they responded accordingly. South Sea stock
    had been at £175 at the end of February, 380 at the end of March, and
    around 520 by May 29. It peaked at the end of June at over £1,000
    (a psychological barrier in that four-digit number).
    With credulity now stretched to the limit and rumors of more and more
    people (including the directors themselves) selling off, the bubble then
    burst according to a slow but steady deflation (not unlike the 60% drop of
    the Japanese Nikkei index after its all-time peak at the end of December
    1989). By mid-August, the bankruptcy listings in the London Gazette
    reached an all-time high, an indication that many people had bought on
    credit or margin. Thousands of fortunes were lost, both large and small.
    The directors attempted to pump up more speculation. They failed. The
    full collapse came by the end of September, when the stock stood at
    £135. The crash remained in the consciousness of the Western world for
    the rest of the eighteenth century, not unlike our cultural memory of the
    1929 Wall Street Crash.
    The Great Crash of October 1929
    The Roaring 20s—a time of growth and prosperity on Wall Street
    and Main Street—ended with the Great Crash of October 1929 (for
    the most thorough and authoritative account and analysis, see [152]).
    (See Figure 1.3.) The Great Depression that followed put 13 million
    Americans out of work. Two thousand investment firms went under, and
    the American banking industry underwent the biggest structural changes
    financial crashes: what, why, and when? 13
    Fig. 1.3. The front page of the October 30, 1929 New York Times exclaimed the
    massive loss on Wall Street. It worked hard to ease fear among panicked investors—
    without success, as history has shown.
    of its history, as a new era of government regulation began. Roosevelt’s
    New Deal politics would follow.
    The October 1929 crash is a vivid illustration of several remarkable
    features often associated with crashes. First, stock market crashes
    are often unforeseen for most people, especially economists. “In a few
    months, I expect to see the stock market much higher than today.”
    Those words were pronounced by Irving Fisher, America’s distinguished
    and famous economist and professor of economics at Yale University,
    14 days before Wall Street crashed on Black Tuesday, October 29, 1929.
    14 chapter 1
    “A severe depression such as 1920–21 is outside the range of probability.
    We are not facing a protracted liquidation.” This was the analysis
    offered days after the crash by the Harvard Economic Society to
    its subscribers. After continuous and erroneous optimistic forecasts, the
    society closed its doors in 1932. Thus, the two most renowned economic
    forecasting institutes in America at the time failed to predict that
    crash and depression were forthcoming and continued with their optimistic
    views, even as the Great Depression took hold of America. The
    reason is simple: the prediction of trend-reversals constitutes by far the
    most difficult challenge posed to forecasters and is very unreliable, especially
    within the linear framework of standard (auto-regressive) economic
    A second general feature exemplified by the October 1929 event is that
    a financial collapse has never happened when things look bad. On the
    contrary, macroeconomic flows look good before crashes. Before every
    collapse, economists say the economy is in the best of all worlds. Everything
    looks rosy, stock markets go up and up, and macroeconomic flows
    (output, employment, etc.) appear to be improving further and further.
    This explains why a crash catches most people, especially economists,
    totally by surprise. The good times are invariably extrapolated linearly
    into the future. Is it not perceived as senseless by most people in a time
    of general euphoria to talk about crash and depression?
    During the build-up phase of a bubble such as the one preceding the
    October 1929 crash, there is a growing interest in the public for the commodity
    in question, whether it consists of stocks, diamonds, or coins.
    That interest can be estimated through different indicators: an increase in
    the number of books published on the topic (see Figure 1.4) and in the
    subscriptions to specialized journals. Moreover, the well-known empirical
    rule according to which the volume of sales is growing during a
    bull market, as shown in Figure 1.5, finds a natural interpretation: sales
    increases in fact reveal and pinpoint the progress of the bubble’s diffusion
    throughout society. These features have been recently reexamined
    for evidence of a bubble, a “fad” or “herding” behavior, by studying
    individual stock returns [455]. One story often advanced for the boom
    of 1928 and 1929 is that it was driven by the entry into the market of
    largely uninformed investors, who followed the fortunes of and invested
    in “favorite” stocks. The result of this behavior would be a tendency for
    the favorite stocks’ prices to move together more than would be predicted
    by their shared fundamental economic values. The co-movement
    indeed increased significantly during the boom and was a signal characteristic
    of the tumultuous market of the early 1930s. These results are
    financial crashes: what, why, and when? 15
    Number of Titles Containing
    Stocks, Stock Market, Speculation
    r = 0.58
    1915 1920 1925 1930
    1935 1940
    Stock Prices
    Fig. 1.4. Comparison between the number of yearly published books about stock
    market speculation and the level of stock prices (1911–1940). Solid line: Books at
    Harvard’s library whose titles contain one of the words “stocks,” “stock market,” or
    “speculation”. Broken line: Standard and Poor’s index of common stocks. The curve
    of published books lags behind the price curve with a time-lag of about 1.5 years,
    which can be explained by the time needed for a book to get published. Source:
    The stock price index is taken from the Historical Abstract of the United States.
    Reproduced from [349].
    thus consistent with the possibility that a fad or crowd psychology played
    a role in the rise of the market, its crash, and subsequent volatility [455].
    The political mood before the October 1929 crash was also optimistic.
    In November 1928, Herbert Hoover was elected president of the United
    States in a landslide, and his election set off the greatest increase in
    stock buying to that date. Less than a year after the election, Wall Street
    Financial markets are not the only systems with extreme events. Financial
    markets constitute one among many other systems exhibiting a complex
    organization and dynamics with similar behavior. Systems with a large
    number of mutually interacting parts, often open to their environment,
    self-organize their internal structure and their dynamics with novel and
    sometimes surprising macroscopic (“emergent”) properties. The complex
    16 chapter 1
    Volume of Sales on the NYSE (mllion shares)
    NYSE closed for 3 months
    1900 1905 1910 1915
    1920 1925 1930 1935 1940
    Fig. 1.5. Comparison between the number of shares traded on the NYSE and the
    level of stock prices (1897–1940). Solid line: Number of shares traded. Broken line:
    Deflated Standard and Poor’s index of common stocks. Source: Historical Statistics
    of the United States. Reproduced from [349].
    system approach, which involves “seeing” interconnections and relationships,
    that is, the whole picture as well as the component parts, is nowadays
    pervasive in modern control of engineering devices and business
    management. It also plays an increasing role in most of the scientific
    disciplines, including biology (biological networks, ecology, evolution,
    origin of life, immunology, neurobiology, molecular biology, etc.), geology
    (plate-tectonics, earthquakes and volcanoes, erosion and landscapes,
    climate and weather, environment, etc.), and the economic and social
    sciences (cognition, distributed learning, interacting agents, etc.). There
    is a growing recognition that progress in most of these disciplines, in
    many of the pressing issues for our future welfare as well as for the
    management of our everyday life, will need such a systemic complex
    system and multidisciplinary approach. This view tends to replace the
    previous “analytical” approach, consisting of decomposing a system in
    components, such that the detailed understanding of each component was
    believed to bring understanding of the functioning of the whole.
    A central property of a complex system is the possible occurrence
    of coherent large-scale collective behaviors with a very rich structure,
    resulting from the repeated nonlinear interactions among its constituents:
    the whole turns out to be much more than the sum of its parts. It is
    financial crashes: what, why, and when? 17
    widely believed that most complex systems are not amenable to mathematical,
    analytic descriptions and can be explored only by means of
    “numerical experiments.” In the context of the mathematics of algorithmic
    complexity [73], many complex systems are said to be computationally
    irreducible; that is, the only way to decide about their evolution
    is to actually let them evolve in time. Accordingly, the “dynamical”
    future time evolution of complex systems would be inherently unpredictable.
    This unpredictability does not, however, prevent the application
    of the scientific method to the prediction of novel phenomena as exemplified
    by many famous cases (the prediction of the planet Neptune by
    Leverrier from calculations of perturbations in the orbit of Uranus, the
    prediction by Einstein of the deviation of light by the sun’s gravitation
    field, the prediction of the helical structure of the DNA molecule by
    Watson and Crick based on earlier predictions by Pauling and Bragg,
    etc.). In contrast, it refers to the impossibility of satisfying the quest
    for the knowledge of what tomorrow will be made of, often filled by
    the vision of “prophets” who have historically inspired or terrified the
    The view that complex systems are unpredictable has recently been
    defended persuasively in concrete prediction applications, such as the
    socially important issue of earthquake prediction (see the contributions
    in [312]). In addition to the persistent failures at reaching a reliable
    earthquake predictive scheme, this view is rooted theoretically in the
    analogy between earthquakes and self-organized criticality [26]. In this
    “fractal” framework (see chapter 6), there is no characteristic scale, and
    the power-law distribution of earthquake sizes reflects the fact that the
    large earthquakes are nothing but small earthquakes that did not stop.
    They are thus unpredictable because their nucleation is not different from
    that of the multitude of small earthquakes, which obviously cannot all
    be predicted.
    Does this really hold for all features of complex systems? Take our
    personal life. We are not really interested in knowing in advance at what
    time we will go to a given store or drive to a highway. We are much more
    interested in forecasting the major bifurcations ahead of us, involving
    the few important things, like health, love, and work, that count for
    our happiness. Similarly, predicting the detailed evolution of complex
    systems has no real value, and the fact that we are taught that it is
    out of reach from a fundamental point of view does not exclude the
    more interesting possibility of predicting phases of evolutions of complex
    systems that really count, like the extreme events.
    18 chapter 1
    It turns out that most complex systems in natural and social sciences
    do exhibit rare and sudden transitions that occur over time intervals that
    are short compared to the characteristic time scales of their posterior evolution.
    Such extreme events express more than anything else the underlying
    “forces” usually hidden by almost perfect balance and thus provide
    the potential for a better scientific understanding of complex systems.
    These crises have fundamental societal impacts and range from large
    natural catastrophes, such as earthquakes, volcanic eruptions, hurricanes
    and tornadoes, landslides, avalanches, lightning strikes, meteorite/asteroid
    impacts (see Figure 1.6), and catastrophic events of environmental degradation,
    to the failure of engineering structures, crashes in the stock
    market, social unrest leading to large-scale strikes and upheaval, economic
    drawdowns on national and global scales, regional power blackouts,
    traffic gridlock, and diseases and epidemics. It is essential to realize
    Fig. 1.6. One of the most fearsome possible catastrophic events, but one with very
    low probability of occurring. A collision with a meteorite with a diameter of 15 km
    with impact velocity of 14 km/s (releasing about the same energy, equal to 100
    Megatons of equivalent TNT, as what is thought to be the dinosaur killer) occurs
    roughly once every 100 million years. A collision with a meteorite with a diameter
    of the order of 1,000 km as shown in this figure occurred only early in the solar
    system’s history. (Creation of the space artist Don Davis.)
    financial crashes: what, why, and when? 19
    that the long-term behavior of these complex systems is often controlled
    in large part by these rare catastrophic events: the universe was probably
    born during an extreme explosion (the “big bang”); the nucleosynthesis
    of all important heavy atomic elements constituting our matter results
    from the colossal explosion of supernovae (stars more heavy than our
    sun whose internal nuclear combustion diverges at the end of their life);
    the largest earthquake in California, repeating about once every two centuries,
    accounts for a significant fraction of the total tectonic deformation;
    landscapes are more shaped by the “millenium” flood that moves
    large boulders than by the action of all other eroding agents; the largest
    volcanic eruptions lead to major topographic changes as well as severe
    climatic disruptions; according to some contemporary views, evolution is
    probably characterized by phases of quasi-stasis interrupted by episodic
    bursts of activity and destruction [168, 169]; financial crashes, which can
    destroy in an instant trillions of dollars, loom over and shape the psychological
    state of investors; political crises and revolutions shape the
    long-term geopolitical landscape; even our personal life is shaped in the
    long run by a few key decisions or happenings.
    The outstanding scientific question is thus how such large-scale patterns
    of catastrophic nature might evolve from a series of interactions
    on the smallest and increasingly larger scales. In complex systems, it
    has been found that the organization of spatial and temporal correlations
    do not stem, in general, from a nucleation phase diffusing across the
    system. It results rather from a progressive and more global cooperative
    process occurring over the whole system by repetitive interactions. For
    instance, scientific and technical discoveries are often quasi-simultaneous
    in several laboratories in different parts of the world, signaling the global
    nature of the maturing process.
    Standard models and simulations of scenarios of extreme events are
    subject to numerous sources of error, each of which may have a negative
    impact on the validity of the predictions [232]. Some of the uncertainties
    are under control in the modeling process; they usually involve trade-offs
    between a more faithful description and manageable calculations. Other
    sources of error are beyond control, as they are inherent in the modeling
    methodology of the specific disciplines. The two known strategies for
    modeling are both limited in this respect: analytical theoretical predictions
    are out of reach for most complex problems. Brute force numerical
    resolution of the equations (when they are known) or of scenarios is reliable
    in the “center of the distribution,” that is, in the regime far from the
    extremes where good statistics can be accumulated. Crises are extreme
    events that occur rarely, albeit with extraordinary impact, and are thus
    20 chapter 1
    completely undersampled and poorly constrained. Even the introduction
    of “teraflop” supercomputers does not qualitatively change this fundamental
    Notwithstanding these limitations, I believe that the progress of science
    and of its multidisciplinary enterprises makes the time ripe for
    a full-fledged effort toward the prediction of complex systems. In particular,
    novel approaches are possible for modeling and predicting certain
    catastrophic events or “ruptures,” that is, sudden transitions from
    a quiescent state to a crisis or catastrophic event [393]. Such ruptures
    involve interactions between structures at many different scales. In the
    present book, I apply these ideas to one of the most dramatic events
    in social sciences, financial crashes. The approach described in this
    book combines ideas and tools from mathematics, physics, engineering,
    and the social sciences to identify and classify possible universal structures
    that occur at different scales and to develop application-specific
    methodologies for using these structures for the prediction of the financial
    “crises.” Of special interest will be the study of the premonitory processes
    before financial crashes or “bubble” corrections in the stock market.
    For this purpose, I shall describe a new set of computational methods
    that are capable of searching and comparing patterns, simultaneously
    and iteratively, at multiple scales in hierarchical systems. I shall
    use these patterns to improve the understanding of the dynamical state
    before and after a financial crash and to enhance the statistical modeling
    of social hierarchical systems with the goal of developing reliable
    forecasting skills for these large-scale financial crashes.
    With the low of 3227 on April 17, 2000, identified as the end of the
    “crash,” the Nasdaq Composite index lost in five weeks over 37% of
    its all-time high of 5133 reached on March 10, 2000. This crash has
    not been followed by a recovery, as occurred from the October 1987
    crash. At the time of writing, the Nasdaq Composite index bottomed at
    1395.8 on September 21, 2001, in a succession of descending waves.
    The Nasdaq Composite consists mainly of stock related to the so-called
    “New Economy,” that is, the Internet, software, computer hardware,
    telecommunications, and similar sectors. A main characteristic of these
    companies is that their price–earning ratios (P/Es), and even more so
    their price–dividend ratios, often come in three digits. Some, such as
    VA LINUX, actually have a negative earning/share (of ?1�68). Yet they
    financial crashes: what, why, and when? 21
    are traded at around $40 per share, which is close to the price of a
    share of Ford in early March 2000. In constrast, so-called “Old Economy”
    companies, such as Ford, General Motors, and DaimlerChrysler,
    have P/E ≈ 10. The difference between Old Economy and New Economy
    stocks is thus the expectation of future earnings as discussed in
    [282] (see also [395] for a new view on speculative pricing): investors
    expect an enormous increase in, for example, the sale of Internet and
    computer-related products rather than of cars and are hence more willing
    to invest in Cisco rather than in Ford, notwithstanding the fact that the
    earning per share of the former is much smaller than for the latter. For a
    similar price per share (approximately $60 for Cisco and $55 for Ford),
    the earning per share in 1999 was $0.37 for Cisco compared with $6.00
    for Ford. Close to its apex on April 14, 2000, Cisco had a total market
    capitalization of $395 billion compared with $63 billion for Ford. Cisco
    has since bottomed at about $11 in September 2001 and traded at around
    $20 at the end of 2001.
    In the standard fundamental valuation formula, in which the expected
    return of a company is the sum of the dividend return and of the growth
    rate, New Economy companies are supposed to compensate for their
    lack of present earnings by a fantastic potential growth. In essence, this
    means that the bull market observed in the Nasdaq in 1997–2000 is
    fueled by expectations of increasing future earnings rather than economic
    fundamentals: the price-to-dividend ratio for a company such as Lucent
    Technologies (LU) with a capitalization of over $300 billion prior to its
    crash on January 5, 2000 (see Figure 1.7) is over 900, which means
    that you get a higher return on your checking account (!) unless the
    price of the stock increases. In constrast, an Old Economy company such
    as DaimlerChrysler gives a return that is more than 30 times higher.
    Nevertheless, the shares of Lucent Technologies rose by more than 40%
    during 1999, whereas the share of DaimlerChrysler declined by more
    than 40% in the same period. Recent crashes of IBM, LU, and Procter &
    Gamble (P&G), shown in Figures 1.7–1.9 correspond to a loss equivalent
    to the national budget of many countries! And this is usually attributed to
    a “business-as-usual” corporate statement of a slightly revised smallerthan-
    expected earnings!
    These considerations suggest that the expectation of future earnings
    (and its perception by others), rather than present economic reality, is an
    important motivation for the average investor. The inflated price may be
    a speculative bubble if the growth expectations are unrealistic (which is,
    of course, easy to tell in hindsight but not obvious at all in the heat of
    the action!). As already alluded to, history provides many examples of
    22 chapter 1
    Fig. 1.7. Top panel: Time series of daily closes and volume of the IBM stock over
    a one-year period around the large drop of October 21, 1999. The time of the crash
    can be seen clearly as coinciding with the peak in volume (bottom panel). Taken
    from http://finance.yahoo.com/.
    bubbles driven by unrealistic expectations of future earnings followed by
    crashes [454]. The same basic ingredients are found repeatedly: fueled
    by initially well-founded economic fundamentals, investors develop a
    self-fulfilling enthusiasm from an imitative process or crowd behavior
    that leads to the building of “castles in the air,” to paraphrase Burton
    Malkiel [282]. Furthermore, the causes of the crashes on the U.S. markets
    in October 1929, October 1987, August 1998, and April 2000 belong
    to the same category, the difference being mainly in which sector the
    bubble was created. In 1929, it was utilities; in 1987, the bubble was
    supported by a general deregulation of the market, with many new private
    investors entering the market with very high expectations about the
    profit they would make; in 1998, it was an enormous expectation for
    the investment opportunities in Russia that collapsed; until early 2000,
    it was the extremely high expectations for the Internet, telecommunications,
    and similar sectors that fueled the bubble. The IPOs (initial
    public offerings) of many Internet and software companies have been followed
    by a mad frenzy, where the share price has soared during the first
    few hours of trading. An excellent example is VA LINUX SYSTEMS
    whose $30 IPO price increased a record 697% to close at $239�25 on its
    financial crashes: what, why, and when? 23
    Fig. 1.8. Top panel: Time series of daily closes and volume of the Lucent Technology
    stock over a one-year period around the large drop of January 6, 2000. The
    time of the crash can be seen clearly as coinciding with the peak in volume (bottom
    panel). Taken from http://finance.yahoo.com/.
    opening day December 9, 1999, only to decline to $28�94 on April 14,

Building on these insights, our hypothesis is that stock market crashes
are caused by the slow build-up of long-range correlations leading to
a global cooperative behavior of the market and eventually ending in a
collapse in a short, critical time interval. The use of the word “critical”
is not purely literary here: in mathematical terms, complex dynamical
systems can go through so-called critical points, defined as the explosion
to infinity of a normally well-behaved quantity. As a matter of fact, as
far as nonlinear dynamical systems go, the existence of critical points is
more the rule than the exception. Given the puzzling and violent nature
of stock market crashes, it is worth investigating whether there could
possibly be a link between stock market crashes and critical points.
� Our key assumption is that a crash may be caused by local selfreinforcing
imitation between traders. This self-reinforcing imitation
process leads to the blossoming of a bubble. If the tendency for traders
to “imitate” their “friends” increases up to a certain point called the
“critical” point, many traders may place the same order (sell) at the
same time, thus causing a crash. The interplay between the progressive
strengthening of imitation and the ubiquity of noise requires a probabilistic
description: a crash is not a certain outcome of the bubble but
24 chapter 1
Fig. 1.9. Top panel: Time series of daily closes and volume of the Procter & Gamble
stock over a one-year period ending after the large drop of March 7, 2000. The
time of the crash can be seen clearly as coinciding with the peak in volume (bottom
panel). Taken from http://finance.yahoo.com/.
can be characterized by its hazard rate, that is, the probability per unit
time that the crash will happen in the next instant, provided it has not
happened yet.
� Since the crash is not a certain deterministic outcome of the bubble,
it remains rational for investors to remain in the market provided they
are compensated by a higher rate of growth of the bubble for taking
the risk of a crash, because there is a finite probability of “landing
smoothly,” that is, of attaining the end of the bubble without crash.
In a series of research articles performed in collaboration with several
colleagues and mainly with Anders Johansen, we have shown extensive
evidence that the build-up of bubbles manifests itself as an overall superexponential
power-law acceleration in the price decorated by log-periodic
precursors, a concept related to fractals, as will become clear later (see
chapter 6). In telling this story, this book will address the following
questions: Why and how do these precursors occur? What do they mean?
What do they imply with respect to prediction?
My colleagues and I claim that there is a degree of predictive skill
associated with these patterns, which has already been used in practice
and has been investigated by us as well as many others, academics and,
financial crashes: what, why, and when? 25
most-of-all, practitioners. The evidence I discuss in what follows arises
from many crashes, including
� the October 1929 Wall Street crash, the October 1987 World crash, the
October 1987 Hong Kong crash, the August 1998 World crash, and
the April 2000 Nasdaq crash;
� the 1985 foreign exchange event on the U.S. dollar and the correction
of the U.S. dollar against the Canadian dollar and the Japanese Yen
starting in August 1998;
� the bubble on the Russian market and its ensuing collapse in 1997–98;
� 22 significant bubbles followed by large crashes or by severe corrections
in the Argentinian, Brazilian, Chilean, Mexican, Peruvian,
Venezuelan, Hong-Kong, Indonesian, Korean, Malaysian, Philippine,
and Thai stock markets.
In all these cases, it has been found that, with very few exceptions,
log-periodic power-laws adequately describe speculative bubbles on the
Western markets as well as on the emerging markets.
Notwithstanding the drastic differences in epochs and contexts, I shall
show that these financial crashes share a common underlying background
as well as structure. The rationale for this rather surprising result is
probably rooted in the fact that humans are endowed with basically the
same emotional and rational qualities in the twenty-first century as they
were in the seventeenth century (or at any other epoch). Humans are
still essentially driven by at least a modicum of greed and fear in their
quest for well-being. The “universal” structures I am going to uncover
in this book may be understood as the robust emergent properties of the
market resulting from some characteristic “rules” of interaction between
investors. These interactions can change in details due, for instance, to
computers and electronic communications. They have not changed at a
qualitative level. As we shall see, complex system theory allows us to
account for this robustness.
chapter 2
fundamentals of
financial markets
Notwithstanding the drama surrounding crashes,
there is a growing body of scholarly work suggesting that they are part
of the family of usual daily price variations; this view, which is rooted
theoretically in some branches of the theory of complex systems, posits
that there is no characteristic scale in stock market price fluctuations
[287]. As a consequence, the very large price drops (crashes) are nothing
but small drops that did not stop [26]. According to this view, since
crashes belong to the same family as the rest of the returns we observe
on normal days, they should be inherently unpredictable because their
nucleation is not different from that of the multitude of small losses
which obviously cannot be predicted at all.
In chapter 3, we examine in detail whether this really holds for the
very largest crashes. In particular, we shall provide strong evidence that
large crashes are in fact in a league of their own: they are “outliers.”
This realization will call for new explanations and hence may suggest a
possibility of predictability. In order to reach this surprising conclusion,
we first need to recall some basic facts about the distribution (also called
the frequency) of price variations or of price returns and their respective
correlation. To this end, we first present the standard view about
price variations and returns on the stock market. A simple toy model
will illustrate why arbitrage opportunities (the possibility to get a “free
lunch”) are usually washed out by the intelligent investment of informed
traders, leading to the concept of the efficient stock market. We shall
fundamentals of financial markets 27
then test this concept in the next chapter, by studying the distribution of
drawdowns, that is, runs of losses over several days, demonstrating that
the largest drawdowns, the crashes (fast or slow), belong to a class of
their own.
Price Trajectories
Stock market prices show changes at all time scales. From the time
scale of “ticks” to that of centuries, prices embroider their complex trajectories.
A tick is the price increment from the last to the next trade,
separated typically by a few seconds or less for major stocks in active
markets. The minimum tick is the smallest increment for which stock
prices can be quoted. Figure 2.1 shows monthly quotes of the Dow Jones
Industrial Average (DJIA) from 1790 to 2000. The great crash of October
1929 followed by the great depression is the most striking pattern
in this figure. In contrast, on this long time scale the crash of October
1987 is barely visible as a small glitch between the two vertical lines.
What is the Dow Jones Industrial Average? The DJIA is an index of
30 “blue-chip” U.S. stocks. It is the oldest continuing U.S. market index.
It is called an “average” because it was originally computed by adding up
stock prices and dividing by the number of stocks (the very first average
price of industrial stocks, on May 26, 1896, was 40.94) and should
ideally represent a correct measure of the state of the economy. The
methodology remains the same today, but the divisor has been changed
to preserve historical continuity. The editors of The Wall Street Journal
select the components of the industrial average by taking a broad view
of what “industrial” means. The most recent changes in the components
of the DJIA occurred Monday, November 1, 1999, when Home Depot
Inc., Intel Corp., Microsoft Corp., and SBC Communications replaced
Union Carbide Corp. (in the DJIA since 1928), Goodyear Tire & Rubber
Co. (in the DJIA since 1930), Sears, Roebuck & Co. (in the DJIA since
1924), and Chevron (in the DJIA since 1984). The previous change
occurred in March 7, 1997, when Hewlett-Packard, Johnson & Johnson,
Traveller’s Group (Now Citigroup), and Wal-Mart Stores replaced
Woolworth, Westinghouse Electric, Texaco and Bethlehem Steel. The
components of the Dow Jones Averages are daily listed on page C3
of the Money and Investing section in The Wall Street Journal. See
http://averages.DowJones.com/about.html. The Dow Jones index shown
28 chapter 2
1800 1840 1880 1920 1960 2000
Dow Jones Industrial Average Jan 1790–Sept 2000
Close Prices
Fig. 2.1. Monthly quotes of the DJIA from September 2000 extrapolated back to
January 1790. The vertical axis uses logarithmic scales such that multiplication
by a fixed factor, for instance 10, corresponds to addition of a constant in this
representation. Mathematically, this corresponds to a mapping from multiplication
to addition and allows us to show on the same graph prices that have changed by
factors of thousands (in the present case, from a value of about 3 in 1790 to a
value above 10�000 in 2000). The thick (respectively, thin) straight line corresponds
to the exponential growth of an initial wealth of $1 in 1780 (respectively, 1880)
invested at the annual rate of return of ≈2�9% (respectively, 6�8%), which would
have transformed into $1�000 (respectively, $10�000) in 2020.
in figure 2.1 is the true Dow Jones index back to 1896 extrapolated back
to 1790 by The Foundation for the Study of Cycles [138].
The thick straight line in Figure 2.1 corresponds to the exponential
growth of an initial wealth of $1 invested in 1780 at the annual rate of
return of ≈2�9%, which will grow to $1�000 in 2020. The thin straight
line corresponds to the exponential growth of an initial wealth of $1
invested in 1880 at the annual rate of return of 6�8%, which will grow
to $10�000 in 2020. They both show the power of compounded interest!
The comparison of these two lines is suggestive of an acceleration of the
growth rate of return of the DJIA, which was on average about 3% per
year 1780 until the 1930s and then shifted to an average of about 7%
fundamentals of financial markets 29
per year. But even this description falls short of capturing adequately the
behavior of the DJIA: the growth of the DJIA is even stronger than given
by the thin straight line and seems to accelerate progressively upward (at
the end of the book, chapter 10 will offer insights one can extract from
this observation).
Figure 2.2 shows the daily close quotes of the DJIA from January 2,
1980 until December 31, 1987. This time period corresponds to a magnification
of the interval bracketed by the two vertical lines in Figure 2.1.
While Figure 2.2 shows only eight years of data compared to the 210
years of data of figure 2.1, the two figures are strikingly similar. Some
caution must be exercised, however, as the scales used in the two figures
are different (logarithmic scale for the ordinate of Figure 2.1 vs. linear
scale for Figure 2.2). We shall perform a detailed comparison in
chapters 7 and 10 of the information provided by these two kinds of
1980 1982 1984 1986 1988
Dow Jones Industrial Average 2 Jan 1980–31 Dec 1987
Close Prices
Fig. 2.2. Daily quotes of the Dow Jones Industrial Average from January 2, 1980
until December 31, 1987. This time period corresponds to a magnification of the
interval bracketed by the two vertical lines in Figure 2.1.
30 chapter 2
Return Trajectories
Figures 2.3, 2.4, and 2.5 show three time series of returns, rather than the
prices themselves, at three very different time scales: the time scale of
minutes over a full day of trading, the time scale of days over eight years
of trading, and the time scale of months over more than two centuries
of trading. For comparison, Figure 2.6 is obtained by randomly tossing
coins, that is, by choosing at random a positive or negative return with
a probability given by the Gaussian bell curve with an average return
amplitude (standard deviation) equal to 1%. Real returns exhibit much
larger variability and clustering of variability compared to the artificial
time series.
What are returns? If your wealth is 100 today, with an interest rate of 5%
per year, it will transform into 105 after one year, since �105 ? 100�/100 =
5%. The one-year return is then equal to �105 ? 100�/100 = 5%; that
is, it is equal to the interest rate. More generally, the return derived
from an asset whose price changed from p�t� at time t to p�t + dt�
0 100 200 300 400
June, 20th, 1995
One Minute Returns
Time (minutes)
Fig. 2.3. Minute by minute returns of the S&P 500 index on June 20, 1995 showing
the highly stochastic nature of the price dynamics. The typical amplitude of the
return fluctuations is large at the beginning of the day, when traders place orders and
discover the price dynamics (mood?) of the day. The fluctuations go through a low
around noon and then increase again at the end of the day, when trading increases
due to the action of strategies trading at the close.
fundamentals of financial markets 31
1-1-80 31-12-81 1-1-84 31-12-85
Dow Jones Index Returns Jan. 2nd 1980–Dec.31st 1987
Daily Returns
Fig. 2.4. Daily returns of the DJIA from January 2, 1980 until December 31, 1987.
The running sum of these series gives approximately the price trajectory shown
in Figure 2.2. Notice the large returns, both positive and negative, associated with
the crash of October 1987. The largest negative daily return (the crash) reached
?22�6% on October 19, 1987. The largest positive return (the rebound after the
crash) reached +9�7% on October 21, 1987. Both are completely off-scale.
at time t + dt is �p�t + dt� ? p�t��/p�t�. Continuously compounding
interest rates amounts to replacing �p�t + dt� ? p�t��/p�t� by the
so-called logarithmic return ln�p�t + dt�/p�t��. In the previous example,
�p�t + dt� ? p�t��/p�t� = 5%, compared to ln�p�t + dt�/p�t�� =
ln�105/100� = 4�88%. Notice that the two ways of calculating the return
give approximately the same results (5% compared to 4�88%) but not
exactly the same result: the logarithmic return is smaller since you need a
smaller return to obtain the same total capital at the end of the investment
period, if the generated interest is continuously reinvested rather than, say,
reinvested annually. Indeed, the interest itself generates interest, which
generates interest, and so forth.
It is striking how both randomness and patterns seem to coexist in
these time series. Figures 2.3, 2.4, and 2.5 show the pervasive variability
of prices at all time scales. These variations are the “pulsations” of the
stock market, the result of investors’ actions. They are fascinating with
their spontaneous motion and they give an appearance of life, akin to
the complexity of the world around us. They condition the future return
of our investment. The price trajectories seen in Figures 2.1 and 2.2 as
32 chapter 2
1800 1850 1900 1950 2000
Dow Jones Index Jan. 1790–Sept. 2000
Fig. 2.5. Monthly returns of the DJIA from January 1790 until September 2000.
The running sum of these series gives approximately the price trajectory shown in
Figure 2.1. Notice the large returns, both positive and negative, associated with the
crashes of October 1929 and of October 1987.
0 200 400 600 1000
Gaussian White Noise
Fig. 2.6. Gaussian white noise time series with a standard deviation of 1% constructed
using a random number generator. The running sum of these numbers define
a random walk as defined in the text (see Figure 2.9).
fundamentals of financial markets 33
well as the returns shown in Figures 2.3, 2.4, and 2.5 have both an aesthetic
and an almost mystical appeal, with their delicate balance between
randomness and apparent order. The many kinds of structures observed
on stock price trajectories, such as trends, cycles, booms, and bursts,
have been the object of extensive analysis by the scientists of the social
and financial fields as well as by professional analysts and traders. The
work of the latter category of analysts has led to a fantastic lexicon of
these patterns with colorful names, such as “head and shoulder,” “doublebottom,”
“hanging-man lines,” “the morning star,” “Elliott waves,” and
so on (see, for instance, [316]).
Investments in the stock market are based on a quite straightforward
rule: if you expect the market to go up in the future, you should buy
(this is referred to as being “long” in the market) and hold the stock
until you expect the trend to change direction; if you expect the market
to go down, you should stay out of it, sell if you can (this is referred to
as being “short” of the market) by borrowing a stock and giving it back
later by buying it at a smaller price in the future. It is difficult, to say
the least, to predict future directions of stock market prices even if we
are considering time scales of the order of decades, for which one could
hope for a negligible influence of “noise.” To illustrate this, even the
widely cited “fact” that in the United States there has been no thirty-year
period over which stocks underperformed bonds turns out to be incorrect
for the period from 1831 to 1861 [378]. If one chooses ten- or twentyyears
periods, the conclusions are much more murky and the evidence
that stocks always outperform bonds over long time intervals does not
exist [375]. The point in comparing stocks and bonds is that bonds are
so-called fixed-income and ensure the capital (in denominated currency
but not in real value if there is inflation) as well as a fixed return. Bonds
thus provide a kind of anchor or benchmark against which to compare
the highly volatile stocks.
Return Distributions and Return Correlation
To decide whether to buy or sell, it seems useful to try to understand
the origin of the price changes, whether prices will go up or down, and
when; more generally, what are the properties of price changes that can
help us guess the future? Two characteristics among many have attracted
attention: the distribution of price variations (or of price returns) and the
correlation between successive price variations (or returns).
34 chapter 2
0 0.02 0 .04 0.06 0.08 0.1
Distribution Function
return DJ>0
return DJ<0
return NAS>0
return NAS<0
Fig. 2.7. Distribution of daily returns for the DJIA and the Nasdaq index for the
period January 2, 1990 until September 29, 2000. The distributions shown here give,
by definition, the number of times a return larger than or equal to a chosen value
on the abscissa has been observed from January 2, 1990 till 29 September 2000.
The distributions are thus a measure of relative frequency of the different observed
returns. The lines corresponds to fits of the data by models discussed in the text.
Figure 2.7 shows the distribution of daily returns of the DJIA and of
the Nasdaq index for the period January 2, 1990 until September 29,

  1. The ordinate gives the number of times a given return larger than
    a value read on the abscissa has been observed. For instance, we read on
    Figure 2.7 that five negative and five positive daily DJIA market returns
    larger than or equal to 4% have occurred. In comparison, fifteen negative
    and twenty positive returns larger than or equal to 4% have occurred
    for the Nasdaq index. The larger fluctuations of returns of the Nasdaq
    compared to the DJIA are also quantified by the so-called volatility,
    equal to 1�6% (respectively, 1�4%) for positive (respectively, negative)
    returns of the DJIA, and equal to 2�5% (respectively, 2�0%) for positive
    (respectively, negative) returns of the Nasdaq index. The lines shown in
    Figure 2.7 correspond to representing the data by a so-called exponential.
    The upward convexity of the trajectories defined by the symbols for the
    fundamentals of financial markets 35
    Nasdaq qualifies a so-called stretched exponential model [253], which
    embodies the fact that the tail of the distribution is “fatter”; that is, there
    are larger risks of large drops (as well as ups) in the Nasdaq compared
    to the DJIA.
    What is the Nasdaq composite index? In 1961, in an effort to improve
    overall regulation of the securities industry, The Congress of the United
    States asked the U.S. Securities and Exchange Commission (SEC) to
    conduct a special study of all securities markets. In 1963, the SEC
    released the completed study, in which it characterized the over-thecounter
    (OTC) securities market as fragmented and obscure. The SEC
    proposed a solution—automation—and charged The National Association
    of Securities Dealers, Inc. (NASD) with its implementation. In 1968,
    construction began on the automated OTC securities system, then known
    as the National Association of Securities Dealers Automated Quotation,
    or “NASDAQ” System. In 1971, Nasdaq celebrated its first official trading
    day on February 8. This was the first day of operation for the completed
    NASDAQ automated system, which displayed median quotes for more
    than 2,500 OTC securities. In 1990, Nasdaq formally changed its name to
    the Nasdaq Stock Market. In 1994, the Nasdaq Stock Market surpassed
    the New York Stock Exchange in annual share volume. In 1998, the
    merger between the NASD and the AMEX created The Nasdaq-AMEX
    Market Group.
    Figure 2.8 shows the minute per minute time correlation function of
    the returns of the Standard & Poors 500 futures for a single day, June 20,
    1995, whose time series is shown in Figure 2.3. The correlation function
    at time lag � is nothing but a statistical measure of the strength with
    which the present price return resembles the price return at � time steps
    in the past. In other words, it quantifies how the future can be predicted
    from the knowledge of a single measure of the past, as we show in
    the following technical inset. The sum of the correlation function over
    all possible time lags (from 1 to infinity) is simply proportional to the
    number of occurrences when future returns will be close to the present
    return for reasons other than pure chance. A correlation function that
    is zero for all nonzero time lags implies that returns are random, as in
    a fair dice game. A correlation of 1 corresponds to perfect correlation,
    which is found only for the return at a given time with itself. (We should
    remark, however, that a zero-correlation function does not rule out completely
    the possibility of predicting future prices to some degree, since
    other quantities constructed using at least three returns [corresponding to
    36 chapter 2
    0 1 2 3 4 5 6 7 8
    S&P 500
    Time Lag (minutes)
    Fig. 2.8. Correlation function of the returns at the minute time scale of the
    Standard & Poors 500 futures for a single day, June 20, 1995, whose time series
    is shown in Figure 2.3. Note the fast decay to zero of the correlations over a few
    minutes with a few oscillations. This curve shows that there is a persistence of a
    price move lasting a little more than one minute. After two minutes, the price tends
    to reverse with a clear anticorrelation (negative correlation) corresponding to a kind
    of price reversal. Beyond, the correlation is indistinguishable from noise.
    so-called “nonlinear” correlations] may better capture the price dynamics.
    However, such dependence is much harder to detect, establish, and
    use [see chapter 3].) As we see in Figure 2.8, the correlation function is
    nonzero only for very short time scales, typically of the order of a few
    minutes. This means that, beyond a few minutes, future price variations
    cannot be predicted by simple (linear) extrapolations of the past.
    Trading strategy to exploit correlations. The reason why, in very liquid
    markets of equities and foreign exchanges, for instance, correlations of
    returns are extremely small is because any significant correlation would
    lead to an arbitrage opportunity that is rapidly exploited and thus washed
    out. Indeed, the fact that there are almost no correlations between price
    variations in liquid markets can be understood from the following simple
    calculation [50, 348]. Consider a return r that occurred at time t and a
    return r� that occurred at a later time t�, where t and t� are multiples of
    some time unit (say 5 minutes). r and r� can each be decomposed into
    fundamentals of financial markets 37
    an average contribution and a varying part. We are interested in quantifying
    the correlation C�t� t�� between the uncertain varying part, which
    is defined as the average of the product of the varying part of r and of
    r� normalized by the variance (volatility) of the returns, so that C�t� t� =
    t� = 1 (perfect correlation between r and itself). A simple mathematical
    calculation shows that the best linear predictor mt for the return at time
    t, knowing the past history rt?1� rt?2� � � � � ri� � � � , is given by
    ≡ 1
    B�t� t�

    B�i� t�ri� (1)
    where each B�i� t� is a factor that can be expressed in terms of the correlation
    coefficient C�t�� t� and is usually called the coefficient �i� t� of the
    inverse correlation matrix. This formula (1) expresses that each past return
    ri impacts on the future return rt in proportion to its value with a coefficient
    B�i� t�/B�t� t� which is nonzero only if there is nonzero correlation
    between time i and time t. With this formula (1), you have the best linear
    predictor in the sense that it will minimize the errors in variance. Armed
    with this prediction, you have a powerful trading strategy: buy if mt > 0
    (expected future price increase) and sell if mt < 0 (expected future price
    Let us consider the limit where only B�t� t� and B�t� t ?1� are nonzero
    and the natural waiting time between transactions is approximately equal
    to the correlation time taken as the time unit, again equal to five minutes
    in this exercise. The point is that you don’t want to trade too much, otherwise
    you will have to pay for significant transaction costs. The average
    return over one correlation time that you will make using this strategy is
    of the order of the typical amplitude of the return over these five minutes,
    say 0�03% (to account for imperfections in the prediction skills, we take
    a somewhat more conservative measure than the scale of 0�04% over one
    minute used before). Over a day, this gives an average gain of 0�59%,
    which accrues to 435% per year when return is reinvested, or 150% without
    reinvestment! Such small correlations would lead to substantial profits
    if transaction costs and other friction phenomena like slippage did not
    exist (slippage refers to the fact that market orders are not always executed
    at the order price due to limited liquidity and finite human execution
    time). It is clear that a transaction cost as small as 0�03%, or $3 per
    $10�000 invested is enough to destroy the expected gain of this strategy.
    The conundrum is that you cannot trade at a slower rate in order
    to reduce the transaction costs because, if you do so, you lose your prediction
    skill based on correlations only present within a five minute time
    38 chapter 2
    horizon. We can conclude that the residual correlations are those little
    enough not to be profitable by strategies such as those described above
    due to “imperfect” market conditions. In other words, the liquidity and
    efficiency of markets control the degree of correlation that is compatible
    with a near absence of arbitrage opportunity.
    Such observations have been made for a long time. A pillar of modern
    finance is the 1900 Ph.D. thesis dissertation of Louis Bachelier, in Paris,
    and his subsequent work, especially in 1906 and 1913 [25]. To account
    for the apparent erratic motion of stock market prices, he proposed that
    price trajectories are identical to random walks.
    The Random Walk
    The concept of a random walk is simple but rich for its many applications,
    not only in finance but also in physics and the description of
    natural phenomena. It is arguably one of the most important founding
    concepts in modern physics as well as in finance, as it underlies the
    theories of elementary particles, which are the building blocks of our
    universe, as well as those describing the complex organization of matter
    around us. In its most simple version, you toss a coin and walk one
    step up if heads and one step down if tails. Repeating the toss many
    times, where will you finally end up standing? The answer is multiple:
    on average, you remain at the same position since the average of one
    step down and one step up is equivalent to no move. However, it is clear
    that there are fluctuations around this zero average, which grow with the
    number of tosses. This is shown in Figure 2.9, where the trajectory of
    a synthetic random market price has been simulated by tossing “computer
    coins” to decide whether to make the price go up or go down.
    In this simulation, the steps or increments have random signs and have
    amplitudes distributed according to the so-called Gaussian distribution,
    the well-known bell curve.
    To the eye, it is rather difficult to see the difference between the
    synthetic and typical price trajectories such as those in Figures 1.7–1.8,
    except at the time of the crash leading to jumps or when there is a strong
    market trend or acceleration as in Figures 2.1 and 2.2. This is bad news
    fundamentals of financial markets 39
    0 200 400 600 800 1000
    Fig. 2.9. Synthetic random market price (or position of the random walk) obtained
    by tossing “computer coins” to decide whether to make the price go up or down.
    In this simulation, the steps or increments have random signs and have amplitudes
    distributed according to the so-called Gaussian distribution with a 1% standard deviation.
    The same increments as in Figure 2.6 have been used: the synthetic price
    trajectory observed here is thus nothing but the running sum of the increments shown
    in Figure 2.6.
    for investment targets: if the price variations are really like tossing coins
    at random, it seems impossible to know what the direction of the price
    will be between today and tomorrow, or between any two other times.
    A qualifying scaling property of random walks. To get a more quantitative
    feeling for how well the random walk model can constitute a
    good model of stock market prices, consider Figures 2.3, 2.4, and 2.5 of
    return time series at three very different time scales (minute, day, and
    month). The most important prediction of the random walk model is that
    the square of the fluctuations of its position should increase in proportion
    to the time scale. This is equivalent to saying that the typical amplitude
    of its position is proportional to the square root of the time scale. This
    means that, for instance, if we look at returns over four minute intervals,
    the typical return amplitude should be twice (and not four times) that at
    the minute time scale. This result is subtle and profound: since a random
    walker has the same probability of making a positive or negative step, on
    average his position remains where he started. However, it is intuitive that,
    40 chapter 2
    as he accumulates steps randomly, his position deviates from the exact
    average, and the longer the time, the larger the deviation of his position
    from the origin. Rather than cruising at a constant speed such that his
    position increases proportionally with time, a random walker describes an
    erratic motion in which the typical fluctuations of his position increase
    more slowly than linearly in time, in fact at the square root of time. This
    slow increase results from the many retracings of his steps upward and
    downward at all scales. Since steps have random ± signs, their square is
    always positive and thus the sum of squares of the steps is increasing in
    proportion to the number of steps, that is to time. Due to the randomness
    in the sign of steps, the square of the total displacement is equal to the
    sum of squares of the steps. Hence we have the result that the square of
    the typical amplitude of the fluctuations in a random walk increases in
    proportion to time.
    Let us see if this prediction is borne out from the data. The underlying
    idea of this test is that a return at the daily scale is the sum of the returns
    over all the minutes constituting the day. Similarly, a monthly return is the
    sum of the daily returns over all the days of this given month. Since the
    returns are close to random steps, the previously discussed “square-root”
    law should apply. To test it, we observe in Figure 2.3 that the typical
    amplitude of the returns at the time scale of 1 minute is about 0�04% (this
    is the ordinate of the level of the majority of the values). In Figure 2.4, by
    the same estimate made by visual inspection, we estimate a typical amplitude
    of the return fluctuations of about 1%. Now, 1% divided by 0�04% is
    25, which is quite close to the square root 20�25 of the number of minutes
    in a trading day (typically 410). Similarly, we estimate from Figure 2.5
    that the typical amplitude of the return fluctuations at the monthly scale
    is about 5%. The ratio of the monthly value 5% by the daily value of 1%
    equal to 5 is not far from the square root of the number of trading days in
    a month, typically equal to 20–24. The random walk model thus explains
    quite well the way typical returns in the stock market change with time
    and with time scale. However, it does not explain the large fluctuations
    that are not “typical,” as can be seen in Figures 2.4 and 2.5.
    The concept that price variations are inherently unpredictable has been
    generalized and extended by the famous economist and Nobel prize
    winner Paul Samuelson [357, 358]. In a nutshell, Bachelier [25] and
    Samuelson and an army of economists after them have observed that
    even the best investors on average seem to find it hard in the long run
    to do better than the comprehensive common-stock averages, such as the
    Standard & Poors 500, or even better than a random selection among
    fundamentals of financial markets 41
    stocks of comparable variability. It thus seems as if relative price changes
    (properly adjusted for expected dividends paid out) are practically indistinguishable
    from random numbers, drawn from a coin-tossing computer
    or a roulette. The belief is that this randomness is achieved through the
    active participation of many investors seeking greater wealth. This crowd
    of investors actively analyze all the information at their disposal and
    form investment decisions based on them. As a consequence, Bachelier
    and Samuelson argued that any advantageous information that may
    lead to a profit opportunity is quickly eliminated by the feedback that
    their action has on the price. Their point is that the price variations in
    time are not independent of the actions of the traders; on the contrary,
    it results from them. If such feedback action occurs instantaneously, as
    in an idealized world of idealized “frictionless” markets and costless
    trading, then prices must always fully reflect all available information
    and no profits can be garnered from information-based trading (because
    such profits have already been captured). This fundamental concept introduced
    by Bachelier, now called “the efficient market hypothesis,” has a
    strong counterintuitive and seemingly contradictory flavor to it: the more
    active and efficient the market, the more intelligent and hard working the
    investors; as a consequence the more random is the sequence of price
    changes generated by such a market. The most efficient market of all is
    one in which price changes are completely random and unpredictable.
    There is an interesting analogy with the information coded in DNA,
    the molecular building block of our chromosomes. Here, our genetic
    information is encoded by the order in which the four constituent bases
    of DNA are positioned along a DNA strand, similarly to words using
    a four-letter alphabet. DNA is usually organized in so-called coding
    sections and noncoding sections. The coding sections contain the information
    on how to synthetize proteins and how to work all our biological
    machinery. Recent detailed analyses of the sequence of these letters have
    shown [444, 286, 14] that the noncoding parts of DNA seem to have
    long-range correlations while, in contrast, the coding regions seem to
    have short-range or no correlations. Notice the wonderful paradox: information
    leads to randomness, while lack of information leads to regularities.
    The reason for this is that a coding region must appear random since
    all bases contain useful, that is, different information. If there were some
    correlation, it would mean that it is possible to encode the information
    in fewer bases and the coding regions would not be optimal. In contrast,
    noncoding regions contain few or no information and can thus be highly
    correlated. Indeed, there is almost no information in a sequence like
    1111111 � � � but there may be a lot in 429976545782 � � � . This paradox,
    42 chapter 2
    that a message with a lot of information should be uncorrelated while a
    message with no information is highly correlated, is at the basis of the
    notion of random sequences. A truly random sequence of numbers or
    of symbols is one that contains the maximum possible information; in
    other words, it is not possible to define a shorter algorithm that contains
    the same information [73]. The condition for this is that the sequence be
    completely uncorrelated so that each new term carries new information.
    It is worthwhile to stop and consider in more detail this extraordinary
    concept, that the more intelligent and hard working the investors,
    the more random is the sequence of price changes generated by such
    a market. In particular, it embodies the fundamental difference between
    financial markets and the natural world. The latter is open to the scrutiny
    of the observer and the scientist has the possibility to construct explanations
    and theories that are independent of his or her actions. In contrast,
    in social and financial systems, the actors are both the observers and
    the observed, which thus create so-called feedback loops. The following
    simple parable is a useful illustration.
    A Parable: How Information Is Incorporated in Prices,
    Thus Destroying Potential “Free Lunches”
    Let us assume that half the population of investors are informed today
    that the price will go up tomorrow from its present value p0, naturally not
    with complete certainty, but still with a rather high probability of 75%
    (there is therefore a 25% probability that the price goes down tomorrow).
    The other half of the population is kept uninformed and we shall call
    them the “noise traders,” after the famous description by Black [40] of
    the individuals who trade on what they think is information but is in
    fact merely noise. These noise traders will buy and sell on grounds that
    are unrelated to the movements of the market, although they believe
    the “information” they have is relevant. For noise traders, selling may
    be triggered by a need for cash for reasons completely unrelated to the
    market. We capture this behavior by tossing coins at random to decide
    the fraction y of noise traders who want to sell. Correspondingly, the
    fraction of noise traders who want to buy is 1 ? y. The important point
    is that noise traders are insensitive, by definition, to the present price or
    to the price offered for the transaction.
    In contrast, the informed traders want to buy because they see an
    opportunity for profit with a high success rate—as high as 3 out of 4.
    In order to buy, they have to make a bid to a central agent, the “market
    fundamentals of financial markets 43
    maker.” The role of the market maker is to compile all buy and sell offers
    and to adjust the price so that the maximum number of transactions can
    be satisfied. This is a form of balance between supply and demand.
    However, informed traders will not buy at any price because they will
    use their special information to estimate what will be their expected gain.
    If the price at which they are offered to buy by the market maker is
    larger than their expectation for the price increase, they will not have
    an incentive to buy. We call � p+� the expected gain conditioned on the
    realization of the tip (i.e., that the price will increase). The fraction of
    informed traders still willing to buy at a price x above the last quoted
    price p0 is clearly a decreasing function of x. Two limits are simple to
    guess: for x = 0, all the informed traders want to buy at price p0 because
    the expected gain is positive. In contrast, for x equal to � p+� or larger,
    the offered buy price is larger than the price expected tomorrow on the
    basis of the prediction, and none of the informed traders wish to buy
    due to the unfavorable probability of a loss. In between, we will for
    simplicity assume a linear relationship fixing the fraction of informed
    traders willing to buy at the price p0
  • x, which interpolates smoothly
    between these two extremes, as shown in Figure 2.10.
    The decision of the informed traders depends on the noise traders. We
    assume for simplicity that each seller (respectively, buyer) sells (buys)
    only one stock. Then two situations can occur.
    Fraction of informed
    traders willing to buy
    p0 p0 + <dp>
    “Ask Price”
    Fig. 2.10. Fraction of informed traders who are willing to buy as a function of the
    “ask price”: if the ask price is the last quote p0, all the informed traders want to
    bid for the stock because their expected return is positive. If the ask price is equal
    to or larger than the last quote plus the expected increase, informed traders are
    not interested in bidding for the stock. This dependence corresponds to so-called
    “risk-neutral” agents.
    44 chapter 2
    � If the fraction y of noise traders who sell is less than 1/2, there is a
    severe undersupply of stocks: both the fraction 1 ?y > 1/2 of noise
    traders and all the informed traders want to buy. The selling noise
    traders cannot even supply enough stocks for their buying counterparts,
    not to mention to the aggressive informed traders. In this situation, the
    market maker increases the price up to the level at which informed
    traders turn down the buying offer. For the noise traders, the price
    does not make a difference since they have no information on what the
    future price will be. In this situation, where y < 1/2, the transaction
    price therefore is equal to the minimum price p0
  • � p+� at which
    all informed traders turn down the buying option. There is no average
    profit from selling later at the expected future price p0
  • � p+�, since
    it equals the buying price! Note in contrast that, in the absence of
    informed traders, the profit opportunity would remain, as the buying
    price is unchanged at p0. It is the presence of the informed traders that
    pushes the price up to the threshold where they do not wish to act.
    While the informed traders do not appear explicitly in this transaction,
    their bid to the market maker has pushed the price up, such that the
    profit opportunity has disappeared.
    � The second situation occurs when the fraction y of noise traders who
    sell is larger than 1/2. They can then supply all their buying counterparts
    as well as a fraction of the informed traders. The price of
    the transaction p0
  • x is then set by the market maker such that the
    fraction of the informed traders willing to buy at this price is equal to
    the remaining available stock after the buying noise traders have been
    served. Counting all possible outcomes for y larger than 1/2 (but of
    course smaller than 1), we see that the average of y, conditioned to be
    larger than 1/2, is 3/4, the middle point between 1/2 and 1. Thus, the
    average transaction price is 1/2 the expected conditional gain � p+�
    (x = � p+�/2), such that 1/2 of the informed traders are still willing
    to buy. In this situation, the balance of supply and demand is upheld:
    the average fraction, 3/4, of noise traders who sell balances exactly
    the other 1/4 of buying noise traders and the 1/2 of the informed
    What, then, is the expected gain for the informed traders? It is (the
    probability 3/4 that the price increases) times (the average gain � p+�?
    x) minus (the probability 1/4 that the price decreases) times (the loss
    amplitude). This loss amplitude is x minus the expected amplitude of
    the price drop, conditioned on its drop. By symmetry of the distribution
    of price variations (very well verified in most stock markets), this is the
    fundamentals of financial markets 45
    same in amplitude as the expected conditional gain � p+�. In sum, the
    total expected gain is
    �3/4� × �� p+� ? x� ? �1/4��� p+� + x�� (2)
    Using the above result, x = � p+�/2, we find that this is in fact zero:
    the action of the noise traders and the response of the informed traders
    to them and to their information makes the buying price increase to a
    level p0
  • x such that the expected gain vanishes!
    Prices Are Unpredictable, or Are They?
    This conclusion remains qualitatively robust against a change of the value
    of the parameters of this toy model or of the buying strategies developed
    by the informed traders. This simple model illustrates the following
    fundamental ideas.
  1. Acting on advantageous information moves the price such that the a
    priori gain is decreased or even destroyed by the feedback of the action
    on the price. This makes concrete the concept that prices are made
    random by the intelligent and informed actions of investors, as put
    forward by Bachelier, Samuelson, and many others. In contrast, without
    informed traders, the profit opportunity remains, since the buying price
    is unchanged at p0.
  2. Noise traders are essential for the function of the stock market. They
    are known under many names: sometimes as speculators, or traders
    basing their strategies on technical indicators or on supposedly relevant
    economic information. All informed traders in our example agree that
    the best strategy is to buy. However, in the absence of noise traders,
    they would not find any counterpart, and there would be no trade: If
    everybody agrees on the price, why trade? No profit can be made. Thus
    the stock market needs the existence of some “noise,” however small,
    which provides “liquidity.” Then, the intelligent traders work hard and,
    according to this theory, will by their investments make the market
    totally and utterly noisy, with no remaining piece of intelligible signal.
  3. The fact that the informed traders are unable on average to make a
    profit notwithstanding their large confidence in an upward move is not
    in contradiction with the notion that, if you alone had this information
    and were willing to be cautious and trade only a few stocks, you would
    on average be able to make a good profit. The reason is simply that
    46 chapter 2
    your small action would not have a significant impact on the market. In
    contrast, if you were bold enough to borrow a lot and buy a significant
    share of the market, you would move the price up, in a way similar to
    the informed traders who constitute half of the total population. Thus,
    the price dynamics becomes random only if there are sufficiently many
    informed traders to affect the dynamics by their active feedback.
    General proof that properly anticipated prices are random. Samuelson
    has proved a general theorem showing that the concept that prices
    are unpredictable can actually be deduced rigorously [357] from a model
    that hypothesizes that a stock’s present price pt is set at the expected
    discounted value of its future dividends dt� dt+1� dt+2� � � � (which are supposed
    to be random variables generated according to any general (but
    known) stochastic process):
    = dt
  • 1 dt+1
  • 1 2 dt+2
  • 1 2 3 dt+3
    +· · · � (3)
    where the factors i
    = 1 ?r < 1, which can fluctuate from one time
    period to the next, account for the depreciation of a future price calculated
    at present due to the nonzero consumption price index r. We see that
    = dt
  • 1pt+1, and thus the expectation E�pt+1� of pt+1 conditioned on
    the knowledge of the present price pt is
    E�pt+1� = pt
    ? dt
    � (4)
    This shows that, barring the drift due to the inflation and the dividend, the
    price increment does not have a systematic component or memory of the
    past and is thus random. Therefore, even when the economy is not free to
    wander randomly, intelligent speculation is able to transform the observed
    stock-price changes into a random process.
    At first glance, these ideas seem to be confirmed by the data. As
    shown in Figure 2.7, the distributions of positive and negative returns are
    almost identical: there is almost the same probability for a price increase
    or a decrease. In addition, Figure 2.8 has taught us that returns are essentially
    decorrelated beyond a few minutes in active and well-organized
    markets. As a consequence, successive returns cannot be predicted by
    linear extrapolations of the past.
    However, as already noted, this does not exclude the possibility that
    there might be other kinds of dependence between price variations of a
    more subtle nature, which might remain either because they have not yet
    fundamentals of financial markets 47
    been detected or taken advantage of by traders or because they are not
    providing significant profit opportunities.
    Asymmetry between positive and negative returns. The distribution of
    price variations may often exhibit a residual bias associated with the overall
    rate of return of the market. For instance, for a 10% annual return, this
    corresponds to an average daily drift of approximately 10%/365 = 0�03%.
    This value is small compared to the typical scale of daily fluctuations of
    the order of 1% for most markets (and more for growth and emergent
    markets which present a larger volatility). Such a drift translates into a
    bias in the frequency of gains versus losses. For the DJIA from 1897 to
    1997, over the 27,819 trading days, the market declined on 13,091 days
    and rose on 14,559 days. This translates into a 47.06% probability of a
    decline and a 52.34% probability of a stock market rise (the probabilities
    do not sum up to 1 because there were some days for which the
    price remained unchanged). In a similar fashion, the decline probability is
    47.27% during the 1946–1997 DJIA period and 46.86% during 1897–1945
    (about 0.5% lower). Preserving the same qualitative pattern, during the
    1897–1997 DJIA period, the weekly decline (rise) probability is 43.98%
    (55.87%). For the Nasdaq from 1962 to 1995, the daily decline (rise)
    probability is 46.92% (52.52%). For the IBM stock from 1962–1996, the
    daily decline (rise) probability is 47.96% (48.25%).
    One of the central insights of modern financial economics is the necessity
    of some trade-off between risk and expected return, and although
    Samuelson’s version of the efficient markets hypothesis places a restriction
    on expected returns, it does not account for risk in any way. In
    particular, if a security’s expected price change is positive, it may be just
    the reward needed to attract investors to hold the asset and bear the associated
    risks. Indeed, if an investor is sufficiently risk averse, he might
    gladly pay to avoid holding a security that has unforecastable returns.
    Grossman and Stiglitz [180] went even further. They argue that perfectly
    informationally, efficient markets are an impossibility, for if markets
    are perfectly efficient, the return on gathering information is nil, in
    which case there would be little reason to trade and markets would eventually
    collapse. Alternatively, the degree of market inefficiency determines
    the effort investors are willing to expend to gather and trade on
    information, hence a nondegenerate market equilibrium will arise only
    when there are sufficient profit opportunities, that is, inefficiencies, to
    48 chapter 2
    compensate investors for the costs of trading and information-gathering.
    The profits earned by these industrious investors may be viewed as economic
    rents that accrue to those willing to engage in such activities. Who
    are the providers of these rents? Black [40] gave us a provocative answer:
    noise traders, individuals who trade on what they think is information
    but is in fact merely noise. More generally, at any time there are always
    investors who trade for reasons other than information (for example,
    those with unexpected liquidity needs), and these investors are willing
    to “pay up” for the privilege of executing their trades immediately.
    chapter 3
    financial crashes
    are “outliers”
    In the spirit of Bacon in Novum Organum about
    400 years ago, “Errors of Nature, Sports and Monsters correct the understanding
    in regard to ordinary things, and reveal general forms. For whoever
    knows the ways of Nature will more easily notice her deviations;
    and, on the other hand, whoever knows her deviations will more accurately
    describe her ways,” we propose in this chapter that large market
    drops are “outliers” and that they reveal fundamental properties of the
    stock market.
    Stock markets can exhibit very large motions, such as rallies and crashes,
    as shown in Figures 2.4 and 2.5. Should we expect these extreme variations?
    Or should we consider them anomalous?
    Abnormality is a relative notion, constrasted to what is considered
    “normal.” Let us take an example. In the Bachelier-Samuelson financial
    world, in which returns are distributed according the Gaussian bell-shape
    distribution, all returns are scaled to a fundamental “ruler” called the
    standard deviation. Consider the daily time scale and the corresponding
    time series of returns of the Dow Jones index shown in Figure 2.4. As
    we indicated in chapter 2, the standard deviation is close to 1%. In this
    Gaussian world, it is easy to quantify the probability of observing a given
    50 chapter 3
    Table 3.1
    X Probability> One in N events Calendar waiting time
    1 0�317 3 3 days
    2 0�045 22 1 month
    3 0�0027 370 1�5 year
    4 6�3 × 10?5 15�787 63 years
    5 5�7 × 10?7 1�7 × 106 7 millenia
    6 2�0 × 10?9 5�1 × 108 2 million years
    7 2�6 × 10?12 3�9 × 1011 1562 million years
    8 1�2 × 10?15 8�0 × 1014 3 trillion years
    9 2�3 × 10?19 4�4 × 1018 17,721 trillion years
    10 1�5 × 10?23 6�6 × 1022 260 million trillion years
    How probable is it to observe a return larger in amplitude (i.e., in absolute value) than some value
    equal to X times the standard deviation? The answer is given in this table for the Gaussian world.
    The left column gives the list of values of X from 1 to 10. The second column gives the probability
    that the absolute value of the return is found larger than X times the standard deviation. The third
    column translated this probability into the number of periods (days in our example) one would
    typically need to wait to witness such a return amplitude. The fourth column translates this waiting
    time into calendar time in units adapted to the value, using the conversion that one month contains
    approximately 20 trading days and one year contains about 250 trading days. For comparison, the
    age of the universe is believed to be (only) of the order of 10–15 billion years.
    return amplitude, as shown in Table 3.1. We read that a daily return
    amplitude of more than 3% should be typically observed only once in
    1�5 years. A daily return amplitude of more than 4% should be typically
    observed only once in 63 years, while a return amplitude of more than
    5% should never be seen in our limited history.
    Armed with this Table 3.1, it is now quite clear what is “normal” and
    what can be considered “abnormal” according to the Gaussian model.
    The drop of ?22�6% on October 19, 1987 and the rebound of +9�7%
    on October 21, 1987 are abnormal: they should not occur according to
    the standard Gaussian model. They are essentially impossible. The fact
    that they occurred tells us that the market can deviate significantly from
    the norm. When it does, the “monster” events that the market creates are
    “outliers.” In other words, they lie “out” and beyond what is possible for
    the rest of the population of returns.
    In reality, the distributions of returns are not Gaussian, as shown in
    Figure 2.7. If they were, they would appear as inverted parabola in this
    semilogarithmic plot. The approximate linear dependence qualifies rather
    financial crashes are “outliers” 51
    as a dependence not far from an exponential law. In this new improved
    representation, we can again calculate the probability of observing a
    return amplitude larger than, say, 10 standard deviations (10% in our
    example). The result is 0�000045, which corresponds to one event in
    22,026 days, or in 88 years. The rebound of October 20, 1987 becomes
    less extraordinary. Still, the drop of 22�6% of October 19, 1987 would
    correspond to one event in 520 million years, which qualifies it as an
    Thus, according to the exponential model, a 10% return amplitude
    does not qualify as an “outlier” in a clear-cut and undisputable manner.
    In addition, we see that our discrimination between normal and abnormal
    returns depends on our choice for the frequency distribution. Qualifying
    what is the correct description of the frequency distribution, especially
    for large positive and negative returns, is a delicate problem that is still a
    hot domain for research. Due to the lack of certainty on the best choice
    for the frequency distribution, this approach does not seem the most
    adequate for characterizing anomalous events.
    Up to now, we have only looked at the distribution or frequency of
    returns. However, the complex time series of returns have many other
    structures not captured by the frequency distribution. We have already
    discussed the additional diagnostic in terms of the correlation function
    shown in Figure 2.8. We now introduce another diagnostic that allows
    us to characterize abnormal market phases in a much more precise and
    nonparametric way, that is, without referring to a specific mathematical
    representation of the frequency distribution.
    Definition of Drawdowns
    One measure going beyond the simple frequency statistics and the
    linear correlations is provided by the statistics of “drawdowns.” A drawdown
    is defined as a persistent decrease in the price over consecutive
    days. A drawdown, as shown in Figure 3.1, is thus the cumulative
    loss from the last maximum to the next minimum of the price. Drawdowns
    are indicators that we care about: they measure directly the
    cumulative loss that an investment may suffer. They also quantify
    the worst-case scenario of an investor buying at the local high and
    selling at the next minimum. It is thus worthwhile to ask if there is
    52 chapter 3
    1987.77 1987.79 1987.81 1987.83 1987.85
    Close Price
    Time (decimal years)
    Fig. 3.1. Definition of drawdowns. Taking the example of the crash that occurred
    on October 19, 1987, this figure shows three drawdowns corresponding to cumulative
    losses from the last maximum to the next minimum of the price. The largest
    drawdown of a total loss of ?30�7% is made of four successive daily drops: on
    October 14, 1987 (1987.786 in decimal years), the DJIA index is down by 3�8%; on
    October 15, the market is down 6�1%; on October 16, the market is down 10�4%.
    The weekend passes and the drop on Black Monday October 19, 1987 leads to a
    cumulative loss or drawdown of 30�7%. In terms of consecutive daily losses, this
    correspond to the series 3�8%, 2�4%, 4�6%, and 22�6% (note that returns are not
    exactly additive, since they are price variations normalized by the price, which itself
    any structure in the distribution of drawdowns absent in that of price
    Drawdowns embody a rather subtle dependence since they are constructed
    from runs of the same sign variations (see below). Their distribution
    thus capture the way successive drops can influence each other
    and construct in this way a persistent process. This persistence is not
    measured by the distribution of returns because, by its very definition, it
    forgets about the relative positions of the returns as they unravel themfinancial
    crashes are “outliers” 53
    selves as a function of time by only counting their frequency. This is
    also not detected by the two-point correlation function, which measures
    an average linear dependence over the whole time series, while the
    dependence may only appear at special times, for instance for very large
    runs, as we shall demonstrate below, a feature that will be washed out
    by the global averaging procedure.
    A nonlinear model with zero correlation but high predictability. To
    understand better how subtle dependences in successive price variations
    are measured by drawdowns, let us play the following game in which
    the price increments p�t� are constructed according to the following
    p�t� =
    �t� +
    �t ? 1�
    �t ? 2�� (5)
    �t� is a white noise process with zero mean and unit variance.
    For instance,
    �t� is either +1 or ?1 with probability 1/2. The definition
    (5) means that the price variation today is controlled by three random
    coin tosses, one for today, yesterday, and the preceeding day, such that a
    positive coin toss today as well as two identical coin tosses yesterday and
    the day before make the price move up. Reciprocally, a negative coin toss
    today as well as two different coin tosses yesterday and the day before
    make the price move down.
    It is easy to check that the average E� p�t�� as well as the two-point
    correlation E� p�t� p�t��� for t = t� are zero and p�t� is thus also
    a white noise process. Intuitively, this stems from the fact that an odd
    number of coin tosses
    enter into these diagnostics, whose average
    is zero (�1/2� × �+1� + �1/2� × �?1� = 0). However, the three-point
    correlation function E� p�t ? 2� p�t ? 1� p�t�� is nonzero and equal
    to 1 and the expectation of p�t� given the knowledge of the two
    previous increments p�t ? 2� and p�t ? 1� is nonzero and equal
    to E� p�t�
    p�t ? 2�� p�t ? 1�� = p�t ? 2� p�t ? 1�. This means
    that it is possible to predict the price variation today with better success
    than 50%, knowing the price variations of yesterday and the day
    While the frequency distribution and the two-point correlation function
    are blind to this dependence structure, the distribution of drawdowns
    exhibits a specific diagnostic. To simplify the analysis and make the message
    very clear, let us again restrict to the case where
    �t� can only take
    two values ±1. Then, p�t� can take only three values 0 and ±2, with
    54 chapter 3
    the correspondence

�t ? 2��
�t ? 1��
�t� → p�t��
+++ → +2�
++? → 0�
+?+ → 0�
+?? → ?2�
?++ → 0�
?+? → ?2�
??+ → +2�
??? → 0�
where the left column gives the three consecutive values
�t ? 2��
�t ?
�t� and the right column is the corresponding price increment p�t�.
We see directly by this explicit construction that p�t� is a white noise
process. However, there is a clear predictability and the distribution
of drawdowns reflects it: there are no drawdowns of duration larger
than two time steps. Indeed, the worst possible drawdown corresponds
to the following sequence for
: ? ? + ? ?. This corresponds to the
sequence of price increments +2�?2�?2, which is either stopped by
a +2 if the next
is + or by a sequence of 0s interrupted by a +2
at the first
= +. While the drawdowns of the process
�t� can in
principle be of infinite duration, the drawdowns of p�t� cannot. This
shows that the structure of the process p�t� defined by (5) has a dramatic
signature in the distribution of drawdowns in p�t�. This illustrates
that drawdowns, rather than daily or weekly returns or any other fixed
time scale returns, are more adequate time-elastic measures of price
Drawdowns and the Detection of “Outliers”
To demonstrate further the new information contained in drawdowns and
contrast it with the fixed time-scale returns, let us consider the hypothetical
situation of a crash of 30% occurring over three days with three
successive losses of exactly 10%. The crash is thus defined as the total
financial crashes are “outliers” 55
loss or drawdown of 30%. Rather than looking at drawdowns, let us
now follow the common approach and examine the daily data, in particular
the daily distribution of returns. The 30% drawdown is now seen
as three daily losses of 10%. The essential point to realize is that the
construction of the distribution of returns amounts to counting the number
of days over which a given return has been observed. The crash will
thus contribute to three days of 10% loss, without the information that
the three losses occurred sequentially! To see what this loss of information
entails, we consider a market in which a 10% daily loss occurs
typically once every four years (this is not an unreasonable number for
the Nasdaq composite index at present times of high volatility). Counting
approximately 250 trading days per year, four years correspond to 1,000
trading days and one event in 1,000 days thus corresponds to a probability
1/1�000 = 0�001 for a daily loss of 10%. The crash of 30% has
been dissected as three events that are not very remarkable (each with a
relatively short average recurrence time of four years). The plot thickens
when we ask, What is, according to this description, the probability for
three successive daily losses of 10%? Elementary probability tells us that
it is the probability of one daily loss of 10% times the probability of
one daily loss of 10% times the probability of one daily loss of 10%.
The rule of products of probability holds if the three events are considered
to be independent. This products gives 0�001 × 0�001 × 0�001 =
0�000�000�001 = 10?9. This corresponds to one event in 1 billion trading
days! We should thus wait typically 4 millions years to witness such
an event!
What has gone wrong? Simply, looking at daily returns and at
their distributions has destroyed the information that the daily returns
may be correlated, at special times! This crash is like a mammoth
that has been dissected in pieces without memory of the connection
between the parts, and we are left with what look like mouses (bear
with the slight exaggeration)! Our estimation that three successive
losses of 10% are utterly impossible relied on the incorrect hypothesis
that these three events are independent. Independence between
successive returns is remarkably well verified most of the time. However,
it may be that large drops may not be independent. In other
words, there may be “bursts of dependence,” that is, “pockets of
It is clear that drawdowns will keep precisely the information relevant
to identifying the possible burst of local dependence leading to
possibly extraordinarily large cumulative losses.
56 chapter 3
Expected Distribution of “Normal” Drawdowns
Before returning to the data, we should ask ourselves what can be
expected on the basis of the random walk hypothesis. If price variations
are independent, positive �+� and negative �?� moves follows each other
like the “heads” and “tails” of a fair coin toss. For symmetric distributions
of price variations, starting from a positive, +, the probability to
have one negative, ?, is 1/2. The probability to have two negatives in a
row is 1/2 × 1/2 = 1/4; the probability to have three negatives in a row
is 1/2 × 1/2 × 1/2 = 1/8, and so on. For each additional negative, we
observe that the probability is divided by two. This defines the so-called
exponential distribution, describing the fact that increasing a drawdown
by one time unit makes it doubly less probable. This exponential law is
also known as the Poisson law and describes processes without memory:
for the sequence+ ? ? ? ?, the fact that four negatives have
occurred in a row does not modify the probability for the new event,
which remains 1/2 for both a positive and a negative. Such a memoryless
process may seem counterintuitive (many people would rather bet
on a tail after a sequence of ten heads than on another head; this is often
refered to as the “gambler’s fallacy”) but it reflects accurately what we
mean by complete randomness: in a fair coin toss, it can happen that ten
heads in a row are drawn. The eleventh event still has the probability
of 1/2 to be head. The absence of memory of such random processes
can be stated as follows: given the past observation of n successive
negatives, the probability for the next one is unchanged from the unconditional
value 1/2 independently of the value of n. Any deviation from
this exponential distribution of drawdowns will signal some correlation
in the process and thus a potential for a prediction of future events.
Since, in the random memoryless model, there are half as many drawdowns
of duration one time step longer, it is convenient to visualize
the empirical distribution of drawdowns on the stock market on a logarithmic
scale, where the expected exponential distribution of drawdowns
becomes a straight line. This is a quite efficient method to test for the
validity of the hypothesis: deviations from the straight line will signal
some deviation from the exponential distribution and thus from the
hypothesis of absence of memory.
The evidence presented below on the presence of “outliers” does
not rely on the validity of this Poisson law. Actually, we have identified
slight deviations from it already in the bulk of the distribution
of drawdowns, suggesting a subtle departure from the hypothesis of
financial crashes are “outliers” 57
independence between successive price returns. This leads us to a quite
delicate point that escaped the attention of even some of our cleverest
colleagues for some time and is still overlooked by many others. This
subtle point is that the evidence for outliers and extreme events does
not require and is not even synonymous in general with the existence
of a break in the distribution of the drawdowns. Let us illustrate this
pictorially and forcefully by borrowing from another domain of active
scientific investigation, namely the search for an understanding of the
complexity of eddies and vortices in turbulent fluid flows, such as in a
mountain river or in atmospheric weather. Since solving the exact equations
of these flows does not provide much insight as the results are
forbidding, a useful line of attack has been to simplify the problem by
studying simple toy models, such as so-called “shell” models of turbulence,
that are believed to capture the essential ingredient of these flows,
while being amenable to analysis. Such “shell” models replace the threedimensional
spatial domain by a series of uniform onion-like spherical
layers with radii increasing as a geometrical series 1� 2� 4� 8� � � � � 2n and
communicating with each other mostly with nearest neighbors.
As for financial returns, a quantity of great interest is the distribution
of velocity variations between two instants at the same position or
between two points simultaneously. Such a distribution for the square of
the velocity variations is shown in Figure 3.2. Notice the approximate
exponential drop-off represented by the straight line and the coexistence
with larger fluctuations on the right for values above 4 up to 7 and
beyond (which are not shown). Usually, such large fluctuations are not
considered to be statistically significant and do not provide any specific
insight. Here, it can be shown that these large fluctuations of the fluid
velocity correspond to intensive peaks propagating coherently over several
shell layers with a characteristic bell-like shape, approximately independent
of their amplitude and duration (up to a rescaling of their size
and duration). When extending the observations to much longer times so
that the anomalous fluctuations beyond the value 4 in Figure 3.2 can be
sampled much better, one gets the continuous curves (apart from some
residual noise always present) shown in Figure 3.3. Here, each of the
three curves corresponds to the measurement of a distribution in a given
shell layer (n = 11� 15, and 18).
In Figure 3.3, a standard transformation has been performed, that is,
contracting or magnifying the abscissa and ordinate for each curve so
that the three curves are collapsed on each other. If one succeeds in doing
so, this means that, up to a definition of units, the three distributions
are identical, which is very helpful for understanding the underlying
58 chapter 3
0 1 2 3 4 5 6 7
Fig. 3.2. Apparent probability distribution function of the square of the fluid velocity,
normalized to its time average, in the eleventh shell of the toy model of hydrodynamic
turbulence discussed in the text. The vertical axis is in logarithmic scale
such that the straight line, which helps the eye, qualifies as an apparent exponential
distribution. Note the appearance of extremely sparse and large bursts of velocities at
the extreme right above the extrapolation of the straight line. Reproduced from [252].
mechanism as well as for future use for risk assessement and control.
Naively, we would expect that the same physics apply in each shell layer
and that, as a consequence, the distributions should be the same, up to
a change of unit reflecting the different scale embodied by each layer.
Here, we observe that the three curves are indeed nicely collapsed, but
only for the small velocity fluctuations, while the large fluctuations are
described by very different heavy tails. Alternatively, when one tries to
collapse the curves in the region of the large velocity fluctuations, then
the portions of the curves close to the origin are not collapsed at all and
are very different. The remarkable conclusion is that the distributions
of velocity increment seem to be composed of two regions, a region of
so-called “normal scaling” and a domain of extreme events.
Here is the message that comes out of this discussion: the concept
of outliers and of extreme events does not rest on the requirement that
the distribution should not be smooth, as shown on the right side of
financial crashes are “outliers” 59
Shell 11
Shell 15
Shell 18
0 50 100 150 200
Fig. 3.3. Probability distribution function of the square of the velocity as in Figure
3.2 but for a much longer time series, so that the tail of the distributions for very
large fluctuations is much better constrained. The hypothesis that there are no outliers
is tested here by “collapsing” the distributions for the three shown layers. While
this is a success for small fluctuations, the tails of the distributions for large events
are very different, indicating that extreme fluctuations belong to a class of their own,
and hence are outliers. The vertical axis is again in logarithmic scale. Reproduced
from [252].
Figure 3.2. Noise and the very process of constructing the distribution
will almost always smooth out the curves. What is found here [252]
is that the distribution is made of two different populations, the body
and the tail, which have different physics, different scaling, and different
properties. This is a clear demonstration that this model of turbulence
exhibits outliers in the sense that there is a well-defined population of
very large and quite rare events that punctuate the dynamics and that cannot
be seen as scaled-up versions of the small fluctuations. It is tempting
to conjecture that the anomalous “scaling” properties of turbulence might
be similarly controlled by the coexistence of normal innocuous velocity
fluctuations and extreme concentrated events, possibly associated with
specific vortex filaments or other coherent structures [371].
As a consequence, the fact that the distribution of small events might
show some curvature or continuous behavior does not say anything
60 chapter 3
against the outlier hypothesis. It is essential to keep this point in mind
in looking at the evidence presented below for the drawdowns.
The Dow Jones Industrial Average
Figure 3.4 shows the distribution of drawdowns for the returns of the
DJIA over this century.
The exponential distribution discussed in the previous section has
been derived on the assumption that successive price variations are independent.
There is a large body of evidence for the correctness of this
assumption for most trading days [68]. However, consider, for instance,
the fourteen largest drawdowns that have occurred in the DJIA in this
century. Their characteristics are presented in Table 3.2. Only three lasted
one or two days, whereas nine lasted four days or more. Let us examine
in particular the largest drawdown. It started on October 14, 1987
(1987.786 in decimal years), lasted four days, and led to a total loss of
?30�7%. This crash is thus a run of four consecutive losses: first day, the
index is down by 3�8%; second day, by 6�1%; third day, by 10�4%; and
0.3 0.25 0.2 0.15 0.1 0.05 0
Log(Cumulative Number)
Draw Down
Null Hypothesis
‘Negative Draw Downs’
Fig. 3.4. Number of times a given level of drawdown has been observed in this
century for the DJIA. Reproduced from [220].
financial crashes are “outliers” 61
Table 3.2
Characteristics of the 14 largest drawdowns of the DJIA in the twentieth century
Rank Starting time Index value Duration (days) Loss
1 87�786 2508�16 4 ?30�7%
2 14�579 76�7 2 ?28�8%
3 29�818 301�22 3 ?23�6%
4 33�549 108�67 4 ?18�6%
5 32�249 77�15 8 ?18�5%
6 29�852 238�19 4 ?16�6%
7 29�835 273�51 2 ?16�6%
8 32�630 67�5 1 ?14�8%
9 31�93 90�14 7 ?14�3
10 32�694 76�54 3 ?13�9%
11 74�719 674�05 11 ?13�3%
12 30�444 239�69 4 ?12�4%
13 31�735 109�86 5 ?12�9
14 98�649 8602�65 4 ?12�4%
The starting dates are given in decimal years. Reproduced from [220].
fourth day by 30�7%. In terms of consecutive losses, this corresponds
to 3�8%, 2�4%, 4�6%, and then 22�6% on what is known as the Black
Monday of October 1987.
The observation of large successive drops is suggestive of the existence
of a transient correlation, as we already pointed out. For the Dow
Jones, this reasoning can be adapted as follows. We use a simple functional
form for the distribution of daily losses, namely an exponential
distribution with decay rate 1/0�63% obtained by a fit to the distribution
of drawdowns shown in Figure 3.4. The quality of the exponential model
is confirmed by the direct calculation of the average loss amplitude equal
to 0�67% and of its standard deviation equal to 0�61% (recall that an
exact exponential would give the three values exactly equal: 1/decay =
average = standard deviation). Using these numerical values, the probability
for a drop equal to or larger than 3�8% is exp�?3�8/0�63� =
2�4 · 10?3 (an event occurring about once every two years); the probability
for a drop equal to or larger than 2�4% is exp�?2�4/0�63� = 2�2 · 10?2
(an event occurring about once every two months); the probability for
a drop equal to or larger than 4�6% is exp�?4�6/0�63� = 6�7 · 10?4
(an event occurring about once every six years); the probability for a
drop equal to or larger than 22�6% is exp�?22�6/0�63� = 2�6 · 10?16 (an
event occurring about once every 1014 years). All together, under the
62 chapter 3
hypothesis that daily losses are uncorrelated from one day to the next,
the sequence of four drops making the largest drawdown occurs with a
probability 10?23, that is, once in about 4 thousands of billions of billions
of years. This exceedingly negligible value 10?23 suggests that the
hypothesis of uncorrelated daily returns is to be rejected: drawdowns,
especially the large ones, may exhibit intermittent correlations in the
asset price time series.
The Nasdaq Composite Index
In Figure 3.5, we see the rank ordering plot of drawdowns for the Nasdaq
composite index, from its establishment in 1971 until April 18, 2000.
The rank ordering plot, which is the same as the (complementary) cumulative
distribution with axes interchanged, puts emphasis on the largest
events. The four largest events are not situated on a continuation of the
distribution of smaller events: the jump between rank 4 and 5 in relative
value is larger than 33%, whereas the corresponding jump between rank
5 and 6 is less than 1%, and this remains true for higher ranks. This
means that, for drawdowns less than 12�5%, we have a more or less
“smooth” curve and then a larger than 33% gap to rank 3 and 4. The
1 10 100 1000
Loss (%)
Apr. 2000
Oct. 1987
Oct. 1987 Aug. 1998
Fig. 3.5. Rank ordering of drawdowns in the Nasdaq composite since its establishment
in 1971 until April 18, 2000. Rank 1 (Apr. 2000) is the largest drawdown,
rank 2 (Oct. 1987, top) is the second largest, etc. Reproduced from [217].
financial crashes are “outliers” 63
four events are, according to rank, the crash of April 2000, the crash
of October 1987, a larger than 17% “aftershock” related to the crash of
October 1987, and a larger than 16% drop related to the “slow crash” of
August 1998, which we shall discuss later, in chapter 7.
To further establish the statistical confidence with which we can conclude
that the four largest events are outliers, we have reshuffled the
daily returns 1,000 times and hence generated 1,000 synthetic data sets.
This procedure means that the synthetic data sets will have exactly the
same distribution of daily returns. However, higher order correlations and
dependence that may be present in the largest drawdowns are destroyed
by the reshuffling. This so-called “surrogate” data analysis of the distribution
of drawdowns has the advantage of being nonparametric, that is,
independent of the quality of fits with a model such as the exponential
or any other model. We will now compare the distribution of drawdowns
for both the real data and the synthetic data. With respect to the synthetic
data, this can be done in two complementary ways.
In Figure 3.6, we see the distribution of drawdowns in the Nasdaq
composite compared with the two lines constructed at the 99% confidence
level for the entire ensemble of synthetic drawdowns, that is,
by considering the individual drawdowns as independent: for any given
drawdown, the upper (respectively, lower) confidence line is such that
-0.25 -0.2 -0.15 -0.1 -0.05 0
Normalised Cumulative Distribution
Draw Down
Nasdaq Composite
99% confidence line
99% confidence line
Fig. 3.6. Normalized cumulative distribution of drawdowns in the Nasdaq
composite since its establishment in 1971 until April 18, 2000. The 99% confidence
lines are estimated from the synthetic tests described in the text. Reproduced from
64 chapter 3
five of the synthetic distributions are above (below) it; as a consequence,
990 synthetic times series out of the 1,000 are within the two confidence
lines for any drawdown value, which defines the typical interval within
which we expect to find the empirical distribution.
The most striking feature apparent in Figure 3.6 is that the distribution
of the true data breaks away from the 99% confidence intervals
at approximately 15%, showing that the four largest events are indeed
“outliers.” In other words, chance alone cannot reproduce these largest
drawdowns. We are thus forced to explore the possibility that an amplification
mechanism and dependence across daily returns might appear at
special and rare times to create these outliers.
A more sophisticated analysis is to consider each synthetic data set
separately and calculate the conditional probability of observing a given
drawdown given some prior observation of drawdowns. This gives a
more precise estimation of the statistical significance of the outliers,
because the previously defined confidence lines neglect the correlations
created by the ordering process which is explicit in the construction of
a cumulative distribution.
Out of 10,000 synthetic data sets that were generated, we find that 776
had a single drawdown larger than 16�5%, 13 had two drawdowns larger
than 16�5%, 1 had three drawdowns larger than 16�5%, and none had 4
(or more) drawdowns larger than 16�5% as in the real data. This means
that, given the distribution of returns, by chance we have an 8% probability
of observing a drawdown larger than 16�5%, a 0�1% probability
of observing two drawdowns larger than 16�5%, and for all practical
purposes, zero probability of observing three or more drawdowns larger
than 16�5%. Hence, we can reject the hypothesis that the four largest
drawdowns observed on the Nasdaq composite index could result from
chance alone with a probability or confidence better than 99�99%, that
is, essentially with certainty. As a consequence, we are led again to
conclude that the largest market events are characterized by a stronger
dependence than is observed during “normal” times.
This analysis confirms the conclusion from the analysis of the DJIA
shown in Figure 3.4 that drawdowns larger than about 15% are to be
considered as outliers with high probability. It is interesting that the same
amplitude of approximately 15% is found for both markets considering
the much larger daily volatility of the Nasdaq composite. This may result
from the fact that, as we have shown, very large drawdowns are more
controlled by transient correlations leading to runs of losses lasting a
few days than by the amplitude of a single daily return.
financial crashes are “outliers” 65
The statistical analysis of the DJIA and the Nasdaq composite suggests
that large crashes are special. In the following chapters, we shall
show that there are other specific indications associated with these
“outliers,” such as precursory patterns decorating the speculative bubbles
ending in crashes.
Further Tests
When one makes observations that deviate strikingly from existing belief
(technically called the “null hypothesis”), it is important to keep a cool
head and scrutinize all possible explanations. As Freeman Dyson eloquently
expressed [116],
The professional duty of a scientist confronted with a new and exciting
theory is to try to prove it wrong. That is the way science works. That is
the way science stays honest. Every new theory has to fight for its existence
against intense and often bitter criticism. Most new theories turn out
to be wrong, and the criticism is absolutely necessary to clear them away
and make room for better theories. The rare theory which survives the
criticism is strengthened and improved by it, and then becomes gradually
incorporated into the growing body of scientific knowledge.
The powerful method of investigation underlying Dyson’s verdict is
the so-called scientific method. In a nutshell, it consists in the following
steps: (1) we observe the data; (2) we invent a tentative description,
called a hypothesis, that is consistent with what we have observed; (3)
we use the hypothesis to make predictions; (4) we test those predictions
by experiments or further observations and modify the hypothesis in
light of our new results; (5) we repeat steps 3 and 4 until there are
only a few or no discrepancies between theory and experiment and/or
observation. When consistency is obtained, the hypothesis becomes a
theory and provides a coherent set of propositions that explain a class of
phenomena. A theory is then a framework within which observations are
explained and predictions are made. In addition, scientists use what is
known as “Occam’s razor,” also known as the law of parsimony, or the
law of simplicity: “When you have two competing theories which make
exactly the same predictions, the one that is simpler is preferable.” There
is a simple, practical reason for this principle: it makes life simpler for
the prediction of the future, as fewer factors have to be determined or
66 chapter 3
More important is the fact that fewer assumptions and fewer parameters
make the prediction of new phenomena more robust. Think, for
instance, of the two competing explanations of Descartes and Newton for
the regularities of planetary motions, such as those of Mercury, Venus,
the Earth, Mars, Jupiter, and Uranus orbiting around the sun. According
to Descartes, the motion of the planets could be explained by a complex
system of vortices moving the Ether (the hypothetical matter filling
space). In contrast, Newton proposed his famous universal inverse square
distance law for the gravitational attraction between any two massive
bodies. Both explanations are a priori valid and they can both explain
the planetary motions. The difference lies in the fact that Descartes’s
explanation could not be extrapolated to predict new observations, while
Newton’s law led to the prediction of the existence of undetected planets,
such as Neptune. The power of a model or a theory thus lies in its
prediction of phenomena that have not served to construct it. Einstein
put it this way: “A theory is more impressive the greater the simplicity
of its premises, the more different the kinds of things it relates and the
more extended its range of applicability.”
Here is where we stand with respect to the scientific method:

  1. We looked at financial data and found it apparently random.
  2. We formed the hypothesis that the time evolution of stock market
    prices are random walks.
  3. We used this hypothesis to make the prediction that the distribution of
    drawdowns should be exponential.
  4. We tested this prediction by constructing this distribution for the DJIA
    and found an apparent discrepancy, especially with respect to the
    largest drawdowns.
    Before rejecting our initial hypothesis and accepting the idea that
    stock market prices are not completely random, we must first verify that
    the observation is “statistically significant.” In plain words, this means
    that the deviation from the exponential could be the result of the smallness
    of the data set or other factors not identified and unrelated to the
    data itself. The apparent deviation from an exponential distribution would
    thus not be genuine but an error, an artifact of our measurements, or simply
    accidental. In order to try to exclude these traps, we thus need tests
    that tell us if the observed deviation is significant and credible. Indeed,
    Occam’s razor imposes that we should prefer the simpler hypothesis
    of randomness as long as the force of the evidence does not impose a
    change of our belief.
    financial crashes are “outliers” 67
    In order to see which one of the two descriptions (random or not
    random) is the most accurate, the following statistical analysis of market
    fluctuations is performed. First, we approximate the distribution
    of drawdowns for the DJIA up to 15% by an exponential and find a
    characteristic drawdown scale of 2%. This characteristic decay constant
    means that the probability of observing a drawdown larger than 2% is
    about 37%. Following the null hypothesis that the exponential description
    is correct and extrapolating this description to, for example, the
    three largest crashes on the U.S. market in this century (1914, 1929, and
    1987), as indicated in Figure 3.4, yields a recurrence time of about fifty
    centuries for each single crash. In reality, the three crashes occurred in
    less than one century. This result is a first indication that the exponential
    model may not apply for the large crashes.
    As an additional test, 10,000 so-called synthetic data sets, each covering
    a time span close to a century, hence adding up to about 1 million
    years, was generated using a standard statistical model used by the financial
    industry [46]. We use the model version GARCH(1,1) estimated
    from the true index with a student distribution with four degrees of freedom.
    This model includes both nonstationarity of volatilities (the amplitude
    of price variations) and the (fat tail) nature of the distribution of
    the price returns seen in Figure 2.7. Our analysis [209] shows that, in
    approximately 1 million years of heavy tail “GARCH-trading,” with a
    reset every century, never did three crashes similar to the three largest
    observed in the true DJIA occur in a single “GARCH-century.”
    Another approach is to use the GARCH model with Student distribution
    of the noise with 4 degrees of freedom fitted to the DJIA to construct
    directly the distribution of drawdowns and compare with real data.
    From synthetic price time series generated by the GARCH model, the
    distribution of drawdowns is constructed by following exactly the same
    procedure as in the analysis of the real time series. Figure 3.7 shows two
    dotted lines defined such that 99% of the drawdowns of the synthetic
    GARCH with noise Student distribution are within the two lines: there
    is thus a 1% probability that a drawdown in a GARCH time series falls
    above the upper line or below the lower line. Notice that the distribution
    of drawdowns from the synthetic GARCH model is approximately exponential
    or slightly subexponential for drawdowns up to about 10% and
    fits well the empirical drawdown distribution shown as the symbol + in
    the DJIA. However, the three largest drawdowns are clearly above the
    upper line. We conclude that the GARCH dependencies cannot (fully)
    account for the dependencies observed in real data, in particular in the
    68 chapter 3
    -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0
    Log (cumulative number)
    Fig. 3.7. The two dashed lines are defined such that 99% of the drawdowns of
    synthetic GARCH(1,1) with noise Student distribution with 4 degrees of freedom
    are within the two lines. The symbols + represent the cumulative distribution of the
    drawdowns for the DJIA. The ordinate is in logarithmic scale, while the abscissa
    shows the drawdowns; for instance, ?0.30 corresponds to a drawdown of ?30%.
    Reproduced from [399].
    special dependence associated with very large drawdowns. This illustrates
    that one of the most used benchmark models in finance fails to
    match the data.
    This novel piece of evidence, adding upon the previous rejection of the
    null hypothesis that reshuffled time series exhibit the same drawdowns
    as the real time series (see also below), strengthens the claim that large
    drawdowns are outliers.
    Of course, these tests do not tell us what the correct model is. They
    only show that one of the standard models of the financial industry and
    of the academic world (which makes a reasonable null hypothesis of random
    markets) is utterly unable to account for the stylized facts associated
    with large financial crashes. It suggests that different mechanisms are
    responsible for large crashes. This conclusion justifies the special status
    that the media and the public in general attribute to financial crashes.
    If the largest drawdowns are outliers, we must consider the possibility
    that they may possess a higher degree of predictability than the smaller
    market movements.
    financial crashes are “outliers” 69
    This is the subject of the present book. The program in front of us is
    to build on this observation that large crashes are very special events in
    order to try understanding how and why, and then test for their potential
    predictability. Before proceeding, we summarize the evidence for the
    existence of outliers in other financial market securities. As outliers will
    be shown to be ubiquitous, this will force us to construct specific models
    for them.
    The data sets that have been analyzed [220] comprise
  5. major world financial indices: the Dow Jones, Standard & Poors, Nasdaq
    composite, TSE 300 Composite (Toronto, Canada), All Ordinaries
    (Sydney stock exchange, Australia), Strait Times (Singapore stock
    exchange), Hang Seng (Hong Kong stock exchange), Nikkei 225
    (Tokyo stock exchange, Japan), FTSE 100 (London stock exchange,
    U.K.), CAC40 (Paris stock exchange, France), DAX (Frankfurt stock
    exchange, Germany), MIBTel (Milan stock exchange, Italy);
  6. currencies: U.S. dollar versus German mark (UD$/DM), U.S. dollar
    versus Japanese yen (UD$/Yen), U.S. dollar versus Swiss franc
  7. gold;
  8. the twenty largest companies in the U.S. market in terms of capitalization,
    as well as nine others taken randomly in the list of the fifty largest
    companies (Coca Cola, Qualcomm, Appl. Materials, Procter & Gamble,
    JDS Uniphase, General Motors, Am. Home. Prod., Medtronic, and
    These different data sets do not have the same time span, largely due to
    different life spans, especially for some recent “new technology” companies.
    This selection of time series is far from exhaustive but is a reasonable
    sample for our purpose: as we shall see, with the exception of
    the index CAC40 (the “French exception”?), all time series exhibit clear
    outlier drawdowns. This suggests that outliers constitute a ubiquitous
    feature of stock markets, independently of their nature.
    70 chapter 3
    0.3 0.25 0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.25 0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.8. Standard & Poor’s (left) and TSE 300 composite (right). Note the isolated
  • at the bottom-left corner of each panel, indicating the largest drawdawn, clearly
    an “outlier.” Its value on the vertical axis is 0 because only one such large event
    was observed and the logarithm of 1 is 0. Indeed, recall that this kind of cumulative
    distribution counts events from bottom to top, sorting them from the largest to the
    smallest when spanning from left to right. Reproduced from [220].
    Main Stock Market Indices, Currencies, and Gold
    The set of Figures 3.8–3.14 tests whether the observations documented
    in the previous section for the U.S. markets is specific to it or is a general
    feature of stock market behavior. We have thus analyzed the main
    stock market indices of the remaining six G7 countries as well as those
    of Australia, Hong Kong, and Singapore and the other important U.S.
    0.3 0.25 0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.3 0.25 0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.9. All Ordinaries (Australian) (left) and Strait Times (Singapore) (right). Note
    again the isolated + at the bottom-left corner of each panel, indicating the largest
    drawdawn, clearly an outlier. Reproduced from [220].
    financial crashes are “outliers” 71
    0.4 0.35 0.3 0.25 0.2 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.15 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.10. Hang Seng (Hong Kong) (left) and Nikkei 225 (Japan) (right). Reproduced
    from [220].
    0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.12 0.1 0.08 0.06 0.04 0.02 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.11. FTSE 100 (United Kingdom) (left) and CAC 40 (France) (right). Reproduced
    from [220].
    0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
    6 Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.12. DAX (Germany) (left) and MIBTel (Italy) (right). Reproduced from
    72 chapter 3
    -0.12 -0.1 -0.06 -0.04 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    -0.08 -0.02 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.13. U.S. dollar/DM currency (left) and U.S. dollar/Yen currency (right).
    Reproduced from [220].
    index called the Standard & Poor’s 500 index. The results of this analysis
    are shown in Figures 3.8–3.12. Quite remarkably, we find that all
    markets except the French market, with the Japanese market being on the
    borderline, show the same qualitative behavior exhibiting outliers. The
    Paris stock exchange is the only exception as the distribution of drawdowns
    is an almost perfect exponential. It may be that the observation
    time used for CAC40 is not large enough for an outlier to have occurred.
    If we compare with the Milan stock market index MIBTel, we see that
    the entire distribution except the single largest drawdown is also close to
    a pure exponential. The presence or absence of this outlier thus makes
    all the difference. In the case of the Japanese stock market, we note that
    it exhibited a general decline from 1990 to early 1999, which is more
    0.12 0.1 0.06 0.04 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.08 0.02 0.25 0.2 0.15 0.1 0.05 0
    8 Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.14. U.S. dollar/CHF currency (left) and gold (right). Reproduced from [220].
    financial crashes are “outliers” 73
    than a third of the data set. The total decline was approximately 60% in
    amplitude. This may explain why the evidence is less striking than for
    the other indices.
    Figures 3.13 and 3.14 show that similar behavior is observed also for
    currencies and for gold. Summarizing, the results of the analysis of different
    stock market indices, the exchange of the U.S. dollar against three
    different major currencies as well as the gold market are that outliers are
    ubiquitous features of major financial markets [220].
    Largest U.S. Companies
    Let us now extend this analysis to the very largest companies in the
    United States in terms of capitalization (market value) [220]. The ranking
    is that of Forbes at the beginning of the year 2000. The top twenty have
    been chosen, with, in addition, a random sample of other companies,
    namely number 25 (Coca Cola), number 30 (Qualcomm), number 35
    (Appl. Materials), number 39 (JDS Uniphase), number 46 (Am. Home
    Prod.), and number 50 (Medtronic). Three more companies have been
    added in order to get longer time series as well as representatives of the
    automobile sector. These are Procter & Gamble (number 38), General
    Motors (number 43), and Ford (number 64). This represents an unbiased
    selection based on objective criteria. We show here only the distribution
    of drawdowns for the six first ranks and refer to [220] for access to the
    full data set.
    From Figures 3.15, 3.16, and 3.17, we can see that the distributions of
    the five largest companies (Microsoft, Cisco, General Electric, Intel, and
    Exxon-Mobil) clearly exhibit the same features as those for the major
    financial markets. Of the remaining 23, for all but America Online and
    JDS Uniphase, we find clear outliers but also a variety of different tails
    of the distributions. It is interesting to note that the two companies,
    America Online and JDS Uniphase, whose distributions did not exhibit
    outliers are also the two companies with by far the largest number per
    year of drawdowns of amplitude above 15% (close to 4).
    Drawups can be similarly defined as runs of positive returns beginning
    after a loss and stopping at a loss. The distributions of drawups also
    exhibit outliers but less strikingly than the distribution of drawdowns
    74 chapter 3
    0.4 0.35 0.2 0.15 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.3 0.25 0.1 0.05 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.15. Microsoft (left) and Cisco (right). Reproduced from [220].
    0.3 0.2 0.15 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.25 0.1 0.05 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.16. General Electric (left) and Intel (right). Reproduced from [220].
    0.3 0.2 0.15 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    0.25 0.1 0.05 0.3 0.25 0.2 0.15 0.1 0.05 0
    Log(Cumulative Number)
    Draw Down
    Null Hypothesis
    Draw Down
    Fig. 3.17. Exxon-Mobil (left) and Oracle (right). Reproduced from [220].
    financial crashes are “outliers” 75
    We have found the following facts [211, 217, 220].
  1. Approximately 1% to 2% of the largest drawdowns are not at all
    explained by the exponential null hypothesis or its extension in terms
    of the stretched exponential [253]. Large drawdowns up to three times
    larger than expected from the null hypothesis are found to be ubiquitous
    occurrences of essentially all the times series that we have investigated,
    the only noticeable exception being the French index CAC40.
    We term these anomalous drawdowns “outliers.”
  2. About half of the time series show outliers for the drawups. The
    drawups are thus different statistically from the drawdowns and constitute
    a less conspicuous structure of financial markets.
  3. For companies, large drawups of more than 15% occur approximately
    twice as often as large drawdowns of similar amplitudes.
  4. The bulk (98%) of the drawdowns and drawups are very well fitted by
    the exponential null hypothesis (based on the assumption of independent
    price variations) or by a slight generalization called the stretched
    exponential model.
    The most important result is the demonstration that the very largest
    drawdowns are outliers. This is true notwithstanding the fact that the
    very largest daily drops are not outliers, except for the exceptional daily
    drop on October 29, 1987. Therefore, the anomalously large amplitude
    of the drawdowns can only be explained by invoking the emergence of
    rare but sudden persistences of successive daily drops, with, in addition,
    correlated amplification of the drops. Why such successions of correlated
    daily moves occur is a very important question with consequences for
    portfolio management and systemic risk, to cite only two applications
    that we will investigate in the following chapters.
    Systemic risks refer to the risk that a disruption (at a firm or bank, in a
    transfer system) causes widespread difficulties at other firms, or in other
    market segments. Systemic risk is the risk that such a failure could cause,
    at the extreme, a complete breakdown in a financial system due to the
    extensive linkages of today’s markets. Such a risk of contagion arising
    from a disruption at a firm or in one market is known as systemic risk.
    That systemic safety can be threatened by the failure of one small institution
    was vividly demonstrated in September 1998 when the U.S. Federal
    Reserve Bank organized a rescue of a hedge fund, Long-Term Capital
    76 chapter 3
    Management, because it feared the fund’s collapse would set off havoc in
    the financial markets. LTCM had market exposures of over $200 billion,
    while its capital base was about $4.8 billion.
    See, for instance, http://riskinstitute.ch/134720.htm for more information
    and a summary of countermeasures used to ensure systemic safety.
    Lillo and Mantegna [267] have recently convincingly documented
    another clear indication that crash and rally days differ significantly
    from typical market days in their statistical properties. Specifically, they
    investigated the return distributions of an ensemble of stocks simultaneously
    traded on the New York Stock Exchange (NYSE) during market
    days of extreme crash or rally in the period from January 1987 to
    December 1998. The total number of assets n traded on the NYSE is
    rapidly increasing and it ranges from 1,128 in 1987 to 2,788 in 1998.
    The total number of data records treated in this analysis thus exceeds
    6 million.
    Figure 3.18 shows 200 distributions of returns, one for each of 200
    trading days, where the ensemble of returns is constructed over the whole
    set of stocks traded on the NYSE. A sectional cut at a fixed trading day
    retrieves the kind of plot shown in Figure 2.7 (except for the absence
    of the folding back of the negative returns performed in Figure 2.7).
    Figure 3.18 clearly shows the anomalously large widths and fat tails
    on the day of the crash of October 19, 1987, as well as during other
    turbulent days.
    Lillo and Mantegna [267] documented another remarkable behavior
    associated with crashes and rallies, namely that the distortion of the
    distributions of returns are not only strong in the tails describing large
    moves but also in their center. Specifically, they show that the overall
    shape of the distributions is modified on crash and rally days. To show
    this, the distributions of the nine trading days with the largest drops and
    of the nine trading days with the largest gains of the Standard & Poors
    500 given in Table 3.3 are shown in Figures 3.19 and 3.20.
    Figure 3.19 shows that on crash days the distribution of returns has a
    peak at a negative value and is skewed with an asymmetric and longer
    tail towards negative return. Not only are there more drops than gains
    among all assets, but the drops are more pronounced. The converse is
    true for rally days, as shown in Figure 3.20. Therefore, on crash and
    financial crashes are “outliers” 77
    Fig. 3.18. Contour and surface plot of the ensemble return distribution in a 200-
    trading-days time interval centered at October 19, 1987 (corresponding to 0 in the
    abscissa). The probability density scale (z-axis) of the surface plot is logarithmic, so
    that a straight decay qualifies exponential distributions. The contour plot at the top is
    obtained for equidistant intervals of the logarithmic probability density. The brightest
    area of the contour plot corresponds to the most probable value. The symbol R
    stands for return. Reproduced from [267].
    rally days, not only the scale but also the shape and symmetry properties
    of the distribution change.
    The change of the shape and of the symmetry properties during the
    days of large absolute returns (crashes and rallies) suggests that, on
    extreme days, the behavior of the market cannot be statistically described
    in the same way as during “normal” periods.
    The realization that large drawdowns and crashes in particular may
    result from a run of losses over several successive days is not without
    consequences for the regulation of stock markets. Following the market
    78 chapter 3
    Table 3.3
    Date S&P 500 return Panel
    19 10 1987 ?0�2041 3.19a
    26 10 1987 ?0�0830 3.19b
    27 10 1997 ?0�0686 3.19c
    31 08 1998 ?0�0679 3.19d
    08 01 1988 ?0�0674 3.19e
    13 10 1989 ?0�0611 3.19f
    16 10 1987 ?0�0513 3.19g
    14 04 1988 ?0�0435 3.19h
    30 11 1987 ?0�0416 3.19i
    21 10 1987 +0�0908 3.20a
    20 10 1987 +0�0524 3.20b
    28 10 1997 +0�0511 3.20c
    08 09 1998 +0�0509 3.20d
    29 10 1987 +0�0493 3.20e
    15 10 1998 +0�0418 3.20f
    01 09 1998 +0�0383 3.20g
    17 01 1991 +0�0373 3.20h
    04 01 1988 +0�0360 3.20i
    List of the eighteen days of the investigated period (from January
    1987 to December 1998) in which the S&P 500 index had the
    greatest return in absolute value. The third column indicates the
    corresponding panel of the ensemble return distribution shown in
    Figures 3.19 and 3.20. Reproduced from [267].
    crash of October 1987, in an attempt to head off future one-day stock
    market tumbles of historic proportions, the Securities and Exchange
    Commission and the three major U.S. stock exchanges agreed to install
    so-called circuit breakers. Circuit breakers are designed to gradually
    inhibit trading during market declines, first curbing NYSE program
    trades and eventually halting all U.S. equity, options, and futures activity.
    Similar circuit breakers are operating in the other world stock markets
    with different specific definitions.
    Circuit breaker values. Effective April 15, 1998, the SEC approved new
    circuit breaker trigger levels for one-day declines in the DJIA of 10%,
    20%, and 30%. The halt for a 10% decline will be one hour if triggered
    before 2:00 p.m. Eastern Standard Time (EST). At or after 2:00 p.m. EST
    but before 2:30 p.m. EST, the halt will be for one half-hour. At or after
    2:30 p.m. EST, the market will not halt at the 10% level and will continue
    financial crashes are “outliers” 79
    a b c
    d e f
    g h i
    -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2
    R R R
    Fig. 3.19. Ensemble return distribution in days of S&P 500 index extreme negative
    return occurring in the investigated time period (listed in the first part of Table 3.3).
    The ordinate is in logarithmic scale. PDF stands for probability distribution function.
    Reproduced from [267].
    trading. The halt for a 20% decline will be two hours if triggered before
    1:00 p.m. EST. At or after 1:00 p.m. EST but before 2:00 p.m. EST,
    the halt will be for one hour. If the 20% trigger value is reached at or
    after 2:00 p.m. EST, trading will halt for the remainder of the day. If
    the market declines by 30%, at any time, trading will be halted for the
    remainder of the day. Previously, the circuit breakers were triggered when
    the DJIA declined 350 points (thirty-minute halt) and 550 points (onehour
    halt) from the previous day’s close. The circuit breakers are based
    on the average closing price of the Dow for the month preceding the start
    of each calendar quarter.
    The argument is that the halt triggered by a circuit breaker will provide
    time for brokers and dealers to contact their clients when there are
    large price movements and to get new instructions or additional margin.
    They also limit credit risk and loss of financial confidence by providing
    a “time-out” to settle up and to ensure that everyone is solvent. This
    inactive period is of further use for investors to pause, evaluate, and
    inhibit panic. Finally, circuit breakers expose the illusion of market liq80
    chapter 3
    a b c
    d e f
    g h i
    -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2
    R R R
    Fig. 3.20. Ensemble return distribution in days of greatest S&P 500 index positive
    return occurring in the investigated time period (listed in the second part of
    Table 3.3). The ordinate is in logarithmic scale. PDF stands for probability distribution
    function. Reproduced from [267].
    uidity by spelling out the economic fact of life that markets have limited
    capacity to absorb massive unbalanced volumes. They thus force large
    investors, such as pension portfolio managers and mutual fund managers,
    to take even more account of the impact of their “size order,” thus
    possibly cushioning large market movements.
    However, others argue that a trading halt can increase risk by inducing
    trading in anticipation of a trading halt. Another disadvantage is that
    they prevent some traders from liquidating their positions, thus creating
    market distortion by preventing price discovery [188].
    As shown in [30] for the October 1987 crash, countries that had stringent
    circuit breakers, such as France, Switzerland, and Israel, also had
    some of the largest cumulative losses. According to our finding that large
    drops are created by transient and rare dependent losses occurring over
    several days, circuit breakers should not be considered reliable crash
    chapter 4
    positive feedbacks
    Human behavior is a main factor in how markets
    act. Indeed, sometimes markets act quickly, violently
    with little warning. � � � Ultimately, history tells us
    that there will be a correction of some significant
    dimension. I have no doubt that, human nature being
    what it is, that it is going to happen again and again.
    — Alan Greenspan, before the Committee on Banking
    and Financial Services, U.S. House of
    Representatives, July 24, 1998.
    The previous chapter 3 documented convincingly
    that essentially all markets exhibit rare but anomalously large runs
    of successive daily losses. How can we explain the existence of these
    exceptionally large drawdown outliers?
    Since it is the actions of investors whose buy and sell decisions move
    prices up and down, any deviation from a random walk has ultimately to
    be traced back to the behavior of investors. We are particularly interested
    in mechanisms that may lead to positive feedbacks on prices, that is, to
    the fact that, conditioned on the observation that the market has recently
    moved up (respectively, down), this makes it more probable to keep it
    moving up (respectively, down), so that a large cumulative move ensues.
    The concept of “positive feedbacks” has a long history in economics
    and is related to the idea of “increasing returns,” which says that goods
    become cheaper the more of them are produced (and the closely related
    82 chapter 4
    idea that some products, like fax machines, become more useful the
    more people use them). “Positive feedback” is the opposite of “negative
    feedback,” a concept well known, for instance, in population dynamics:
    the larger the population of rabbits in a valley, the less grass there is
    per rabbit. If the population grows too much, the rabbits will eventually
    starve, slowing down their reproduction rate, which thus reduces
    their population at a later time. Thus negative feedback means that the
    higher the population, the slower the growth rate, leading to a spontaneous
    regulation of the population size; negative feedbacks thus tend to
    regulate growth towards an equilibrium. In contrast, positive feedback
    asserts that the higher the price or the price return in the recent past, the
    higher will be the price growth in the future. Positive feedbacks, when
    unchecked, can produce runaways until the deviation from equilibrium is
    so large that other effects can be abruptly triggered and lead to ruptures
    or crashes. Youssefmir, Huberman, and Hogg [460] have stressed the
    importance of positive feedback in a dynamical theory of asset price bubbles
    that exhibits the appearance of bubbles and their subsequent crashes.
    The positive feedback leads to speculative trends which may dominate
    over fundamental beliefs and which make the system increasingly susceptible
    to any exogenous shock, thus eventually precipitating a crash.
    There are many mechanisms in the stock market and in the behavior
    of investors that may lead to positive feedbacks. Figure 4.1 provides a
    humorous account of trader folklore on the many influences and factors
    active in the stock market. Some of these influences lead to negative
    feedbacks, others to amplification.
    We first sketch the evolution of economic thinking in relation to
    feedback and self-organization, then describe how positive feedback on
    prices can result from hedging of derivatives and from insurance portfolio
    strategies. We follow by turning to a general mechanism for positive
    feedback, which is now known as the “herd” or “crowd” effect, based
    on imitation processes. We present a simple model of the best investment
    strategy that an investor can develop based on interactions with
    and information taken from other investors. We show how the repetition
    of these interactions may lead to a remarkable cooperative phenomenon
    in which the market can suddenly “solidify” a global opinion, leading to
    large price variations.
    The recognition of the importance of feedbacks to fathom the sheer complexity
    of economic systems has been at the root of economic thinking
    positive feedbacks 83
    Crash here
    Equity Markets Overview
    Bears bail
    yield curve
    TV ads
    30yr bond
    P/E’s of 2000
    breakup New Era
    is in”
    16 year olds
    beat market vets
    Dollar goes every
    which way
    Buy on dips
    Flight to safety
    Buy and
    hold forever
    CNBC guest
    new metric
    tiny jobs
    big jobs
    Asia up Telecom Fuel Cell 4:1 splits Stewart
    From “Hold”
    to “Strong Buy”
    NASDAQ special
    GDP at 7%
    growth rate
    Greenspan speaks
    What bubble?
    Wireless! Maria excited
    Cash is
    Question basic
    Fund trade
    Back into
    8 week
    bear market!?
    Bull market
    Better off
    in cash?
    Looking good!
    Any gains lost
    in next day rally
    Finally, fundamentals
    reassert themselves Slippage Insults
    Bulls sneer NAV angst
    Academics agree,
    market is crazy
    Internet Bio-tech Shorts killed New price targets
    Fig. 4.1. Cartoon illustrating the many factors influencing traders, as well as the
    psychological and social nature of the investment universe (source: anonymous).
    for a long time. Indeed, the general equilibrium theory is nothing but a
    formalization of the idea that “everything in the economy affects everything
    else” [244]. The historical root and best pictorial synthesis of this
    idea is found in the work of 18th-century Scotsman Adam Smith. Smith’s
    masterpiece [384], An Inquiry into the Nature and Causes of the Wealth
    of Nations, introduced the then-radical notion that selfish, greedy individuals,
    if allowed to pursue their interests largely unchecked, would
    interact to produce a wealthier society as if guided by an “invisible hand.”
    Smith never worked out a proof that this invisible hand existed. Not all
    subsequent economists agreed with his optimistic assessment. T. Malthus
    thought people would have too many children and overpopulate the
    world. Karl Marx thought capitalists would be so greedy they would
    84 chapter 4
    bring down the system. But they all shared Smith’s view of economics
    as the study of people trying to maximize their material well being. In
    1954, K. Arrow and G. Debreu [16] published an article that in essence
    mathematically proved the existence of Adam Smith’s invisible hand.
    This “general equilibrium” proof, which relies on a set of very restricted
    assumptions of an idealized world, has been a mainstay of graduate-level
    economics training ever since.
    The most important tool in this analysis was game theory: the study
    of situations, like poker or chess games, in which players have to make
    their decisions based on guesses about what the other player is going
    to do next. Game theory was first adapted to economics in the 1940s
    by mathematician John von Neumann (the same von Neumann whose
    theoretical insights made the computer possible) and economist O. Morgenstern.
    Since then, the standard economics and social science model
    of a human agent is that it is like a general-purpose logic machine. All
    decision tasks, regardless of context, constitute optimization problems
    subject to external constraints whether from the physical environment or
    from the reaction functions of other agents. This central dogma is the
    core of economics courses taught in universities and is often found very
    difficult to “swallow” by students, many of whom give up, unable to
    learn it. This idealization both is convenient for the development of a
    coherent theoretical framework and has many rich consequences. However,
    it is a poor representation of reality, as most of us are actually not
    versed in economic optimization reasoning! The remarkable insight of
    Adam Smith is that this does not mean we shall fail to function effectively
    in social and economic exchanges in life. This is because people
    have natural intuitive mechanisms—mind modules that serve them
    well in daily interchanges—enabling them to “read” situations and the
    intentions and likely reactions of others without deep, tutored, cognitive
    analysis. This fact has been established by “experiments” performed
    by a large school of economics researchers (the bibliography of which
    contains 1500 entries [197]) in the fields of “experimental economics”
    These experimental approaches to economics, started in the midtwentieth
    century, were developed to examine propositions implied by
    economic theories of markets. An untested theory is simply a hypothesis,
    and science seeks to expand our knowledge of things by a process of testing
    hypotheses. In contrast, much of traditional economic theory can be
    called, appropriately, “ecclesiastical theory”; it is accepted (or rejected)
    on the basis of authority, tradition, or opinion about assumptions, rather
    than on the basis of having survived a rigorous falsification process
    positive feedbacks 85
    that can be replicated. Hundreds of experiments on artificial markets
    constructed and performed with students from economics classes and
    with professionals have shown the crucial importance of repeating
    interactions in the presence of unconscious decisions in order to lead
    to an apparent rationality in rule-governed problems [390]. In these socalled
    continuous double auction experiments, which attempt to mimick
    real market situations, subjects have private information on their own
    willingness-to-pay or willingness-to-accept schedules which bound the
    prices at which each can profitably trade. No subject has information
    on market supply and demand. After an experiment, upon interrogation,
    the participants deny that they could have maximized their monetary
    earnings or that their trading results could be predicted by a theory.
    Yet despite these conditions, the subjects tend to converge quickly over
    time to the competitive equilibrium. Thus “the most common responses
    to the market question were unorganized, unstable, chaotic, and confused.
    Students were both surprised and amazed at the conclusion of
    the experiment when the entrusted student opened a sealed envelope
    containing the correctly predicted equilibrium price and quantity” [157].
    The fact that economic agents can achieve efficient outcomes that are
    not part of their intentions was the key principle formulated by Adam
    Smith [384], as we already stressed. Indeed, “in many experimental markets,
    poorly informed, error-prone, and uncomprehending human agents
    interact through the trading rules to produce social algorithms which
    demonstrably approximate the wealth maximizing outcomes traditionally
    thought to require complete information and cognitively rational actors”
    In much of the literature on experimental economics [101, 226, 143],
    the rational expectations model has been the main benchmark against
    which to check the informational efficiency of experimental markets.
    The research generally falls into two categories: information dissemination
    between fully informed agents (“insiders”) and uninformed agents,
    and information aggregation among many partially informed agents. The
    former experiments investigate the common intuition that market prices
    reflect insider information, hence uninformed traders should be able to
    infer the true price from the market. The latter experiments explore the
    aggregation of diverse information by partially informed agents, a more
    challenging objective because none of the agents possesses full information
    (traders identify the state of the world with certainty only by
    pooling their private information through the process of trading). Experiments
    on markets with both insiders and uninformed traders [333, 334]
    86 chapter 4
    show that equilibrium prices do reveal insider information after several
    trials of the experiments, suggesting that the markets disseminate
    information efficiently. The success of the rational expectations model
    can be attributed to the fact that traders learn about the equilibrium
    price and the state of the world simultaneously from market conditions
    However, these results are not always present if the following conditions
    are not fulfilled [334, 137]: identical preferences, common knowledge
    of the dividend structure, and complete contingent claims (i.e.,
    existence of a full spectrum of derivative instruments allowing one to
    probe the expectation of future risks). These studies provide examples of
    the failure of the rational expectations model and suggest that information
    aggregation is a more complicated situation. In particular, it seems
    that market efficiency, defined as full information aggregation, depends
    on the “complexity” of the market, as measured by market parameters
    such as the number of stocks and the number of trading periods in the
    market [319]. For instance, overreaction of people to trades that are
    uninformative may create self-generated information “mirages,” which
    may provide an explanation for the apparent excess volatility of asset
    prices [67]. Furthermore, there is evidence from market experiments
    about two types of judgment errors: errors in judging exogeneous events
    that affect the value of assets and errors in judging variables that are
    endogeneously created by market activity, such as prices in future trading
    periods. Notwithstanding ideal learning conditions, individual errors
    are not eliminated, but are, at best sometimes reduced [65]. Another
    idiosyncrasy of human beings highlighted by experiments is the so-called
    “disposition effect,” corresponding to the tendency to sell assets that
    have gained value and to keep assets that have lost value [446]. Disposition
    effects can be explained by the idea that people value gains
    and losses relative to a reference point and have a tendency to seek risk
    when faced with possible losses but to avoid risk when a certain gain is
    possible. Another important psychological trait is that most people are
    overconfident about their own relative abilities and unreasonably optimistic
    about their futures. This has been shown to influence economic
    behavior, such as entry into competitive games or investment in stock
    It is in this context that the concept of the “emergence” of a macroscopic
    organization from the repeated action of simple rules at the microscopic
    level is particularly intriguing. The main question concerns the
    qualities of agents that are crucial to shape the properties of this emergence.
    This question is now at the center of an exciting and vigorous
    positive feedbacks 87
    body of research aimed at understanding “complex systems” as a result
    of self-organization mechanisms [8]. Philip W. Anderson, a condensedmatter
    physicist and Nobel laureate in physics at Princeton University,
    contended in “More Is Different” [7] an essay published in 1972, that
    particle physics and indeed all reductionist approaches have only a limited
    ability to explain the world. Reality has a hierarchical structure,
    Anderson argued, with each level independent, to some degree, of the
    levels above and below. “At each stage, entirely new laws, concepts and
    generalizations are necessary, requiring inspiration and creativity to just
    as great a degree as in the previous one,” Anderson noted. “Psychology
    is not applied biology, nor is biology applied chemistry.”
    This “emergence” principle does not imply, however, that the “market”
    will always be equivalent to an efficient and global optimization machine.
    Actually, empirical economics in particular has taught us that market
    forces may lead to plenty of imperfections, problems, and paradoxes,
    depending on many different ingredients that are indeed present in reallife
  5. Trading rules of market institutions seem to matter significantly in the
    realization of efficient markets. Inadequate methods of pricing may
    lead to a slow and inefficient convergence to the equilibrium price or
    event to a divergence from it.
  6. Providing subjects with complete information, far from improving market
    competition, may tend to make it worse. Indeed, when people have
    complete information, they can identify more self-interested outcomes
    than competitive equilibria and use punishing strategies in an attempt
    to achieve them, which delays reaching equilibrium.
  7. There is no assurance that a public announcement will yield common
    expectations among the players, since each person may still be uncertain
    about how others will use the information.
  8. According to survey studies reported by Kahneman, Knetsch, and
    Thaler [227], people indicate that it is unfair for firms to raise prices
    and increase profits in response to certain changes in the environment
    that are not justified by an increase in costs. Thus, respondents report
    that it is “unfair” for firms to raise the price of snow shovels after
    a snowstorm or to raise the price of plywood after a hurricane. In
    these circumstances, economic theory predicts shortages, an increase
    in prices toward the new market clearing levels, and, eventually, an
    increase in output. In other words, the increase of price is the equilibrium
    solution associated with the new supply–demand relationship,
    88 chapter 4
    but this is considered unfair by people. How this perception impacts
    the real dynamics of the price and the behavior of firms and buyers
    to give rise to efficient or inefficient markets remains a subject of
  9. Prices in experimental asset markets tend to bubble and then crash
    to their dividend value at the end of the asset’s useful life [335].
    The introduction of a futures market, that allows participants to obtain
    information on future share prices, is found to reduce the bubbles in
  10. The experience of traders is paramount to the appearance of bubbles
    and crashes in these synthetic experimental markets. Providing full
    information on the future dividend flow, which should give full information
    on the equilibrium price of the corresponding asset, had little
    effect on the character of bubbles with inexperienced traders [335].
    Repeating the market game several times, the bubbles tend to decrease
    in amplitude.
  11. The phenomenon of “herding,” discussed at length in the remainder of
    this chapter, can also be considered an example of market failure, as
    it leads to important deviations from “fundamental” or “equilibrium”
    This research has fertilized many novel approaches that are working
    out ways in which rational behavior could lead to less-than-optimal
    market outcomes. Another important step has been the introduction of
    so-called “information asymmetry,” which describes situations in which
    different parties to a transaction possess different amounts of information.
    Such “asymmetric information,” the fact that people are not equal
    with respect to the quality and quantity of information they use to make
    decisions, blossomed in the seventies as a way to explain the behavior of
    financial markets, which are indeed extremely susceptible to information
    The present situation is that economics has moved away from the
    dead certainties of the past into a much more interesting universe of
    research possibilities including, as we shall see, imperfection, bounded
    rationalities, behaviors, and even psychology. The mathematical models
    that had come to form the basis of academic economics are shifting from
    general equilibrium, in which everything would work out for the best,
    to multiple equilibriums and out-of-equilibrium, in which it might not.
    The resulting encompassing concept is that the economy and the stock
    market are self-organizing systems.
    positive feedbacks 89
    Consider, for instance, a so-called call or buy option, which is a financial
    instrument issued by, say, a bank on an underlying stock such as
    IBM. An option gives the buyer the right, but not the obligation, to buy
    an IBM share in the future at a predefined price xc (usually called the
    “strike”). It is clear that, if the IBM price goes up above this predefined
    price xc, the option acquires a value equal to the difference between the
    IBM price and the predefined price xc, since the owner of the option
    can always buy at xc from the bank and sell immediately at market
    value, pocketing the difference. In order to be able to provide the IBM
    stock to the option holder, the bank has to buy the stock at the market
    value, if it has not taken the precautionary measure of holding some
    stock in reserve. This means that the bank has a potential maximum
    loss equal to the potential gain of the option holder. But the bank is
    not weaponless in this situation, as it can cover its risks against such a
    possibility by buying the stock in advance at a cheaper price, a procedure
    called “hedging.” Such hedging strategy leads to positive feedback:
    if the price increases, the option issuer should buy more of the underlying
    stock to hedge its position and prepare to deliver to the option
    buyer. Buying the stock obviously provides a driving force for further
    increase of the price, hence the positive feedback. This is only one example
    among many cases associated with derivative products in financial
    A related phenomenon is the increase in market volatility of asset
    prices that have been observed and analyzed in recent years (see, for
    instance, Table 4.1 for a striking illustration) and its cause has often been
    attributed to the popularity of hedging strategies for derivative securities.
    It can indeed be shown that optimal hedging strategies (using improvements
    of the famous Black and Scholes methodology) not only provide
    a positive feedback on prices, they also increase the price volatility
    [381]. As Miller [298] noted, the view is widespread and is expressed
    almost daily in the financial press: stock market volatility has been rising
    in the last decade mainly due to the introduction of low-cost speculative
    vehicles such as stock index futures and options. It is, however,
    naive to attribute the increase in volatility only to this origin. As
    we shall see, there are many other causes, and disentangling them is
    90 chapter 4
    Table 4.1
    Date High ? low Close(t) ? close(t ? 1)
    27 Oct 97 8% ?8%
    28 Oct 97 12% +6%
    31 Aug 98 12% ?11%
    1 Sept 98 6% +8%
    4 Apr 00 15% ?1%
    12 Apr 00 9% ?8%
    14 Apr 00 12% ?11%
    17 Apr 00 12% +9%
    27 Apr 00 8% +5%
    23 May 00 9% ?7%
    24 May 00 9% +5%
    13 Oct 00 8% +8%
    Daily highs minus lows larger than 5% for the Nasdaq composite index
    over the time period from 1991 to October 2000. Out of the twelve moves
    of more than 5% since 1991, none occurred before 1997 and eight have
    occurred in the time interval from April to October 2000! Notice that the
    variation of the close close�t� ? close�t ? 1� from one day to the next
    day is not always a good signature of the excitement of the day, as can
    be seen for instance on April 4.
    A second mechanism is provided by investment strategies with an
    “insurance portfolio.” Indeed, the initial assessment of the origins of
    the October 1987 crash pointed to the then-popular hedging strategies
    deriving from portfolio insurance models. In a nutshell, such strategies
    consist of selling when price decreases below a threshold (stop loss)
    and in buying when price increases. It is clear that by increasing the
    volume of sell orders following a price decrease, this may lead to further
    price decreases, possibly cascading in a downward spiral. The 1988
    Brady Commission appointed to investigate the cause of the 1987 crash
    has indeed named portfolio insurance as a major factor contributing to
    the downward pressure on stock prices that led to the crash of October
  12. Recent works, for instance, Barlevy and Veronesi [28], show that
    uninformed traders can behave as insurance portfolios and precipitate
    a price crash because, as price declines, they reasonably surmise that
    better informed traders could have received negative information which
    leads them to reduce their own demand for assets, driving the price of
    stocks even lower.
    positive feedbacks 91
    Behavioral Economics
    In debates and research on the social sciences, the sciences dealing with
    human societies, it is customary to oppose two approaches, the first striving
    for objectivism, the second being more interpretative.
    � The first approach attempts to view “social facts” as “material things,”
    looking for examples where human groups appear to behave as much
    as possible as inanimate matter, such as in crowds, queues, traffic jams,
    competition, attraction, perturbations, and markets.
    � In contrast, the second approach attempts as much as possible to distinguish
    the behavior of social agents from that of inanimate matter.
    In this framework, it is believed that human endowments such as conscience,
    reflection, intention, morality, and history forbid the use and
    transfer of quantitative methods developed in the physical, material,
    and more generally natural sciences to the humanities.
    In recent economic and finance research, there is a growing interest in
    marrying the two viewpoints, that is, in incorporating ideas from social
    sciences to account for the fact that markets reflect the thoughts, emotions,
    and actions of real people as opposed to the idealized economic
    investor who underlies the efficient market and random walk hypotheses.
    This was captured by the now-famous pronouncement of Keynes [235]
    that most investors’ decisions “can only be taken as a result of animal
    spirits—of a spontaneous urge to action rather than inaction, and not
    the outcome of a weighed average of benefits multiplied by the quantitative
    probabilities” (see the section entitled “Is Prediction Possible?”
    in chapter 1 and the section entitled “Prices Are Unpredictable, or Are
    They?” in chapter 2). A real investor may intend to be rational and may
    try to optimize his or her actions, but that rationality tends to be hampered
    by cognitive biases, emotional quirks, and social influences. “Behavioral
    finance” [424, 372, 376, 163, 104] is a growing research field that uses
    psychology, sociology, and other behavioral theories to explain the behavior
    of investors and money managers. The behavior of financial markets is
    thought to result from varying attitudes toward risk, the heterogeneity in
    the framing of information, cognitive errors, self-control and lack thereof,
    regret in financial decision making, and the influence of mass psychology.
    Assumptions about the frailty of human rationality and the acceptance of
    92 chapter 4
    such drives as fear and greed are underlying the recipes developed over
    decades by so-called technical analysts.
    Prof. Thaler, now at the University of Chicago, was one of the earliest
    and strongest proponents of behavioral economics [424] and has made a
    career developing a taxonomy of anomalies that embarrass the standard
    view from neoclassical economics that markets are efficient and people
    are rational. According to accepted economic theory, for instance, a person
    is always better off with more rather than fewer choices. One day,
    Thaler noticed that a few of his supposedly rational colleagues who were
    over at his house were unable to stop themselves from gorging on some
    cashew nuts he had put out. Why, then, did Thaler’s colleagues thank
    him for removing the tempting cashews from his living room? Another
    case-in-point was when a friend admitted to Thaler that, although he
    mowed his own lawn to save $10, he would never agree to cut the lawn
    next door in return for the same $10 or even more. According to the
    concept of “opportunity cost,” foregoing a gain of $10 to mow a neighbor’s
    lawn “costs” just as much as paying somebody else to mow your
    own. According to theory, you prefer either the extra time or the extra
    money—it cannot be both. Still another example reported in [272] is
    when Thaler and another friend decided to skip a basketball game in
    Rochester because of a swirling snowstorm. His friend remarked that if
    they had bought the tickets already, they would have gone. The problem
    refers to “sunk costs.” Similarly, there is no sense going to the health club
    just because you have paid your dues. After all, the money is already
    paid: sunk. And yet, Thaler observed that we do, in general. People, in
    short, do not behave like rational economics would like them to. Even
    economics professors are not as rational as the people in their models.
    For instance, a bottle of wine that sells for $50 might seem far too
    expensive to buy for a casual dinner at home. But if you already owned
    that bottle of wine, having purchased it earlier for far less, you would
    be more likely to uncork it for the same meal. To an economist, this
    makes no sense, but Thaler culled that anecdote from Richard Rosett, a
    prominent neoclassicist [272]. The British economist K. Binmore once
    proclaimed at a seminar that people evolve toward rationality by learning
    from mistakes. Thaler retorted that people may learn how to shop for
    groceries sensibly because they do it every week, but the big decisions—
    marriage, career, retirement—do not come up very often. So Binmore’s
    highbrow theories, he concluded, were good for “buying milk” [272].
    In his doctoral thesis on the economic “worth” of a human life, Thaler
    proposed quantifying it by measuring the difference in pay between lifethreatening
    jobs and safer lines of work. He came up with a figure of
    positive feedbacks 93
    $200 a year (in 1967 dollars) for each 1-in-1,000 chance of dying. When
    he asked friends about it, most insisted that they would not accept a
    1-in-1,000 mortality risk for anything less than a million dollars. Paradoxically,
    the same friends said they would not be willing to forgo any
    income to eliminate the risks that their jobs already entailed. Thaler concluded
    that rather than rationally pricing mortality, people had a cognitive
    disconnect; they put a premium on new risks and casually discounted
    familiar ones [272]. In experiments designed to test his ideas, Thaler
    found that subjects would usually agree to pay more for a drink if they
    were told that the beer is being purchased from an exclusive hotel rather
    than from a rundown grocery. It strikes them as unfair to pay the same.
    This violates the law-of-one-price that one drink is worth the same as
    another, and it suggests that people care as much about being treated
    fairly as they do about the actual value of what they are paying for
    [227, 228]. An important discovery, extending the framing principle of
    Kahneman and Tversky, was “mental accounting” [423, 373]. “Framing”
    says that the positioning of choices prejudices the outcome, an issue that
    received a lot of publicity in the 2000 U.S. presidential election. “Mental
    accounting” says that people draw their own frames, and that where
    they place the boundaries subtly affects their decisions. For instance,
    most people sort their money into accounts like “current income” and
    “savings” and justify different expenditures from each [425]. Applied
    to the stock market, Thaler noticed that some behavioral patterns like
    “categorization” may provide arbitrage opportunities: for instance, when
    Lucent Technologies was riding high, people categorized it as a “good
    stock” and mentally coded news about it in a favorable way. Later, when
    Lucent had become a “bad stock,” similar news was interpreted more
    gloomily. Another anomaly, called “hyperbolic discounting” [254, 255],
    refers to preference reversals: when people expect money but have not
    yet received it, they are capable of planning, quite rationally, how much
    of it to spend immediately and how much to save. This is in agreement
    with economics theory, which argues that for a modest incentive,
    people are willing to save and put off spending. But when the money
    actually arrives, willpower breaks down and the money is often spent
    right away. In other words, when sacrifices are distant, patience predominates:
    I want/plan/intend to start exercising next month. But next month,
    the designated sacrifice is often avoided. Such preferences, neglected
    by neoclassical economics, have important implications, in particular for
    investors’ life-cycle savings decisions.
    One of the most robust findings in the psychology of judgment is that
    people are overconfident (see the review [104] and references therein).
    94 chapter 4
    A significant manifestation in the context of herding is that people overestimate
    the reliability of their knowledge and of their abilities: one
    famous finding is that 90% of the automobile drivers in Sweden consider
    themselves “above average” [417], while of course by definition (for a
    symmetric distribution) 50% are below average and 50% are above average!
    Most people also consider themselves above average in their ability
    to get along with others. Such overconfidence is enhanced in domains
    where people have self-declared expertise, holding their actual predictive
    ability comparable [190]. This seems to have important implications
    for understanding managers’ decisions concerning corporate growth and
    external acquisition and why most funds are actively managed [104].
    Overconfidence implies that managers all think they can pick winners.
    There is growing empirical evidence of the existence of herd or “crowd”
    behavior in speculative markets as carefully documented in the recent
    book of Shiller [375] and references therein. Herd behavior is often
    said to occur when many people take the same action, because some
    mimic the actions of others. The term “herd” obviously refers to similar
    behavior observed in animal groups. Other terms such as “flocks”
    or “schools” describe the collective coherent motion of large numbers
    of self-propelled organisms, such as migrating birds and gnus, lemmings
    and ants [426]. In recent years, physicists have shown that much of the
    observed herd behavior in animals can be understood from the action of
    simple laws of interactions between animals. With respect to humans,
    there is a long history of analogies between human groups and organized
    matter [64, 305]. More recently, extreme crowd motions such as in panic
    situations have been remarkably well quantified by models that treat the
    crowd as a collection of individuals interacting as a granular medium
    with friction, like the familiar sand of beaches [191].
    Herding has been linked to many economic activities, such as investment
    recommendations [364, 171], price behavior of IPOs [450], fads
    and customs [39], earnings forecasts [427], corporate conservatism [463],
    and delegated portfolio management [290]. Researchers are investigating
    the incentives investment analysts face when deciding whether to
    herd and, in particular, whether economic conditions and agents’ individual
    characteristics affect their likelihood of herding. Although herding
    behavior appears inefficient from a social standpoint, it can be rational
    from the perspective of managers who are concerned about their
    positive feedbacks 95
    reputations in the labor market. Such behavior can be rational and may
    occur as an information cascade [450, 107, 39], a situation in which
    every subsequent actor, based on the observations of others, makes the
    same choice independent of his or her private signal. Herding among
    investment newsletters, for instance, is found to decrease with the precision
    of private information [171]: the less information you have, the
    stronger is your incentive to follow the consensus.
    Research on herding in finance can be subdivided in the following
    non-mutually exclusive manner [107, 171].
  13. Informational cascades occur when individuals choose to ignore or
    downplay their private information and instead jump on the bandwagon
    by mimicking the actions of individuals who acted previously. Informational
    cascades occur when the existing aggregate information becomes
    so overwhelming that an individual’s single piece of private information
    is not strong enough to reverse the decision of the crowd. Therefore, the
    individual chooses to mimic the action of the crowd, rather than act on his
    private information. If this scenario holds for one individual, then it likely
    also holds for anyone acting after this person. This domino-like effect is
    often referred to as a cascade. The two crucial ingredients for an informational
    cascade to develop are: (i) sequential decisions with subsequent
    actors observing decisions (not information) of previous actors; and (ii) a
    limited action space.
  14. Reputational herding, like cascades, takes place when an agent
    chooses to ignore his or her private information and mimic the action of
    another agent who has acted previously. However, reputational herding
    models have an additional layer of mimicking, resulting from positive
    reputational properties that can be obtained by acting as part of a group
    or choosing a certain project. Evidence has been found that a forecaster’s
    age is positively related to the absolute first difference between his
    forecast and the group mean. This has been interpreted as evidence that
    as a forecaster ages, evaluators develop tighter prior beliefs about the
    forecaster’s ability, and hence the forecaster has less incentive to herd
    with the group. On the other hand, the incentive for a second-mover to
    discard his private information and instead mimick the market leader
    increases with his initial reputation, as he strives to protect his current
    status and level of pay [171].
  15. Investigative herding occurs when an analyst chooses to investigate
    a piece of information he or she believes others also will examine. The
    analyst would like to be the first to discover the information but can only
    96 chapter 4
    profit from an investment if other investors follow suit and push the price
    of the asset in the direction anticipated by the first analyst. Otherwise, the
    first analyst may be stuck holding an asset that he or she cannot profitably
  16. Empirical herding refers to observations by many researchers of
    “herding” without reference to a specific model or explanation. There
    is indeed evidence of herding and clustering among pension funds,
    mutual funds, and institutional investors when a disproportionate share
    of investors engage in buying, or at other times selling, the same stock.
    These works suggest that clustering can result from momentum-following,
    also called “positive feedback investment,” for example, buying past
    winners or perhaps repeating the predominant buy or sell pattern from
    the previous period.
    There are many reported cases of herding. One of the most dramatic
    and clearest in recent times is the observation by G. Huberman and
    T. Regev [204] of a contagious speculation associated with a nonevent
    in the following sense. A Sunday New York Times article on the potential
    development of a new cancer-curing drug caused the biotech company
    EntreMed’s stock to rise from 12 at the Friday, May 1, 1998 close to
    open at 85 on Monday, May 4, close near 52 on the same day, and remain
    above 39 in the three following weeks. The enthusiasm spilled over to
    other biotechnology stocks. It turns out that the potential breakthrough
    in cancer research had already been reported in one of the leading scientific
    journals, Nature, and in various popular newspapers (including the
    Times) more than five months earlier. At that time, market reactions were
    essentially nil. Thus the enthusiastic public attention induced a longterm
    rise in share prices, even though no genuinely new information had
    been presented. The very prominent and exceptionally optimistic Sunday
    New York Times article of May 3, 1998 led to a rush on EntreMed’s
    stock and other biotechnology companies’ stocks, which is reminiscent
    of similar rushes leading to bubbles in historical times (see chapter 1). It
    is to be expected that information technology, the Internet, and biotechnology
    are among the leading new frontiers on which sensational stories
    will lead to enthusiasm, contagion, herding, and speculative bubbles.
    Empirical Evidence of Financial Analysts’ Herding
    A recently published empirical work by Ivo Welch [451] shed new
    light on the important question of whether herding is more rational or
    positive feedbacks 97
    “irrational.” He considered the buy and sell recommendations of security
    analysts and asked whether previous recommendations as well as
    the prevailing consensus influence the recommendations of the following
    analyses. This is one of the rare studies where a scientific approach
    can be developed to gain insight into this delicate question. Welch studied
    more than 50,000 stock recommendations made between 1989 and
    1994 by hundreds of U.S. security analysts from the Zacks database,
    which is a commercially compiled database of analysts’recommendation,
    used, for instance, by The Wall Street Journal to publish regular performance
    reviews of major brokerage houses. To formulate the problem in
    a langage suitable for a rigorous statistical analysis, the recommendations
    are divided into five classes: 1: “strong buy,” 2: “buy,” 3: “hold,”
    4: “sell,” 5: “strong sell.” From this numerical coding of the recommendation,
    Welch started by constructing Table 4.2 or “transition matrix,” in
    which an entry denoted Ni→j represents the number of recommendations
    j, given that the previous recommendation was i. Thus, for example,
    = 92 is the number of recommendations “sell” following the previous
    recommendation “strong buy”; N4→3
    = 1�826 is the number of
    recommendations “hold” following the previous recommendation “sell,”
    and so on. As can be seen from the table, the transition matrix is highly
    irregular: the numbers of recommendations vary strongly from one recommendation
    to another. The total number of recommendations (of any
    direction) starting from a previous “strong buy” is 14,682, compared to
    only 1,584 recommendations starting from a previous “strong sell.” It
    is thus clear that there is a rather strong bias toward “buy” and “strong
    buy”: the total number of such recommendations is 25,784 compared to
    only 4,951 “sell” and “strong sell” recommendations, that is, more than
    five times more “buy” and “strong buy” recommendations than “sell”
    and “strong sell” recommendations.
    To test for herding, Welch first defined the global consensus as T0
    � =
    j=1 j total�j�/N = �1 × total�1� + 2 × total�2� + 3 × total�3�+ 4 ×
    total�4� + 5 × total�5��/N , which gives a value close to 2.5, where
    total�j� is the total number of recommendation of type j following any
    previous recommendation as defined in Table 4.2. Since the value 2.5 is
    less than 3, which would be the expected result in the absence of bias,
    this confirms the bias toward “buy” positions corresponding to smaller
    coding numbers (1 and 2). The second step is to extract the subset of
    recommendations on a given day t and recalculate the transition matrix
    for this day. The entries will be smaller, but what is important are the
    proportions (i.e., normalized by total�i�), which will probably be different
    from those shown in Table 4.2. To quantify how different, one again
    98 chapter 4
    Table 4.2
    From ↓ �i� to → �j� 1 2 3 4 5 total�i�
    1 : Strong buy 8,190 2,234 4,012 92 154 14,682
    2 : Buy 2,323 4,539 3,918 262 60 11,102
    3 : Hold 3,622 3,510 13,043 1,816 749 22,740
    4 : Sell 115 279 1,826 772 375 3,367
    5 : Strong sell 115 39 678 345 407 1,584
    Total�j� 14,365 10,601 23,477 3,287 1,745 53,475
    The “transition matrix” giving the number of recommendations j, given that the previous recommendation
    was i, where the numbers i and j are taken from five values defined by classifying the
    recommendations into five classes: 1: strong buy, 2: buy, 3: hold, 4: sell, 5: strong sell. The total
    number of recommendations used in the construction of this table is N = 53�475. Reproduced from
    computes the consensus T �t� for this day t. If T �t� = T0, this day is like
    any other day and there is no special difference from the point of view
    of the analysts. More interesting are the days when T �t� is significantly
    different from T0. The question is, then, What is the origin of this difference?
    The answer is given by calculating how this difference depends on
    different factors, such as the recommendations made the previous day or
    the prevailing consensus. Welch introduced for this a “herding” parameter
    measuring the tendency to herd, that is, when recommendations are
    influenced by the prevailing consensus. The first result is that analysts
    do indeed bias their recommendations towards the prevailing consensus.
    He then measures the probability of making one of the five recommendations
    when herding is absent and compares it to that when herding is
    present: a “hold” recommendation, for instance, occurs 42% of the time
    when herding is absent and 47% when it is present. While this impact
    appears small, any statistically significant change in behavior indicates
    herding, given that analysts rarely agree on a stock pick when acting in
    isolation, and in a sense it is their job to disagree.
    What is the cause of this herding? If all analysts receive new information
    about a stock at the same time and interpret it in the same
    way, rational herding could ensue. Alternatively, analysts could simply
    be mimicking their colleagues blindly, even when no new fundamental
    information is released, leading to “irrational” herding. In order to
    distinguish these two hypotheses, Welch measured the propensity to follow
    a consensus when the herd proves to be correct. The idea is that if
    herding is rationally based on fundamental information, it should lead to
    positive feedbacks 99
    better recommendations, on average, than when it is irrationally based on
    mimicking behavior. The data shows that “analysts are more inclined to
    follow the prevailing consensus when it later on turns out to be wrong.”
    Since there does not seem to be any informational advantage to consensus
    herding, one can conclude that it is of the irrational kind. It also
    constitutes evidence that analysts follow the prevailing consensus based
    on limited information, if any.
    However, as is often the case in this difficult subject, there are alternative
    explanations. The fact that the prevailing consensus among analysts
    turns out to be wrong can also be interpreted as the fact that investors,
    who are not the same population as analysts, do not follow the recommendations
    of the latter! This situation is then similar to a natural
    system having its own dynamics, which are independent of the existence
    of observers or analysts trying to forecast, its dynamics being created by
    the aggregate investment actions of the investors.
    Another important fact outlined by the research of Welch is that the
    strength of herding is different in bull and bear markets. Analysts tend to
    follow the consensus more strongly (1) in up-markets and (2) following
    recent revisions in down-markets. Behavior (1) tends to create “bubbles”:
    price inflations deconnected from fundamental values. Behavior (2) suggests
    that revision from an optimistic to a pessimistic outlook can be
    amplified by herding, a mechanism that can amplify losses and may lead
    to brutal drops and crashes.
    It Is Optimal to Imitate When Lacking Information
    All the traders in the world are organized into a network of family,
    friends, colleagues, contacts, and others who are sources of opinion, and
    influence each other locally through this network [48]. We call “neighbors”
    of agent Anne on this worldwide graph the set of people in direct
    contact with Anne. Other sources of influence also involve newspapers,
    Web sites, TV stations, and similar media. Specifically, if Anne is
    directly connected with k “neighbors” in the worldwide graph of connections,
    then there are only two forces that influence Anne’s opinion:
    (a) the opinions of these k people together with the influence of the
    media; and (b) an idiosyncratic signal that she alone receives (or generates;
    see Figure 4.2). According to the concept of herding and imitation,
    the assumption is that agents tend to imitate the opinions of their
    100 chapter 4
    X Y
    Fig. 4.2. A message path running through a block of agents. The signals are the
    idiosyncratic noise received at the previous time, which then combines with the state
    of each agent. Each agent sends a signal to neighbors. A given agent then makes
    a decision based on the signal of her neighbors and her own private information
    (reproduced from [383]).
    “neighbors,” not contradict them. It is easy to see that force (a) will tend
    to create order, while force (b) will tend to create disorder, or in other
    words, heterogeneity. The main story here is the fight between order
    and disorder, and the question we are now going to investigate is, What
    behavior can result from this fight? Can the system go through unstable
    regimes, such as crashes? Are crashes predictable? We show that the science
    of self-organizing systems (sometimes also referred to as “complex
    systems”) bears very significantly on these questions: the stock market
    and the web of traders’ connections can be understood in large part from
    the science of critical phenomena (in a sense that we are going to examine
    in some depth later in this chapter and in chapter 5), from which
    important consequences can be derived.
    To make progress, we formalize the problem a bit and consider a network
    of investors: each one can be named by an integer i = 1� � � � � I, and
    positive feedbacks 101
    N�i� denotes the set of the agents who are directly connected to agent
    i according to the worldwide graph of acquaintances. If we isolate one
    trader, Anne, N�Anne� is the number of traders in direct contact with her,
    who can exchange direct information with her and exert a direct influence
    on her. For simplicity, we assume that any investor such as Anne can
    be in only one of several possible states. In the simplest version, we
    can consider only two possible states: sAnne
    = ?1 or sAnne
    = +1. We
    could interpret these states as “buy” and “sell,” “bullish” and “bearish,”
    “optimistic” and “pessimistic.” Now, the section entitled “Explanation of
    the Imitation Strategy” shows that, based only on the information of the
    actions sj �t ? 1� performed yesterday (at time t ? 1) by her N�Anne�
    “neighbors,” Anne maximizes her return by having taken yesterday the
    decision sAnne�t ?1� given by the sign of the sum of the actions of all her
    “neighbors.” In other words, the optimal decision of Anne, based on the
    local polling of her “neighbors,” who she hopes represents a sufficiently
    faithful representation of the market mood, is to imitate the majority
    of her neighbors. This is, of course, open to some possible deviations
    when she decides to follow her own idiosyncratic “intuition” rather than
    being influenced by her “neighbors.” Such an idiosyncratic move can be
    captured in this model by a stochastic component independent of the
    decisions of the neighbors or of any other agent. Intuitively, the reason
    why it is generally optimal for Anne to follow the opinion of the majority
    is simply because prices move in that direction, forced by the law
    of supply and demand. Later in this chapter and in chapter 5, we shall
    show that this apparently innocuous evolution law produces remarkable
    self-organizing patterns.
    Explanation of the imitation strategy. Consider N traders in a network,
    whose links represent the communication channels through which the
    traders exchange information. The graph describes the chain of intermediate
    acquaintances between any two people in the world. We denote by
    N�i� the number of traders directly connected to a given trader i on the
    graph. The traders buy or sell one asset at price p�t� which evolves as a
    function of time assumed to be discrete and measured in units of the time
    step �t. In the simplest version of the model, each agent can either buy or
    sell only one unit of the asset. This is quantified by the buy state si
    = +1
    or the sell state si
    = ?1. Each agent can trade at time t ? 1 at the price
    p�t ? 1� based on all previous information, including that at t ? 1. The
    a�sset price variation is taken simply proportional to the aggregate sum
    i=1 si�t ? 1� of all traders’ actions: indeed, if this sum is zero, there are
    as many buyers as there are sellers and the price does not change since
    there is a perfect balance between supply and demand. If, on the other
    102 chapter 4
    hand, the sum is positive, there are more buy orders than sell orders and
    the price has to increase to balance the supply and the demand, as the
    asset is too rare to satisfy all the demand. There are many other influences
    impacting the price change from one day to the next, and this can usually
    be accounted for in a simple way by adding a stochastic component to the
    price variation. This term alone would give the usual log-normal random
    walk process [92], while the balance between supply and demand together
    with imitation leads to some organization, as we show below.
    At time t ? 1, just when the price p�t ? 1� has been announced, the
    trader i defines her strategy si�t ?1� that she will hold from t ?1 to t, thus
    realizing the profit (or loss) equal to the price difference �p�t�?p�t ?1��
    times her position si�t ? 1�. To define her optimal strategy si�t ? 1�, the
    trader should calculate her expected profit PE, given the past information
    and her position, and then choose si�t ? 1� such that PE is maximum.
    Since the price moves with the general opinion

    i=1 si�t ? 1�, the best
    strategy is to buy if it is positive and sell if it is negative. The difficulty
    is that a given trader cannot poll the positions sj that will take all
    other traders, which will determine the price drift according to the balance
    between supply and demand. The next best thing that trader i can do is to
    poll her N�i� “neighbors” and construct her prediction for the price drift
    from this information. The trader needs additional information, namely the
    a priori probability P+ and P? for each trader to buy or sell. The probabilities
    P+ and P? are the only information that she can use for all the traders
    that she does not poll directly. From this, she can form her expectation of
    the price change. The simplest case corresponds to a market without drift
    where P+ = P? = 1/2.
    Based on the previously stated rule that the price variation is proportional
    to the sum of actions of traders, the best guess of trader i is that
    the future price change will be proportional to the sum of the actions of
    her neighbors who she has been able to poll, hoping that this provides a
    sufficiently reliable sample of the total population. Traders are indeed constantly
    sharing information, calling each other to “take the temperature,”
    effectively polling each other before taking actions. It is then clear that
    the strategy that maximizes her expected profit is such that her position is
    of the sign given by the sum of the actions of all her “neighbors.” This is
    exactly the meaning of the following expression:
    si�t ? 1� = sign


  • i

    positive feedbacks 103
    such that this position si�t ? 1� gives her the maximum payoff based on
    her best prediction of the price variation p�t� ? p�t ? 1� from yesterday
    to today. The function sign�x� is defined by being equal to +1 (to ?1) for
    positive (negative) argument x, K is a positive constant of proportionality
    between the price change and the aggregate buy/sell orders. It is inversely
    proportional to the “market depth”: the larger the market, the smaller is
    the relative impact of a given unbalance between buy and sell orders,
    hence the smaller is the price change. i is a noise and N�i� is the number
    of neighbors with whom trader i interacts significantly. In simple terms,
    this law (6) states that the best investment decision for a given trader is to
    take that of the majority of her neighbors, up to some uncertainty (noise),
    capturing the possibility that the majority of her neighbors might give an
    incorrect prediction of the behavior of the total market.
    Expression (6) can be thought of as a mathematical formulation of
    Keynes’s beauty contest. Keynes [235] argued that not only are stock
    prices determined by the firm’s fundamental value, but, in addition, mass
    psychology and investors’ expectations influence financial markets significantly.
    It was his opinion that professional investors prefer to devote
    their energy, not to estimating fundamental values but rather, to analyzing
    how the crowd of investors is likely to behave in the future. As a result,
    he said, most persons are largely concerned not with making superior
    long-term forecasts of the probable yield of an investment over its whole
    life, but with foreseeing changes in the conventional basis of valuation a
    short time ahead of the general public. Keynes used his famous beauty
    contest as a parable for stock markets. In order to predict the winner of
    a beauty contest, the ability to recognize objective beauty is not nearly
    as important as the ability to predict others’ recognition of beauty. In
    Keynes’s view, the optimal strategy is not to pick those faces the player
    thinks are the prettiest, but those the other players are likely to think the
    average opinion will be, or those the other players will think the others
    will think the average opinion will be, or even further along this iterative
    loop. Expression (6) precisely captures this concept: the opinion si at
    time t of an agent i is a function of all the opinions of the other “neighboring”
    agents at the previous time t ? 1, which themselves depend on
    the opinion of the agent i at time t ?2, and so on. In the stationary equilibrium
    situation in which all agents finally form an opinion after many
    such iterative feedbacks have had time to develop, the solution of (6) is
    precisely the one taking into account all the opinions in a completely
    self-consistent way compatible with the infinitely iterative loop.
    104 chapter 4
    Mimetic Contagion and the Urn Models
    Orléan [323]–[328] has captured the paradox of combining rational and
    imitative behavior under the name “mimetic rationality” (rationalité
    mimétique). He has developed models of mimetic contagion of investors
    in the stock markets that are based on irreversible processes of opinion
    forming. In the simplest version, called the Urn model, which has a
    long history in the mathematical literature dating from Polya [269], let
    us assume that, at some time, there are M white balls and N black
    balls in an urn. Then, we draw one ball at random from the urn. Here,
    “random” means that any ball has the same probability 1/�M + N� to
    be chosen. Then, we return the winner as well as another additional ball
    of the same color to the set of balls from which it was drawn. Thus,
    after this experiment, if white is the winner, there will be M + 1 white
    balls in the white set and N black balls in the black set. On the other
    hand, if a black was chosen, there would be M white balls in the white
    set and N + 1 black balls in the black set. We repeat this experiment on
    and on. This simple model describes the process in which a newcomer
    (the added ball) mimicks in his action (his color) one of the existing
    investors. This irreversible process of aggregation is clearly based on
    imitation, but it also has a strong stochastic component.
    Consider the initial fair state M = N = 1 at time t = 0. At the next
    time step t = 1, after application of the rules of the game, the urn contains
    either M = 2 white balls and N = 1 black balls with probability
    1/2 or M = 1 white balls and N = 2 black balls with probability 1/2.
    At the next time step t = 2, the urn contains one of three possible populations:
    (1) M = 3 white balls and N = 1 black balls with probability
    �1/2� × �2/3� = 1/3. (2) M = 2 white balls and N = 2 black balls with
    probability �1/2� × �1/3� + �1/2� × �1/3� = 1/3. There are indeed two
    paths to achieve this final state, and we have thus to sum over them to
    obtain the correct probability. (3) M = 1 white balls and N = 3 black
    balls with probability �1/2� × �2/3� = 1/3. It is easy but becomes more
    and more cumbersome to continue counting the different possibilities and
    their associated probabilities as time goes on. A typical trajectory of the
    fraction fw of white and fb of black balls in the urn may be as follows.
    Time (t = 0, fw
    = 1/2� fb
    = 1/2); (t = 1� fw
    = 1/3� fb
    = 2/3); (t =
    3� fw
    = 1/4� fb
    = 3/4); (t = 4� fw
    = 2/5� fb
    = 3/5�� � � � In the limit
    where the game is repeated a large number of times, one obtains a truly
    remarkable result [269], whose two sides are enticingly paradoxical: on
    one hand, the fractions M/�M + N� of white balls and N/�M + N� of
    positive feedbacks 105
    black balls eventually converge towards well-defined numbers fW and
    = 1 ? fW , which do not fluctuate anymore; on the other hand, fW
    and thus fB
    = 1 ? fW can take any arbitrary value between 0 and 1
    with equal uniform probability. This means that, restarting the game several
    times, the final fraction of white and black balls will be different,
    with no relationship between one play and the next! This irreversible
    model describes an imitation process that can lead to a continuum of
    states; in other words, many different possible states coexist and compete.
    Phrased in the context of imitation between agents that successively
    enter the market and imitate at random one of the already active investor,
    a bull or bear market may emerge completely at random as the volume
    of investors progressively grows. What controls the long-term value of
    fW and fB
    = 1 ? fW is the initial fluctuation of the random drawing
    process: if, for instance, a white ball is drawn four times in a row, this
    gives a probability 4/5 to continue drawing a white ball at the next time
    step, compared to only 1/5 for a black ball. If at the tenth time step,
    there are 11 white and 1 black balls, the probability of reinforcing the
    dominance of white balls is 11/12 compared to a probability of only
    1/12 to get a black ball. This progressive freezing of the probabilities
    and its feedback on the fraction of the two populations is the underlying
    mechanism. We thus see that the fractions of the two populations and
    their corresponding probabilities become progressively frozen, simply by
    the law of large numbers.
    The urn model can be generalized by changing the rules of addition
    of the new balls; that is, how many new investors come into play, how
    do they do so, and how do they imitate the existing players so as to
    include more complex nonlinear behaviors [20, 19, 325].
    This class of models also offers a mechanism for curious facts in economics
    and history. Two well-cited examples are the dominance of the
    VHS over the Betamax standard in the video industry and the blossoming
    of concentrations of high-tech companies such as Silicon Valley in
    California. In both cases, it is argued that some slight advantage due
    to chance or other factors, such as a few more buys and movies favoring
    the VHS standard, has progressively been amplified and frozen by
    the urn mechanism. Similarly, if two valleys are competing in order to
    attract high-tech companies, the one that initially has a few more companies
    than the other will be more attractive to new start-ups, as they will
    get a slighty more active business environment. Again, this slight initial
    advantage may be amplified and lead to a major advantage in the end.
    The urn mechanism also provides a natural framework for reanalyzing
    historical facts, in particular the often tortuous paths of human societies.
    106 chapter 4
    Accordingly, the urn mechanism may cast some doubts on the view often
    constructed in retrospect that history is following a deterministic trajectory.
    In contrast, the Urn process suggests that some major historical
    facts may have resulted from progressive freezing of stochastic events
    that accumulated to finally put the balance on one side.
    This class of models provides an alternative to the “influence” model
    summarized by the expression (6), putting more emphasis on the irreversibility
    of the decision processes. In contrast, the imitation model (6)
    is more in tune with a kind of “equilibrium,” allowing changes of opinion
    for any of the investors. Notwithstanding these differences, the important
    message is that apparently anomalous bubble phases of the market are
    robust consequences of the imitative behavior of agents.
    Imitation from Evolutionary Psychology
    Beyond the rationale to imitate discussed before, justification for imitative
    tendencies can be found in evolutionary psychology [93]. The
    point is that humans are rarely at their best when they use rational reasoning.
    It can indeed be demonstrated that “rational” decision-making
    methods (i.e., the usual methods drawn from logic, mathematics, and
    probability theory) are incapable of solving the natural adaptive problems
    our ancestors had to solve reliably in order to survive and reproduce.
    Because biological evolution is a slow process, and the modern
    world has emerged in an evolutionary eye-blink, our present abilities
    are inherited from the past and remain functionally specialized to solving
    the particular problems facing the hunter-gatherers of the past. This
    poor performance on most natural problems is the primary reason why
    problem-solving specializations were favored by natural selection over
    general-purpose problem solvers. Despite widespread claims to the contrary,
    the human mind is not worse than rational, but may often be better
    than rational! On evolutionarily recurrent computational tasks, such as
    object recognition, grammar acquisition, or speech comprehension, the
    human mind exhibits impressive skills of a quality often comparable to
    or better than the best artificial problem-solving systems that decades of
    research have produced.
    General-purpose systems are constrained to apply the same problemsolving
    methods to every problem and make no special assumption about
    the problem to be solved. Specialized problem solvers are not handicapped
    by these limitations. From this perspective, the human mind is
    powerful and intelligent primarily because it comes equipped with a large
    positive feedbacks 107
    array of what one might call “reasoning instincts.” Although instincts
    are often thought of as the polar opposite of reasoning, a growing body
    of evidence indicates that humans have many reasoning, learning, and
    preference circuits that are complexly specialized for solving the specific
    adaptive problems our hominid ancestors regularly encountered. These
    circuits are developed without conscious effort and are applied without
    any awareness of their underlying logic. In other words, these reasoning,
    learning, and preference circuits have all the hallmarks of what people
    usually think of as “instincts.” They make certain kinds of inferences
    just as easy and natural to humans as spinning a web is to a spider or
    building a dam is to a beaver. For example, humans do not seem to have
    available on-line circuits that perform many logic operations. On the
    other hand, experimental evidence indicates that humans have evolved
    circuits dedicated to a more specialized task of equal or greater complexity:
    detecting cheaters in situations of exchange. Equally important,
    humans have specialized circuits for understanding threats, as well as
    recognizing bluffs and double-crosses. Such skills allowed the emergence
    of coercive coalitions, governments, and other social arrangements, and
    probably the stock market. The large risks of failure involved in hunting
    game and gathering food led hunter-gatherers to cooperate in small
    tribes and share food in order to smooth out the otherwise wildly fluctuating
    feast-or-famine cycles that prevailed for individuals and families.
    In more modern contexts, upon stress under sufficiently large risks and
    uncertainties, humans may switch on some of these adaptive sharing
    Experiments show that a lucky event can lead to overconfidence [100].
    In the experiments of Darke and Freedman [100], some subjects experienced
    a lucky event, whereas others did not. All subjects then completed
    an unrelated decision task, rated their confidence, and placed a bet. After
    the lucky event, those who believed in luck (i.e., thought of luck as a
    stable, personal attribute) were more confident and bet more. Subjects
    who did not believe in luck (i.e., thought luck was random) were less
    confident and bet less. Studies have also compared decisions made alone
    to decisions made following interactions with others [189]. Results show
    that, while interaction did not increase decision accuracy or metaknowledge,
    subjects frequently showed stable or increasing confidence when
    they interacted with others, even with those who disagreed with them
    [189, 361, 382, 346, 347]. A possible interpretation is that the interaction
    serves the role of rationalizing the subjects’ decisions rather than that
    of collecting valuable information. There is also a herding effect. In the
    same spirit, exposing to others the rationale behind decisions has been
    108 chapter 4
    shown to markedly increase subjects’ confidence that their choices were
    appropriate [377]. This is reminiscent of a well-known fact established
    in education studies that writing enhances comprehension. It has also
    been demonstrated that feedback concerning the appropriateness of confidence
    judgments improves calibration and resolution skills [369]. The
    effect is significantly stronger in men compared to women, as men often
    exhibit stronger confidence in situations in which they are wrong [291].
    More to the point, psychological experiments [10] have been conducted
    in which subjects are shown real stock prices from the past and
    asked to forecast subsequent changes while performing trades consistent
    with these forecasts and, by so doing, accumulating wealth. These subjects,
    of course, were asked to trade only based on past prices and were
    not exposed to external “fundamental” news. It was found that subjects
    track the past average when the stock prices are stable, thus trading
    against price fluctuations when they arise. However, as prices began to
    show consistent trends, they began to switch to a trend-chasing strategy,
    buying more when prices increase and selling when prices decrease. Perhaps
    even more compelling evidence of the presence of trend-chasing
    strategies is the wide prevalence of “technical analysis” that tries to spot
    trends and trend reversals by using technical indicators associated with
    past price movements [53].
    Many on Wall Street think that rumors move stocks (see Figure 4.3).
    The old Wall Street saying, “buy on the rumor, sell on the news,” is alive
    and well, as can be seen from numerous sources in the media and the
    Internet. Rumors can drive herding behavior strongly.
    Rumors are most easily documented for extraordinary events. Here
    are a few remarkable examples. The Y2K bug is one of the most famous
    recent rumors during which misinformation was rampant. Rumors, assertions,
    predictions, demagoguery, bluster, cover-up, and denial abounded,
    such that, for the layman, it was almost impossible to sort fact from
    fiction. Another example is the completely false rumor concerning the
    U.S. Postal Service that was being circulated on Internet e-mails. The
    e-mail message claimed that a “Congressman Schnell” has introduced
    “Bill 602P” to allow the federal government to impose a ¢5 surcharge on
    each e-mail message delivered over the Internet. The money would be
    collected by Internet service providers and then turned over to the Postal
    Service. No such proposed legislation exists. In fact, no “Congressman
    positive feedbacks 109
    Fig. 4.3. Cartoon of the impact of rumors in stock market behavior taken from the
    front page of The Economist, November 1–7, 1997, commenting on the turmoil
    following the 7% loss of October 27, 1997 on the DJIA. Creation of KAL.
    Schnell” exists. And the U.S. Postal Service denied having any authority
    to surcharge e-mail messages sent over the Internet [430].
    Large-scale rumors have also developed on the scale of nations [259].
    Hideo Ibe, previous president of the Research Institute for Policies on
    Aging, declared in a press release on February 14, 1996: “It has been
    brought to my attention that Deng Xiaoping has said: Since Japanese do
    not have enough children, we could send them fifty million Chinese.”
    This statement seemed strange given that Japan had 340 inhabitants
    per square kilometer while China had only 100, and also inprobable in
    view of the strong control exerted by the immigration service of Japan.
    110 chapter 4
    Had Deng Xiaoping pronounced this sentence, or was it expected from
    Japanese public opinion? To determine the truth, the source of the information
    should be checked, which implies checking all Chinese newspapers,
    radio, and TV recordings during the months and perhaps the few
    years preceding this announcement. This would be a difficult task that
    could well fail, as occurred in the case of an alleged declaration to the
    Washington Post by Algerian president Houari Boumediene: “One day,
    millions of men and women will leave the meridonial and poor parts of
    the world to erupt in the relatively accessible regions of the north hemisphere
    in search of their survival.” Cited by famous French demographers
    and amplified by important media managers, the declaration, which fed
    a fear of invasion, has never been documented, notwithstanding a careful
    investigation by the Washington Post over several years.
    Circulation of such rumors calls for epidemiological studies such
    as the one performed by Edgar Morin to investigate the rumor that
    spread through Orléans, France, that young women were disappearing
    from fashion shops owned by Jews. Morin showed how all social layers
    participated in the diffusion of this rumor. On the other hand, in the
    two previous examples, the contagion was maintained, justified, and
    probably even created by elites, either scientists or people in charge of
    the media. These rumors do not circulate in all directions, but essentially
    from the top to the bottom of society. The rather sophisticated presentations,
    the apparently serious references that seem to justify their origins,
    and their distinguished proponents provide food for amplifications
    serving diverse interests and psychological biases in all layers of
    Notwithstanding the probable confusion it may bring to the mind of
    readers, it seems appropriate to mention here a recent book by P. M.
    Garber that reexamined the tulip mania and the Law and South Sea
    bubbles described in chapter 1 with a fresh and close look at the historical
    record [153]. His main conclusion is that the fabled elements
    ritually invoked as underlying speculative bubbles with herding and irrational
    behavior are just not true. Instead, he defends the view that these
    events have a possible explanation in terms of fundamental valuation.
    The interesting part is that Garber views the tulip mania “myth” as originating
    from a rumor that was progressively strengthened by successive
    authors using it for their own agenda, such as to support moralistic
    attacks against “excessive speculation” and, in modern times, to plead for
    government regulation: “the tulipmania episode � � � is simply a rhetorical
    device used to put forward an argument that � � � the existence of
    positive feedbacks 111
    tulipmania proves that markets are crazy. A curious disturbance in a
    particular modern market can then be attributed to crazy behavior, so
    perhaps the market needs to be more severely regulated” [153, p. 11.],
    While Garber’s book has been hailed by a series of financial economists
    with high reputations, economist C. P. Kindleberger pointed out some
    of the work’s shortcoming and concludes [237]: “The debate between
    those who believe markets are always rational and efficient, resting on
    fundamentals, and historians who call attention to a series of financial
    crises going back to at least 1550 is likely to continue. Parsimony calls
    for making a choice for or against financial crises; complexity permits
    one to say that markets are mostly reliable but occasionally get caught
    up in untoward activities.”
    The Survival of the Fittest Idea
    The drive of humans to share ideas and behaviors can be tracked back
    to a more fundamental level, according to the theory of “memes” introduced
    by Richard Dawkins [102, 42]. A meme is to thinking what a gene
    is to evolution. A meme is defined as any idea, behavior, or skill. Like
    a gene, it can replicate by transferrring from one person to another by
    imitation: stories, fashions, inventions, recipes, songs, ways of plowing a
    field or throwing a baseball or making a sculpture. Like a gene, it competes
    with other memes, as ideas and behavior compete in a culture and
    between cultures. The memes come to us from all the speakers who are
    vocal wherever we happen to grow up: parents, siblings, friends, neighbors,
    teachers, preachers, bosses, coworkers, and everyone involved in
    producing things like textbooks, novels, comic books, movies, television
    shows, newspapers, magazines, Internet sites, and so on. All these people
    are constantly repeating to each other (and of course to their children,
    their students, their employees, and so on) the memes they have received
    during their lifetime. All these voices taken together constitute the voice
    of Mother Culture [339]. According to the meme theory, “just as the
    design of our bodies can be understood only in terms of natural selection,
    so the design of our minds can be understood in terms of memetic
    selection” [42]. For instance, Blackmore [42] showed that once our distant
    ancestors acquired the crucial ability to imitate, a second kind of
    natural selection began, a survival of the fittest among competing ideas
    and behaviors. Ideas that proved most adaptive—making tools, for example,
    or using language—survived and flourished, replicating themselves
    in as many minds as possible. These memes then passed themselves on
    112 chapter 4
    from generation to generation by helping to ensure that the genes of
    those who acquired them also survived and reproduced. Applying this
    theory to many aspects of human life, this offers new perspectives for
    why we live in cities, why we talk so much, why we can’t stop thinking,
    why we behave altruistically, how we choose our mates, and much
    more. According to Blackmore, “When we look at religions” or other
    nonscientific beliefs such as astrology,
    from a meme’s eye view, we can understand why they have been so successful.
    These religious memes did not set out with an intention to succeed.
    They were just behaviors, ideas and stories that were copied from
    one person to another in the long history of human attempts to understand
    the world. They were successful because they happened to come
    together into mutually supportive gangs that included all the right tricks
    to keep them safely stored in millions of brains, books and buildings, and
    repeatedly passed on to more. They evoked strong emotions and strange
    experiences. They provided myths to answer real questions and the myths
    were protected by untestability, threats, and promises. They created and
    then reduced fear to create compliance, and they used the beauty, truth
    and altruism tricks to help their spread. [42, p. 192]
    In a similar vein, it is tempting to interpret within the same theory
    some behaviors observed on stock markets, for instance, the use of technical
    analysis (for a large collection of free technical analysis materials,
    see http://decisionpoint.com/) for which a genuine “culture” is striving,
    even if technical analysis has not been really established from a firm
    scientific point of view (see, however, [53, 36, 6]).
    Gambling Spirits
    Investing in the stock market is a kind of lottery or gambling to many
    investors, at least if one follows some of the popular press, which coined
    the expression “casino stock market.” The gambling spirit, usually
    exerted in lotteries and in casinos, has become a prominent state of mind
    in many states of the United States of America and may be an important
    psychological factor at work in the stock market as well. Gambling
    is more than taking risks. There is, of course, risk in gambling, but
    gambling is something more. The word “gambling” is related to the
    word “game” and comes from an old English word gammon. Gambling
    is thus associated with the idea of a game. Gambling is a game. It is not
    a game based on skill or on reason; it is a game based on sheer chance.
    positive feedbacks 113
    Gambling is an appeal to sheer chance: random luck without skill or
    one’s personal involvement [277]. Gambling is an activity in which a
    person risks something of value to forces of chance completely beyond
    his or her control, or any rational expectation, in hopes of winning
    something of greater value, usually more money.
    The lottery has become a major American fantasy. Estimates of the
    total amount wagered are difficult to obtain, but about $500 billion are
    wagered every year legally in America, and estimates run as high as $1
    trillion total when illegal gambling is added in. The best statistics indicate
    that there are about 10 million compulsive gamblers in the United
    States, more than the number of alcoholics. It is interesting to realize
    that gambling also played a prominent role in early American history. In
    1612, the British government ran a lottery to assist the new settlement
    at Jamestown, Virginia. In 1776, the First Continental Congress of the
    United States sold lottery tickets to finance the American Revolution.
    President Washington himself bought the first lottery ticket to build the
    new capital, called Federal City—now known as Washington, D.C. The
    United States was founded on a lottery, the revolution was financed by
    a lottery, and the capital city was financed by a lottery.
    From 1790 to 1860, 24 of the 36 states sponsored government-run
    lotteries. Many schools, universities, colleges, and hundreds of churches
    conducted their own lotteries to raise funds for their own buildings.
    Through this period of early American history and involvement with
    lotteries and government-sponsored gambling, because of the increasing
    corruption of the gambling, by 1894 it had disappeared from America.
    By 1894, there was no more government-sponsored gambling—it ended
    in corruption and in a financial fiasco. Public gambling at any level was
    stopped completely. Between 1894 and 1964, there was no governmentsponsored
    gambling in America. In 1964, it was reintroduced by the state
    of New Hampshire, which became the first state to offer a lottery, and
    now there are 37 states that have government-sponsored lotteries, and
    Washington D.C. makes 38 entities. There are over 500 casinos across
    the nation.
    In 1974, thus 10 years later, a poll indicated that 61% of Americans
    gambled, wagering $47.4 billion annually. In 1989, 71% were wagering
    $246 billion. In 1992, $330 billion was being wagered. By 1995, studies
    indicate that 95% of Americans gamble, 82% play the lottery, 75% play
    slot machines, 50% bet on dogs and horses, 44% on cards, 34% on bingo,
    26% on sporting events, 74% frequented casinos, and 89% approved of
    gambling. One cannot help but compare this growth of enthusiasm for
    114 chapter 4
    gambling with the bullish stock market and the remarkable growth of
    the number of households owning stocks in the last decades.
    Gambling expenditures each year exceed the amount spent on films,
    books, amusements, music, and entertainment combined. People spend
    more money gambling than they do buying tickets to all national athletic
    events put together (baseball, football, and everything else). In 1993,
    people spent $400 billion legally, $482 billion in 1994, and well over
    $500 billion in 1999! Five billion is spent every year just in the slot
    machines in Nevada alone! Ninety-two million households visit the casinos,
    and 10% of all money earned by people in America is thrown away
    in gambling!
    It is difficult to assess how much this gambling spirit is active in the
    minds of individual investors. If it is, even to a small degree, it is relevant
    to our discussion since it makes investors prone to imitation and herding
    because they invest on little information. It may also explain the anomalously
    large volatility of prices [374] and their potential instabilities.
    Why It May Pay to Be in the Minority
    In a practical implementation of a trading strategy, it is not sufficient to
    know or guess the overall direction of the market. There are additional
    subtleties governing how the trader is going to enter (buy or sell) the
    market. For instance, Anne will want to be slightly ahead of the herd
    to buy at a better price, before the price is pushed up for the bullish
    consensus. Symmetrically, she will want to exit the market a bit before
    the crowd, that is, before a trend reversal. In other words, she would like
    to be a little bit contrarian by buying when the majority is still selling
    and by selling when the majority is still buying, slightly before a change
    of opinion of the majority of her “neighbors.” This means that she will
    not always want to follow the herd, at least at short time scales. At
    this level, Anne cannot rely on the polling of her “neighbors” because
    she knows that they, as well as the rest of the crowd, will have similar
    ideas to try to out guess each other on when to enter the market. More
    generally, Anne would ideally like to be in the minority when entering
    the market, in the majority while holding her position, and again in the
    minority when closing her position.
    This leads to another class of behaviors, very different from those
    based on imitation and herding. Here, the problem for Anne is to use past
    positive feedbacks 115
    information to make her decision to buy the market when she believes
    that the majority of the others will not yet do it. She thus has to be in
    the minority. Profiting from being in the minority leads to interesting
    paradoxes. Rather diabolically, if all traders use the same set of rules,
    they will end up doing the same thing at the same time and cannot
    therefore be in the minority. This leads to a wonderful paradox: contrary
    to imitative behavior that gets reinforced when everybody does it, to be
    in the minority implies striving to be different and, thus, cannot result
    from using the same rules for all. By adaptation, Anne and her colleagues
    will thus learn and be forced to differentiate their enter strategies based
    on past successes and failures.
    El-Farol’s Bar Problem
    This issue has recently been formalized in the framework of so-called
    “minority games.” A minority game is a repeated game where N players
    have to choose one out of two alternatives (say A and B) at each time
    step. Those who happen to be in the minority win. Although being rather
    simple at first glance, this game is subtle in the sense that, as we have
    already said, if all players analyze the situation in the same way, they all
    will choose the same alternative and lose. Moreover, there is a frustration
    since not all the players can win at the same time. Minority games are
    abstractions of the famous El-Farol’s bar problem [17]. In that model,
    100 people decide independently each week whether to go to a bar that
    offers entertainment on a certain night. Space is limited, and the evening
    is only enjoyable if the bar is not too crowded—specifically, if fewer
    than 60% of the possible 100 are present. There is no way to tell the
    numbers coming for sure in advance, therefore a person goes, that is,
    deems it worth going, if she expects fewer than 60 to show up; she
    stays home if she expects more than 60 to go. Choices are unaffected by
    previous visits; there is no collusion or prior communication among the
    people; and the only information available is the numbers who came in
    past weeks. What is the dynamics of the numbers attending from week
    to week?
    To answer this, Arthur [17] assumed that the 100 persons can each
    individually form several predictors or hypotheses in the form of functions
    that map the past d weeks’ attendance figures into next week’s.
    Such predictors are the analog of technical trading recipes that investors
    116 chapter 4
    use to help form their decisions. For example, following the example of
    Arthur, recent attendance numbers might be
    44 78 56 15 23 67 84 34 45 76 40 56 22 35�
    Particular hypotheses or predictors to predict next week’s number might
    be [17]
    � the same as last week’s, giving 35 at the prediction for the attendence
    of next week,
    � a mirror image around 50 of last week’s, giving 65,
    � a (rounded) average of the last four weeks, giving 49,
    � the trend in the last eight weeks, bounded by 0 and 100, giving 29,
    � the same as two weeks ago (two-period cycle detector), giving 22,
    � the same as five weeks ago (five-period cycle detector), giving 76,
    � etc.
    Arthur assumes that each person possesses and keeps track of an individualized
    set of k such focal predictors. She decides to go or stay according
    to the currently most accurate predictor in her set. Once decisions
    are made, each agent learns the new attendance figure and updates the
    accuracies of her monitored predictors. In this bar problem, the set of
    hypotheses currently most credible and acted upon by the person determines
    the attendance. But the attendance history determines the set of
    active hypotheses. This is an analog to an important mechanism at work
    in stock markets: the use of predictors and their impact on attendance
    is indeed similar to the use of “technical indicators” used by technical
    analysts to forecast the market.
    Using artificial persons who choose at random k (6 or 12 or 23, say)
    different predictors among several dozen focal predictors replicated many
    times, a computer simulation allows us to investigate what happens. Each
    artificial person then possesses k predictors or hypotheses she can draw
    upon, and at each time step, she chooses the one that has performed best
    in the past (even if it has not been used). This deterministic dynamics
    gives the bar attendance shown in Figure 4.4. The remarkable result
    is that the predictors self-organize into an equilibrium pattern in which
    the most accurate predictors, on average, are forecasting 40% of the
    time above 60, and 60% of the time below 60. While the population
    of best predictors splits into this 60/40 average ratio, it keeps changing
    positive feedbacks 117
    20 30 40 50
    Numbers Attending
    10 60 70 80 90 100
    Fig. 4.4. Bar attendance in El Farol’s bar problem posed by B. Arthur as a paradigm
    for “minority games.” Reproduced from [17].
    in membership forever. These results appear throughout the experiments
    robust to changes in types of predictors created and in numbers assigned
    [17]. The pattern shown in Figure 4.4 is reminiscent of the patterns of
    price variations observed for a typical stock (see chapter 2). This suggests
    a mechanism for the “noisy” structure of price variations and returns
    whose origin may be rooted in the fact that investors cannot all win at
    the same time and have to choose different strategies if they want to win.
    Minority Games
    Many variants of this minority game have been introduced which generalize
    the phenomenon and capture an essential feature of systems where
    agents compete for limited resources. In minority games, artificial agents
    with partial information and bounded rationality base their decision only
    on the knowledge of the M (for memory) last winning alternatives, called
    histories. Take all the histories and fix a choice (A or B) for each of
    them: you get a strategy, which is like a theory of the world. Each strategy
    has an intrinsic value, called virtual value, which is the total number
    of times the strategy has predicted the right alternative, A or B. At the
    beginning of the game, every player gets a limited set of S strategies.
    She uses them inductively; that is, she uses the strategy with the highest
    118 chapter 4
    virtual value (ties are broken by coin tossing). It must be emphasized that
    a player does not know anything about the others; all her information
    come from the virtual values of the strategies.
    The more striking properties of the minority game (MG) are: (1) it is a
    model that addresses the interaction between agents and information; (2)
    the agents are able to cooperate (but without direct exchanges); (3) the
    agents minimize the available information; (4) there is a critical transition
    between a symmetric phase with no information available to agents and
    an asymmetric phase with available information to agents. The control
    parameter is the ratio
    = P/N of the number P of the different possible
    states of fundamental information divided by the number N of agents.
    is less than
    c, where
    c is a special value of the order of 1,
    the market is efficient and there is no information that can be used for
    prediction. In contrast, for
    larger than
    c, a new agent could profit
    from the existence of predictive structure in the dynamics: there are not
    enough agents to exploit and remove all information. We recover here the
    insight already discussed in the section titled “A Parable,” in chapter 2.
    An intuitive and qualitative understanding of minority games can be
    obtained by using the insight obtained from expression (6) in the section
    titled “Explanation of the Imitation Strategy” for the imitative strategy.
    Indeed, in (6), a positive coefficient K quantifies the force of imitation.
    Contrarian behavior corresponds to the case where K is negative. In the
    analogy with spins of magnetic materials, imitation (K > 0) leads to
    the ferromagnetic phase (magnet) or global cooperative behavior that we
    describe in the following section, titled “Cooperative Behaviors Resulting
    from Imitation.” Contrarian behavior (K < 0) corresponds to the
    so-called “antiferromagnetic” interaction. In the physics of material sciences,
    anti-ferromagnetic interactions are known to lead to weird behavior
    and often complex phases resulting from the frustration induced by
    not being able to satisfy all pairs of interacting elements simultaneously.
    This problem has the same qualitative paradoxical properties that we
    have described for the minority games.
    Imitation versus Contrarian Behavior
    Real markets result from agents’ behaviors, which are neither fully imitative
    nor fully anti-imitative, in contrast with the claims of presently
    available reductionist models and theories. A better representation of real
    markets requires a combination of the two. Indeed, one should distinguish
    the “buy” and “sell” actions from the “holding” period.
    positive feedbacks 119
  1. The price of an asset at any given time is fundamentally determined
    from the balance between supply and demand: more “buy” than “sell”
    orders will drive the price up and vice versa. If Anne wants to buy
    (sell), she wants to be in the minority such that the price tends to
    decrease (increase) and she thus gets a better instantaneous bargain.
    The “buy” and “sell” actions are optimized when Anne is able to be
    in the minority.
  2. Once she is invested in the market, she gains if her investment agrees
    with the opinion of the majority: if she bought (sold), she would gain
    in a book-to-market measure only if the price goes up (down). The
    gain in the “holding” period is thus optimized when Anne belongs to
    the majority.
    To fix these ideas, let us assume that the time it takes for a transaction
    to be concluded is �t, equal to, say, one minute (most of the
    time, not-too-large transactions can be performed much faster through
    the Internet). The first minority optimization thus concerns this short
    time interval and amounts to minimizing the possible difference between
    an order price and its concrete implementation: Anne gives a “buy” order
    at 100 but the transaction is concluded at 101 because many others are
    buying, driving the price up during the short time interval between her
    order and its concrete implementation. She thus pays more than what
    she intended. This is what she wants to avoid by being in the minority,
    that is, buying before the crowd of buyers. In contrast to what happens at
    this short time scale, the holding period can last much longer, say n�t.
    The relative impact of the contrarian behavior on the imitation forces
    is thus of the order of 1/n, the ratio of the time to enter in position to
    the holding time. For “intraday” traders who are very active, this ratio
    may not be small at all. The large amount of works on minority games
    [77, 78, 76, 75] suggests that changing one’s strategy often may be profitable
    in that situation. It also suggests that only when the information
    complexifies or when the number of traders decreases will the traders be
    able to make consistent profits. In contrast, the buy-and-hold strategies
    profit as long as the information remains simple, such as when a trend
    remains strong. The problem then boils down to exit/reverse before or at
    the reversal of the trend.
    The difficulty however, as everyone who has tried to invest in the stock
    market will know, is that trends and trend reversals occur at all time
    scales. Figure 4.5 illustrates this observation by a construction based on
    the insertion of a succession of trends and trend reversals at all scales.
    This geometric construction, which improves and generalizes the random
    120 chapter 4
    Generator Trend Line
    Piece 1
    Piece 2
    Piece 3
    Fig. 4.5. Simple chart that inserts price changes from time 0 to a later time 1
    in successive steps to illustrate the concept of trends occurring at all time scales.
    The intervals are chosen arbitrarily and may represent a minute, an hour, a day, or
    a year. The process begins with a trend from the bottom-left corner (0,0) to the
    right-up corner (1,1). Next, a broken line called a generator is used to create the
    up-and-down pattern piece 1–piece 2–piece 3. Then, each of the three pieces are
    themselves replaced by three smaller pieces obtained by a suitable scale reduction
    of the initial generator (the interpolated generator is inverted for each descending
    piece). Repeating these steps reproduces the shape of the generator, or price curve,
    but at compressed scales. Both the horizontal axis (time scale) and the vertical axis
    (price scale) are squeezed to fit the horizontal and vertical boundaries of each piece
    of the generator. Reproduced from [285] Courtesy of Laurie Grace.
    positive feedbacks 121
    walk model, reproduces quite closely the structure of price trajectories
    shown in chapter 2. These scale-invariant patterns are made of building
    blocks of up-and-down trends that can be observed and reproduce themselves
    at all scales and almost everywhere. These patterns belong to the
    geometry of fractals [284], a rough or fragmented geometric shape that
    can be subdivided into parts, each of which is (at least approximately) a
    reduced-size copy of the whole. The concept of fractals, introduced by
    Mandelbrot, captures the rough, broken, and irregular characteristics of
    many phenomena in nature, present at all scales. We shall come back to
    this construction, shown in Figure 4.5, and its implications in chapter 6.
    We borrow and adapt the following tale on the slime mold from Steven
    Johnson [223] and Evelyn Fox Keller [233]. The slime mold (Dictyostelium
    discoideum) is a reddish orange mass of cells that can be
    found, among other places, coating rotting wood in damp sections
    of forests. Most of the time, the slime mold’s motions are barely
    perceptible, except when the weather conditions grow wetter and cooler,
    when suddenly it “decides” to “walk away.” Indeed, the slime mold
    spends much of its life as thousands of distinct single-celled units, each
    moving separately from its other comrades. Under the right conditions,
    those myriad cells will coalesce into a single, larger organism, which
    then begins its leisurely crawl across the forest floor, consuming rotting
    leaves and wood as it moves about.
    When the environment is less hospitable, the slime mold acts as a single
    organism: when the mold enjoys a large food supply, “it” becomes a
    “they.” The slime mold oscillates between being a single creature and a
    swarm. How do all these cells manage to work so well together? Slime
    cells have been shown to emit a common substance called acrasin (also
    known as cyclic AMP), through which they exchange information. For
    many years, scientists believed that the aggregation process was coordinated
    by specialized slime-mold cells, known as “pacemaker” cells.
    According to this theory, each pacemaker cell sends out a chemical signal,
    telling other slime-mold cells to gather around it, resulting in a
    However, while scientists agreed that waves of cyclic AMP do indeed
    flow through the slime-mold community before aggregation, all the cells
    122 chapter 4
    in the community are effectively interchangeable. None of them possess
    any distinguishing characteristics that might elevate them to pacemaker
    status. In the late 1960s, Evelyn Fox Keller and Lee Segel developed a
    mathematical model [234] (now called the Keller–Segel model in chemotaxis)
    of how slime cells could self-organize into a coherent organism
    by continuous release and exchange of cyclic AMP. The model only
    assumes that every individual cell follows the same set of simple rules,
    involving the emission and sensing of chemicals. Altering the amount of
    cyclic AMP each cell releases individually as a function of the amount
    of cyclic AMP present in the environment, each cell can follow trails of
    the pheromone that they encounter as they wander through their environment.
    When the slime cells pump out enough cyclic AMP, clusters of
    cells start to form spontaneously. Cells can then better follow the trails
    created by other cells, creating a positive feedback loop that encourages
    more cells to join the cluster.
    Slime mold aggregation is now recognized as a classic case study
    in bottom-up behavior and self-organization, similar in a sense to that
    occurring in stock markets. Spontaneous pattern formation has been
    and is still a very active domain of study, allowing us to understand,
    for instance, the origins of the patterns on the furs of zebras and
    leopards [409, 410]. The general concept works similarly in many
    distinct fields: pattern and evolving organization result from the competition
    between at least one disordering and one ordering force. In
    the case of the slime-mold, the disordering force is the spontaneous
    tendency of cells to wander on their own. The ordering force stems from
    the interactions mediated through the release and reaction of cells to
    cyclic AMP. The relative strength of these two forces decides whether
    the slime-mold cells self-organize into a single unit or live their own
    distinct lives. A similar fight between ordering and disordering forces
    between financial agents will be described in chapter 5. The concept that
    cooperative behavior leads to the emergence of self-organization into
    novel patterns is at the core of the take-home message of this book. The
    force derived from self-organization is nicely illustrated in the cartoon
    of Figure 4.6.
    The Ising Model of Cooperative Behavior
    The imitative behavior discussed in the section titled “It Is Optimal to
    Imitate When Lacking Information” in the present chapter and captured
    by the expression (6) on page 102 belongs to a very general class of
    positive feedbacks 123
    Fig. 4.6. Illustration of the concept that cooperative behavior is a strong force for
    self-organization. Created by and courtesy of B. A. Huberman.
    so-called stochastic dynamical models developed to describe interacting
    elements, particles, and agents in a large variety of contexts, in particular
    physics and biology [265, 266]. The tendency or force towards imitation
    is governed by the parameter K, which can be called the “coupling
    strength”; the tendency towards idiosyncratic (or noisy) behavior is governed
    by the amplitude � of the noise term. Thus the value of K relative
    to � determines the outcome of the battle between order and disorder,
    and eventually the structure of the market prices. More generally, the
    coupling strength K could be heterogeneous across pairs of neighbors,
    and it would not substantially affect the properties of the model. Some
    of the Kij ’s could even be negative, as long as the average of all Kij ’s
    was strictly positive.
    The expression (6) on page 102 only describes the state of an agent
    at a given time. In the next instant, new i’s are realized, new influences
    propagate themselves to neighbors, and agents can change their
    decision according to Figure 4.2. The system is thus constantly changing
    and reorganizing, as shown in Figure 4.7. The model does not assume
    instantaneous opinion interactions between neighbors. In real markets,
    opinions indeed tend not to be instantaneous, but are formed over a
    period of time by a process involving family, friends, colleagues, newspapers,
    web sites, TV stations, and so on. Decisions about the trading
    124 chapter 4
    Fig. 4.7. Four snapshots at four successive times of the state of a planar system of
    64 × 64 agents put on a regular square lattice. Each agent placed within a small
    square interacts with her four nearest neighbors according to the imitative rule (6)
    of page 102. White (respectively, black) squares correspond to “bull” (respectively,
    “bear”). The four cases shown here correspond to the existence of a majority of buy
    orders, as white is the predominant color.
    activity of a given agent may occur when the consensus from all these
    sources reaches a trigger level. This is precisely this feature of a threshold
    reached by a consensus that expression (6) captures: the consensus is
    quantified by the sum over the N�i� agents connected to agent i, and the
    threshold is provided by the sign function. The delay in the formation of
    the opinion of a given trader as a function of other traders’ opinions is
    captured by the progressive spreading of information during successive
    updating steps (see, for instance, [265, 266]).
    The simplest possible network is a two-dimensional grid in the
    Euclidean plane. Each agent has four nearest neighbors: one to the
    North, the South, the East, and the West. The tendency K towards
    imitation is balanced by the tendency � towards idiosyncratic behavior.
    positive feedbacks 125
    Fig. 4.8. K < Kc : Buy (white squares) and sell (black squares) configuration in
    a two-dimensional Manhattan-like planar network of 256 × 256 agents interacting
    with their four nearest neighbors. There are approximately the same number of white
    and black sells; that is, the market has no consensus. The size of the largest local
    clusters quantifies the correlation length, that is, the distance over which the local
    imitations between neighbors propagate before being significantly distorted by the
    “noise” in the transmission process resulting from the idiosyncratic signals of each
    In the context of the alignment of atomic spins to create magnetization
    (magnets), this model is identical to the so-called two-dimensional Ising
    model, which has been solved explicitly by Onsager [321]. Only its
    formulation is different from what is usually found in textbooks [164],
    as we emphasize a dynamical viewpoint.
    In the Ising model, there exists a critical point Kc that determines the
    properties of the system. WhenK < Kc (see Figure 4.8), disorder reigns:
    the sensitivity to a small global influence is small, the clusters of agents
    who are in agreement remain of small size, and imitation only propagates
    between close neighbors. In this case, the susceptibility � of the system
    126 chapter 4
    Fig. 4.9. Same as Figure 4.8 for K close to Kc. There are still approximately the
    same number of white and black sells; that is, the market has no consensus. However,
    the size of the largest local clusters has grown to become comparable to the
    total system size. In addition, holes and clusters of all sizes can be observed. The
    “scale-invariance” or “fractal”-looking structure is the hallmark of a “critical state”
    for which the correlation length and the susceptibility become infinite (or simply
    bounded by the size of the system).
    to external news is small, as many clusters of different opinions react
    incoherently, thus more or less cancelling out their responses.
    When the imitation strength K increases and gets close to Kc (see
    Figure 4.9), order starts to appear: the system becomes extremely sensitive
    to a small global perturbation, agents who agree with each other form
    large clusters, and imitation propagates over long distances. In the natural
    sciences, these are the characteristics of so-called critical phenomena.
    Formally, in this case the susceptibility � of the system goes to infinity.
    The hallmark of criticality is the power law, and indeed the susceptibility
    goes to infinity according to a power law � ≈ A�Kc
    ? K�?�, where A
    is a positive constant and � > 0 is called the critical exponent of the
    susceptibility (equal to 7/4 for the two-dimensional Ising model). This
    positive feedbacks 127
    Fig. 4.10. Same as Figure 4.8 for K > Kc. The imitation is so strong that the
    network of agents spontaneously breaks the symmetry between the two decisions and
    one of them predominates. Here, we show the case where the “buy” state has been
    selected. Interestingly, the collapse into one of the two states is essentially random
    and results from the combined effect of a slight initial bias and of fluctuations during
    the imitation process. Only small and isolated islands of “bears” remain in an ocean
    of buyers. This state would correspond to a bubble: a strong bullish market.
    kind of critical behavior is found in many other models of interacting
    elements [265, 266] (see also [310] for applications to finance, among
    others). The large susceptibility means that the system is unstable: a
    small external perturbation may lead to a large collective reaction of the
    traders who may drastically revise their decision, which may abruptly
    produce a sudden unbalance between supply and demand, thus triggering
    a crash or a rally. This specific mechanism will be shown to lead to
    crashes in the model described in chapter 5.
    For even stronger imitation strengthK > Kc , the imitation is so strong
    that the idiosynchratic signals become negligible and the traders selforganize
    into strong imitative behavior, as shown in Figure 4.10. The
    selection of one of the two possible states is determined from small and
    128 chapter 4
    subtle initial biases as well as from the fluctuations during the evolutionary
    These behaviors apply more generically to other network topologies.
    Indeed, the stock market constitutes an ensemble of interacting investors
    who differ in size by many orders of magnitude ranging from individuals
    to gigantic professional investors, such as pension funds. Furthermore,
    structures at even higher levels, such as currency influence
    spheres (U.S.$, DM, Yen, � � � ), exist and with the current globalization
    and deregulation of the market one may argue that structures on the
    largest possible scale, that is, the world economy, are beginning to form.
    This observation and the network of connections between traders show
    that the two-dimensional lattice representation used in the Figures 4.7,
    4.8, 4.9, and 4.10 is too naive. A better representation of the structure
    of the financial markets is that of hierarchical systems with “traders” on
    all levels of the market. Of course, this does not imply that any strict
    hierarchical structure of the stock market exists, but there are numerous
    examples of qualitatively hierarchical structures in society. In fact, one
    may say that horizontal organizations of individuals are rather rare. This
    means that the plane network used in our previous discussion may very
    well represent a gross oversimplification.
    One of the best examples of a hierarchy is found in the army. At
    the lowest level of a military force is a single soldier. Ten soldiers produce
    a squad. Three squads produce a regiment; three regiments produce
    a brigade; three brigades give a division; three divisions give a corps.
    An army might have several corps and a country might have several
    armies. In hierarchical networks, information can flow from the top down
    and from bottom up, as shown in Figure 4.11. Notwithstanding the large
    variety of topological structures, the qualitative conclusion of the existence
    of a critical transition between a mostly disordered state and an
    ordered one, separated by a critical point, survives by-and-large for most
    possible choices of the network of interacting investors, including for
    hierarchical networks.
    Even though the predictions of these models are quite detailed, they
    are very robust to model misspecification. We indeed claim that models
    that combine the following features would display the same characteristics,
    in particular apparent coordinate buying and selling periods, leading
    eventually to several financial crashes. These features are:
  3. a system of traders who are influenced by their “neighbors”;
  4. local imitation propagating spontaneously into global cooperation;
  5. global cooperation among noise traders causing collective behavior;
    positive feedbacks 129
  6. prices related to the properties of this system;
  7. system parameters evolving slowly through time.
    As we shall show in the following chapters, a crash is most likely when
    the locally imitative system goes through a critical point.
    In physics, critical points are widely considered to be one of the
    most interesting properties of complex systems. A system goes critical
    when local influences propagate over long distances and the average
    state of the system becomes exquisitely sensitive to a small perturbation;
    that is, different parts of the system become highly correlated. Another
    characteristic is that critical systems are self-similar across scales: in
    Figure 4.9, at the critical point, an ocean of traders who are mostly
    bearish may have within it several continents of traders who are mostly
    bullish, each of which in turns surrounds seas of bearish traders with
    islands of bullish traders; the progression continues all the way down to
    the smallest possible scale: a single trader [458]. Intuitively speaking,
    critical self-similarity is why local imitation cascades through the scales
    into global coordination.
    Critical points are described in mathematical parlance as singularities
    associated with bifurcation and catastrophe theory. Catastrophe theory
    studies and classifies phenomena characterized by sudden shifts in behavior
    arising from small changes in circumstances. Catastrophes are bifurcations
    between different equilibria, or fixed point attractors of dynamical
    systems. Due to their restricted nature, catastrophes can be classified
    Node A
    Hierarchical Structure
    Node B
    Node E
    Node C Node D
    Node F
    Node I
    Node G Node H
    Node A
    Ancestor Structure
    Node B
    Node E
    Node C Node D
    Node F
    Node I
    Node G Node H
    Fig. 4.11. In a hierarchical structure, the messages can move from the top of the
    hierarchy to the bottom (left panel) or from the bottom to the top (right panel), as
    in the ancestor structure. The difference between the two is that, in the hierarchical
    structure, the nodes have to make a decision (as to which node to pass the message
    on to) before they pass the message on, while in the ancestor structure there is
    no need to make such a decision because there is only the single choice available
    (reproduced from [383]).
    130 chapter 4
    based on how many control parameters are being simulataneously varied.
    For example, if there are two controls, then one finds the most
    common type, called a “cusp” catastrophe. Catastrophe theory has been
    applied to a number of different phenomena, such as the stability of
    ships at sea and their capsizing and bridge collapse. It has also been
    used to describe situations in which agents with similar characteristics
    and objectives and facing identical or similar environments make choices
    that are considerably different. The use of catastrophe theory relies on
    the desire to model many of the situations that lead to sudden changes
    in decisions on the part of policy makers and individuals, polarity of
    opinion, and group conflict [385, 47]. In essence, this book attempts to
    provide mechanisms for the spontaneous occurrence of bifurcations and
    “catastrophes” in the behavior of investors and of financial markets.
    Complex Evolutionary Adaptive Systems
    of Boundedly Rational Agents
    The previous Ising model is the simplest possible description of cooperative
    behaviors resulting from repetitive interactions between agents.
    Many other models have recently been developed in order to capture
    more realistic properties of people and of their economic interactions.
    These multiagent models, often explored by computer simulations,
    support the hypothesis that the observed characteristics of financial
    prices described in chapter 2, such as non-Gaussian “fat” tails of distributions
    of returns, mostly unpredictable returns, clustered and excess
    volatility, may result endogenously from the interaction between agents.
    This relatively new school of research, championed in particular by
    the Santa Fe Institute in New Mexico [8, 18] and being developed
    now in many other institutions worldwide, views markets as complex
    evolutionary adaptive systems populated by boundedly rational agents
    interacting with each other. El-Farol’s bar problem and the minority
    games discussed previously are examples of this general class of
    models. We now briefly review some representative works to illustrate
    the variety and power but also the limitations of these approaches.
    These agent-based models owe a great intellectual debt to the work
    of Herbert Simon [379], whose notion of “bounded rationality,” based
    on his contributions at the intersection of economics, psychology, and
    computer science, is the foundation on which much of the recent
    behavioral economics literature is built. The principal concern of this
    school of research applied to economic modeling [2] is to understand
    positive feedbacks 131
    why certain global regularities have been observed to evolve and persist
    in decentralized market economies despite the absence of top-down
    planning and control such as trade networks, socially accepted monies,
    market protocols, business cycles, and the common adoption of technological
    innovations. The challenge is to demonstrate constructively
    how these global regularities might arise from the bottom up, through
    the repeated local interactions of autonomous agents. A second concern
    of researchers is to use this framework as computational laboratories
    within which alternative socioeconomic structures can be studied and
    tested with regard to their effects on individual behavior and social
    Typical of the Sante Fe school, Palmer et al. [329, 21, 258] modelled
    traders as so-called “genetic algorithms,” which are computer software
    creatures mimicking the adaptative and evolving biological genes
    that compete for survival and replication. These intelligent algorithms
    make predictions about the future, and buy and sell stock as indicated
    by their expectations of future risk and return. With certain characteristics,
    these computer agents are found to be able to collectively learn to
    create a homogeneous rational expectations equilibrium, that is, to discover
    dynamically the economic equilibrium imagined by pure theoretical
    economists. In this highly competitive artificial world, a trader-gene
    taking some “vacation” loses his “shirt” when returning back in the stock
    market arena, because he is no longer adapted to the new structures that
    were developed by the market in his absence! Farmer [123] has simplified
    this approach using the analogy between financial markets and an
    ecology of strategies. In a variety of examples, he shows how diversity
    emerges automatically as new strategies exploit the inefficiencies of old
    The Laboratory for Financial Engineering at the Massachusetts Institute
    of Technology [251, 341] is another noteworthy example of such
    pursuits. The artificial market project in particular focuses on the dynamics
    arising from interactions between human and artificial agents in a
    stochastic market environment in which agents learn from their interactions,
    using recently developed techniques in large-scale simulations,
    approximate dynamic programming, computational learning, and tapping
    insights in and resources from mathematics, statistics, physics, psychology,
    and computer science. This laboratory recently constructed an artificial
    market, designed to match those in experimental-market settings
    with human subjects, to model complex interactions among artificially
    intelligent (AI) traders endowed with varying degrees of learning capabilities
    [79]. The use of AI agents with simple heuristic trading rules and
    132 chapter 4
    learning algorithms shows that adding trend-follower traders to a population
    of empirical fundamentalists has an adverse impact on market
    performance, and the trend-follower traders do poorly overall. However,
    this effect diminishes over time as the market becomes more effcient.
    In numerical experiments in which “scalper” traders, who simply trade
    on patterns in past prices, are added to a population of fundamentalists,
    the “scalpers” are relatively successful free riders, not only matching the
    performance of fundamentalists in the long run, but outperforming them
    in the short run.
    Brock and Hommes and coworkers [54, 58, 55, 56, 57, 200, 257]
    have developed models of financial markets seen as “adaptative belief”
    systems of boundedly rational agents using different, competing trading
    strategies. The terms “rational” and “adaptative” refer to the fact that
    agents tend to follow strategies that have performed well, according to
    realized profits or accumulated wealth, in the recent past; the adjective
    “boundedly” refers to the fact that they can only use one among a set
    of relatively simple strategies. Price changes are explained by a combination
    of economic fundamentals and “market psychology,” that is, by
    the interplay between several coexisting heterogeneous classes of trading
    strategies. Most of the systems considered by Brock and Hommes
    and their coworkers have specialized to the case of a small number
    of competing strategies leading to dynamical trajectories of prices governed
    by so-called low-dimensional strange attractors, exemplifying the
    importance of chaos, of the simultaneous importance of different attractors,
    and of the existence of local bifurcations of steady states in these
    models. This theoretical approach explains why simple technical trading
    rules may survive evolutionary competition in a heterogeneous world
    where prices and beliefs coevolve over time. These evolutionary models
    account for stylized facts of real markets, such as the fat tails and
    volatility clustering described in chapter 2.
    Several works have modelled the epidemics of opinion and speculative
    bubbles in financial markets from an adaptative agent point of view
    [238, 273, 274, 275, 276]. The main mechanism for bubbles is that above
    average returns are reflected in a generally more optimistic attitude that
    fosters the disposition to overtake others’ bullish beliefs and vice versa.
    The adaptive nature of agents is reflected in the alternatives available to
    agents to choose between several classes of strategies, for instance, to
    invest according to fundamental economic valuation or by using technical
    analysis of past price trajectories. Other relevant works put more
    emphasis on the heterogeneity and threshold nature of decision making,
    which lead in general to irregular cycles [421, 460, 262, 360, 263, 154].
    positive feedbacks 133
    These approaches are to be constrasted with the efficient market
    hypothesis that assumes that the movement of financial prices is an
    immediate and unbiased reflection of incoming news about future earning
    prospects. Under the efficient market hypothesis, the deviations
    from the random walk observed empirically would simply reflect similar
    deviations in extraneous signals feeding the market. The simulations
    performed on computers allow us to test this hypothesis in artificial
    stock markets. Notwithstanding the fact that the news arrival processes
    are constructed as random walk processes, non-random-walk price
    characteristics emerge spontaneously as a result of the nonlinear and
    imitative interactions between investors. This shows that one does not
    need to assume a complex information flow to account for the complexity
    of price structures: the self-organization of the market dynamics is
    sufficient to create it endogenously.
    In conclusion, we see that there is a plethora of models that account
    approximately for the usual main stylized facts observed in stock markets
    (fat tail of the distribution of returns, absence of correlation between
    returns, long-range dependence between successive return amplitudes,
    and volatility clustering). However, these models do not predict the characteristic
    bubble structures discussed in this book (see chapters 6–10).
    In the next chapter, we therefore turn to models aimed specifically at
    capturing these important patterns.
    chapter 5
    modeling financial
    bubbles and market
    The purpose of models is not to fit the data but to
    sharpen the questions.
    — S. Karlin, 11th R. A. Fisher Memorial Lecture,
    Royal Society, April 20, 1983.
    Knowledge is encoded in models. Models are
    synthetic sets of rules, pictures, and algorithms providing us with useful
    representations of the world of our perceptions and of their patterns. As
    argued by philosophers and shown by scientists, we do not have access
    to “reality,” only to some of its manifestations, whose regularities are
    used to determine rules, which when widely applicable become “laws
    of nature.” These laws are constantly tested in the scientific march, and
    they evolve, develop and transmute as the frontier of knowledge recedes
    further away.
    Like a novel, a model may be convincing—it may ring true if it is consistent
    with our experience of the natural world. But just as we may wonder how
    much the characters in a novel are drawn from real life and how much
    is artifice, we might ask the same of a model: how much is based on
    observation and measurement of accessible phenomena, how much is based
    on informed judgment, and how much is convenience? Verification and
    modeling bubbles and crashes 135
    validation of numerical models of natural systems is impossible. The only
    propositions that can be verified, that is, proved true, are those concerning
    closed systems, based on pure mathematics and logic. Natural systems are
    open: our knowledge of them is always partial, approximate, at best. [322]
    Models are usually formulated with mathematics. Mathematics is
    nothing but a language, with its own grammar and syntax—arguably
    the simplest, clearest, and most concise language of all. It allows us to
    articulate efficiently and guide our trains of thought. It gives us logical
    deductions, flowing from the premises that we imagine to their forceful
    consequences. Learning and using mathematics is like striving to master
    Kung-Fu, both a technique and a way of life that enhances your skills
    and awareness. As with Kung-Fu, mathematics may be frightening or
    incomprehensible to many. As with any foreign language or combat
    technique, you have to learn it and practice it to be fluent and comfortable
    with it. The two models presented in what follows are also based
    on mathematics, and their rigorous treatment requires its use. Here,
    however, we shall strive to remove all the unnecessary technicalities and
    present only the main concepts with illustrations and pictures.
    Basic Principles
    The consistent modeling of financial markets remains an open and
    challenging problem. A simple, economically plausible mathematical
    approach to market modeling is needed which captures the essence of
    reality. The existing approaches to financial market modeling are quite
    diverse, and the literature is rather extensive. Significant progress in
    our understanding of financial markets was acquired, for instance, by
    Markowitz with the mean-variance portfolio theory [288], the capital
    asset pricing model of Sharpe [370] and its elaboration by Lintner,
    Merton’s [293] and Black and Scholes’s option pricing and hedging theory
    [41], Ross’s arbitrage pricing theory [353], and Cox, Ingersoll, and
    Ross’s theory of interest rates [95], to cite a few of the major advances.
    Economic models differ from models in the physical sciences in
    that economic agents are supposed to anticipate the future. Each one’s
    decision depends on the decisions of others (strategic interdependence)
    and on expectations about the future. This is illustrated by the following
    pictorial analogy [113]. Suppose that in the middle ages, before
    Copernicus and Galileo, the Earth really was stationary at the center of
    136 chapter 5
    the universe, and only began moving later on. Imagine that during the
    nineteenth century, when everyone believed classical physics to be true,
    it really was true, and quantum phenomena were nonexistent. These
    are not philosophical musings, but an attempt to portray how physics
    might look if it actually behaved like the financial markets. Indeed, the
    financial world is such that any insight is almost immediately used to
    trade for a profit. As the insight spreads among traders, the “universe”
    changes accordingly. As G. Soros has pointed out, market players are
    “actors observing their own deeds.” As E. Derman, head of quantitative
    strategies at Goldman Sachs, puts it, in physics you are playing against
    God, who does not change his mind very often. In finance, you are
    playing against God’s creatures, whose feelings are ephemeral, at best
    unstable, and the news on which they are based keeps streaming in.
    Value clearly derives from human beings, while mass, electric charge
    and electromagnetism apparently do not. This has led to suggestions
    that a fruitful framework for studying finance and economics is to use
    evolutionary models inspired from biology and genetics, to which we
    alluded in chapter 4.
    Perhaps the most profound synthesis of physical sciences came from
    the realization that everything could be understood from “conservation
    laws” and symmetry principles. For instance, Newton’s law that the
    acceleration, that is, the rate of change of velocity of a body of mass m,
    is proportional to the total force applied to it divided by m, follows from
    the conservation of momentum in free space (the law of inertia associated
    with Galilean invariance). Another example is that the fundamental
    equations of motion of so-called “strings,” formulated to describe the
    fundamental particles such as quarks and electrons, derive from global
    symmetry principles and dualities between descriptions at long-range and
    short-range scales. Are there similar principles that can guide the determination
    of the equations of motion of the more down-to-earth financial
    The Principle of Absence of Arbitrage Opportunity
    One such organizing principle is the condition of absence of arbitrage
    opportunity, which we have already visited in chapter 2. Recall that
    no-arbitrage, also known as the Law of One Price, states that two assets
    with identical attributes should sell for the same price, and so should
    the same asset trading in two different markets. If the prices differ,
    a profitable opportunity arises to sell the asset where it is overpriced
    modeling bubbles and crashes 137
    and to buy it where it is underpriced. The basic idea is that, if there are
    arbitrage opportunities, they cannot live long or must be quite subtle,
    otherwise traders would act on them and arbitrage them away. The
    no-arbitrage condition is an idealization of a self-consistent dynamical
    state of the market resulting from the incessant actions of the traders
    (arbitragers). It is not the out-of-fashion equilibrium approximation
    sometimes described; rather, it embodies a very subtle cooperative
    organization of the market. We take this condition as the first-order
    approximation of reality. We shall see that it provides strong constraints
    on the structure of the model and allows us to draw interesting and
    surprising predictions. The idea to impose the no-arbitrage condition
    is in fact the prerequisite of most models developed in the academic
    finance community. Modigliani and Miller [302, 299], for instance, have
    indeed emphasized the critical role played by arbitrage in determining
    the value of securities.
    It is important here to stress again that the no-arbitrage condition
    together with rational expectations is not a mechanism. It does not
    explain its own origin. It is a principle describing the emergent large-scale
    organization of market participants. It does not tell us what its underlying
    specific mechanisms are. Assuming the validity of the no-arbitrage
    condition together with rational expectations amounts to postulating
    that a fraction of the population of traders behave in such a way that
    prices tend to reflect available information and that risk is adequately
    and approximately fairly remunerated. In order to understand the specific
    manners with which this is attained would require a level of modeling
    not yet available at present and whose achievement is at the heart of a
    very active domain of research that we only glimpsed in chapter 4.
    As we pointed out in chapter 2, the existence of transaction costs and
    other imperfections of the market should not be used as an excuse for disregarding
    the no-arbitrage condition but rather should be constructively
    invoked to study its impacts on the models. In other words, these market
    imperfections are considered as second-order effects.
    Existence of Rational Agents
    Mainstream finance and economic modeling add a second overarching
    organizing principle, namely that investors and economic agents are
    rational. Contrary to an oft-quoted perception in the popular press and
    in certain circles of the stock market as populated by irrational herds
    (see chapter 4), a significant fraction of the traders most of the time do
    138 chapter 5
    exhibit a rational behavior in which they try to optimize their strategies
    based on the available information. One may refer to this as “bounded
    rationality” since not only is the available information in general incomplete,
    but stock market traders also have limited abilities with respect
    to analyzing the available information. In addition, investors are uncertain
    about the characteristics and preferences of other investors in the
    market. This means that the process of decision making is essentially a
    “noisy process” and, as a consequence, a probabilistic approach in stock
    market modeling is unavoidable since there are no certainties. Clearly, a
    noise-free stock market with all information available occupied by fully
    rational traders of infinite analysis abilities would have a very small
    trading volume, if any.
    The assumption of perfectly rational, maximizing behavior won out
    until recently in the art of modeling, not because it often reflects reality,
    but because it was useful. It enabled economists to build mathematical
    models of behavior and to give their discipline a rigorous, scientific
    air. This process started in the mid-1800s, evolving by the end of the
    century into the approach known today as neoclassical economics. And
    while twentieth-century critics like the University of Chicago’s T. Veblen
    and Harvard’s J. K. Galbraith argued that people are also motivated
    by altruism, envy, panic, and other emotions, they failed to come up
    with a way to fit these emotions into the models that economists had
    grown accustomed to—and thus had little impact, until recently. As we
    showed in chapter 4, the field is being enriched with revisitations and
    extensions of these approaches based on novel research encompassing
    the sciences of human behavior, psychology, and social interactions and
    This long list of irrational or anomalous behavior shown by human
    beings in certain specific systematic ways should not confuse us: the
    relevant task for understanding stock markets is not so much to focus
    on these irrationalities but rather to study how they aggregate in the
    complex, long-lasting, repetitive, and subtle environment of the market.
    This extension requires us to put aside the description of the individual
    in favor of the search for emerging collective behaviors. The market
    may have many special features that protect it from aggregating the
    irrationalities of individuals into prices. In other instances, the aggregation
    may stigmatize this irrationality in what we shall refer to as “speculative
    Market rationality should thus be understood in the sense that asset
    prices are set as if all investors are rational [354]. Clearly, markets can
    be rational even if not all investors are actually rational, as discussed
    modeling bubbles and crashes 139
    extensively in chapter 4. The “minority game” described in chapter 4
    taught us in particular that the market becomes rational if there are sufficiently
    many heterogeneous agents acting on limited information. This
    is consistent with the view of M. Rubinstein from the University of California
    at Berkeley, who argued that the most important trait of investor
    irrationality, to the extent that it affects prices, is particularly likely to
    be manifest through overconfidence, which in turn is likely to make the
    market “hyperrational” [354]. Indeed, overconfidence leads investors to
    believe they can beat the market, causes them to spend too much time
    on research, and causes many to trade too quickly on the basis of their
    information without recovering in benefits what they pay in trading costs.
    Thus, overconfidence leads to extensive analysis of the scarse available
    information and its incorporation into stock prices, which is consistent
    with the conclusions of the “minority games.”
    Therefore, the machinery behind market rationality is that each
    investor, using the market to serve his or her own self-interest, unwittingly
    makes prices reflect that investor’s information and analysis. It
    is as if the market were a huge, relatively low-cost continuous polling
    mechanism that records the updated votes of millions of investors in
    continuously changing current prices. In light of this mechanism, for
    a single investor (in the absence of inside information) to believe that
    prices are significantly in error is almost always folly [354]. Let us
    quote Rubinstein:
    Remember the chestnut about the professor and his student. On one of
    their walks, the student spies a $100 bill lying in the open on the ground.
    The professor assures the student that the bill cannot be there because
    if it were, someone would already have picked it up. To this attempt
    to illustrate the stupidity of believing in rational markets, my colleague
    Jonathan Berk asks: How many times have you found such a hundred
    dollar bill? He implies, of course, that such a discovery is so rare that the
    professor is right in a deeper sense: It does not pay to go out looking for
    money lying around.
    “Rational Bubbles” and Goldstone Modes of the Price
    “Parity Symmetry” Breaking
    Blanchard [43] and Blanchard and Watson [45] originally introduced
    the model of rational expectations (RE) bubbles to account
    for the possibility, often discussed in the empirical literature and by
    practitioners, that observed prices may deviate significantly and over
    140 chapter 5
    extended time intervals from fundamental prices. While allowing for
    deviations from fundamental prices, rational bubbles keep a fundamental
    anchor point of economic modeling, namely that bubbles must obey
    the condition of rational expectations and of no-arbitrage opportunities.
    Indeed, for fluid assets, dynamic investment strategies rarely perform
    better than simple buy-and-hold strategies [282]; in other words, the
    market is not far from being efficient and few arbitrage opportunities
    exist as a result of the constant search for gains by sophisticated
    investors. The conditions of rational expectations and of no-arbitrage are
    useful approximations. The rationality of both expectations and behavior
    does not imply that the price of an asset is equal to its fundamental
    value. In other words, there can be rational deviations of the price from
    this value, called “rational bubbles.” A rational bubble can arise when
    the actual market price depends positively on its own expected rate of
    change, as sometimes occurs in asset markets, which is the mechanism
    underlying the models of [43] and [45].
    Price Parity Symmetry.
    Recall that pricing of an asset under rational expectations theory is
    based on the two following hypotheses: the rationality of the agents and
    the “no-free lunch” condition. In addition, the “firm-foundation” theory
    asserts that a stock has an intrinsic value determined by careful analysis
    of present conditions and future prospects. Developed by S. Eliot Guild
    [183] and John B. Williams [457], it is based on the concept of discounting
    future dividend incomes. In the words of Burton G. Malkiel [282],
    discounting refers to the following concept:
    Rather than seeing how much money you will have next year (say $1.05 if
    you put $1 in a saving bank at 5% interest), you look at money expected
    in the future and see how much less it is currently worth (thus next year’s
    $1 is worth today only about 95 ¢, which would be invested at 5% to
    produce $1 at that time).
    The discounting process thus captures the usual concept that something
    tomorrow is less valuable than today: a given wealth tomorrow has
    a little less value than the same wealth today, as we have to wait to use it.
    In practice, the intrinsic value approach is a quite reasonable idea that is,
    however, confronted with slippery estimations: the investor has to estimate
    future dividends, their long-term growth rates as well as the time
    horizon over which the growth rate will be maintained. Notwithstanding
    modeling bubbles and crashes 141
    these problems, this approach has been promoted by Irving Fisher [134]
    and Graham and Dodd [170] so that generations of Wall Street security
    analysts have been using some kind of “firm-foundation” valuation to
    pick their stocks.
    Therefore, under the rational expectation condition, the best estimation
    at time t of the price pt+1 of an asset at time t + 1 viewed from time t
    is given by the expectation of pt+1 given the knowledge of all available
    information accumulated up to time t. The “no-free-lunch” condition
    then imposes that the expected returns of all assets are equal to the
    return r of the risk-free asset, such as a return on CD bank accounts.
    From this condition, one obtains the “fundamental” price today as equal
    to the sum of the price tomorrow discounted by a discount factor acting
    from today to tomorrow and of the dividend served today. The dividend
    is added to express the fact that the expected price tomorrow has to be
    decreased by the dividend since the value before giving the dividend
    incorporates it into the pricing. The standard “forward” or “fundamental”
    value pf
    t at time t is thus the sum over all future dividends discounted to
    the present t. According to this rule, if interest rates are 4%, a promise
    to pay (dividend) $4 per year forever is worth $100, but a promise to
    pay $4 this year, $4.12 next year, and $4.24 the year after (the payout
    increases each year at the same rate as GDP, say 3%) should be worth
    $400—100 times the current payment.
    It turns out that this fundamental price is not the full solution of this
    valuation problem. It is easy to show that the most general solution is
    the sum of the fundamental solution plus an arbitrary “bubble” component
    Xt . This bubble component has to obey the single no-free-lunch
    condition; that is, its value today is equal to its expected value tomorrow
    discounted by the discount factor. In the bubble component, there is no
    dividend! It is important to note that the speculative bubbles appear as a
    natural consequence of the fundamental “firm-foundation” valuation formula,
    that is, as a consequence of the no-free-lunch condition and of the
    rationality of the agents. Thus, the concept of bubbles is not an addition
    to the theory but is entirely embedded in it.
    It is interesting to pause a bit to ponder this result and deepen
    our understanding by developing an analogy with another deep result
    from particle and condensed-matter physics. The novel insight [403] is
    that the arbitrary bubble component Xt of an asset price plays a role
    analogous to the so-called “Goldstone mode” in nuclear, particle, and
    condensed-matter physics [59, 62]. Goldstone modes are the zero-energy
    infinite-wavelength mode fluctuations that attempt to restore broken
    142 chapter 5
    For instance, consider a “Bloch” wall between two large magnetic
    domains of opposite magnetization within a magnet, for instance,
    selected by opposite magnetic fields at boundaries far away. The broken
    symmetry is the fact that the two domains separated by the wall have
    opposite magnetization. A full symmetry would be that both domains
    have the same magnetization or both have magnetization with equal
    It turns out that, at nonzero temperature, “capillary” waves propagating
    along the wall are excited by thermal fluctuations. The limit of very
    long-wavelength capillary modes corresponds to arbitrary translations of
    the wall, an embodiment of the concept of Goldstone modes, which
    tend to restore the translational symmetry broken by the presence of the
    “Bloch” wall.
    What could be the symmetry-breaking acting in asset pricing? The
    answer may be surprising. It is the so-called “parity symmetry” between
    positive and negative prices [395],
    p→?p parity symmetry� (7)
    where both positive and negative prices quantify our liking or disliking
    of the commodity. Indeed, it makes perfect sense to think of negative
    prices. We are ready to pay a (positive) price for a commodity that we
    need or like. However, we will not pay a positive price to get something
    we dislike or which disturbs us, such as garbage, waste, a broken and
    useless car, chemical and industrial hazards, and so on. Consider a chunk
    of waste. We will be ready to buy it for a negative price; in other words,
    we are ready to take the unwanted commodity if it comes with cash.
    This exchange of waste for income is the basis for the industry of waste
    management. Nuclear waste from some countries, such as Japan, are
    shipped to La Hague reprocessing complex in France, which is ready to
    store the unwanted wastes for income. The Japanese are thus paying a
    price to get rid of their waste, that is, La Hague is paying a negative price
    to get the nuclear waste commodity! As a matter of fact, this exchange
    of wastes is at the basis of a huge business for the present and future
    management of industrial and nuclear waste that counts in the hundreds
    of billions of dollars. A less obvious example is the case of electricity
    companies in California, for instance, which sell surplus electricity in
    exceptional cases for negative prices; it is expensive for them to shut
    down a power plant and to restart it again [452]. My German colleague,
    Prof. D. Stauffer, humorously points out that the page charges some
    authors pay to journals to get rid of their manuscripts are an example of
    modeling bubbles and crashes 143
    Desired good
    or service
    or payment
    Positive Prices
    Undesired good
    or service
    or payment
    Negative Prices
    Fig. 5.1. Graphic showing that the sign of price is defined by the relative direction
    of the flow of cash or payment compared to the flow of goods or services; a positive
    price corresponds to the more commonly experienced situation where the cash or
    payment flow is with a direction opposite to the flow of goods or services; a negative
    price corresponds to the reverse situation where the cash or payment flow has the
    same direction as the flow of goods or services. Reproduced from [395].
    negative prices. Actually, this is not correct, but this example illustrates
    the subtlety of the concept: authors pay to get published, not to get rid
    of their paper but to buy fame; that is, cash leaves the authors but fame
    comes to them (hopefully), hence the positivity of the price in this case.
    In sum, we pay a positive price for something we like and a negative
    price for something we would rather be spared of; that is, we pay a
    positive price to get rid of it or we need a remuneration to accept this
    unwanted commodity. This concept is illustrated in Figure 5.1.
    In the economy, what makes a share of a company desirable? Answer:
    Its earnings, which provide dividends, and its potential appreciation,
    which gives rise to capital gains. As a consequence, in the absence of
    dividends and of speculation, the price of share must be nil. The earnings
    leading to dividends d thus act as a symmetry-breaking “field,” since a
    positive d makes the share desirable and thus develops a positive price.
    This is, as we have seen, at the basis of the “firm-foundation” fundamental
    pricing of assets. It is clear that a negative dividend, a premium
    that must be paid regularly to own the share, leads to a negative price,
    that is, to the desire to get rid of that stock if it does not provide other
    benefits. For a share of a company that is providing neither utility nor
    a waste, there is no intrinsic value for it if it does not give you more
    buying power for something you desire. Hence, its price is p = 0 for a
    vanishing dividend d = 0. In this case, we can allow for both positive
    and negative price fluctuations, but there is a priori nothing that breaks
    the symmetry (7).
    We stress that the price symmetry (7) is distinct from the gain/loss
    symmetry of stock holders, before the advent of limited liability companies
    in the middle of the nineteenth century. With the present limited
    144 chapter 5
    liability of stock holders, owning a stock is akin to holding an option:
    gain is accrued from dividend and capital gains; on the downside, losses
    are limited at the buying price of the stock. This asymmetry, which
    is a relatively recent phenomenon and led to the full development of
    capitalism, is also conceptually distinct from the breaking of the parity
    symmetry (7) of prices induced by a positive dividend.
    It is now clear that there are no restrictions on the nature of the bubble
    Xt added to the fundamental price pf
    t , except for the no-free-lunch
    condition. The bubble is thus playing the role of the Goldstone modes,
    restoring the broken parity symmetry: the bubble price can wander up
    or down and, in the limit where it becomes very large in absolute value,
    dominate over the fundamental price, restoring the independence of the
    price with respect to dividend. Moreover, as in condensed-matter physics,
    where the Goldstone mode appears spontaneously since it has no energy
    cost, the rational bubble itself can appear spontaneously with no dividend.
    A similar point of view has been advocated in [27] to explain the
    dynamics of money.
    Speculation as Spontaneous Symmetry Breaking.
    When the dividends are not constant and grow with time, the fundamental
    price is larger since it must incorporate the additional expected
    value of the future cash flow. There is thus a competition between the
    increasing growth of the dividends far in the expected future resulting
    from the expected growth of the company and the decreasing impact of
    dividends further in the future due to the effect of the discount factor (for
    instance, inflation). The increasing growth of dividends tends to increase
    the fundamental price. The decreasing impact of dividends further in the
    future tends to decrease the fundamental price. In the example in which
    the Interest rate is 4% and the growth rate of dividend is 3%, and if there
    were no risks, stocks would be worth 100 times the current cash flow
    to stockholders. But a stock is not riskless, and the future dividend flow
    is only a hope, not a promise. Thus, investors require a “risk premium”
    to compensate them for the risk. This amounts to reducing the dividend
    growth rate to a so-called risk-adjusted growth rate r�
    Now, when this risk-adjusted growth rate r�
    d becomes equal to or larger
    than the discount rate r, the fundamental valuation formula becomes
    meaningless, as it predicts an infinite price: the effect of discounting
    the future dividends is perfectly balanced by the dividend growth rate
    and, with an infinite time horizon, the price is just the sum of all future
    presently adjusted dividends. In the economic literature, this regime is
    known as the growth stock paradox [44]. This valuation problem was
    modeling bubbles and crashes 145
    posed in 1938 by Von Neumann [442], who demonstrated that, in an
    economy with balanced growth, the growth rate is always identical to the
    interest rate and thus equal to the discount rate. Zajdenweber [461] later
    pointed out that the value of a share is, as a consequence, always infinite
    since it is based on an infinite sum of nondecreasing future dividends
    (this reasoning neglects the finiteness of human life and therefore the
    finiteness of the utility of an asset for a given investor). The intuition
    is that when r�
    d becomes equal to (and this is all the more true when
    it is larger than) r, the price of money is not enough to stabilize the
    economy: it becomes favorable to borrow money to buy shares and earn
    an effective rate of return, which is positive for all values of the dividend.
    This is exactly what happened on the U.S. market in the rally preceeding
    the October 1929 crash [152]. Note that a negative r ? r�
    d is similar to a
    negative interest rate r in the absence of growth and risks: it leads to an
    arbitrage opportunity since you can borrow $1 now, keep it under your
    mattress, and give back $1 × �1 ?
    � at a later time, pocketing 100

cents in the process.
The existence of the parity symmetry of the price and the breakdown
of the fundamental pricing formula when the risk-adjusted growth rate r�
of the dividend becomes equal to or larger than the discount rate r suggests
a novel interpretation of speculative regimes and of bubble formations:
the price can become nonzero or develop an important component
decoupled from the dividend flow by a mathematical mechanism known
as “spontaneous symmetry breaking.”
Spontaneous symmetry breaking is one of the most important concepts
in modern science as it underpins our present understanding of
the universe, of its interactions, and of matter—nothing less! Its basic
principle can be illustrated by a very simple dynamical system whose stationary
solutions are represented in Figure 5.2 as a function of a control
parameter � = ?�r ? r�
d�. This dynamical system possesses a priori the
parity symmetry (7), since both the prices p and ?p are solutions of the
same equation. A solution respecting this symmetry obeys the symmetry
condition p = ?p whose unique solution p = 0 is called the symmetryconserving
solution. There is a critical value �c such that for � < �c,
p is attracted to zero and the asymptotic solution p�t → +�� is zero,
which, as we said, is the only solution respecting the parity symmetry.
However, a solution of the dynamical evolution may not always respect
the parity symmetry of its equation. This occurs for � > �c for which the
dynamical system possesses two distinct solutions, each of them being
related to the other by the action of the parity transformation p →?p:
the set of solutions respects the parity symmetry as an ensemble but each
146 chapter 5
μc μ
Fig. 5.2. Bifurcation diagram, near the threshold �c, of a “supercritical” bifurcation.
The “order parameter” that is, the price p bifurcates from the symmetrical state zero
to a nonzero value ±ps��� represented by the two branches, as the control parameter
crosses the critical value �c. The parity symmetry preserving value p = 0, shown
as the dashed line, becomes unstable for � > �c. Reproduced from [395].
solution separately does not respect this symmetry. This phenomenon is
called “spontaneous symmetry breaking.” More generally, the concept
of spontaneous symmetry breaking describes the situation in which a
solution has a lower symmetry than its equation. The so-called “supercritical
bifurcation” diagram near the threshold � = �c, representing the
transition from a symmetric solution p = 0 to a spontaneous symmetrybreaking
solution is shown in Figure 5.2. Spontaneous symmetry breaking
refers to the fact that the dynamical system will choose only one
of the two branches, as its evolution is unique (you cannot be at two
places at the same time) and will thus have a lower symmetry as a
The concept of spontaneous symmetry breaking takes its full meaning
in the presence of a small external perturbation or “field” H. In
the spontaneous symmetry-breaking regime � > �c, p jumps from one
branch to the other when the perturbation H goes from positive to negative,
as illustrated in Figure 5.3: any infinitesimal field is enough to
flip the price p abruptly from one of its two symmetry-broken solutions
to the other. It cannot be stressed sufficiently how important this concept
of spontaneous symmetry breaking is. For instance, it is invoked
for unifying fundamental interactions: weak, strong, and electromagnetic
interactions are now understood as the result of a more fundamental
spontaneous symmetry-broken interaction [448]. In another sweeping
application, particles and matter in this universe seem to be the
spontaneous symmetry-broken phases of a fundamental vacuum state
[448], similar to the nonvanishing price emerging in the spontaneous
modeling bubbles and crashes 147
Fig. 5.3. “Order parameter” or price p as a function of the external field for different
values of the control parameter �. The two thin lines correspond to two different
values of � < �c. The thick line is the spontaneous symmetry broken phase occurring
for � > �c. Reproduced from [395].
symmetry-breaking phase � > �c out of the symmetry-conserved “vacuum”
solution p = 0. Critical phase transitions are also understood as
spontaneous symmetry-breaking phenomena [164].
In the context of the asset valuation problem, we propose [395] that,
when the risk-adjusted growth rate r�
d of the dividend becomes equal
to or larger than the discount rate r, assets acquire a spontaneous valuation
as a result of this spontaneous symmetry-breaking mechanism.
When r ? r�
d becomes negative, money is not a desirable commodity.
You lose money by keeping it. Other commodities become valuable
in comparison with money, hence the spontaneous price valuation in
the absence of a dividend. We thus propose that, for r ? r�
d < 0, the
price becomes spontaneously positive (or possibly negative depending
on initial conditions or external constraints), and this spontaneous valuation
is nothing but the appearance of a speculation regime or bubble:
investors do not look at or care for dividends; the increase of price is
According to this theory, the regime r < r �
d is a self-sustained growth
regime where prices become unrelated to earnings and dividends: prices
can go up independently of the dividends due to the spontaneous symmetry
breaking, where a company’s shares spontaneously acquire value
without any earnings. This situation is similar to the spontaneous magnetization
of iron at sufficiently low temperature, which acquires a spontaneous
magnetization under zero magnetic field. This regime could be
relevant to understanding periods of bubbles such as in the so-called
New Economy, where price increases result in high price-over-dividend
ratios with debatable economic rationalization.
148 chapter 5
The self-sustained growth regime r < r �
d, where the expected growth
rate of the dividends is larger than the discount rate, accounts for a
number of stylized facts observed during speculative bubbles:
� The sentiment is broadly shared that the “run” will last indefinitely.
� There is a large increase in the price-over-dividend ratio;
� So-called “growth companies” are present: each speculative move has
had its growth companies: in 1857, the railways; in 1929, the utilities
(electricity production); in the 1960s, the office equipment companies
(e.g., IBM) and the rubber companies (car makers); today, we have the
Internet, software companies, banks, and investment companies. These
companies have a fast growth rate (usually larger than 30% per year)
and investors thus expect a large growth rate, rd, for their earnings.
� Speculative phases are often stopped by successive increases of the
discount rate; this occurred in 1929 (increase from 3.25% up to 6%),
in 1969, and in 1990 in Japan (increase from 2.5% to 6%).
� The high sensitivity of valuation close to the critical point r ? r�
= 0
and the spontaneous speculative valuation below it suggest that crashes
and rallies can also be interpreted as reassessments of expected riskadjusted
returns and their growth rates.
This leads to the following avenue for future research: new technologies,
such as Internet, wireless communication, and wind power,
should be compared to old technologies, such as cars, shipping, and
mining. We expect that stocks in the new technology class have high
prices and low earnings and thus high price-over-dividend and priceover-
earnings ratios, while stocks in the old technology class have lower
prices and higher earnings and then lower price-over-dividend and priceover-
earnings ratios. This is indeed what is observed. If one goes back
in time, present “old technology” was new technology and a similar pattern
of high price-over-dividend and price-over-earnings ratios should be
seen. This has indeed been documented, for instance during the 1929
and 1962 bubbles.
Basic Ingredients of the Two Models
We now describe two models, which provide two extreme views of the
relationship between returns and risks associated with crashes. These
models use the no-arbitrage condition to link stock market returns during
modeling bubbles and crashes 149
bubbles and the risk associated with potential crashes. Bounded rationality
is used to obtain a simple specification of price dynamics. These two
models recognize as essential the coexistence of and interplay between
two distinct populations of traders: the “noise” traders on one hand and
the “rational” traders on the other hand.
In the first “risk-driven” model, by their imitative and cooperative
behavior, the exhuberant noise traders may make the market more and
more unstable at certain times, as they can sometimes change opinion
abruptly on a large scale. As the risk of a crash looms stronger, rational
traders are enticed to stay invested only because of the higher accelerating
returns, which provide an adequate compensation for the increasing
risks. The fundamental point in this model is that a crash is not certain
and there is a finite chance that the bubble ends and lands smoothly, thus
making it rational for traders to stay invested in the market and to profit
from (risky) gains.
The second “price-driven” model, discussed in this chapter, is also
based on the interplay between two distinct and complementary groups
of traders. The first population of noise traders drives the price volatility
up in an accelerating but stochastic spiral by their collective behavior,
allowing the emergence of price bubbles. The rational investors then
recognize that such a bubble is unsustainable and identify the existence
of an associated risk for a crash or of a severe correction that may
drive the price back to its fundamental value. This behavior, embodied
by the condition of no-arbitrage, leads to the following consequence:
anomalous sky-rocketing prices imply an increasing crash hazard rate,
defined as the probability that a crash will occur the next day, conditioned
on the fact that it has not yet happened. This increasing risk of a crash
is the unavoidable dark side of the market gains. Again, crashes are
stochastic events quantified by this hazard rate, which diverges when
the market valuation blows up. In this model, the long-term stationary
behavior of the market is a succession of normal random-walk phases,
with interpersed bubble phases ending in crashes bringing the market
back closer to fundamental valuation, like a springy young dog running
along with his mistress and receiving bolts that bring him back each time
he reaches the end of the rope. The remarkable property of this model
is that a crash may never happen if prices remain reasonable. This is
because the crash hazard rate is a strongly nonlinear amplifying function
of the price level. The probability of a crash is therefore very low at
modest price deviations from the fundamental value but becomes larger
and larger as the price increases. Even if the market price blows up, it
is always possible that the price will reverse smoothly without a crash,
150 chapter 5
a scenario that, however, becomes less and less probable the higher the
price is.
Summary of the Main Properties of the Model
The rational expectation model of bubbles and crashes discussed below
is an extension [221, 209, 212] of the Blanchard model [43] and of the
Blanchard and Watson model [45]. It finds justifications in microscopic
models of investor behaviors, developed to formalize herd behavior or
mutual mimetic contagion in speculative markets [273]. In such a class
of models, the emergence of bubbles is explained as a self-organizing
process of “infection” among traders, leading to equilibrium prices that
deviate from fundamental values. Assuming that the speculators’ readiness
to follow the crowd may depend on an economic variable, such as
actual returns, above-average returns are reflected in a generally more
optimistic attitude that fosters the disposition to overtake others’ bullish
beliefs, and vice versa. This economic influence makes bubbles transient
phenomena and leads to repeated fluctuations around fundamental
Here, we stress the salient features that will be useful for the analysis
of the market data sets presented in chapters 7–10. Our model has two
main components.
� Its key assumption is that a crash may be caused by local selfreinforcing
imitation between traders. This self-reinforcing imitation
process leads to the blossoming of a bubble. If the tendency for
traders to “imitate” their “friends” increases up to a certain point
called the “critical” point, many traders may place the same order
(sell) at the same time, thus causing a crash. The interplay between
the progressive strengthening of imitation and the ubiquity of noise
requires a stochastic description: a crash is not certain but can be
characterized by its hazard rate h�t�, that is, the probability per unit
time that the crash will happen in the next instant provided it has not
happened yet.
� Since the crash is not a certain deterministic outcome of the bubble, it
remains rational for traders to remain invested provided they are compensated
by a higher rate of growth of the bubble for taking the risk
of a crash, because there is a finite probability of “landing smoothly,”
modeling bubbles and crashes 151
that is, of attaining the end of the bubble without crash. In this model,
the ability to predict the critical date is perfectly consistent with the
behavior of the rational agents: they all know this date, the crash
may happen anyway, and they are unable to make any abnormal riskadjusted
profits by using this information.
The model distinguishes between the end of the bubble and the time of
the crash: the rational expectation constraint has the specific implication
that the date of the crash must have some degree of randomness. The
theoretical death of the bubble is not the time of the crash, because the
crash could happen at any time before, even though this is not very
likely. The death of the bubble is the most probable time for the crash.
The model does not impose any constraint on the amplitude of the
crash. If we assume that it is proportional to the current price level, then
the natural variable is the logarithm of the price. If, instead, we assume
that the crash amplitude is a finite fraction of the gain observed during
the bubble, then the natural variable is the price itself [212]. The standard
economic proxy is the logarithm of the price and not the price itself,
since only relative variations should play a role. However, different price
dynamics give both possibilities.
In the construction of a model, it is convenient to retain only the
essential aspects of reality and simplify by forgetting all the gory details
that are immaterial for the purpose of the model and that would blur the
demonstration. We thus neglect or incorporate dividends in the price, we
neglect the risk-free interest rate such as the interest you get on a CD
bank account (which can easily be reincorporated by a simple modification
of the argument), and we assume that investors are neutral with
respect to risks (again, this can be easily relaxed with some complication
of the model without changing the main conclusions) and that
all have the same information. Then, the no-arbitrage condition together
with rational expectations are simply equivalent to the statement that the
average of the price tomorrow based on all present knowledge and all
information revealed until the present is equal to the price today. In other
words, the average of the total price variation is zero. The same principle
is used when it is sometimes claimed that the best forecast for the
weather tomorrow is the weather today. This principle is a message of
complete randomness or, equivalently, of complete absence of knowledge
of the future. This condition is illustrated geometrically in Figure 5.4 and
corresponds to imposing that the average over all scenarios, shown as the
dark circle, be at the same price level as the empty circle representing
the price at the present time.
152 chapter 5
Present Future
Fig. 5.4. A price trajectory ending at the present, at the position of the open circle.
The six trajectories from present to future delineated by the vertical lines constitute
six possible scenarios. Averaging over all possible scenarios, given the present price,
gives a price shown as the dark circle.
The Crash Hazard Rate Drives the Market Price
For each period, for instance a day, the model assumes that two components,
and only two, compete to determine the price increment from one
day to the next: (1) a daily market return that may change and fluctuate
from day to day; (2) the possibility that a crash will occur.
In this framework, the no-arbitrage condition together with rational
expectations tell us that the price variation due to the market return
should compensate exactly the average loss due to the possibility of a
crash. The average loss is performed by considering all possible scenarios,
most of them having no crash and thus no loss. Only those scenarios
that lead to a crash yield a loss. We can group all scenarios that give a
crash and count them. Their proportion among all possible scenarios is
nothing but the hazard rate previously defined, that is, the probability that
a crash occurs knowing that it has not yet happened. Then, the average
loss is simply equal to the market drop due to a crash times the probability
that such a crash will occur on this day, since all other scenarios that
do not give a crash do not contribute to a loss. For instance, suppose that,
on a given day, a crash of 30% has a probability of 0�01 (a chance of one
in one hundred) to occur and a probability of 0�99 not to happen. Then,
the loss averaged over all possible scenerios is 30% × 0�01 = 0�3%.
The no-arbitrage condition together with rational expectations hold true,
under the condition that the market remunerates investors by a return
of 0�3%. In this presentation of the argument, we have assumed, to simmodeling
bubbles and crashes 153
plify the discussion, that all crashes have the same amplitude. The results
are essentially the same when one takes into account the variability of
crash sizes. We would then need to perform an additional average over
all possible crash amplitudes.
This line of reasoning provides us with the following important result:
the market return from today to tomorrow is proportional to the crash
hazard rate. As we announced, we have derived that the higher the risk
of a crash, the larger is the price return. In essence, investors must be
compensated by a higher return in order to be induced to hold an asset
that might crash. This is the only effect that we wish to capture in this
part of the model. This effect is fairly standard, and it was pointed out
earlier in a closely related model of bubbles and crashes under rational
expectations by Blanchard [43]. It may go against the naive preconception
that price is adversely affected by the probability of the crash, but
this result is the only one consistent with rational expectations.
Let us stress an interesting subtlety that this reasoning allows us to
unearth. The no-arbitrage condition together with rational expectations
imposes that the total average return at any time is exactly zero. The zero
average return embodies the unrealized risks of a looming crash. This
return is not what investors actually experience but would correspond to
the average gain that a pool of many investors would get by aggregating
their portfolios when living over many repetitions of history, some with
a crash and most without a crash. In contrast, knowing that the crash has
not yet occurred, the return is not zero and may indeed exhibit all features
of a speculative bubble with inflating prices. We cannot stress enough
that there is no contradiction between the two ways of quantifying market
returns. Some might question the validity of the averaging procedure
over all possible scenarios. The point is that, in the absence of advanced
knowledge of the future, its best predictor is the average of all possible
scenarios. This market price reflects the equilibrium between the greed
of buyers who hope the bubble will inflate and the fear of sellers that it
may crash. A bubble that goes up is just one that could have crashed but
did not.
The situation can perhaps be clarified further with the following analog
example. Suppose you are given the possibility to play a casino game
with a rotating wheel with 100 numbers, such that you lose $30 if the
number comes out as 1 and you gain $x otherwise. What is the minimum
value of the gain $x that can make this a game fair and entice
you to play? The simplest idea is to request that you should obtain at
least a nonnegative gain, on average, over many repetitions of the game.
This average is $x × 99 ? $30 × 1 divided by the total number 100 of
154 chapter 5
outcomes of the casino wheel. We thus see that the minimum value of x
that makes the average gain positive is $30/99, which is close to $0�3. A
minimum gain of $0�3 for any of the numbers 2 to 100 is thus required
to make the gain at least fair from your point of view (and profitable on
average if $x is larger). Thus, as long as the number 1 does not come
up, each game remunerates you with a gain of $0�3, which thus gives
the impression of an anomalous bias in your favor. Indeed, since the
number 1 has only one chance in one hundred to come out, the typical
number of games one needs to play to encounter it once is 100. One
may thus be attracted to this game and reason that it is safe to play the
game for a while, say n < 100 times, and thus accumulate a profit equal
to n times $0�3. As in the stock market, the gambler needs to decide
when to stop (exit) and be happy with her gains. Otherwise, she will
eventually get the number 1 and suddenly lose the gain of 100 games.
This example illustrates how a return can be large, conditioned on the
fact that the crash has not occurred. This return actually compensates for
the risk that the number 1 may come up at any time.
Now, suppose that you knew in advance that the number 1 was not
going to come out in the next game. It is clear that you would play
the game even if the gain $x is smaller than $0�3 as long as it remains
positive. It is the absence of knowledge of the future that requires a remuneration
for taking risks precisely associated with the lack of knowledge
of the future. If we knew the specific future exactly, risk would vanish
(which does not mean that bad news would disappear).
To be complete, we should add that most people would not play this
game if the gain $x for the numbers 2 to 100 were only $0�3 because they
are “risk averse”: this means that most people do not like to gain zero on
average while facing the possibility of losing at some times. Most people
need a positive bias above $0�3 to play such a game. This subject of risk
aversion and its consequences for economic modeling is an important
subject of its own, which refers to a large body of scholarly work
dating back at least from the founding book [443] of Von Neumann and
Morgenstern, which introduced the concept of a utility function to
address this problem specifically. Risk aversion is a central feature
of economic theory, and it is generally thought to be stable within
a reasonable range, associated with slow-moving secular trends like
changes in education, social structures, and technology. For our purpose
here, it suffices to say that the market return may be larger than the
minimum value imposed by the no-arbitrage condition together with the
rational expectations discussed above. The important message is thus
the existence of this minimum. Risk aversion is easily incorporated into
modeling bubbles and crashes 155
our model, for instance by saying that the probability of a crash in
the next instant is perceived by traders as being some factor F times
bigger than it objectively is. This amounts to multiplying our hazard
rate by this same factor F . This makes no substantive difference to our
conclusion as long as F is bounded away from zero and infinity (a very
weak restriction indeed).
Imitation and Herding Drive the Crash Hazard Rate
The crash hazard rate quantifies the probability that a large group of
agents place sell orders simultaneously and create enough of an imbalance
in the order book for market makers to be unable to absorb the
other side without lowering prices substantially. Most of the time, market
agents disagree with one another and submit roughly as many buy orders
as sell orders (these are all the times when a crash does not happen).
The key question is, By what mechanism did they suddenly manage to
organize a coordinated sell-off?
As discussed in the last section of chapter 4, titled “Cooperative
Behavior Resulting from Imitation,” all the traders in the world are organized
into a network (of family, friends, colleagues, etc.) and they influence
each other locally through this network. For instance, an active
trader is constantly on the phone exchanging information and opinions
with a set of selected colleagues. In addition, there are indirect interactions
mediated, for instance, by the media and the Internet. Our working
hypothesis is that agents tend to imitate the opinions of their connections
according to the mechanism detailed in the section titled “It Is Optimal
to Imitate,” in chapter 4. The interaction between connections will tend
to create order, while personal idiosynchrasis will tend to create disorder.
Disorder represents the notions of heterogeneity or diversity as opposed
to uniformity.
The main story here is a fight between order and disorder. As far as
asset prices are concerned, a crash happens when order wins (a majority
has the same opinion: selling), and normal times are when disorder wins
(buyers and sellers disagree with each other and roughly balance each
other out). This mechanism does not require an overarching coordination,
since macro-level coordination can arise from micro-level imitation and
it relies on a realistic model of how agents form opinions by constant
Many models of interaction and imitation between traders have been
developed. We have described some of them in chapter 4. To make a
156 chapter 5
long story short, the upshot is that the fight between order and disorder
often leads to a regime where order may win. When this occurs, the
bubble ends. Models that contain the imitation mechanism undergo this
transition in a “critical” manner: the sensitivity of the market reaction
to news or external influences increases in an accelerated manner on the
approach to this transition. This was shown in chapter 4 in the set of
Figures 4.8–4.10 representing the configurations of buyers and sellers in
a simple space of investors arranged on a square Manhattan-like lattice.
When the imitation strength K gets close to a special critical value Kc
(whose specific value is not important and depends on details of the
models), very large groups of investors share the same opinion and may
act in a coordinate manner. This leads to a remarkable and very specific
precursory “power law” signature, which we now explain.
Let us assume that the imitation strength K changes smoothly with
time, as will be shown later in Figure 5.7, as a result, for instance, of the
varying confidence level of investors, the economic outlook, and similar
factors. The simplest assumption, which does not change the nature of
the argument, is that K is proportional to time. Initially, K is small and
only small clusters of investors self-organize, as shown in Figure 4.8.
As K increases, the typical size of the clusters increases as shown in
Figure 4.9. These kinds of systems exhibiting cooperative behavior are
characterized by a broad distribution of cluster sizes s (the size of the
black islands, for instance) up to a maximum s?, which itself increases in
an accelerating fashion up to the critical value Kc as shown in Figure 5.5.
As explained in chapter 4, right at K = Kc, the geography of clusters
of a given kind becomes self-similar with a continuous hierarchy of
sizes from the smallest (the individual investor) to the largest (the total
system). Within this phenomenology, the probability for a crash to occur
is constructed as follows.
First, a crash corresponds to a coordinated sell-off of a large number
of investors. In our simple model, this will happen as soon as a single
cluster of connected investors, which is sufficiently large to set the
market off-balance, decides to sell off. Recall indeed that “clusters” are
defined by the condition that all investors in the same cluster move in
concert. When a very large cluster of investors sells, this creates a sudden
unbalance, which triggers an abrupt drop of the price, and hence
a crash. To be concrete, we assume that a crash occurs when the size
(number of investors) s of the active cluster is larger than some minimum
value sm. The specific value sm is not important, only the fact that
sm is much larger than 1, so that a crash can only occur as a result of
a cooperative action of many traders who destabilize the market. At this
modeling bubbles and crashes 157
0.2 0.4 0.6 0.8
Fig. 5.5. Power law acceleration of the size s? (in arbitrary units) of the typical
largest cluster as a function of the imitation strength K. As K approaches Kc, s?
diverges. This divergence embodies the observation that infinitely large clusters form
at the critical point Kc. In practice, s? is bounded by the system size.
stage, we do not specify the amplitude of the crash, only its triggering
as an instability. In general, investors change opinion and send market
orders only rarely. Therefore, we should expect only one or few large
clusters to be simultaneously active and able to trigger a crash.
For a crash to occur, we thus need to find at least one cluster of size
larger than sm and to verify that this cluster is indeed actively selling
off. Since these two events are independent, the total probability for a
crash to occur is thus the product of the probability of finding such a
cluster of size larger than the threshold sm by the probability that such
a cluster begins to sell off collectively. The probability ns of finding a
cluster of size s is a well-known characteristic of critical phenomena
[164, 414]: it is a power law distribution truncated at a maximum s?;
this maximum increases without bound (except for the total system size)
on the approach to the critical value Kc of the imitation strength, as we
see in Figure 5.5.
If the decision to sell off by an investor belonging to a given cluster
of size s was independent of the decisions of all the other investors in
the same cluster, then the probability per unit time that such a cluster of
size s would become active would be simply proportional to the number
s of investors in that cluster. However, by the very definition of a cluster,
investors belonging to a given cluster do interact with each other. Therefore,
the decision of an investor to sell off is probably quite strongly
coupled with those of the other investors in the same cluster. Hence, the
158 chapter 5
Probability of a Crash
0.2 0.4 0.6 0.8
Crash Hazard Rate
0.2 0.4 0.6 0.8
Fig. 5.6. Left panel: Probability for a crash to occur. In this example, the probability
reaches its maximum equal to 0�7 at the critical point K = Kc with an infinite slope.
Right panel: Crash hazard rate. The crash hazard rate is proportional to the slope of
the probability shown in the left panel and goes to infinity at K = Kc. Equivalently,
the area under the curve of the hazard rate of the right panel up to a given K/Kc is
proportional to the probability shown in the left panel for this same value K = Kc.
probability per unit time that a specific cluster of s investors becomes
active is a function of the number s of investors belonging to that cluster
and of all the interactions between these investors. Clearly, the maximum
number of interactions within a cluster is s × �s ? 1�/2; that is, for large
s, it becomes proportional to the square of the number of investors in
that cluster. This occurs when each of the s investors speaks to each of
his or her s ? 1 colleagues. The factor 1/2 accounts for the fact that if
investor Anne speaks to investor Paul, then in general Paul also speaks
to Anne, and their two-ways interactions must be counted only once. Of
course, one can imagine more complex situations in which Paul listens
to Anne but Anne does not reciprocate, but this does not change the
results. Notwithstanding these complications, one sees that the probability
h�t��t per unit time �t that a specific cluster of s investors becomes
active must be a function growing with the cluster size s faster than s but
probably slower than the maximum number of interactions (proportional
to s2). A simple parameterization is to take h�t��t proportional to the
cluster size s elevated to some power
larger than 1 but smaller than 2.
This exponent
captures the collective organization within a cluster of
size s due to the multiple interactions between its investors. It is deeply
related to the concept of fractal dimensions, explained in chapter 6.
modeling bubbles and crashes 159
The probability for a crash to occur, which is the same as the probability
of finding at least one active cluster of size larger than the minimum
destabilizing size sm, is therefore the sum over all sizes s larger than sm of
all the products of probabilities ns to find a cluster of a specific size s by
their probability per unit time to become active (itself proportional to s
as we have argued). With mild technical conditions, it can then be shown
that the crash hazard rate exhibits a power law acceleration as shown in
Figure 5.6. Intuitively, this behavior stems from the interplay between
the existence of larger and larger clusters as the interaction parameter K
approached its critical value Kc and from the nonlinear accelerating probability
per unit time for a cluster to become active as its typical size s?
grows with the approach of K to Kc. In sum, the risk of a crash per unit
time, knowing that the crash has not yet occurred, increases dramatically
when the interaction between investors becomes strong enough that the
network of interactions between traders self-organizes into a hierarchy
containing a few large, spontaneously formed groups acting collectively.
If the hazard rate exhibits this behavior, the previous section convinced
us that the return must exhibit the same behavior in order for the
no-arbitrage condition together with rational expectations to hold true.
We find here our first prediction of a specific pattern of the approach to
a crash: returns increase faster and faster; that is, they accelerate with
time. Since prices are formed by summing returns, the typical trajectory
of a price as a function of time, which is expected on the approach to a
critical point, is parallel to the dependence of the probability of a crash
shown in the left panel of Figure 5.6.
We stress that Kc is not the value of the imitation strength at which
the crash occurs, because the crash could happen for any value before
Kc, though this is not very likely. Kc is the most probable value of the
imitation strength for which the crash occurs. To translate these results
as a function of time, it is natural to expect that the imitation strength K
is changing slowly with time as a result of several factors influencing the
tendency of investors to herd. A typical trajectory K�t� of the imitation
strength as a function of time t is shown in Figure 5.7. The critical time
tc is defined as the time at which the critical imitation strength Kc is
reached for the first time starting from some initial value. tc is not the
time of the crash, it is the end of the bubble. It is the most probable
time of the crash because the hazard rate is largest at that time. Due
to its probabilistic nature, the crash can occur at any other time, with
the likelihood changing with time following the crash hazard rate. In a
given time history, the evolution of K as a function of time follows a
trajectory like that shown in Figure 5.7. For each value of K, we read on
160 chapter 5
tc t
Fig. 5.7. A typical evolution of the imitation strength K�t� as a function of time t
showing its smooth and slow variation. As time goes on, K may approach and even
cross the critical value Kc at a critical time tc at which very large clusters of investors
are created spontaneously and may trigger a crash. Around tc, the dependence of
K�t� is approximately linear, as shown by the thick linear segment tangent to the
the right panel of Figure 5.6 the corresponding value of the crash hazard
rate. Since K may go up and down, so does the crash hazard rate.
As shown in the left panel of Figure 5.6, there is a residual finite
probability (0�3 in this example) of attaining the critical time tc without
a crash. This residual probability is crucial for the coherence of the story,
because otherwise the whole model would unravel since rational agents
would anticipate the crash with certainty.
Intuitive explanation of the creation of a finite-time singularity at tc.
The faster-than-exponential growth of the return and of the crash hazard
rate correspond to nonconstant growth rates, which increase with the
return and with the hazard rate. The following reasoning allows us to
understand intuitively the origin of the appearance of an infinite slope or
infinite value in a finite time at tc, called a finite-time singularity.
Suppose, for instance, that the growth rate of the hazard rate doubles
when the hazard rate doubles. For simplicity, we consider discrete-time
intervals as follows. Starting with a hazard rate of 1 per unit time,
we assume it grows at a constant rate of 1% per day until it doubles.
We estimate the doubling time as proportional to the inverse of the
growth rate, that is, approximately 1/1% = 1/0�01 = 100 days. There
is a multiplicative correction term equal to ln 2 = 0�69 such that the
doubling time is ln 2/1% = 69 days. But we drop this proportionality
modeling bubbles and crashes 161
factor ln 2 = 0�69 for the sake of pedagogy and simplicity. Including
it just multiplies all time intervals below by 0�69 without changing the
When the hazard rate turns 2, we assume that the growth rate doubles
to 2% and stays fixed until the hazard rate doubles again to reach 4. This
new doubling time is only approximately 1/0�02 = 50 days at this 2%
growth rate. When the hazard rate reaches 4, its growth rate is doubled
to 4%. The doubling time of the hazard rate is therefore approximately
halved to 25 days, and the scenario continues with a doubling of the
growth rate every time the hazard rate doubles. Since the doubling time
is approximately halved at each step, we have the following sequence:
(time = 0, hazard rate = 1, growth rate = 1%), (time = 100, hazard
rate = 2, growth rate = 2%), (time = 150, hazard rate = 4, growth rate =
4%), (time = 175, hazard rate = 8, growth rate = 8%), and so on. We
observe that the time interval needed for the hazard rate to double is
shrinking very rapidly by a factor of 2 at each step. In the same way that

  • 1
  • 1
  • 1
    +· · · = 1�
    which was immortalized by the ancient Greeks as Zeno’s paradox, the
    infinite sequence of doubling thus takes a finite time and the hazard
    rate reaches infinity at a finite “critical time” approximately equal to
    100 + 50 + 25+· · · = 200 (a rigorous mathematical treatment requires a
    continuous-time formulation, which does not change the qualitative content
    of the example). A spontaneous singularity has been created by the
    increasing growth rate! This process is quite general and applies as soon as
    the growth rate possesses the property of being multiplied by some factor
    larger than 1 when the hazard rate or any other observable is multiplied
    by some constant larger than 1. We shall revisit this example in chapter 10
    when we analyze the world demography, major financial indices, and the
    World Gross Economic product over several centuries to look ahead and
    attempt to predict what is coming next.
    To sum up, we have constructed a model in which the stock market
    price is driven by the risk of a crash, quantified by its hazard rate. In
    turn, imitation and herding forces drive the crash hazard rate. When the
    imitation strength becomes close to a critical value, the crash hazard rate
    162 chapter 5
    diverges with a characteristic power law behavior. This leads to a specific
    power law acceleration of the market price, providing our first predictive
    precursory pattern anticipating a crash. The imitation between agents
    leading to an accelerating crash hazard rate may result, for instance,
    from a progressive shift in the belief of investors about market liquidity,
    without invoking asymmetric information, and independently of the price
    behavior and its deviation from its fundamental value [132].
    The price-driven model inverts the logic of the previous risk-driven
    model: here, again as a result of the action of rational investors, the price
    is driving the crash hazard rate rather than the reverse. The price itself is
    driven up by the imitation and herding behavior of the “noisy” investors.
    As before, a stochastic description is required to capture the interplay
    between the progressive strengthening of imitation controlled by
    the connections and interactions between traders and the ubiquity of
    idiosyncratic behavior as well as the influence of many other factors that
    are impossible to model in detail. As a consequence, the price dynamics
    are stochastic and the occurrence of a crash is not certain but can
    be characterized by its hazard rate h�t�, defined as the probability per
    unit time that the crash will happen in the next instant if it has not
    happened yet.
    Imitation and Herding Drive the Market Price
    Hsieh has stressed that the evidence documented in chapter 2 of an
    absence of correlation of price changes and a strong persistence of
    volatility (i.e., the amplitude of the price variations), when taken together,
    cannot be explained by any linear model [201, 202]. Recall that a linear
    model is a description in which the consequence or output is proportional
    to the cause. Nonlinearity generalizes tremendously the quite special
    “linear” behavior by allowing the output to depend on the cause in a
    more complicated way. Nonlinearity is an ingredient of chaos, a theory of
    complex systems that have been studied intensely in the last few decades
    as a possible origin of complexity. Chaos has been widely popularized
    and has even been advocated by some as a useful description of stock
    markets. This, however, remains too simplistic, as chaos theory relies
    on the assumption that only a few major variables interact nonlinearly
    modeling bubbles and crashes 163
    and create complicated trajectories. In reality, the stock market needs
    many variables to obtain a reasonably accurate description. In technical
    jargon, the stock market has many degrees of freedom, while chaos theory
    requires only a few. The existence of many degrees of freedom is
    precisely the ingredient used by the models of collective behavior that
    exhibit critical points described in the previous section and in chapter 4.
    Here, we retain only the more general observation that effects are not
    proportional to causes, that is, that the world and the stock market are
    nonlinear systems.
    A well-known joke among scientists in this field is to compare “nonlinearity”
    with a “non-elephant”: all creatures, except the elephants, are
    non-elephant; similary, all systems and phenomena are nonlinear, except
    the very special subsystems that are linear. Notwithstanding the fact that
    we are educated at school in a “linear” framework of thoughts, this
    ill-prepares us for the intrinsic nonlinearity of the universe, be it physical,
    biological, psychological, or social. Nonlinearity is at the origin
    of the most profound difficulties in disentangling the causes of a given
    observation: since effects are not in general proportional to causes, two
    causes do not add up their impacts. Indeed, the output resulting from
    the presence of two causes acting simultaneously is not the sum of the
    outputs obtained in the presence of each cause in the absence of the
    other one.
    It is customary among modelers of financial markets to represent the
    price variation over an elementary time period as resulting from two contributions:
    a certain instantaneous return and a random return. The first
    constribution embodies the remuneration due to estimated risks as well
    as the effect of imitation and herding. The second contribution embodies
    the noise component of the price dynamics with an amplitude called the
    volatility. The volatility can also present a systematic component controlled
    by imitation as well as many other factors. If the first contribution
    is absent and the volatility is constant, the second term alone creates the
    random walk trajectories described in chapter 2. Reinserting the ubiquitous
    property of nonlinear dependence of the volatility and of the certain
    instantaneous return on past values of the volatility and the returns provides
    a rich universe of possible trajectories. Here, I am interested in
    the many possible mechanisms leading to a nonlinear positive feedback
    of prices on themselves. For instance, imperfect information and risk
    shifting from investors to lending banks may lead investors to bid up
    asset prices far above what they would be willing to pay if they were
    fully exposed to all potential losses [3]. We shall return to an intuitive
    description of other mechanisms in chapters 7 and 8.
    164 chapter 5
    The Price Return Drives the Crash Hazard Rate
    Earlier in this chapter, we showed that the no-arbitrage condition together
    with the rational expectations imposes that the price variations from one
    day to the next should compensate exactly for the average loss due to
    the possibility of a crash. We now view this balance in the reverse logic:
    noisy investors look at the market price going up, they speak to each
    other, develop herding, buy more and more of the stock, thus pushing
    prices further up. As the price variation speeds up, the no-arbitrage condition,
    together with rational expectations, then implies that there must
    be an underlying risk, not yet revealed in the price dynamics, which justifies
    this apparent free ride and free lunch. The fundamental logic here
    is that the no-arbitrage condition, together with rational expectations,
    automatically implies a dramatic increase of a risk looming ahead each
    time the price appreciates significantly, such as in a speculative frenzy or
    in a bubble. This is the conclusion that rational traders will reach. This
    phenomenon can be summarized by the following proverb applied to an
    accelerating bullish market: “It’s too good to be true.”
    In the goal of capturing the phenomenon of speculative bubbles, we
    focus on the class of models with positive feedbacks, as discussed in
    chapter 4. In the present context, this means that the instantaneous return
    as well as the volatility become larger and larger when past prices and/or
    past returns and/or past volatilities become large. As explained in the
    technical insert entitled “Intuitive Explanation of the Creation of a Finite-
    Time Singularity at tc” earlier in this chapter, such positive feedbacks
    with increasing growth rate may lead to singularities in a finite time.
    Here, this means that, unchecked, the price would blow up without
    bounds. However, two effects compete to tamper with this divergence.
    First, the stochastic component impacting the price variations makes
    the price much more erratic, and the convergence to the critical time
    becomes a random, uncertain event. This is represented in Figure 5.8,
    illustrating the variability of the price trajectory preceding the singularity
    of B�t�.
    Figure 5.8 shows a typical trajectory of the bubble component of the
    price generated by the nonlinear positive feedback model [396], starting
    from some initial value up to the time just before the price starts
    to blow up. The simplest version of this model consists in a bubble
    price B�t� being essentially a power of the inverse of a random walk
    W�t� in the following sense. Starting from B�0� = W�0� = 0 at the
    origin of time, when the random walk approaches some value Wc, here
    modeling bubbles and crashes 165
    500 1000 1500 2000
    500 1000 1500 2000
    500 1000 1500 2000
    500 1000 1500 2000
    Fig. 5.8. Top panel: Realization of a bubble price B�t� as a function of time constructed
    from the “singular inverse random walk.” This corresponds to a specific
    realization of the random numbers used in generating the random walks W�t� represented
    in the second panel. The top panel is obtained by taking a power of the
    inverse of a constant Wc, here taken equal to 1 minus the random walk shown in
    the second panel. In this case, when the random walk approaches 1, the bubble
    diverges. Notice the similarity between the trajectories shown in the top (B�t�) and
    second (W�t�) panels as long as the random walk W�t� does not approach the
    value Wc
    = 1 too much. It is free to wander, but when it approaches 1, the bubble
    price B�t� shows much greater sensitivity and eventually diverges when W�t�
    reaches 1. Before this happens, B�t� can exhibit local peaks, that is, local bubbles,
    which come back smoothly. This corresponds to realizations of when the random
    walk approaches Wc without touching it and then spontaneously recedes away from
    it. The third (respectively, fourth) panel shows the time series of the increments
    dB�t� = B�t� ? B�t ? 1� of the bubble (respectively, dW�t� = W�t� ? W�t ? 1�
    of the random walk). Notice the intermittent bursts of strong volatility in the bubble
    compared to the featureless constant level of fluctuations of the random walk
    (reproduced from [396]).
    166 chapter 5
    taken equal to 1, B�t� increases and vice versa. In particular, when
    W�t� approaches 1, B�t� blows up and reaches a singularity at the
    time tc when the random walk crosses 1. This process generalizes in
    the random domain the finite-time singularities described earlier in this
    chapter, such that the monotonously increasing process culminating at
    a critical time tc is replaced by the random walk that wanders up and
    down before eventually reaching the critical level. This nonlinear positive
    feedback bubble process B�t� can thus be called a “singular inverse
    random walk.” In absence of a crash, the process B�t� can exist only
    up to a finite time: with probability 1 (i.e., with certainty), we know
    from the study of random walks that W�t� will eventually reach any
    level, in particular the value Wc
    = 1 in our example, at which B�t�
    The second effect that tampers with the possible divergence of the
    bubble price, by far the most important one in the regime of highly
    overpriced markets, is the impact of the price on the crash hazard rate
    discussed above: as the price blows up due to imitation, herding, speculation,
    and randomness, the crash hazard rate increases even faster, so
    that a crash will occur and drive the price back closer to its fundamental
    value. The crashes are triggered in a random way governed by the crash
    hazard rate, which is an increasing function of the bubble price. In the
    present formulation, the higher the bubble price, the higher is the probability
    of a crash. In this model, a crash is similar to a purge administered
    to a patient.
    Determination of the crash hazard rate. Concretely, a simulation using
    a computer program proceeds as follows. First, we choose a discretization
    of the time in steps on size t. Then, knowing the value of the random
    walk W�t ? t� and the price B�t ? t� at the previous time t ? t, we
    construct W�t� by adding an increment taken from the centered Gaussian
    distribution with variance t. From this, we construct the price B�t� by
    taking the inverse of �Wc
    , where
    is a positive exponent defined
    in the model. We then read off from the no-arbitrage condition together
    with the rational expectations what the probability h�t� t is for a crash
    to occur during the next time step, where h�t� is the crash hazard rate.
    We compare this probability with a random number ran uniformly drawn
    in the interval �0� 1� and trigger a crash if ran ≤ h�t� t. In this case,
    the price B�t� is changed into B�t��1 ? ��, where � is drawn from a
    prechosen distribution. For instance, the crash drop � can be fixed to,
    say, 20%. It is straightforward to generalize to an arbitrary distribution of
    jumps. After the crash, the dynamics proceeds incrementally as before,
    modeling bubbles and crashes 167
    starting from this new value for time t after a proper translation of W�t�
    to ensure continuity of prices. If ran > h�t� t, no crash occurs and the
    dynamics can be iterated another time step.
    This model thus proposes two scenarios for the end of a bubble: either
    a spontaneous deflation or a crash. These two mechanisms are natural
    features of the model and have not been artificially added. These two
    scenarios are indeed observed in real markets, as will be described in
    chapters 7–9.
    This model has an interesting and far-reaching consequence in terms
    of the repetition and organization of crashes in time. Indeed, we see
    that each time the random walk approaches the chosen constant Wc,
    the bubble price blows up and, according to the no-arbitrage condition
    together with rational expectations, this implies that the market enters
    “dangerous waters” with a crash looming ahead. The random walk model
    provides a very specific prediction of the waiting times between successive
    approaches to the critical value Wc, that is, between successive
    bubbles. The distribution of these waiting times is found to be a very
    broad power law distribution [394], so broad that the average waiting
    time is mathematically infinite. In practice, this leads to two interrelated
    phenomena: clustering (bubbles tend to follow bubbles at short times)
    and long-term memory (there are very long waiting times between bubbles
    once a bubble has deflated for a sufficiently long time). In particular,
    amusing paradoxes follow, such as “the longer since the last bubble, the
    longer the waiting time till the next” [402]. Anecdotally, this property
    of random walks also explains the overwhelming despair of frustrated
    drivers on densely packed highways that neighboring lanes always go
    faster than their lane because they often do not notice catching up to a
    car that was previously adjacent to them: assuming that we can model
    the differential motion of lanes in a global traffic flow by a random walk,
    this impression is a direct consequence of the divergence of the expected
    return time of a random walk! To summarize, the “singular inverse random
    walk” bubble model predicts very large intermittent fluctuations in
    the recurrence time of speculative bubbles.
    An additional layer of refinement can easily be added. Indeed, following
    [184], which introduced so-called Markov switching techniques
    for the analysis of price returns, many scholarly works have documented
    the empirical evidence of regime shifts in financial data sets [432, 175,
    63, 431, 363, 24, 80, 110]. For instance, Schaller and Van Norden [363]
    have proposed a Markov regime-switching model of speculative behavior
    whose key feature is similar to ours, namely overvaluation of the price
    168 chapter 5
    above the fundamental price increases the probability and expected size
    of a stock market crash.
    This evidence, taken together with the fact that bubbles are not
    expected to permeate the dynamics of the price all the time, suggests
    the following natural extension of the model. In the simplest and most
    parsimonious extension, we can assume that only two regimes can
    occur: bubble and normal. The bubble regime follows the previous
    model definition and is punctuated by crashes occuring with the hazard
    rate governed by the price level. The normal regime can be, for
    instance, a standard random walk market model with constant small
    drift and volatility. The regime switches are assumed to be completely
    random. This dynamical and very simple model recovers essentially all
    the stylized facts of empirical prices, that is, no correlation of returns,
    long-range correlation of volatilities, a fat tail on return distributions,
    apparent fractality and multifractality, and sharp peak–flat trough pattern
    of price peaks. In addition, the model predicts and we confirm by
    empirical data analysis that times of bubbles are associated with nonstationary
    increasing volatility correlations. This will be further elaborated
    in our empirical chapters 7–10. The apparent long-range correlation of
    volatility is proposed to result from random switching between normal
    and bubble regimes. In addition, and perhaps most importantly, the
    visual appearance of price trajectories is very reminiscent of real ones,
    as shown in Figure 5.9. The remarkably simple formulation of the
    price-driven “singular inverse random walk” bubble model is able to
    reproduce convincingly the salient properties and appearance of real
    price trajectories, with their randomness, bubbles, and crashes.
    Together, the risk-driven model and the price-driven model presented in
    this chapter describe a system of two populations of traders, the “rational”
    and the “noisy” traders. Occasional imitative and herding behaviors
    of the noisy traders may cause global cooperation among traders, causing
    a crash. The rational traders provide a direct link between the crash
    risks and the bubble price dynamics.
    In the risk-driven model, the crash hazard rate determined from herding
    drives the bubble price. In the price-driven model, imitation and
    herding induce positive feedbacks on the price, which itself creates an
    increasing risk for a looming yet unrealized financial crash.
    modeling bubbles and crashes 169
    Fig. 5.9. Top panel: The Hang Seng index (thick line) from July 1, 1991 to February
    4, 1994 (denoted “bubble II” in Figure 7.8 and analyzed in Figure 7.10) as well
    as ten realizations of the “singular inverse random walk” bubble model generated
    by the nonlinear positive feedback model [396]. Each realization corresponds to an
    arbitrary random walk whose drift and variance have been adjusted so as to best fit
    the distribution of the Heng Seng index returns. Bottom panel: The Nasdaq composite
    index bubble (thick line) from October 5, 1998 to March 27, 2000 analyzed
    in Figure 7.22 as well as ten realizations of the “singular inverse random walk”
    bubble model generated by the nonlinear positive feedback model [396]. Each realization
    corresponds to an arbitrary random walk whose drift and variance have been
    adjusted so as to best fit the distribution of the Nasdaq index returns. Reproduced
    from [396].
    We believe that both models capture a part of reality. Studying them
    independently is the standard strategy of dividing-to-conquer the complexity
    of the world. The price-driven model appears as perhaps the
    most natural and straightforward, as it captures the intuition that skyrocketing
    prices are unsustainable and announce endogeneously a significant
    correction or a crash. The risk-driven model captures a very
    subtle self-organization of stock markets, related to the ubiquitous balance
    between risk and returns. Both models embody the notion that the
    170 chapter 5
    market anticipates the crash in a subtle, self-organized, and cooperative
    fashion, hence releasing precursory “fingerprints” observable in stock
    market prices. In other words, this implies that market prices contain
    information on impending crashes. The next chapter 6 explores the origin
    and nature of these precursory patterns and prepares the road for a
    full-fledged analysis of real stock market crashes and their precursors.
    Chapter 6 also provides a description of price dynamics incorporating
    the interplay between trend-followers (who replace the noisy traders
    considered here) and value-investors (who replace the rational traders
    envisioned here). Recognizing the importance of their nonlinear (close
    to threshold-like) behavior leads to regimes similar to but richer than
    those described until now. This approach pertains to a body of literature
    taking a middle ground between fully rational and irrational behavior
    [239]: stock prices can rationally change as information is released and
    revealed through the trading process itself. As the market conditions do
    not allow the complete aggregation of individuals’ information in a fully
    revealing rational expectation equilibrium, prices may deviate substantially
    from their fundamental value. Lack of common knowledge about
    traders’ preferences or beliefs has been shown to create crashes in models
    (see [239] and references therein). The mechanism is that some external
    news may provide the trigger that reveals internal news (among traders)
    through the trading process.
    chapter 6
    hierarchies, complex
    fractal dimensions,
    and log-periodicity
    The previous chapter 5 put forward the concept
    that a critical point in the time domain, or equivalently a finite-time singularity,
    underlies stock market crashes. A crash is not the critical or
    singular point itself, but its triggering rate is strongly influenced by the
    proximity of the critical point: the closer to the critical time, the more
    probable is the crash. We have seen that the hallmark of critical behavior
    is a power law acceleration of the price, of its volatility, or of the
    crash hazard rate, as the critical time tc is approached. The purpose of
    the present chapter is to extend this analysis and suggest that additional
    important ingredients and patterns beyond the simple power law acceleration
    should be expected. An important motivation is that a power law
    acceleration is notoriously difficult to detect and to qualify in practice in
    the presence of the ubiquitous noise and irregularities of the trajectories
    of stock market prices.
    As we already emphasized, the stock market is made of actors that
    differ in size by many orders of magnitudes, ranging from individuals
    to gigantic professional investors such as pension funds. Structures
    at even higher levels, such as currency influence spheres (U.S.$, Euro,
    Yen, � � � ), exist and with the current globalization and deregulation of the
    market one may argue that structures on the largest possible scale—that
    172 chapter 6
    of the world economy—are beginning to form. This means that the structure
    of the financial markets has features that resemble that of hierarchical
    systems with “agents” on all levels of the market. Of course, this
    does not imply that any strict hierarchical structure of the stock market
    exists. However, critical phenomena induced by imitation forces in these
    conditions may often exhibit a rather nonintuitive phenomenon, called
    “log-periodicity,” in which, for instance, the probability or the hazard
    rate are not monotonously accelerating as shown in Figure 5.6 but are
    decorated by oscillations with frequencies accelerating as the critical
    time is approached. In the present chapter, we explore this novel phenomenon
    and explain its possible origins. The main message is that these
    oscillatory structures provide a complementary signature of impending
    criticality which is more robust with respect to noise. These patterns will
    turn out to be instrumental in the analysis performed on past crashes and
    in the prediction of future crashes presented in chapters 7–10.
    In this chapter, we first show how models of cooperative behaviors
    resulting from imitation between agents organized within a hierarchical
    structure exhibit the announced critical phenomena decorated with “logperiodicity.”
    Log-periodicity turns out to be a direct and general signature
    of the existence of a preferred scaling factor of similarity (which is then
    called discrete scale invariance), corresponding to the magnifying factor
    linking one level of the hierarchy to the next. We then formalize this idea
    a bit and show how a remarkable technique, called the “renormalization
    group,” capitalizes on the existence of multiscale self-similar properties
    of critical phenomena to derive a fundamental and concise description
    of these patterns. We provide several graphical examples, including the
    generalized Weierstrass function, a fractal model of stock market price
    trajectories that is continuous but exhibits jerky structures at all scales
    of magnification.
    Even more interesting and surprising is the discovery that logperiodicity
    and discrete scale invariance in critical phenomena may
    emerge spontaneously from a purely dynamical origin, without a preexisting
    hierarchy. To show this, we discuss a simple model exhibiting
    a finite-time singularity due to a positive feedback induced by trendfollowing
    investment strategies. Without any additional ingredients, it
    does not introduce a significant novelty compared to the models presented
    in chapter 5. The novel idea is to add the impact of fundamental
    analysts who tend to restore the price back to its fundamental value.
    When this restoring force is a nonlinear function of the difference
    between the bubble price and the fundamental value, the dynamics of the
    price exhibits a competition between power law acceleration culminating
    hierarchies and log-periodicity 173
    in a finite-time singularity, as shown in chapter 5, and accelerating
    log-periodic oscillations decorating this power law acceleration. The
    interplay between these two patterns is shown to be robust as a function
    of model specification. Intuitively, the strategies based on fundamental
    analysis introduce a restoring “force” on the price, which constantly
    overshoots the target, that is, the fundamental price. In the presence
    of trend-following strategies, which provide a positive feedback, the
    overshots tend to accelerate and follow the acceleration of the price,
    leading to ever accelerating oscillations.
    The Underlying Hierarchical Structure of Social Networks
    Investors are organized into social/professional networks, defined as a
    collection of people, each of whom is acquainted with some subset
    of the others. Social networks have been studied intensively because
    they embody patterns of human interactions and because their structure
    controls the spread of information (and of diseases), as we showed in
    chapters 4 and 5.
    Stanley Milgram [297] conducted one of the first empirical studies
    of the structure of social networks. He asked test subjects, chosen at
    random from a Nebraska telephone directory, to get a letter to a target
    subject in Boston, a stockbroker friend of Milgram’s. The instructions
    were that the letters were to be sent to their addressee (the stockbroker)
    by passing them from person to person, but that they could be passed
    only to someone whom the passer knew on a first-name basis. Since it
    was not likely that the initial recipients of the letters were on a firstname
    basis with a Boston stockbroker, their best strategy was to pass
    their letter to someone whom they felt was nearer to the stockbroker in
    some sense, either social or geographical—perhaps someone they knew
    in the financial industry, or a friend in Massachusetts.
    A moderate number of Milgram’s letters did eventually reach their
    destination, and Milgram discovered that the average number of steps
    taken to get there was only about six, a result which has since passed
    into folklore and was immortalized by John Guare in the title of his 1990
    play Six Degrees of Separation [182]. Milgram’s result is usually taken as
    evidence of the “small world hypothesis” [445] that most pairs of people
    in a population can be connected by only a short chain of intermediate
    174 chapter 6
    acquaintances, even when the size of the population is very large. This
    result has been shown to apply to essentially all social networks that have
    been investigated, including affiliation networks such as clubs, teams,
    or organizations. Examples include women and the social events they
    attend, company CEOs and the clubs they frequent, company directors
    and the boards of directors on which they sit, and movie actors and the
    movies in which they appear. Recently, M. E. J. Newman has studied the
    affiliation networks of scientists in which a link between two scientists
    is established by their coauthorship of one or more scientific papers
    [313, 314]. This network may represent a good proxy for professional
    networks such as traders and, to a lesser degree, investors. The idea is
    that most pairs of people who have written a scientific paper together
    are genuinely acquainted with one another, as they are supposed to have
    conducted together the research reported in the paper.
    The idea of networks of coauthorship is not new. Most practicing
    mathematicians are familiar with the definition of the Erd?s number
    [178]. Paul Erd?s (1913–1996), the widely traveled and incredibly
    prolific Hungarian mathematician, wrote at least 1,400 mathematical
    research papers in many different areas, many in collaboration with
    others. His Erd?s number is 0 by definition. Erd?s’s coauthors have
    Erd?s number 1. There are 507 people with Erd?s number 1. People
    other than Erd?s who have written a joint paper with someone with
    Erd?s number 1 but not with Erd?s have Erd?s number 2 and so on.
    There are currently 5�897 people with Erd?s number 2. If there is
    no chain of coauthorships connecting someone with Erd?s, then that
    person’s Erd?s number is said to be infinite. The present author has
    Erd?s number 3; that is, I have published with a colleague who has
    published with another colleague who has written a paper with Erd?s.
    There is a mathematical conjecture that the graph of mathematicians
    organized around the vertex defined by Erd?s himself and connected to
    him contains almost all present-day publishing mathematicians and has
    a not very large diameter; that is, the largest finite Erd?s number is 15,
    while the average value is about 4�7 [179, 33].
    The explanation of the “small world” effect is illustrated in Figure 6.1,
    which shows all the collaborators of the author of [313, 314] and all
    the collaborators of those collaborators, that is, all his first and second
    neighbors in the collaboration network of scientists. As the figure shows,
    M. E. J. Newman has 26 first neighbors and 623 second neighbors. As
    the increase in numbers of neighbors with distance continues at this
    impressive rate, it takes only a few steps to reach a size comparable to
    the whole population of scientists, hence the “small-world” effect.
    hierarchies and log-periodicity 175
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    Ronis, D.
    Siu-ka, Chan
    Smailer, I.
    Tremblay, A.-M.S.
    Zwanzig, R.
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    Girvin, S.M.
    Grannan, E.R.
    Grigoriu, M.
    Gronlund, L.D.
    Hodgdon, J.A.
    Holzer, M.
    Ho, W.
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    Ingraffea, A.R.
    Jacobsen, J.
    Jacobsen, K.W.
    Kleman, M.
    Knaak, W.
    Krishnamachari, B.
    Langer, S.A.
    Louis, A.A.
    MacDonald, A.H.
    McLean, J.G.
    Meiboom, S.
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    Mermin, N.D.
    Miller, M.P.
    Min, Huang
    Morgan, J.D.,
    Mungan, C.E.
    Muttalib, K.A.
    Myers, C.R.
    Nagel, S.R.
    Nielsen, O.H.
    Norskov, J.K.
    Ostlund, S.
    Peale, D.R.
    Pohl, R.O.
    Ramakrishnan, T.V.
    Rand, D.
    Richter, L.J.
    Rokhsar, D.S.
    Shore, J.D.
    Shukla, P.
    Shumway, S.
    Sievers, A.J.
    Soo-Jon, Rey
    Spitzer, R.C.
    Stoltze, P.
    Trugman, S.A.
    Umrigar, C.J.
    Wawrzynek, P.A.
    Wickham, L.K.
    Wright, D.C.
    Alstrom, P.
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    Brandt, M.
    Coppersmith, S.N.
    Heinz, K.H.
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    Littlewood, P.B.
    Meintrup, T.
    Middleton, A.A.
    Pedersen, A.
    Pedersen, B.
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    Pedersen, J.M.
    Salamon, P.
    Schmidt, M.R.
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    Bohr, T.
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    Gregoire, C.
    Gross, D.H.E.
    Hansen, A.
    Hansen, F.S.
    Heik, Leschhorn
    Hennino, T.
    Huber, G.
    Iljinov, A.S.
    Jakobsson, B.
    Jayaprakash, C.
    Jensen, M.H.
    Jonsson, G.
    Jourdain, J.C.
    Karlsson, L.
    Koiller, B.
    Kopljar, V.
    Krug, J.
    Kuchni, Fygenson,
    Larsen, J.S.
    Lauritsen, K.B.
    Leibler, S.
    Lei-Ha, Tang
    Leray, S.
    Libchaber, A.
    Lopez, J.
    Lozhkin, O.V.
    L’vov, V.S.
    Markosova, M.
    Mishustin, I.N.
    Murin, Y.A.
    Nakanishi, H.
    Ngo, C.
    Nifenecker, H.
    Noren, B.
    Nybo, K.
    Olberg, E.
    Pethick, C.J.
    Pollarolo, G.
    Priezzhev, V.B.
    Ramstein, B.
    Randrup, J.
    Robinson, M.M.
    Sams, T.
    Schultz, H.
    Schussler, F.
    Soederstroem, K.
    Souza, S.R.
    Thorsteinsen, T.F.
    Thrane, U.
    Vinet, L.
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    Paul, R.
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    Richardson, G.S.
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    Yeung, M.K.S.
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    Babalievski, F.
    Barshad, Y.
    Bingli, Lu
    Brosilow, B.J.
    Campbell, L.J.
    Cohen, E.G.D.
    Cummings, P.T.
    Ernst, M.H.
    Fichthorn, K.A.
    Finch, S.R.
    Gulari, E.
    Hendriks, E.M.
    Hovi, J.-P.
    Kac, M.
    Kincaid, J.M.
    Kleban, P.
    Knoll, G.
    Kong, X.P.
    Lorenz, C.D.
    Maragos, P.
    McGrady, E.D.
    Meakin, P.
    Merajver, S.D.
    Mizan, T.I.
    O’Connor, D.
    Sander, E.
    Sander, L.M.
    Sapoval, B.
    Savage, P.E.
    Savit, R.
    Stauffer, D.
    Stell, G.
    Suding, P.N.
    Uhlenbeck, G.E.
    Vigil, R.D.
    Voigt, C.A.
    Roth, J.
    Schilling, R
    Schulz, H.J.
    Shaw, L.J.
    Siggia, E.D.
    Sompolinsky, H.
    Tin, Lei
    Trebin, H.-R.
    vonDelft, J.
    Widom, M.
    Zhang, N.-G.
    Zhu, W.-J.
    Candela, D.
    Cao, M.S.
    Chan, S.
    Choi, Y.S.
    Cohen, S.M.
    Condat, C.A
    Duxbury, PM
    Fig. 6.1. The point in the center of the figure represents the author of two articles
    [313, 314] studying the network of scientists, the first ring his collaborators, and
    the second ring their collaborators. Collaborative ties between members of the same
    ring, of which there are many, have been omitted from the figure for clarity. Courtesy
    of M. E. J. Newman [313, 314]. A similar construction holds for most scientists,
    including the author of this book. However, being older than the author of [313, 314],
    the present author has fifty-five (instead of twenty-six) nearest neighbor collaborators
    in the first ring, and many more in the second ring, counting only his collaborators
    from 1996 to 2000. The corresponding figure would not be as aesthetically pleasing
    at the present one, being too crowded to appeal to the eye.
    176 chapter 6
    p=0 p=1 p=2
    Fig. 6.2. First three steps of the iterative construction of the hierarchical diamond
    lattice. p refers to the index of the iteration.
    Indeed, in most networks, the average distance between any pair of
    vertices (scientists or traders in our example below) is proportional to
    the logarithm of the number of vertices. Recall that the logarithm of a
    number is nothing but the exponent in the exponential representation of
    that number; that is, it is roughly equal to the number of digits minus
    one (the logarithm of 1�000 in base 10 is 3 because 1�000 = 103). The
    logarithm is thus a very slowly varying function, since multiplying the
    number by 10 corresponds to adding 1 to its logarithm. A hierarchical
    network provides a simple justification of this point. Let us therefore
    consider a simplified hierarchical structure, called the diamond hierarchy,
    whose construction is represented in Figure 6.2. Let us start with a pair
    of investors who are linked to each other (p = 0). Let us replace this link
    by a diamond, where the two original traders occupy two diametrically
    opposed vertices, and where the two other vertices are occupied by two
    new traders (p = 1). This diamond contains four links. For each one
    of these four links, let us replace it by a diamond in exactly the same
    way (p = 2). Iterating the operation a large number of times gives the
    hierarchical diamond lattice. After p iterations, we have N = 2
    3 �2 + 4p�
    traders and L = 4p links between them. Since N and L are essentially
    proportional to 4p for large p, reciprocally the order p of the iteration is
    proportional to the logarithm of the number of traders and of the number
    of links between them. The logarithm of a number N is thus nothing but
    a quantity proportional to the exponent of the power of a given reference
    number (here 4), providing a representation of the number N.
    Most traders have only two neighbors, a few traders (the original
    ones) have 2p neighbors, and the others are in between. Note that the
    least-connected agents have 2p?1 times fewer neighbors than the mostconnected
    ones, who themselves have approximately 2p fewer neighbors
    hierarchies and log-periodicity 177
    than there are agents in total. Averaging over all traders, we retrieve the
    result that the average distance between any pair of traders is proportional
    to the index p of iteration, that is, to the logarithm of the number of
    Such a hierarchical network may be a more realistic model of the
    complicated network of communications between financial agents than
    the grid in the Euclidean plane used in chapters 4 and 5 with Figures 4.7–
    Critical Behavior in Hierarchical Networks
    Consider a network of agents positioned at the nodes of the hierarchical
    diamond lattice shown in Figure 6.2 and interacting with their nearest
    neighbors through the links in a noisy imitative fashion according to
    expression (6) on page 102. We recall that this expression (6) embodies
    the competition between the ordering effect of imitation and the disordering
    forces of idiosyncratic signals modeled as random noise. This
    network as well as different extensions turns out to be exactly solvable
    [106, 9]. The extensions of the network shown in Figure 6.2 comprise
    networks constructed by substituting any link of a given generation by a
    set of q branches, each containing a series of r bonds. The construction
    of Figure 6.2 corresponds to q = r = 2.
    The basic properties obtained for these networks are similar to the
    ones described in the risk-driven model of chapter 5 using the grid in the
    Euclidean plane shown in Figures 4.7–4.10. There exists a critical point
    Kc for the imitation strength. As shown in the left panel of Figure 6.3, the
    probability P�K� for a crash to occur goes to a constant P�Kc� (=0�7 in
    this example), by accelerating upward, reaching an infinite acceleration
    right at the critical point K = Kc. Recall that the mechanism underlying
    this behavior stems from the existence of larger and larger clusters of
    traders as K approaches Kc and from the larger and larger probability for
    the collective activation of a very large cluster, thus triggering the coordinated
    sell-off of the group. The novel pattern shown in the left panel
    of Figure 6.3 compared to that of Figure 5.6 is the existence of an oscillation
    decorating the overall acceleration. Notice that these oscillations
    also accelerate, as can be seen from the fact that the distances between
    successive crossings with the dashed line become smaller and smaller as
    Kc is approached. To visualize the nature of these oscillations, the left
    panel of Figure 6.4 shows the difference P�Kc� ? P�K� of the accelerating
    part of the probability, again as a function of the reduced distance
    178 chapter 6
    Probability of a Crash
    0.2 0.4 0.6 0.8
    Crash Hazard Rate
    0.2 0.4 0.6 0.8
    Fig. 6.3. Left panel: Probability for a crash to occur in the hierarchical diamond
    network. In this example, the probability reaches its maximum, equal to 0�7, at
    the critical point K = Kc with an infinite slope after accelerating with log-periodic
    oscillations. The dashed line is the same as in the left panel of Figure 5.6 obtained
    for the Euclidean lattice. Right panel: Crash hazard rate for the hierarchical diamond
    network. The dashed line is the same as in the right panel of Figure 5.6, obtained
    for the Euclidean lattice. The crash hazard rate is proportional to the slope of the
    probability shown in the left panel.
    ? K�/Kc to the critical point Kc. This novel representation uses a
    logarithmic scale both for the abscissa and for the ordinate, such that a
    power law acceleration is seen as the straight dashed line. Decorating
    this, we see periodic oscillations. Since these oscillations are periodic in
    the logarithm of the variable �Kc
    ? K�/Kc, we refer to them as “logperiodic.”
    The strength of these log-periodic oscillations depends on the
    nature of the interactions between traders within the hierarchical lattice
    and on the choice of the observable. When one utilizes these models
    for other purposes, such as in models of magnetic materials for which
    the traders on the nodes are replaced by tiny magnets, called spins, the
    relevant physical observables such as the energy or the magnetization
    usually exhibit log-periodic oscillations with quite tiny amplitudes. For
    the sake of pedagogy, we have thus artificially enhanced their amplitude
    compared to what they would be in the physical problem, in order to
    obtain a clearer visual appearance. However, this enhancement is not
    really artificial in the financial context. It can be justified by the fact that
    financial crashes are not characterized by the same observables as physical
    quantities. As explained in chapter 1, market crashes are more like
    ruptures, which are sensitive to extreme fluctuations in the distribution
    hierarchies and log-periodicity 179
    P(Kc) - P(K)
    (Kc - K)/Kc
    0.1 0.01
    1.0 0.1 0.01 0.001 0.001
    Crash Hazard Rate
    (Kc - K)/Kc
    Fig. 6.4. Left panel: Logarithm scale of the difference P�Kc� ? P�K� of the accelerating
    part of the probability shown in the left panel of Figure 6.3 as a function of
    the reduced distance �Kc
    ? K�/Kc, also in logarithmic scale. The grid on the two
    axis are inverted to obtain the correct visual impression that the closer we get to
    Kc, the larger is the probability. Right panel: Logarithmic scale of the crash hazard
    rate shown in the right panel of Figure 6.3 as function of the reduced distance
    ? K�/Kc, also in logarithmic scale. The dashed line corresponds to the pure
    power law acceleration obtained for the Euclidean lattice and shown on the right
    panel of Figure 5.6. The grid on the horizontal axis has been inverted to obtain the
    correct visual impression that the closer we get to Kc, the larger is the crash hazard
    of clusters of imitative traders. The log-periodic signals can be much
    stronger when the largest fluctuations are emphasized. This is illustrated
    in Figure 6.5 in another context, corresponding to a model of chaotic and
    turbulent dynamics [462]. The log-periodicity is clearly seen as regular
    steps for the values of the parameter m = 3 and 4, whose increasing
    value corresponds to putting more and more emphasis on the largest fluctuations.
    This figure illustrates that log-periodicity may not be detectable
    in some observables, while being a strong feature of others for the same
    The diverging acceleration of the crash probability shown in
    Figure 6.3 again implies that the crash hazard rate, which is nothing
    but the rate of change of the probability of a crash as a function of
    time, increases without bounds as K goes to Kc. The novel feature is
    the existence of the log-periodic oscillations. They are accelerating as
    the critical point is approached, while their arches are represented as
    equidistant in the double-logarithmic representation of the right panel
    180 chapter 6
    Fig. 6.5. The ordinate is a measure of the average amplitude of fluctuations in the
    dynamical evolution in a simple model of hydrodynamic turbulence, weighted more
    and more towards large amplitudes as m increases. The abscissa is the time window
    in which the measures are calculated. This figure illustrates that log-periodicity may
    not be detectable in some observables (here, for m = 1), while being a strong feature
    of others (for m = 3 and 4). Reproduced from [462].
    of Figure 6.4. The oscillations are more pronounced for the hazard rate
    than for the (cumulative) probability of the crash, because constructing
    a rate (derivative quantity) enhances local features. This implies that the
    risk of a crash per unit time, knowing that the crash has not yet occurred,
    increases dramatically when the interaction between investors becomes
    strong enough, but this acceleration is interrupted by and mixed with an
    accelerating sequence of quiescient phases (the decreasing parts of the
    log-periodic oscillations) in which the risk decreases.
    If the hazard rate exhibits this behavior, we have seen in chapter 5 that
    the return must, as a consequence, possess the same qualitative properties
    in order for the no-arbitrage condition together with rational expectations
    to hold true. We obtain our second prediction of a specific pattern of the
    approach to a crash: returns increase faster and faster in an intermittent
    fashion; that is, they alternatively accelerate and decelerate with time,
    with a pattern converging to the critical point. Since prices are formed
    by summing returns, the typical trajectory of a price as a function of
    time, which is expected on the approach to a critical point, is parallel to
    hierarchies and log-periodicity 181
    the dependence of the probability of a crash shown in the left panel of
    Figure 6.3.
    A Hierarchical Model of Financial Bubbles
    It is useful to illustrate further the impact of a hierarchical structure
    of imitative behavior between traders on observable signatures in the
    stock market. We thus assume the hierarchical organization shown in
    Figure 6.6 such that a trader influences only a limited number of traders
    at the same level of the hierarchy and below. Due to a cascade effect,
    the decisions of the lower levels in turn influence the higher levels. For
    instance, the position of a bank within a country will be highly sensitive
    to the position of the currency block as a whole and to that of its country
    and of other banks from which it can get information. On the other hand,
    the position of the currency block will be an aggregate of that of the
    constituting countries.
    The model formalizes the hierarchical organization and refers to the
    individual traders as traders of order 0. According to the hierarchical
    organization, these traders are organized in groups of m traders and we
    consider each such group as a single “trader” of order 1. These groups
    (or “traders”) of order 1 are also organized in groups of m to form a
    group of order 2 and so forth. In this way, a hierarchical organization is
    obtained, where a group of order n is made of mn individual traders. For
    simplicity, but without loss of generality, we take m = 2. The analysis
    Fig. 6.6. Schematic representation of a simple dichotomous hierarchical structure
    of influences between the traders. Reproduced from [398].
    182 chapter 6
    for other values of m is the same and only detailed numerical values are
    At time 0, all individual traders of the zeroth level of the hierarchy are
    assumed to start gathering and processing information to form a decision
    on whether or when to enter the market. The traders are thought to be
    heterogeneous in the sense that the time they need to perform their analysis
    of the situation is different for each of them and hence each trader
    has a characteristic time to form his decision and enter the market. The
    behavior of the traders thus differs with respect to the timing of their
    action [437]. Assume that the trader i has a preferred time ti to buy
    the stock (assumed to be unique in this toy-market model) and that the
    ti’s are distributed according to some distribution, say a Poisson (exponential)
    distribution. Trader i’s time to buy ti should not be confused
    with his reaction time once his decision is made. The latter is almost
    instantaneous, as it is to the trader’s benefit that his order be executed
    efficiently. In contrast, the time to buy ti reflects the need for the trader
    to accumulate data, to carry out his analysis, and to be convinced that
    he has to enter the market. In a sense, this is the time that he needs
    for strengthening his confidence that his decision is correct. Establishing
    this confidence can be a long learning process, also rooted in different
    psychologies and past experiences. The characteristic time scales ti are
    expected to range from minutes (or less) to years. These are the time
    scales for new information to be gathered and analyzed.
    One trader’s move in the market can be interpreted by another trader
    as relevant additional information due to the uncertainty he faces. To
    be specific, consider the hierarchical organization with m = 2 shown in
    Figure 6.6. Suppose that, at the zeroth level, one of the two traders of a
    group reaches the end of his time-to-buy period and enters the market.
    The rule of the model is that the other trader of the group, and only him,
    has the privilege of incorporating this information. In principle, a trader
    would probably benefit from a survey of other traders’ actions. However,
    getting the information on the different levels of the hierarchy is not easy
    and may not be possible for all. Furthermore, this would have a cost,
    which introduces a severe limitation. Our simplifying assumption thus
    corresponds to the limit of a cost-effective minimal information strategy.
    After the analysis of this information, the second trader in general is
    influenced positively towards the decision of entering the market. The
    model specifies that the remaining waiting time is reduced by a fixed
    “influence” factor � less than 1. This is basically an imitation rule and is
    devised in order to model the highly nonlinear (threshold-like) behavior
    of traders, with positive and negative feedback patterns, as discussed in
    hierarchies and log-periodicity 183
    chapter 4. If � is close to 1, then the interaction is weak and a trader
    does not significantly modify his strategy after receiving the information
    on the action of his neighboring trader. On the contrary, in the limit
    � → 0, the second trader almost instantaneously enters the market on
    the knowledge of the action of the first trader; this is the regime where
    traders are strongly influenced by the other traders in the same group
    and will amplify the actions of the other traders by their own decision.
    A strong “crowd” effect is expected in this regime.
    The model assumes that the imitation process works at all levels of
    the hierarchy. When two “traders” of order m belonging to the same
    group have finally bought the stock, this information is transferred to the
    next level in the hierarchy. Since the two traders of order m have bought,
    the trader of order m + 1, defined as their sum, has also bought, and this
    information is reported to the other trader of order m+1 of the pair. This
    will then change the time-to-buy of this “trader” of order m + 1. As a
    consequence, the remaining waiting time of the two traders of this neighboring
    group of order m + 1 is also multiplied by � at this next level
    of the hierarchy. This process may continue to increasingly high levels
    and lead to a complex superposition of actions and influences starting at
    the lowest level of the hierarchy and progressively overlapping as more
    groups get linked at higher levels. This cascade process of information
    is illustrated geometrically in Figure 6.7.
    The price of the stock is strongly influenced by the behavior of the
    traders in a nontrivial way. This drastically simplified description does
    not provide a specific formula for the price. Instead, the model uses the
    very weak assumption that the price is a nondecreasing function of the
    total number of buy positions taken by the traders up to time t. In other
    words, the demand curve is positive. The idea is simply that demand
    has a direct influence on the price and tends to appreciate it. Another
    important simplification is that the traders are only interested in buying
    the stock, an assumption which, taken at face value, would obviously
    be in contradiction with the balance between sellers and buyers: to be
    able to buy, some traders must sell! The model assumes, in fact, that the
    sellers are necessarily a homogeneous group that remains fixed and neutral
    throughout the period in which the progressive cooperative activity
    between the buyers develops. The problem thus reduces to determining
    quantitatively the temporal behavior of the total number of buy positions.
    This model allows for a rigorous definition of a crash. Indeed, in the
    limit of an infinite number of traders (and therefore of hierarchical levels),
    the existence of a crash occurring at some time tc is defined by the
    fact that at times much before tc the number of buyers remains small
    184 chapter 6
    Fig. 6.7. Spatiotemporal evolution of the system. The abscissa represents 512
    traders who are linked hierarchically, as shown in Figure 6.6. The ordinate is time,
    which flows from bottom to top. Those buyers who have entered the markets are
    represented by the wells. The widening of the wells depicts the progressive “invasion”
    among neighboring agents of a buy order spreading from an agent. Notice
    the cascade of doubling that can be observed at many different scales and along
    many different branches. There are many competing cascades starting from different
    traders and at different times, which lead to the noisy structure observed in
    Figure 6.8. Reproduced from [398].
    and their mutual influences are small. As time goes on, these quantities
    accelerate progressively until tc, at which a finite fraction of the traders
    have made buy orders and already entered the position, thus saturating
    the market with no more buyer to be found. The model describes
    the preparation phase, called a bubble, ending in a crash, which is not
    specifically modeled itself.
    Figure 6.8 shows the number of traders who have made buy orders as
    a function of time. The left panel corresponds to one specific realization
    of the initial population of traders’ waiting times at the zeroth order. The
    right panel shows five realizations with different initial configurations
    of waiting times, in a double logarithmic scale, such that a power law
    acceleration of the form shown in Figures 6.3 and 6.4 is represented
    as a straight line. One can indeed observe a characteristic power law
    acceleration, which is decorated by log-periodic structures at many different
    scales as the critical time is approached. It turns out to be possible
    to explicitly solve this model and demonstrate rigorously the existence
    of these log-periodic structures decorating the average power law [398].
    hierarchies and log-periodicity 185
    Fig. 6.8. Left panel: Number of traders who have made buy orders as a function of
    time. Notice the power law acceleration and the log-periodic structure of step-like
    jumps decorating this acceleration. Right panel: Same as the left panel in a double
    logarithmic representation of the number of traders who have entered the market as
    a function of the distance to the critical time. Five different trajectories are shown,
    each one corresponding to a different but statistically equivalent initial condition,
    reflecting the variability of the real world. The approximate linear dependence decorated
    by large and complicated log-periodic structures qualifies as a power law, as
    discussed in the text. Reproduced from [398].
    The log-frequencies of the log-periodic oscillations are determined by
    the “influence” factor �, which quantifies the change of the remaining
    waiting time of a trader observing the action of his companions and of
    traders at the higher levels of the hierarchy.
    Note the strength of the log-periodic oscillations seen in Figure 6.8.
    This can be traced back to the fundamental threshold nature of the cascade
    of traders’ influences. As we have indicated with Figure 6.5, observables
    that put an emphasis on extreme and abrupt behavior will enhance
    the effect of log-periodicity. To make the argument even clearer, let us
    mention that this model can be mapped exactly onto a model of material
    failure according to a cascade of abrupt ruptures [355].
    The acceleration of the number of traders buying the market in the
    inflating bubble captures the oft-quoted observation that bubbles are
    times when the “greater fool theory” applies. In financial circles, this
    refers to buyers of stocks buying confidently irrespective of the dividends
    or other underlying fundamental factors, expecting to sell to someone
    186 chapter 6
    else at an even higher price in some future. As an illustration, it has been
    reported [142] that Henry Ford was taking the elevator to his penthouse
    one day in 1929, and the operator said, “Mr. Ford, a friend of mine who
    knows a lot about stocks recommended that I buy shares in X, Y, and Z.
    You are a person with a lot of money. You should seize this opportunity.”
    Ford thanked him, and as soon as he got into his penthouse, he called
    his broker, and told him to sell everything. He explained afterwards: “If
    the elevator operator recommends buying, you should have sold long
    ago.” More generally, this refers to the following cascade [309]: New
    demographic, technological, or economic developments prompt spontaneous
    innovation in financial markets and the first wave of investors and
    innovators become wealthy. Then, imitators arrive and overdo the new
    techniques. In the ensuing crises, latecomers lose big before regulators
    and academics put out the fires.
    Discrete Scale Invariance
    What is the origin of the novel log-periodic oscillations decorating the
    overall acceleration of the probability of a crash, of the crash hazard
    rate, and of the returns and price trajectories themselves, documented in
    the previous section?
    The answer turns out to be quite simple: hierarchical networks such
    as the diamond lattice shown in Figure 6.2 or the tree lattice shown in
    Figure 6.6 possess a fundamental symmetry property, called “discrete
    scale invariance.” A symmetry refers to the property of a geometrical
    figure, a system, or an observable to remain invariant under a specific
    transformation, such as a translation, rotation, inversion, or dilation. For
    instance, regular tilings paving floors are endowed with discrete translational
    symmetries, because they are invariant with respect to discrete
    translations of a multiple of one motif, since the same motif is repeated
    periodically. Symmetries are aesthetically pleasing as in tiling, carpets,
    furnitures, diamond stones, and antique churches. Nature seems to have
    organized its laws around a core set of fundamental symmetries, such as
    the symmetry under translation, under rotation, under a change between
    two frames moving at different constant velocities (called Galilean
    invariance), as well as under a set of more esoteric symmetries called
    gauge symmetries (which refer to other internal variables describing
    hierarchies and log-periodicity 187
    fundamental particles). All the phenomena, matter, and energy of our
    universe seem to emerge as slight deviations (resulting from spontaneous
    symmetry breaking) from these fundamental symmetries [448]. Thus,
    we cannot emphasize enough the overarching importance of symmetries
    in helping us understanding the organization and complexity of the
    The diamond and tree lattices of Figures 6.2 and 6.6 also enjoy a
    symmetry, called the symmetry of “scale invariance”: in the limit where
    both geometrical constructions are extrapolated up to an infinite number
    of iterations, replacing a diamond by a single link and vice versa does
    not change the diamond network. Similarly, replacing a branch by two
    subbranches and vice versa does not change the tree lattice of Figure 6.6.
    In other words, the hierarchical diamond and tree networks have the
    property of reproducing themselves exactly on different magnifications.
    Such a property has been coined “fractal” by Mandelbrot [284], who recognized,
    based on the pioneering work of Richardson [343], that many
    natural and social phenomena are endowed, at least approximately, with
    the scale invariance symmetry. Many of us have met fractals through
    their beautiful, delicately complex pictures, which are usually computer
    generated. Modern Hollywood movies use landscapes, mountain ranges,
    cloud structures, and other artificial constructions that are computer generated
    according to recipes devised to obtain fractal geometries. It turns
    out that many of the natural structures of the world are approximately
    fractal [29, 126, 88, 31, 292, 394] and that our aesthetic sense resonates
    with fractal forms.
    In most simple fractal constructions and textbook examples, the scale
    invariance property does not hold for arbitrary magnification. This is
    also true for the two hierarchical lattices of Figures 6.2 and 6.6. For
    the diamond lattice, only magnifications that multiply the number of
    links by a factor of 4, or more generally by any power of 4, leave the
    network invariant. In the tree lattice, only magnifications that multiply
    the number of branches for a factor of 2, or more generally by any
    power of 2, leave the tree invariant. These special magnification factors
    4 and 2, respectively, are the direct consequence of the construction
    scheme of the two hierarchical networks. Such systems, which are selfsimilar
    only under magnifications by arbitrary integer powers 4n or 2n
    or any other fixed factor �n, where n = · · · ? 3�?2�?1� 0� 1� 2� 3� � � �
    is an integer number, are said to enjoy the symmetry of discrete scale
    invariance [392]. Discrete scale invariance is a weaker symmetry than
    the general scale invariance: it is the latter restricted to special discrete
    choices of magnification factors, here integer powers of 4 or of 2.
    188 chapter 6
    Fractal Dimensions
    During the third century before the Christian epoch, Euclid and his students
    introduced the concept of “dimension,” an exponent that can take
    positive integer values equal to the number of independent directions.
    The dimension d, for instance, is used as an exponent linking the volume
    V to the length L: V = Ld, where V is the generalized volume of a generalized
    cube of side length L. For a real cube in our three-dimensional
    space, d = 3 and the volume is the cube L3 = L × L × L of a side L.
    For a square, d = 2, and its surface is the square L2 = L × L of its
    side. For a segment, d = 1, its length L1 = L is equal to its length L.
    A line has a dimension 1, a plane has a dimension 2, and a volume has
    a dimension 3. The surface of a sphere also has a dimension 2, since
    any point on it can be located with two coordinates, the latitude and
    longitude. Another way of realizing that the surface of a sphere has a
    dimension 2 is that its area is proportional to the square of its radius.
    In the second half of the nineteenth century and in the first quarter
    of the twentieth century, mathematicians imagined geometrical figures
    that are endowed with dimensions that can take fractional values, for
    instance, d = 1�56 or d = 2�5 or any other number. The remarkable
    discovery was the understanding that this generalization of the notion of
    a dimension from integers to real numbers reflects the conceptual jump
    from translational invariance to continuous scale invariance. A line and a
    plane are unchanged when viewed from different points translated from
    one to another, a property called translational invariance. Objects with
    fractional dimensions turn out to possess the property of scale invariance.
    To capture this novel concept, we already mentioned that the word
    “fractal” was coined by Mandelbrot [284], from the Latin root fractus
    to capture the rough, broken, and irregular characteristics of the objects
    presenting at least approximately the property of scale invariance. This
    roughness can be present at all scales, which distinguishes fractals from
    Euclidean shapes. Mandelbrot worked actively to demonstrate that this
    concept is not just a mathematical curiosity but has strong relevance to
    the real world. The remarkable fact is that this generalization, from integer
    dimensions to fractional dimensions, has a profound and intuitive
    interpretation: noninteger dimensions describe irregular sets consisting
    of parts similar to the whole.
    There are many examples of (approximate) fractals in nature, such as
    the distribution of galaxies at large scales, certain mountain ranges, fault
    networks and earthquake locations, rocks, lightning bolts, snowflakes,
    hierarchies and log-periodicity 189
    Fig. 6.9. Synthetic fractal coastline (courtesy of P. Trunfio).
    river networks, coastlines, patterns of climate change, clouds, ferns and
    trees, mammalian blood vessels, and so on.
    In his pioneering paper [283], Mandelbrot revisited and extended the
    investigation launched by Richardson [343] concerning the regularity
    between the length of national boundaries and scale size. He dramatically
    summarized the problem by the question written in the title of his article
    [283], “How Long Is the Coast of Britain?” This question is at the core
    of the introduction of fractal geometry. Figure 6.9 shows a synthetically
    generated coastline that has a corrugated structure reminiscent of the
    coastline of Brittany in France.
    Such a coastline is irregular, and therefore a measure with a straight
    ruler, as in Figure 6.10, provides only an estimate. The estimated length
    � equals the length of the ruler
    multiplied by the number N�

    of such rulers needed to cover the measured object. In Figure 6.10, the
    length of the coastline is measured twice with two rulers of length
    1 and

2, where the length of the second ruler is approximately half that of the
first one:
1/2. It is clear that the estimate of the length L�
2� using
the smaller ruler
2 is significantly larger than the length L�
1� using the
larger ruler

  1. For very corrugated coastlines exhibiting roughness at all
    length scales, as the ruler becomes very small, the length grows without
    190 chapter 6
    Fig. 6.10. Implementation of the ruler method consisting in covering the rough line
    by segments of fixed size. As the ruler length decreases, finer details are captured,
    and the total length of the line increases. Courtesy of G. Ouillon.
    bound. The concept of (intrinsic) length begins to make little sense and
    has to be replaced by the notion of (relative) length measured at two
    resolutions. To the question, “What is the length of the coast of Britain?”
    the wise one should thus reply either “it is a function of the ruler” or
    “infinity” (obtained for an infinitely small ruler capable of detecting the
    smallest details of the irregular coastline).
    The fractal dimension d quantifies precisely how the relative length
    � changes with the ruler length
    (which we also call “resolution”
    as details smaller than
    are not seen by definition). By construction,
    � is proportional to
    to the power 1 ? d: L�
    � ~
    1?d. The fact that
    1 ? d and not d appears in this expression comes from the definition
    of the fractal dimension in terms of the number of elements identified
    at a given resolution: for a resolution
    , one typically sees M�
    � =
    elements. The number of elements resolved with a ruler
    inversely proportional to
    to the power d. For Great Britain, d = 1�24,
    which is a fractional value. In constrast, the coastline of South Africa
    is very smooth, virtually an arc of a circle, and d = 1. In general, the
    “rougher” the line, the larger the fractal dimension, that is, the closer
    is the line to filling a plane (of dimension 2). When d = 1, the length
    � ~
    1?d becomes independent of the resolution
    0 = 1:
    only when the fractal dimension is equal to the topological dimension
    can the measure be independent of the scale of the ruler. This is the
    situation with which we are most familiar from our school lessons on
    Euclidean geometry. However, as this discussion shows, this constitutes
    an exceptionally special case: the general situation is when any measure
    performed on an object depends on the scale at which the measure is
    Let us apply this definition of a fractal dimension to the two hierarchical
    networks of Figures 6.2 and 6.6. For the diamond lattice of
    hierarchies and log-periodicity 191
    Figure 6.2, let us assume that the ratio of the length of the four bonds
    replacing one link to the length of the link is r, equal to, say, 2/3. Then,
    each time the resolution is magnified by a factor 1/r = 3/2, four new
    links are observable. In other words, the number of links is multiplied
    by 4 when the resolution is multiplied by 3/2. By the definition of the
    fractal dimension, 3/2 raised to the power d must give 4. This implies
    that d = ln 4/ ln 3/2 = 3�42. This object thus has a dimension larger
    than that of our familiar space. The fact that a high-dimensional object
    can be represented in a (two-dimensional) plane is not a problem; it
    just means that the hierarchical construction will cross itself many many
    times and, in the present case where the dimension is smaller than 4,
    only by unfolding it in a space of at least four dimensions shall we avoid
    crossings and overlaps. Notice that the fractal dimension increases when
    r increases, that is, when the ratio of the size of each of the four “daughter”
    links to the “mother” link increases (while still being less than 1).
    This simply reflects the fact that the fractal object fills more and more
    The same calculation can be repeated for the tree lattice of Figure 6.6.
    Let us assume that the length of the vertical segments separating each
    branching shrinks by the same factor r = 2/3. Now, each time the resolution
    increases by the factor 1/r = 3/2, twice as many branches can be
    detected. The number of branches thus doubles when the resolution is
    multiplied by 3/2. By the definition of the fractal dimension, 3/2 raised
    to the power d must give 2. This implies that d = ln 2/ ln 3/2 = 1�71.
    This hierarchical network of dimension 1�71 is thus intermediate in some
    sense between a line and a plane. Notice again that the fractal dimension
    increases when r increases, that is, when the four links become not
    much shorter than the initial bond.
    Scale invariance and scaling law. The concept of (continuous) scale
    invariance means reproducing oneself on different time or space scales.
    More precisely, an observable � which depends on a “control” parameter
    x is scale invariant under the arbitrary change x → �x, if there is a
    number ���� such that
    ��x� = ����x�� (8)
    Expression (8) defines a so-called homogeneous function and is encountered
    in the theory of critical phenomena of liquid-gas and magnet phase
    transitions, in hydrodynamic turbulence, and many other systems [112].
    Its solution is simply a power law ��x� = x
    , where the exponent

192 chapter 6
(which plays the same role as the fractal dimension d discussed before)
is given by

= ?ln �
ln �
� (9)
This solution can be verified directly by insertion in expression (8). Power
laws are the hallmark of scale invariance, as the ratio ���x�
= �
does not
depend on x; that is, the relative values of the observable at two different
scales only depend on the ratio of the two scales. This is the fundamental
property that associates power laws to scale invariance, self-similarity, and
Organization Scale by Scale: The Renormalization Group
Principle and Illustration of the Renormalization Group.
The expression (8) describes the system precisely standing at the critical
point at which the scale invariance symmetry is exact. For concrete
applications, we would like to have a fuller description of the properties
of the system in the vicinity of the critical point, and not just right at the
critical point. The obvious reason is that precursors of the critical point
may be deciphered before reaching it. The question is to determine how
much of expression (8) remains valid and how much of it must be modified.
In other words, how much of the exact scale invariance symmetry
is conserved when not standing right at the critical point.
The answer to this question is provided by a calculation technique
called the “renormalization group,” whose invention is mainly attributed
to K. Wilson, who received the Nobel prize for physics in 1982 for
it, but its maturation owes a lot to other physicists such as B. Widom,
M. Gellman, L. Kadanoff, A. Migdal, M. Fisher, and others. The
renormalization group has been invented to tackle critical phenomena
which, as we have stressed, already correspond to a class of behaviors
characterized by structures on many different scales [458] and by power
law dependences of measurable quantities on the control parameters. It
is a very general mathematical tool, which allows one to decompose the
problem of finding the “macroscopic” behavior of a large number of
interacting parts into a succession of simpler problems with a decreasing
number of interacting parts, whose effective properties vary with the
scale of observation. The renormalization group thus follows the proverb
“divide to conquer” by organizing the description of a system scale-byscale.
It is particularly adapted to critical phenomena and to systems
hierarchies and log-periodicity 193
close to being scale invariant. The renormalization group translates into
mathematical language the concept that the overall behavior of a system
is the aggregation of an ensemble of arbitrarily defined subsystems,
with each subsystem defined by the aggregation of sub-subsystems, and
so on.
It works in three stages. To illustrate it, let us consider a population of
agents, each of whom has one out of two possible opinions (bull or bear,
yes or no, vote for A or vote for B, etc.). The renormalization group then
works as follows.

  1. The first step is to group neighboring elements into small groups. For
    instance, in a two-dimensional square lattice, we can group agents in
    clusters of size equal to nine agents corresponding to squares of side
    3 by 3.
  2. The second step is to replace the cacophony of opinions within each
    group of nine agents by a single representative opinion, resulting from
    a chosen majority rule. Doing this “decimation” procedure obviously
    lowers the complexity of the problem since there are nine times fewer
    opinions to keep track of.
  3. The last step is to scale down or shrink the superlattice of squares of
    size 3 by 3 to make them of the same size as the initial lattice. Doing
    this, each cluster is now equivalent to an effective agent endowed with
    an opinion representing an average of the opinions of the nine constitutive
    One loop involving the three steps applied to a given system transforms
    it into a new system that looks quite similar but is different in one
    important aspect: the distribution and spatial organization of the opinions
    have been modified as shown in Figures 6.11, 6.12, and 6.13.
    Three situations can occur that are illustrated in Figures 6.11, 6.12,
    and 6.13. Let us discuss them in the context of the model of imitative
    behavior presented in chapter 4 and summarized by the evolution equation
    (6) on page 102. Let us recall that, in this model, agents tend to
    imitate each other according to an inclination strength K that quantifies
    the relative force of imitation compared to idiosynchratic judgment.
    A large K leads to strong organization where most of the agents share
    the same opinion. A small K corresponds to a population that is split
    in half between the two opinions such that the spatial organization of
    agents is disorganized. In between, we have shown in chapter 4 that there
    exists a critical value Kc separating these two extreme regimes at which
    194 chapter 6
    Fig. 6.11. This figure illustrates the effect of renormalization for K < Kc of the
    Ising model (6) on page 102, which corresponds to the disordered regime. The two
    discording opinions are encoded in white and black. Starting from a square lattice
    with some given configuration of opinions on the left, two successive applications of
    the renormalization group are shown in the right panels. Repeated applications of the
    renormalization group change the structure of the lattice with more and more disorganization.
    All the shorter range correlations, quantified by the typical sizes of the
    black and white domains, are progressively removed by the renormalization process
    and the system becomes less and less ordered, corresponding to an effective decrease
    in the imitation strength K. Eventually, upon many iteration of the renormalization
    group, the distribution of black and white squares becomes completely random. The
    system is driven away from criticality by the renormalization. The renormalization
    group thus qualifies this regime as disordered under change of scales.
    the system is critical, that is, scale invariant. The renormalization group
    makes these statements precise, as shown in Figures 6.11, 6.12, and 6.13.
    Except for the special critical value Kc, application of the renormalization
    group drives the system away from the critical value. It is possible to
    use this “flow” in the space of systems to calculate precisely the critical
    exponents characterizing the divergence of observables when approaching
    the critical points. Critical exponents play the role of control functions
    of this flow; that is, they describe the speed of separation from the
    critical point.
    hierarchies and log-periodicity 195
    Fig. 6.12. This figure illustrates the effect of renormalization for K > Kcof the
    Ising model (6) on page 102, which corresponds to the ordered regime in which
    one opinion (white) dominates (the two discording opinions are encoded in white
    and black). Starting from a square lattice with some given configuration of opinions,
    two successive applications of the renormalization group are shown in the right
    panels. We observe a progressive change of the structure of the lattice with more
    and more organization (one color, i.e., opinion, dominates more and more). All the
    shorter range correlations are removed by the renormalization process and the system
    becomes more and more ordered, corresponding to an effective increase in the imitation
    strength K. The system is driven away from criticality by the renormalization.
    The renormalization group thus qualifies this regime as ordered under change
    of scales.
    The Fractal Weierstrass Function: A Singular Time-Dependent
    Solution of the Renormalization Group.
    Right at the critical point, scale invariance holds exactly. It is only broken
    at either the smallest scale, if there is a minimum unit scale, and/or the
    largest scale corresponding to the finite system size. In between these
    two limiting scales, the system is fractal.
    When not exactly at the critical point, the same description holds true,
    but only up to a scale, called the correlation length, which now plays
    the same role as did the finite size of the system at the critical point.
    Figure 4.8 showed us that the correlation length is the size of largest
    196 chapter 6
    Fig. 6.13. This figure illustrates the effect of renormalization for K = Kc of the
    Ising model (6) on page 102, which corresponds to the critical point. The two
    discording opinions are encoded in white and black. Repeated applications of the
    renormalization group leave the structure of the lattice invariant statistically. All the
    shorter range correlations are removed by the renormalization process; nevertheless,
    the system keeps the same balance between order and disorder and the effective
    imitation strength remains unchanged and fixed at the critical value Kc. The system
    is kept fixed at criticality by the renormalization. The renormalization group thus
    qualifies this regime as critical, which is characterized by the symmetry of scale
    invariance. In other words, the system of clusters of opinions is fractal.
    clusters, that is, the distance over which the local imitations between
    neighbors propagate before being significantly distorded by the “noise”
    in the transmission process resulting from the idiosynchratic signals of
    each agent. This means that the mathematical expression (8) on page 191
    expressing exact scale invariance is no longer exactly true and must be
    slightly modified. The renormalization group provides the answer and
    shows that a new term must be added to the right-hand side of expression
    (8). This new term captures the effect of the degrees-of-freedom
    leftover in the coarse-graining procedure of the renormalization group
    when going from one scale to the next larger one.
    With the choice of this new term equal to the simple cosine function
    cos x, corresponding to regular oscillations, the solution of the
    hierarchies and log-periodicity 197
    Weierstrass Function
    0.85 0.9
    0 0.2 0.4 0.6 0.8 1.0 0.95 1.0
    Weierstrass Function
    Fig. 6.14. The Weierstrass function defined as the solution of the renormalization
    group equation obtained from the exact self-similar critical expression (8) on
    page 191 by adding a simple cosine embodying the effect of the degrees of freedom
    at small scales on the next larger scale. The Weierstrass function exhibits the property
    of self-similarity as demonstrated by comparing a magnified portion in the right
    panel to the left panel. There is an infinitely ramified set of structures accumulating
    as the critical time tc
    = 1 is approached. This self-similarity is captured by a fractal
    dimension equal to 1�5. The power law singularity at tc
    = 1 is described by an
    = 1/2. The slowly oscillating dashed line, which captures the large scale
    structure of the Weierstrass function, is a simple power law 3�4 ? �tc
    ? t�1/2 with
    critical exponent 1/2 decorated by a log-periodic oscillation cos�2� ln�tc
    ? t�/ ln 2�
    showing that the dominating discrete scale factor is � = 2 in this example. The repetition
    of spiky structures thus occurs in a regular geometrical log-periodic manner
    with a main log-periodicity given by � = 2 in the present example. A mathematical
    transformation (called the Mellin transform) furthermore shows that there is an
    infinite hierarchy of harmonics of this major log-periodicity with all integer powers
    of � = 2, which are responsible for the delicately corrugated structure at all scales.
    renormalization group equation turns out to be a famous function, called
    the Weierstrass function [447] (see [117] for an English translation).
    This function, shown in Figure 6.14, has the remarkable property of
    being continuous but nowhere differentiable. Intuitively, continuity
    means that there are no holes. Nondifferentiability means that we
    cannot define a local tangent slope; that is, the curve is rough at
    all length scales. The Weierstrass curve is critical at tc
    = 1 in the
    example shown in Figure 6.15. In addition, it is characterized by a selfsimilar
    hierarchy of log-periodic structures accumulating at the critical
    time tc
    = 1.
    198 chapter 6
    Quasi-Weierstrass Function
    0.2 0.4 0.6 0.8
    Fig. 6.15. Same as Figure 6.14 but for the replacement of the cosine function by
    an exponentially attenuated cosine function whose decay rate is equal to 1 minus
    the number indicated by the arrows. This “quasi-Weierstrass” function is no longer
    exactly fractal, as it becomes smooth at small scales. Note that the log-periodicity
    is conserved at large scales but is destroyed at the smaller scales.
    Complex Fractal Dimensions and Log-Periodicity
    We are now in position to make intuitive sense of the description of
    discrete scale invariance presented earlier in the chapter. As we said
    previously, discrete scale invariance is nothing but a weaker kind of scale
    invariance according to which the system or the observable obeys scale
    invariance as defined above only for specific choices of the magnification
    or resolution factor �, which form in general an infinite but countable
    set of values �1� �2� � � � that can be written as integer powers �n
    = �n.
    � is the fundamental scaling ratio.
    It is obvious that the two hierarchical networks of Figures 6.2 and
    6.6 obey discrete scale invariance but not (continuous) scale invariance.
    Indeed, by construction, the diamond lattice is recovered exactly only
    under a discrete set of successive magnifications replacing each link by
    four links, each of the four links by four new links, and so on. In the
    hierarchies and log-periodicity 199
    same vein, the dichotomous tree is invariant only under a discrete set of
    magnifications under which each branch is doubled in a discrete hierarchy.
    Indeed, all regular constructions of fractals are endowed with the
    discrete scale invariance symmetry. Other famous examples are the Cantor
    set, the Sierpinsky gasket, and the Koch snowflake, among many
    others [284].
    We have seen that the hallmark of scale invariance is the existence of
    power laws reflecting the absence of preferred scales. The exponents of
    these power laws define the fractal dimensions. The signature of discrete
    scale invariance turns out to be the existence of log-periodic oscillations
    decorating the power laws. As we shall see, these log-periodic structures
    can be represented mathematically by the fact that the exponents
    or equivalently the dimensions d, are not only noninteger but become
    complex numbers.
    We have seen that continuous scale invariance gives rise to noninteger
    (real) fractal dimensions. We now claim that discrete scale invariance
    is characterized by complex fractal dimensions. Before backing up this
    claim, let us reflect a bit on this wonderful example of the incredible adequacy
    of mathematics for describing natural phenomena: the search for a
    more “aesthetically” pleasing generality and consistency in mathematics
    turns out to capture the generalization of a deep concept. E. P. Wigner,
    a Nobel prize winner in physics for his work on symmetries of nuclear
    physics and quantum mechanics, put it this way [456]: “The enormous
    usefulness of mathematics in the natural sciences is something bordering
    on the mysterious. � � � The miracle of the appropriateness of the language
    of mathematics for the formulation of the laws of physics is a wonderful
    gift, which we neither understand nor deserve.”
    Complex numbers form the most general set of numbers obeying the
    standard rules of addition/subtraction and multiplication/division. They
    contain in particular the integers numbers 0� 1� 2� 3� � � � and the real
    numbers, such as any number with an integer and decimal part like
    876�34878278 � � � . Fractions of two integers like 13/8 are special real
    numbers called rational because they are characterized by either a finite
    decimal part (13/8 = 1�625) or an infinite but periodic decimal part, for
    instance, 13/11 = 1�181818181818 � � � , where the motif 18 is repeated
    ad infinitum. Most of the real numbers allowing engineers to perform
    calculations of length, weight, force, resistance, and so on, are characterized
    in principle by an infinite nonrepetitive decimal part. The set of all
    real numbers can be represented as a continuous line, each point on the
    line in exact correspondence with a single real number. The real numbers
    200 chapter 6
    Complex Plane
    -4 -3 -2 -1 0 1
    2 3 4
    Fig. 6.16. Complex plane: The horizontal line represents the real numbers, which
    include in particular the integers ?3�?2�?2� 0� 1� 2� 3� � � � . The vertical line represents
    the purely imaginary numbers, products of i by arbitrary real numbers. The
    rest of the plane is the set of nonreal complex numbers. The terms “complex” and
    “imaginary” are suggestive of the fact that these numbers lie above the real numbers
    and are observed as projections or “shadows” on the real axis.
    are thus the marks pinpointing the position along the line, as shown in
    Figures 6.16 and 6.17.
    Any complex number is equivalent to a pair of real numbers. The
    first member of the pair is called the real part of the complex number.
    The second member of the pair is called the imaginary part. If this second
    member is 0, the complex number reduces to a pure real number.
    While a real number can be viewed as a point on a line, a complex
    number is nothing but a representation of a point in the plane, as shown
    in Figure 6.16, such that the pair of numbers constituting the complex
    number corresponds to the two coordinates or projections, respectively,
    onto the horizontal and vertical axes. “Imaginary” numbers are proportional
    to their fundamental representative denoted “i”, which is such that
    its square i2 = i × i is equal to ?1. To the nonexpert, this property
    may seem unnatural, almost like a magical trick, but mathematicians
    like to define objects that have the most general properties and that are
    still consistent with the previous rules, here the standard rules of addition/
    subtraction and multiplication/division. The property i2 = ?1 turns
    out to be natural when interpreted as an operation in the plane, rather
    than only along the line of real numbers. While the multiplication by
    a real number corresponds to a shrinkage or a dilation along the real
    line, in contrast, a multiplication by i corresponds to a rotation by a
    rectangular angle (equal to 90 degrees or �/2 radians) in the plane. A
    multiplication by an arbitrary complex number is thus the combination
    hierarchies and log-periodicity 201
    Fig. 6.17. Geometrical representation of the multiplication of a complex number z
    by another complex number w: the multiplication is equivalent to the combination
    of a stretching operation and of rotation.
    of two transformations, a contraction or dilation as for a real number
    and a rotation (of an angle not necessarily equal to 90 degrees).
    It turns out that introducing numbers like i does not lead to any inconsistency
    and all the standard calculations apply. More than a pure creation
    of imagination, complex numbers have found a fantastically large
    role for understanding properties of telecommunications by electromagnetic
    and acoustic waves, which we use daily in our modern civilization,
    because they conveniently encode the dual information of a wave, namely
    its amplitude (the loudness) and its frequency and phase (pitch). Complex
    numbers are also essential elements to formulating in a simple way
    one of the most fundamental theories of particles, quantum mechanics,
    for instance in the famous Schr?dinger equation. The nonintuitive novel
    phenomena captured by quantum mechanics, such as the superposition
    principle made famous with Schr?dinger’s cat, which is both alive and
    dead as long as no one observes it, result technically from the fact that
    quantum mechanics is a theory of complex numbers; or, to be technically
    more specific, quantum mechanics is a theory of their immediate
    (noncommutative) generalization, called the quaternions.
    We can now attempt to explain intuitively how a complex fractal
    dimension may lead to log-periodic oscillations, as claimed above. First,
    202 chapter 6
    -0.8 -0.4
    0 4 8 12 16 0 0.4 0.9 1.2
    -1.2 -0.8 -0.4 0 0.4 0.8 1.2
    Fig. 6.18. Illustration of the fact that a circular motion in the x–y plane corresponds
    to oscillatory motions along each of the coordinates x and y, respectively.
    recall the general result illustrated by Figure 6.17 that multiplication by a
    complex number correspond to the combination of a contraction/dilation
    and a rotation in the plane. For our purpose, let us forget the contraction/
    dilation and focus only on the rotation. Now consider a point
    rotating around a center as in Figure 6.18. For instance, consider the
    tip of the second-hand of a watch making a full circle in exactly one
    minute. The direction of rotation is not important for our discussion.
    This perfect periodic circular motion can actually be seen as the combination
    of two simultaneous and ordered oscillatory motions going back
    and forth between two extreme positions. The first motion is horizontal
    and goes from 9:00 to 3:00; the second motion is vertical and spans
    hierarchies and log-periodicity 203
    the interval from 6:00 to 12:00. Seen only as a projection along the
    horizontal axis, the circular motion of the tip of the second-hand is transformed
    into an oscillation similar to that shown in Figure 6.18. Seen as
    a projection along the vertical axis, the circular motion of the tip of the
    second-hand is transformed into another oscillation similar to that shown
    in Figure 6.18. This is a general result: any circular or locally curved
    motion can be transformed into a combination of oscillatory motions
    along straight lines.
    Coming back to the complex fractal dimensions, we need in addition
    to recall the intuitive meaning of an exponent. As the notations L3 = L ×
    L × L and L2 = L × L used previously suggest, the exponents 3 and 2
    used here indicate that L is multiplied with itself 3 and 2 times, respectively.
    The beauty of mathematics is often in generalizing such obvious
    notions to enlarge their use and meaning considerably. Here, the generalization
    from integer exponents to real exponents, for instance in L1�5,
    means that L is multiplied with itself somehow 1�5 times! This curious
    statement can actually be made rigorous and makes perfect sense. Similarly,
    we can take the power of a complex number with a real exponent:
    the result is shown in Figure 6.19. Stretching the imagination even more,
    we can also take the power of L with a complex exponent. Since, as
    we just said, taking L to some power corresponds to multiplying it with
    itself a certain number of times, here we have to multiply L with itself a
    “complex number of times.” Since complex numbers are pairs of numz6
    Powers of z
    Fig. 6.19. Geometrical representation of successive n = 1� 2� 3� � � � powers of a
    complex number z for z inside the circle of radius 1. Varying the exponent n continuously
    as a real number gives the continuous spiraling curve. Courtesy of David
    E. Joyce, Clark University.
    204 chapter 6
    bers, the way to make sense of this curious statement is to decompose
    the action of the complex exponent into two transformations, as in case
    of multiplication. Focusing on the rotational component of the multiplication
    of complex numbers, we can guess (correctly) that the complex
    exponent of L will also correspond to a rotation. Now comes the last step
    of the reasoning: since we observe real numbers, such as stock market
    prices, this corresponds to seeing only the projection on the real line of
    the complex set of operations. As we said and showed with Figure 6.18,
    a rotation is projected on a line as an oscillation. Therefore, constructing
    Ld, where d is a complex number, corresponds to performing an
    oscillatory kind of multiplication, which turns out to be the log-periodic
    In order to understand the log-periodic structure, we need to recall
    the basic property of the logarithm function, used in many of the figures
    in this book, namely that the logarithm transforms multiplications
    into translations and thus powers into additions. As we have said and
    used several times, the logarithms (in base 10) of 10� 100� 1�000� � � � ,
    noted log 10� log 100� log 1�000� � � � are, respectively, 1� 2� 3� � � �. In
    other words, they correspond to the exponent of the powers of 10:
    10 = 101� 100 = 102� 1�000 = 103� � � � . Therefore, an “oscillatory kind
    of multiplication,” induced by taking the power of a number with a complex
    exponent, should be seen as a regular oscillation in the logarithm
    of the number, hence the log-periodicity.
    We illustrate this surprising phenomenon in Figures 6.20 and 6.21,
    which show a measure of the fractal dimensions in the presence of discrete
    scale invariance of the fractal objects. Specifically, we consider
    so-called Cantor sets, which are among the simplest geometrical fractal
    constructions. Figure 6.22 shows the first five iterations of the construction
    of the so-called triadic Cantor set. At the zeroth level, the construction
    of the Cantor set begins with the unit interval, that is, all points
    on the line between 0 and 1. This unit interval is depicted by the filled
    bar at the top of the figure. The first level is obtained from the zeroth
    level by deleting all points that lie in the middle third, that is, all points
    between 1/3 and 2/3. The second level is obtained from the first level
    by deleting the middle third of each remaining interval at the first level,
    that is, all points from 1/9 to 2/9 and 7/9 to 8/9. In general, the next
    level is obtained from the previous level by deleting the middle third
    of all intervals obtained from the previous level. This algorithm can be
    encoded by the following symbolic rule: 1 → 101 and 0 → 000. This
    process continues ad infinitum, and the result is a collection of points
    that are tenuously cut out from the unit interval. At the nth level, the
    hierarchies and log-periodicity 205
    Log2 C( l) - d Log2 l
    Log2 l Log2 l
    -10 -8 -6 -4 -2 0
    Log2 C( l) - d Log2 l
    -10 -8 -6 -4 -2 0
    Fig. 6.20. Oscillatory residuals of the fractal dimensions obtained from the slope
    of the curve shown in Figure 6.22 for (a) the triadic Cantor set constructed with
    the iterative rule 1 → 101 and (b) the Cantor set constructed with the iterative rule
    1 → 101010001. Both Cantor sets have the same real fractal dimension. They differ
    via the imaginary part of their fractal dimensions, which is reflected in the different
    log-periodic structures shown in the two panels. Reproduced from [387].
    set consists of Nn
    = 2n segments, each of which has length �n
    = 1/3n,
    so that the total length (i.e., measure in a mathematical sense) over all
    segments of the Cantor set is �2/3�n. This result is characteristic of a
    fractal set: as n goes to infinity, the number of details (here the segments)
    grows exponentially to infinity, while the total length goes to zero also
    Log2 C( l)
    Log2 l
    -10 -8 -6 -4 -2 0
    Fig. 6.21. Measure of the fractal dimension of the triadic Cantor set by the correlation
    method. The figure plots the logarithm of the correlation integral as a function
    of the logarithm of the separation. Reproduced from [387].
    206 chapter 6
    Fig. 6.22. The initial unit interval and the first five iterations of the construction of
    the so-called triadic Cantor set are shown from top to bottom.
    exponentially fast. In the limit of an infinite number of recursions, we
    find the Cantor set made of an infinite number of dots of zero size. Since
    there are twice as many segments each time the resolution increases by
    a factor 3 the fractal dimension d of this triadic Cantor set is such that 2
    to the power d should be equal to 3, hence d = ln 2/ ln 3 = 0�6309 � � � .
    The Cantor set, which is an infinite dust of points, is more than a point
    (of dimension 0) but less than a line.
    The difference between a measure of the fractal dimension of this triadic
    Cantor set and the theoretical value 0�6309 � � � is shown in the left
    panel of Figure 6.20 as a function of the (logarithm) of the resolution
    scale �. Rather than a constant value 0 which should be obtained if the
    fractal dimension was exactly d = ln 2/ ln 3 = 0�6309 � � � , we observe
    instead a complex oscillatory structure around the expected 0. The naive
    result d = ln 2/ ln 3 = 0�6309 � � � correctly embodies a part of the information
    on the Cantor set structure, but only a part, as it turns out. As
    we explained, these log-periodic (i.e., periodic in the logarithm of the
    scale �) oscillations reflect the fundamental symmetry of discrete scale
    invariance of the triadic Cantor set. The fundamental period seen in the
    graph is ln 3, corresponding to the preferred scaling factor 3 of the discretely
    self-similar construction of the Cantor set. It is obvious to see
    that, by construction, the triadic Cantor set is geometrically identical to
    itself only under magnification by factors �p
    = 3p, which are arbitrary
    integer powers of 3. If you take another magnification factor, say 1�5,
    you will not be able to superimpose the magnified part on the initial
    Cantor set. We must thus conclude that the triadic Cantor set does not
    possess the property of continuous scale invariance but only that of discrete
    scale invariance under the fundamental scaling ratio 3. It is this
    property that is captured by the log-periodic oscillations.
    hierarchies and log-periodicity 207
    Note that the oscillations are more complex than just a single smooth
    sinusoidal structure. Actually, this reflects the presence of all the other
    scaling ratios 32 = 9� 33 = 27� � � � under which the Cantor set is invariant.
    The delicate structure observed in the left panel of Figure 6.20 is
    the result of the superposition of all the pure log-periodic oscillations,
    one for each of the admissible scaling factors. This is similar to a chord,
    composed by combining a set of pure tunes with different loudnesses.
    The right panel of Figure 6.20 gives the same information as the left
    panel for another Cantor set obtained under a slightly different construction
    rule: the unit interval is divided into nine intervals of length 1/9 and
    only the first, third, fifth, and last are kept. This is then iterated on each
    of the four remaining intervals. This construction is encoded symbolically
    into the rule 1 → 101010001. Notice that, each time the resolution
    is increased by a factor 9, four new segments appear. Hence the fractal
    dimension d of this new Cantor set should be such that 9 raised to the
    power d is equal to 4, hence d = ln 4/ ln 9 = 2 ln 2/2 ln 3 = ln 2/ ln 3 =
    0�6309 � � � . This new Cantor set has the same real fractal dimension as
    the triadic Cantor set, but its structure is very different. The log-periodic
    oscillations seen in the right panel of Figure 6.20 make this point clear
    and show how they can embody important information on the construction
    rule beyond the simple self-similar properties captured by the real
    fractal dimension.
    Figure 6.21 illustrates the dependence of a measure of the fractal
    structure of the triadic Cantor set as a function of the resolution scale.
    This measure is called a correlation and counts the number of pairs of
    points on the Cantor set separated by less than the resolution. In this
    double logarithmic representation, the slope should be equal to the real
    fractal dimension d = ln 2/ ln 3 = 0�6309 � � � , as the correlation function
    grows according to the power d of the resolution. Here we again see
    the log-periodic oscillations decorating an average linear trend with the
    correct average slope. These log-periodic structures reflect the discrete
    scale invariance of the Cantor set.
    To summarize, we have shown that the signature of discrete scale
    invariance is the presence of a power law with a complex exponent,
    which manifests itself in data by log-periodic oscillations providing corrections
    to the simple power law scaling. In addition to the existence
    of a single preferred scaling ratio and its associated log-periodicity discussed
    up to now, there can be several preferred ratios corresponding to
    several log-periodicities that are superimposed. This can lead to a richer
    behavior such as log-quasi-periodicity [400].
    208 chapter 6
    As a last illustration, the Weierstrass function shown in Figure 6.14
    has a real fractal dimension equal to 1�5. As it exhibits a strong discrete
    scale invariance with preferred scaling ratio (the scale ratio of the successive
    spiky structures) equal to 2, it is endowed with an infinite number of
    complex fractal dimensions given by 1�5 + i2�n/ ln 2 = 1�5 + i9�06n,
    where n takes any possible integer value. As the integer n increases to
    larger and larger values, the corresponding complex dimensions describe
    smaller and smaller discrete scale invariant patterns.
    As we have seen, going from integer dimensions to real dimensions
    (with fractional parts) corresponds to a generalization of the translational
    symmetry to the scaling symmetry. It may come as a surprise to observe
    that further generalizing the concept of dimension to the set of complex
    numbers is, in contrast, reducing the scale symmetry into a subgroup, the
    discrete scale symmetry. This results from the fact that the imaginary part
    of the complex dimension is actually introducing an additional constraint
    that the symmetry must obey.
    Importance and Usefulness of Discrete Scale Invariance
    Existence of Relevant Length Scales.
    Suppose that a given analysis of some data shows log-periodic structures.
    What can we get out of it? First, as we have seen, the period in log-scale
    of the log-periodicity is directly related to the existence of a preferred
    scaling ratio. Thus, log-periodicity must immediately be seen and interpreted
    as the existence of a set of preferred characteristic scales forming
    altogether a geometrical series � � � � �?p� �?p+1� � � � � �� �2� � � � � �n� � � �
    Log-periodic structures in the data thus indicate that the system and/or
    the underlying physical mechanisms have characteristic length scales.
    This is extremely interesting, as it provides important constraints on the
    underlying mechanism. Indeed, simple power law behaviors are found
    everywhere, as seen from the explosion of the concepts of fractals, criticality,
    and self-organized criticality [26]. For instance, the power law
    distribution of earthquake energies known as the Gutenberg–Richter law
    can be obtained by many different mechanisms and described by a variety
    of models and is thus extremely limited in constraining the underlying
    physics (one fact, many competing explanations). Its usefulness as
    a modeling constraint is even doubtful, in contradiction with the common
    belief held by many scientists on the importance of this power law.
    In contrast, the presence of log-periodic features would teach us that
    hierarchies and log-periodicity 209
    important physical structures, hidden in the fully scale invariant description,
    Let us mention a remarkable application of log-periodicity used by
    bats and dolphins. It turns out that the amplitude of the ultrasound signals
    sent by animal sonars for echolocation, such as by bats and dolphins,
    is remarkably well described by a mathematical function called
    the Altes wavelet [5]. Having little real-time high-performance computer
    processing in their brain, these animals compensate for this limitation by
    using a very special waveform for their ultrasound signals that turns out
    to be the optimal shape with respect to distortion under Doppler shifts
    (the pitch of a sound depends on the relative velocity between the listener
    and emitter; for instance, an approaching car is heard at a higher
    frequency (pitch) than a receeding one). The Altes wavelet, which has a
    log-periodic structure with a local frequency varying hyperbolically, also
    has the remarkable property of minimizing the time-scale uncertainty,
    in the same sense that the Gaussian law minimizes the time-frequency
    uncertainty. It also has the nice property that differentiating it corresponds
    to dilating it by a fixed factor. Such a typical waveform is shown
    in Figure 6.23.
    It is important to stress the practical consequence of log-periodic structures.
    For prediction purposes, it is more constrained and thus reliable
    to fit a part of an oscillating data than a simple power law, which can
    be quite degenerate especially in the presence of noise. This is well
    known, for instance, in electronics and in signal processing in the presence
    of a controlled oscillatory wave carrier, on which one can “lock
    in” to extract a tiny signal from a large noise. This property that logperiodicity
    provides more reliable fits to data has been used and is vigorously
    investigated in several applied domains, such as rupture prediction
    [13, 12, 210, 215] and earthquakes [405, 355, 222], and will be discussed
    in depth in its application to financial crashes in its following chapters.
    We shall show that log-periodicity is very useful from an empirical
    point of view in analyzing financial data because such oscillations are
    much more strikingly visible in actual data than a simple power law: as
    we said, a fit can lock in on the oscillations which contain information
    about the critical date tc. If they are present, they can be used to predict
    the critical time tc simply by extrapolating frequency acceleration. Since
    the probability of the crash is highest near the critical time, this can be an
    interesting forecasting exercise. Note, however, that for rational traders in
    the models of chapter 5, such forecasting is useless because they already
    210 chapter 6
    ω [v = -1, k = 3, λ = 2, (211 points)]
    -10 -8 -6 -4 -2 0 2 4 6 8 10
    -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
    Fig. 6.23. Typical waveform of an Altes wavelet, a mathematical function exhibiting
    log-periodic symmetry, used by bats and dolphins to optimize their ultrasonic
    signals. The upper panel shows the Fourier transform as a function of angular frequency
    of the Altes wavelet represented in the lower panel as a function of time.
    Recall that the Fourier transform of a signal is nothing but the quantification of the
    sinusoidal components constituting the signal. Notice that both the Altes wavelet
    and its Fourier transform exhibit log-periodic oscillations.
    know the crash hazard rate h�t� at every point in time (including at tc),
    and they have already reflected this information in prices through the
    rational expectation condition!
    Scenarios Leading to Discrete Scale
    Invariance and Log-Periodicity
    After the rather abstract description of discrete scale invariance given
    above, let us briefly discuss the mechanisms that may be found at its origin.
    It turns out that there is not a unique cause but several mechanisms
    that may lead to discrete scale invariance. Since discrete scale invariance
    is a partial breaking of a continuous symmetry, this is hardly surprising,
    as there are many ways to break a symmetry. Some mechanisms
    have already been unravelled, while others are still under investigation.
    hierarchies and log-periodicity 211
    For a list of mechanisms, we refer to [392]. Discrete scale invariance is
    found in particular in chaotic systems, especially in the way they transit
    from order to chaos and respond to external perturbations. Discrete
    scale invariance is also a profound property of numbers and of the arithmetic
    system, symbolized by the so-called Newcomb–Benford law of
    first digits [195]. Before turning to a general dynamical system description
    of spontaneously generated log-periodic singularities in financial
    time series, we review the fascinating Newcomb–Benford law of first
    digits and its profound link with log-periodicity. Our motivation is that
    reducing a problem to number theory is like stripping it down to its sheer
    Newcomb–Benford Law of First Digits
    and the Arithmetic System
    In this section and in the next one, we discuss two remarkable occurrences
    of log-periodicity which, by their breadth and generality, suggest
    that discrete scale invariance may be a very important organizing
    Maybe the simplest example of log-periodicity occurs in the frequencies
    of first digits in natural numbers, which provides a mechanism for
    the Newcomb–Benford law. Newcomb in 1881 and Benford in 1938 [38]
    noticed that the pages of much-used tables of logarithms show evidence
    of a selective use of the natural numbers. The pages containing the logarithms
    of the low numbers 1 and 2 were more stained and worn out
    than those of the higher numbers 8 and 9. Benford compiled more than
    20�000 first digits taken from widely different sources, including river
    areas, population, constants of physics, newspapers, addresses, molecular
    weights, death rates, and so on, and showed that the frequency p�n�
    with which the digit n appears is given by
    p�n� = log10
    n + 1
    for n = 1� � � � � 9� (10)
    This gives p�1� = 0�301� p�2� = 0�176� p�3� = 0�125� p�4� = 0�0969,
    p�5� = 0�0792� p�6� = 0�0669� p�7� = 0�0580� p�8� = 0�0512, p�9� =
    0�0458. The number 1 thus appears in the first position more than six
    times more frequently than the number 9! These frequencies p�n� mean
    that out of 100 numbers drawn at random in a representative population
    of numbers, approximately 30 should start with 1 as the first digit, about
    212 chapter 6
    18 should start with 2 as the first digit, about 12 should start with 3
    as the first digit, about 10 should start with 4 as the first digit, about 8
    should start with 5 as the first digit, about 7 should start with 6 as the
    first digit, about 6 should start with 7 as the first digit, about 5 should
    start with 8 as the first digit, and about 4 should start with 9 as the first
    To explain this law, Benford constructed the running frequency Fn�R�
    of the first digits n = 1 to 9 of natural numbers from 1 to R. In other
    words, for instance, F1�R� = N1�R�/R is the ratio of the number N1�R�
    of occurrences of natural numbers between 1 and R with first digit 1
    divided by the total number R. We thus have
  4. R = 19, N1
    = 11, and F1
    = 11/19 = 0�5789, while for R = 99, N1
    11, and F1
    = 11/99 = 1/9 = 0�1111;
  5. R = 199, N1
    = 111, and F1
    = 111/199 = 0�5578, while for R = 999,
    = 111 and F1
    = 111/999 = 1/9 = 0�1111;
  6. R = 1� 999, N1
    = 1�111, and F1
    = 1�111/1999 = 0�5557, while for
    R = 9� 999, N1
    = 1�111 and F1
    = 1�111/9�999 = 1/9 = 0�1111;
    and so on. We thus see that F1�R� grows monotonically from 1/9 =
    0�1111 at R = 10r ? 1 to ≈0�5555 at 2 10r ? 1. Then, F1�R� decays
    monotonically from this maximum down to 1/9 = 0�1111, reached again
    at R = 10r+1 ? 1. Notice that F1�R� is always larger than or equal to
    1/9. It is thus clear that F1�R� does not have a limit as R goes to infinity
    but endlessly oscillates as a log-periodic function of R, for large R, with
    log-period log10 10, that is, preferred scaling ratio 10 (see Figure 4 in
    [38]). This result of course generalizes to arbitrary counting systems, say
    in base b. Then, the scaling ratio controlling the log-periodicity is b. The
    log-periodicity is expressing simply the hierarchical rule of the numbering
    system. Thus log-periodicity is at the very root of our arithmetic
    Generalization and statistical mechanism of the Newcomb–Benford
    law. A similar analysis can be performed for each of the digits. Let us
    discuss here n = 9 and its frequency F9�R�:
  7. R = 89, N9
    = 1, and F9
    = 1/89 = 0�0112, while for R = 99,
    = 11 and F9
    = 11/99 = 1/9 = 0�1111;
  8. R = 899, N9
    = 11, and F9
    = 11/899 = 0�0122 while for R = 999,
    = 111 and F9
    = 111/999 = 1/9 = 0�1111;
    hierarchies and log-periodicity 213
  9. R = 8999, N9
    = 111 and F9
    = 111/8999 = 0�0123 while for R =
    9999, N1
    = 1111 and F9
    = 1111/9999 = 1/9 = 0�1111;
    and so on. We thus see that F9�R� decreases monotonically from 1/9 =
    0�1111 at R = 10r ? 1 to ≈0�01234 at 10r+1 ? 10r ? 1. Then, F9�R�
    increases monotonically from this minimum up to 1/9 = 0�1111, reached
    again at R = 10r+1 ? 1. It is thus clear that F9�R� tends again to a (different)
    log-periodic function of R, for large R, again with log-period log10 10,
    that is, preferred scaling ratio 10. Notice that, in contrast to F1�R�, F9�R�
    is always smaller than or equal to 1/9. For the other digits, n = 2� � � � � 8,
    the corresponding Fn�R�’s oscillate log-periodically by crossing the value
    Averaging these frequencies Fn�R� over a log-period (i.e., a factor of
  1. can then be shown [38] to lead to the Newcomb–Benford law (10).
    This provides one possible mechanism. It turns out that there are other
    mechanisms and that, more deeply, the Newcomb–Benford law can be seen
    to be the only law that is invariant with respect to a change of scale, that
    is, to an arbitrary multiplication of all numbers by a common factor. Hill
    established in 1995 [194] that a probability measure is scale invariant if and
    only if the probability measure of the set of all intervals �1� t� × 10n for all
    n integers is log10 t, for any t in �1� 10�. Benford’s law is then easily seen
    to result from this theorem since the probability of having a given first digit
    d corresponds to the difference log10�d + 1� ? log10 d of the probability
    measure, thus giving expression (10). Notice that this does not provide a
    mechanism for Benford’s law but rather relates it to a symmetry principle.
    In [196], Hill gives a statistical mechanism: if distributions are selected
    at random and random samples are then taken from each of these distributions,
    the significant digits of the combined sample will converge to the
    logarithmic Benford distribution.
    The Log-Periodic Law of the Evolution of Life?
    The founding concept of evolution was introduced by Darwin and Wallace
    in a paper presented in 1858 to the Linnean Society of London
    entitled “On the Tendency of Species to Form Varieties by Natural
    Selection.” According to this now well-established theory, new biological
    species are created by direct mutuation and selection from existing
    species. The complexity of the universe of vegetals and animals in
    particular can then be seen to be a beautiful “tree of life,” as shown
    in Figure 6.24, whose branching or bifurcation structure reflects the
    cascades of jumps between species in the history of life. The evolu214
    chapter 6
    Fig. 6.24. The dates (in millions of years, denoted by the symbol “My”) with
    respect to the origin of time taken to be the present (hence the negative dates that
    refer to the past), of major evolutionary events of seven lineages (common evolution
    from life origin to viviparity, theropod and sauropod dinosaurs, rodents, equidae,
    primates including hominidae, and echinoderms) are plotted as black points. The
    scale of the time axis is such that the black dots should be equidistant in the logarithm
    of the distance in time from the epoch of a dot to a critical time Tc which lies
    beyond the end of the sequence. The dates indicated close to each dot correspond to
    the exact numerical values predicted by the log-periodic model, computed from the
    hierarchies and log-periodicity 215
    tionary view of biological species allows one to construct a taxonomy
    represented by this “tree of life,” organized from trunk to the smallest
    leaves, from the superkingdoms (Archaea, Bacteria, Eukaryota, Viroids,
    Viruses), kingdoms, phyla, subphyla, orders, suborders, families, genus,
    and finally the species. These different levels correspond to the main
    nodes of bifurcations, such that, going from the species to the genus,
    then to the families, and so on, up to the superkingdoms, is similar to
    going backward in time and identifying the creation of new species in
    successive steps. For instance, the domestic cat has the following lineage:
    Eukaryota, Metazoa, Chordata, Craniata, Vertebrata, Euteleostomi,
    Mammalia, Eutheria, Carnivora, Fissipedia, Felidae, Felis domesticus
    (see http�//www�ncbi�nlm�nih�gov/Taxonomy/tax�html/).
    There is much evidence that evolution is characterized by phases of
    quasi-statis, where species remain stable for a long time, interrupted by
    episodic bursts of activity, with destruction of species and creation of
    new ones [168, 169]. There are thus rather precise dates for speciation
    events, and one can therefore define the length of branches between
    nodes in the “tree of life,” which represent time intervals between such
    major evolutionary events. Can this tree be described by a mathematical
    structure, at least at a statistical level? Remarkably, Nottale, Chaline,
    and Grou [74, 317, 318] have recently suggested that a self-similar logperiodic
    law seems to characterize the tree of life. True or false, this
    example provides a simple and fascinating application of log-periodicity.
    Recall that log-periodicity in the present context is synonymous
    with the existence of periodicity of some observable in the logarithm
    ? T
    of the time T to some critical time Tc. Periodicity of some
    observable in the variable ln
    ? T
    implies the existence of a hierarchy
    of characteristic time scales T0 < T1 < · · · < Tn < Tn+1 < · · · ,
    corresponding, for instance, to the periodic maxima of the observable as
    a function of time, given by
    = Tc
    ? �Tc
    ? T0�g?n� (11)
    where g is the preferred scaling factor of the underlying discrete scale
    invariance (which has also been denoted � above). This formula (11)
    Fig. 6.24 continued. best fit to each time series. Perfect log-periodicity is qualified
    by equidistant black points. The adjusted critical time Tc and scale ratio g are
    indicated for each lineage after the arrows. In the case of the echinoderms, the logperiodicity
    is inverted; that is, Tc is in the past and the characteristic times Tn are
    more and more spaced as time flows from past to future. Reproduced from [318].
    216 chapter 6
    turns out to fit the dates of the major evolutionary events shown in
    Figure 6.24 well.
    Notice that the spacings Tn+1
    ? Tn between successive values of Tn
    approach zero as n becomes large and Tn converges to the critical time
    Tc. From three successive observed values of Tn, say Tn, Tn+1, and Tn+2,
    the critical time Tc can be determined by the formula
    = T 2
    ? Tn+2Tn
    ? Tn
    ? Tn+2
    � (12)
    This relation is invariant with respect to an arbitrary translation in time.
    In addition, the next time Tn+3 is predicted from the preceding Tn by
    = T 2
  • T 2
    ? TnTn+2
    ? Tn+1Tn+2
    ? Tn
    � (13)
    These formulas are reproduced in chapter 9 as (23) and (24) in the
    section titled “A Hierarchy of Prediction Schemes.”
    Nottale, Chaline, and Grou [74, 317, 318] have found that the fossil
    equine of North America, the primates, the rodents, and other lineages
    have followed an evolution path punctuated by major events that follow
    the geometrical time series (11). The critical time Tc is roughly
    the present for the equines, in agreement with the known extinction of
    this species in North America 10,000 years ago (but this could be a
    coincidence, as North American horses went extinct when humans put
    foot on the continent and hunted them down). The critical time Tc is
    approximately 2 million years in the future for the primates and about
    12 million years in the future for the rodents. Tc is the end (respectively,
    beginning) of the evolutionary process for an accelerating (respectively,
    decelerating) lineage. For an accelerating lineage, the critical time Tc
    may tentatively be interpreted as the end of the evolutionary ability of
    this lineage, not necessarily as a pre-set extinction age of the group.
    There are, of course, many methological issues as well as fundamental
    biological problems associated with this proposed log-periodic law (11).
    It is far from impossible that this regularity could be an artifact of the
    data (which has many deficiencies) and of the method of analysis. In
    particular, the further back one goes in trying to reconstruct the past,
    the coarser and sparser the information becomes. It is possible that this
    uneven sampling may create an apparent log-periodicity in the manner
    discussed in [203], but several tests seem to exclude this possibility. If
    the “log-periodic law of the evolution of life” given by (11) turns out to
    be genuine, this would then call for a profound explanation. In any case,
    hierarchies and log-periodicity 217
    it provides a vivid example of the power of the discrete scale invariance
    symmetry to organize complex data in a more transparent way, maybe
    guiding us towards a possible deeper understanding.
    This section presents an alternative understanding of the emergence
    of critical points (finite-time singularities) decorated by accelerating
    oscillations, which complements the previous description. It is based on
    a “dynamical system” description in which these characteristics emerge
    dynamically. The main ingredient is the coexistence of two classes of
    investors, the “fundamentalists,” or “value-investors,” and the trendfollowers
    (often called chartists, technical analysts, or noise traders in
    the jargon of academic finance). The second essential ingredient is to
    recognize that both classes of investors behave in a “nonlinear” way.
    These two ingredients produce a finite-time singularity with accelerating
    oscillations. The power law singularity results from the nonlinear accelerating
    growth rate due to trend-following. The oscillations, which are
    approximately log-periodic with remarkable scaling properties, result
    from the nonlinear restoring force exerted on price by value-investors,
    which tends to bring it back to its fundamental value. As a function
    of the degree of non-linearity of the growth rate and of the restoring
    term, a rich variety of behavior can be observed. We shall see that
    the dynamical behavior is traced back fundamentally to the self-similar
    spiral structure of the dynamics in a (price, price variation) space
    representation, unfolding around a central fixed point [205].
    The price variation of an asset on the stock market is controlled by
    supply and demand, in other words by the net order size � equal to the
    number of buy orders minus the number of sell orders. It is clear that
    the price increases (respectively, decreases) if � is positive (respectively,
    negative). If the ratio of the price ?p at which the orders are executed over
    the previous quoted price p is solely a function of the net order size �
    and assuming that it is not possible to make profits by repeatedly trading
    through a closed circuit (i.e., by buying and selling with final net position
    equal to zero), one can show that the difference in the logarithm of
    the price between tomorrow and today is directly proportional to the net
    order size � [123]. The net order size � resulting from the action of all
    traders is continuously readjusting with time so as to reflect the information
    flow in the market and the evolution of the traders’ opinions and
    218 chapter 6
    moods. Various derivations have related the price variation or the variation
    of the logarithm of the price to factors that control the net order size
    itself [123, 49, 330]. Three basic ingredients are thought to be important
    in determining the price dynamics: trend following, reversal to the
    estimated fundamental value, and risk aversion.
    Trend Following: Positive Nonlinear Feedback
    and Finite-Time Singularity
    Trend following (in various elaborated forms) was (and probably still is)
    one of the major strategies used by so-called technical analysts (see [6]
    for a review and references therein). In its simplest form, trend following
    amounts to taking the net order size � as proportional to the past trend,
    that is, to the difference between the logarithm of the price today and the
    logarithm of the price yesterday. Trend-following strategies thus exert a
    positive feedback on prices, since previous price increases (decreases)
    lead to buy (sell) orders, thus enhancing the previous trend. Taken alone,
    this implies that the difference in the logarithm of the price between
    tomorrow and today is proportional to the logarithm of the price between
    today and yesterday. This simple relationship expresses the existence of
    a constant growth rate, leading to an exponential growth of the logarithm
    of the price. This means that the price increases as the exponential of
    the exponential of time.
    This linear relationship between past price variation and net order size
    is usually chosen by modelers. Here, we depart from this convention and
    consider it more realistic to assume that the net order size may grow
    faster than the previous price change; that is, that they are nonlinearly
    related. Indeed, a small price change from time t ? 1 to time t may not
    be perceived as a significant and strong market signal. Since many of the
    investment strategies are nonlinear, it is natural to consider an average
    trend-following order size which increases in an accelerated manner as
    the price change increases in amplitude. Usually, trend-followers increase
    the size of their order faster than just proportionally to the last trend. This
    is reminiscent of the argument [6] that traders’ psychology is sensitive
    to a change of trend (“acceleration” or “deceleration”), not simply to
    the trend (“velocity”). The fact that trend-following strategies have an
    impact on price proportional to the price change over the previous period
    raised to some power m > 1 means that trend-following strategies are
    not linear when averaged over all of them: they tend to underreact for
    small price changes and over-react for large ones. Note that the value
    hierarchies and log-periodicity 219
    m = 1 retrieves the linear case. Figure 6.25 explains the concept of a
    nonlinear response.
    When the sum of all trend-following behaviors is expressed in a nonlinear
    form so that the net order size � is proportional to a power of the
    difference between the logarithm of the price today and the logarithm of
    the price yesterday with an exponent larger than 1, by the same reasoning
    as in the technical inset entitled “Intuitive explanation of the creation of
    a finite-time singularity at tc” in chapter 5, the price exhibits a finite-time
    n=1 n=10
    0 0.5 1 1.5
    Net Order Size
    Observed Price - Fundamental Price
    Fig. 6.25. We illustrate the different response of a system (the net order size shown
    on the ordinate) as a function of a stimulus (the distance between market price and
    the fundamental price shown in the abscissa) for different nonlinear dependences
    quantified by the parameter n: response = stimulusn. For n = 1, the response is
    proportional to the stimulus, as shown with the straight continuous line: this is the
    linear description. For n > 1, for instance, n = 4, the response is very small for
    small stimulii but starts to shoot up when the stimulus increases above some characteristic
    value here normalized to 1, as shown with the curved continuous line. This
    is the case discussed here. The dotted-dashed line corresponds to a stronger nonlinearity
    with exponent n = 10, showing even more strongly the almost threshold-like
    nature of the response of the system. The thin dashed line illustrates the opposite
    nonlinear situation with a parameter n < 1 for which the response accelerates fast
    for small stimulii but saturates at large stimulii.
    220 chapter 6
    singularity. This effect is just a rephrasing of the phenomenon already
    described by the price-driven model discussed in that chapter.
    Reversal to the Fundamental Value:
    Negative Nonlinear Feedback
    Fundamental value trading is based on an estimation of the financial
    value of the company based on objective economic and accounting criteria
    such as assets, earnings, and growth potential. The fundamental analyst
    thus establishes her estimation for the “correct” fundamental value
    of a given firm and then compares it with the price quoted in the stock
    market. If the latter is smaller than the fundamental price, this is a buy
    opportunity as the analyst expects that the stock market will soon realize
    that the stock is underpriced compared to its real value. The ensuing
    wave of preferential buy orders will drive the price up until the fundamental
    value is reached. In these circumstances, the buy decision is
    based on the belief that you are among the first to realize that the corresponding
    stock is underpriced. The reverse is expected to occur if the
    market price is larger than the fundamental value.
    However, in practice, there are severe difficulties in obtaining a precise
    estimation of the fundamental value, as it is not clear how to value some
    of the important intangible assets of a company such as the quality of
    its managers, its position in its market niche, and so on. In addition,
    predicting future earnings and their growth is an inexact science, to say
    the least. This has a very important consequence that we now discuss.
    An important feature of our model is the nonlinear dependence of the
    net order size � as a function of the difference between the logarithm of
    the price and the logarithm of the fundamental value. The nonlinearity
    allows one to capture the following effect. In principle, as we said above,
    the fundamental value p0 is determined by the discounted expected future
    dividends and is thus dependent upon the forecast of their growth rate
    and of the riskless interest rate, both variables being very difficult to
    predict. The fundamental value is thus extremely difficult to quantify
    with high precision and is often estimated within relatively large bounds
    [282, 85, 260, 69]: all of the methods determining intrinsic value rely on
    assumptions that can turn out to be far off the mark. For instance, several
    academic studies have disputed the premise that a portfolio of sound,
    cheaply bought stocks will, over time, outperform a portfolio selected by
    any other method (see, for instance, [256]). As a consequence, a trader
    trying to track fundamental value has no incentive to react when she
    hierarchies and log-periodicity 221
    feels that the deviation is small since this deviation is more or less within
    the uncertainty of her estimations. Only when the departure of price
    from fundamental value becomes relatively large will the trader act. The
    strongly nonlinear dependence of the net order size � as proportional
    to the amplitude of the difference between the logarithm of the price
    and the logarithm of the fundamental value raised to a power n larger
    than 1 precisely accounts for this effect, as shown in Figure 6.25: for
    an exponent n larger than 1, �n remains small for � < 1 and shoots up
    rapidly only when it becomes larger than 1, approximating a threshold
    behavior of all or nothing.
    Such a nonlinear sensitivity is not just a theoretical construction; it
    has been recently documented in the context of the sensitivity of the
    money demand to interest rate. Using a survey of roughly 2,700 households,
    Mulligan and Sala-i-Martin [311] estimated the interest elasticity
    of money demand (the sensitivity or log-derivative of money demand to
    interest rate) to be very small at low interest rates. This is due to the
    fact that few people decide to invest in interest-producing assets when
    rates are low, due to “shopping” costs. In constrast, for large interest
    rates or for those who own a significant bank account, the interest elasticity
    of money demand is significant. This is a clear-cut example of a
    threshold-like behavior characterized by a very nonlinear response. This
    can be captured by e ≡ d lnM/d ln r = �r/rinfl�m with m > 1 such that
    the elasticity e of money demand M is negligible when the interest r
    is not significantly larger than the inflation rate rinfl, and becomes large
    From the fact that a low (large) price is driven upwards (downwards)
    towards the fundamental value, we see that the class of investment strategies
    based on fundamental valuation lead to a reversal of the price. This
    reversal force can be linear, that is, the corresponding net order size � is
    proportional to the difference between the logarithm of the price and the
    logarithm of the fundamental value. In the case n = 1, since the difference
    in the logarithm of the price between tomorrow and today is directly
    proportional to the net order size �, this implies that the difference in the
    logarithm of the price between tomorrow and today is proportional to the
    difference between the logarithm of the price today and the logarithm of
    the fundamental value. The relationship is an exact analog to the equation
    of an oscillator such as a pendulum: starting from a position away from its
    equilibrium immobile position, it undergoes endless oscillations around
    this equilibrium point, as shown by the thick line trajectory in Figure 6.26.
    Similarly, with this term alone the price oscillates endlessly around the
    fundamental value. The reason for the oscillations is that there is an inertia
    222 chapter 6
    0 0.4 0.6
    0.2 0.8
    Fig. 6.26. Time dependence of the logarithm of the price normalized by the fundamental
    price resulting from the interplay between the reversal “force” created
    by fundamental value investing and the “inertia” stemming from the fact that the
    decision to invest from today to tomorrow is based on information from yesterday
    to today. Four different values of the exponent n = 1� 3� 5, and 15 are shown.
    Compared to the linear case n = 1 whose solution is a pure sine y�n=1�
    1 �t� =
    10 sin�

    10 t�, increasing the nonlinear exponent n has three effects: (i) decrease
    of the amplitude; (ii) increase of the frequency; and (iii) production of a
    saw-tooth profile with increasingly sharper corners as n increases. Reproduced
    from [205].
    in the reversal force which does not vanish sufficiently rapidly and leads to
    overshooting. This overshooting then triggers a price motion in the opposite
    direction which itself overshoots and so on. When the fundamental
    reversal term is nonlinear, the oscillations persist but change shape. Their
    main properties is that their frequency (reciprocal of their period, which
    is the time interval between two successive maxima) becomes dependent
    on the amplitude of the deviation between market price and fundamental
    value. This property is very important because, if there are other effects
    or perturbations that tend to modify this amplitude, the frequency will
    be modified accordingly. This nonlinear frequency dependence upon the
    amplitude provides a mechanism for accelerating frequencies when the
    amplitude shoots up.
    hierarchies and log-periodicity 223
    Some Characteristics of the Price Dynamics
    of the Nonlinear Dynamical Model
    Let us now put all the ingredients together:
    � “inertia” resulting from the fact that a decision today to invest will
    bear its fruit in the future while it is based on past analysis;
    � nonlinear trend following which, together with “inertia,” creates a
    finite-time singularity in the amplitude of the deviation between market
    price and fundamental price;
    � nonlinear fundamental value investing which, together with “inertia,”
    produces nonlinear oscillations dependent on the amplitude of the
    deviation between market price and fundamental price.
    Figure 6.27 shows the time evolution of the logarithm of the market
    price normalized by the fundamental value, which we shall refer to as
    3.2 3.4 3.5
    α = 1 γ = 10 m = 1.3 n = 3 D(0) = 1
    3.3 3.6 3.7 3.8 3.9
    Fig. 6.27. Solution of the dynamical equation incorporating “inertia,” nonlinear
    trend-following, and nonlinear fundamental value investing for the parameters m =
    1�3, n = 3. The envelope of the “reduced price” y1�t� grows faster than exponentially
    and approximately as �tc
    ? t�?1�5, where tc
    ≈ 4. A negative value of the
    reduced price y1 just means that the observed price is below the fundamental value.
    This stems from the definition of reduced price as the logarithm of the ratio between
    observed price and fundamental value. Reproduced from [205].
    224 chapter 6
    the “reduced price” for the choice �m = 1�3� n = 3� of the exponents
    controlling, respectively, the nonlinear trend following (or elasticity) and
    the fundamental reversal (or sharpness of the threshold response) terms.
    Two main features are apparent. First, the reduced price diverges on
    the approach of the critical time tc as �tc
    ? t�?�. Note that the specific
    value of the critical time is dictated by initial conditions. Second, this
    acceleration is decorated by accelerating oscillations. As we mentioned
    in the previous section, the acceleration of the oscillations results from
    their nonlinear dependence upon the accelerating amplitude.
    Figure 6.28 shows the same data as in Figure 6.27, but using scales
    such that a pure power law behavior qualifies as a straight line: the
    logarithm of the reduced price is plotted as a function of the logarithm
    of the distance from the critical time. We observe that the envelop is
    indeed well qualified by the power law shown as the straight dashed line.
    In addition, the oscillations are approximately equidistant in this representation,
    which, as we showed several times in the previous sections,
    0.2 0.3 0.4
    4 - t
    α = 1 γ = 10 m = 1.3 n = 3 D(0) = 1
    0.5 0.6 0.7 0.8
    Fig. 6.28. Same data as in Figure 6.27: The absolute value
    of the “reduced
    price” is shown as a function of tc
    ? t, where tc
    = 4 in double logarithmic coordinates,
    such that a linear envelope qualifies the power law divergence �tc
    ? t�?1�5.
    The slope of the dashed line is ?1�5. Notice also that the oscillations are approximately
    equidistant in the variable ln�tc
    ? t�, resembling log-periodic behavior of
    accelerating oscillations on the approach to the singularity. Reproduced from [205].
    hierarchies and log-periodicity 225
    qualify as an approximate log-periodicity. The dynamics involving
    “inertia,” nonlinear trend following, and nonlinear fundamental reversal
    behaviors is thus able to create a quasi-log-periodic behavior of
    accelerating oscillations on the approach to a finite-time singularity.
    Figure 6.29 shows the reduced price for a larger value of the trendfollowing
    exponent m = 2�5. In this case, the reduced price goes to
    a constant at tc with an infinite slope (the singularity is thus on its
    derivative, or “velocity”). We can also observe accelerating oscillations,
    somewhat reminiscent of log-periodicity. The novel feature is that the
    oscillations are only transient, leaving place to a pure final accelerating
    trend in the final approach to the critical time tc.
    Figure 6.18 has taught us that an oscillatory motion can be seen
    as the projection of a rotation occurring in a plane on one axis. We
    now extend this logic and show with Figure 6.30 that an oscillation
    with varying frequency and amplitude as in Figures 6.27 and 6.29 is
    nothing but the projection on one axis of a spiraling structure in the
    plane. Actually, Figure 6.30 shows more than that: in the plane of the
    2 4 5
    3 6 7
    m = 2.5 n = 3 y(0) = 0.02 D(0) = 0.3
    Fig. 6.29. “Reduced price” as a function of time for a trend-following exponent
    m = 2�5 with n = 3, with two amplitudes � = 10 and � = 1�000 of the fundamental
    reversal term. Reproduced from [205].
    226 chapter 6
    b- Bp-
    B+ b+
    Fig. 6.30. Geometrical spiral showing two special trajectories (the continuous and
    dashed lines) in the “reduced price”–“velocity” plane �y1� y2� that exactly connect
    the origin y1
    = 0� y2
    = 0 to infinity. This spiraling structure, which exhibits scaling
    or fractal properties, is at the origin of the accelerating oscillations decorating the
    power law behavior close to the finite-time singularity. The different segments of
    curves and domains pointed out by the arrows are mapped from one to another
    throughout the dynamics of the model. Reproduced from [205].
    reduced price y1 and its “velocity” y2, it shows two special trajectories
    that connect exactly the origin y1
    = 0� y2
    = 0 to infinity. From
    general mathematical theorems of dynamical systems, one can then show
    that any trajectory starting close to the origin will never be able to
    cross any of these two orbits. As a consequence, any real trajectory
    will be guided within the spiraling channel, winding around the central
    hierarchies and log-periodicity 227
    point 0 many times before exiting towards the finite-time singularies.
    The approximately log-periodic oscillations result from the oscillatory
    structure of the fundamental reversal term associated with the acceleration
    driven by the trend-following term. The conjunction of the two
    leads to the beautiful spiral, governing a hierarchical organization of the
    spiralling trajectories around the origin in the price-velocity space [205].
    chapter 7
    autopsy of major
    crashes: universal
    exponents and
    As discussed in chapter 1, the crash of October
    1987 and its Black Monday on October 19 remains one of the most
    striking drops ever seen in stock markets, both by its overwhelming
    amplitude and its encompassing sweep over most markets worldwide. It
    was preceded by a remarkably strong “bull” regime epitomized by the
    following quote from The Wall Street Journal on August 26, 1987, the
    day after the 1987 market peak: “In a market like this, every story is a
    positive one. Any news is good news. It’s pretty much taken for granted
    now that the market is going to go up.” Investors were thus largely
    unaware of the forthcoming risk happenings [174]. This surprise change
    in risk view of October 19, 1987 is supported by the time-series behavior
    of implied risk estimates calculated on the Standard & Poor’s (S&P)
    500 Index Option for the September–November daily trading period.
    The highest implied risk estimate for stocks in the pre-crash period was
    18�5% and occurred on October 15, 1987 [174]. This was still below
    the 22% annualized standard deviation of return calculated during 1974,
    the most volatile recent year before 1987, and significantly below the
    autopsy of major crashes 229
    46% recorded on Monday, October 19, 1987, and the 88% recorded
    on Monday, October 26, 1987. As we shall show in Figure 7.4, during
    November 1987, market volatility as measured by the implied annual
    return standard deviation fell to about 30%, which was still much higher
    than the highest implied risk value observed immediately prior to Black
    Monday [174].
    The October 19, 1987, stock-market crash stunned Wall Street professionals,
    hacked about $1 trillion off the value of all U.S. stocks,
    and elicited predictions of another Great Depression. On Black Monday,
    the Dow Jones industrial average plummeted 508 points, or 22.6%, to
    1,738.74. It was the largest one-day point and percentage loss ever for
    the blue-chip index. The broader markets followed the Dow downward.
    The S&P 500 index lost more than 20%, falling 57�86 to 224�84. The
    Nasdaq composite index dived 46�12 to 360�21. No Dow components
    emerged unscathed from Black Monday. Even market stalwarts suffered
    massive share losses. IBM shed 31 ? 3/4 to close at 103 ? 1/3, while
    USX lost 12 ? 1/2 to 21 ? 1/2 and Eastman Kodak fell 27 ? 1/4 to
    62?7/8. The crash splattered technology stocks as well. On the Nasdaq,
    Apple Computer lost 11 ? 3/4 to close at 36 ? 1/2, while Intel dropped
    10 to 42.
    Stocks descended quickly on Black Monday, with the Dow falling
    200 points soon after the opening bell to trade at around 2,046. Yet by
    10 a.m., the index had crept back up above 2,100, beginning a pattern of
    rebound and retreat that would continue for most of the day. Later, with
    75 minutes left in the trading day, it looked like the Dow would escape
    with a loss of “only” about 200 points. But the worst was yet to come.
    Starting at about 2:45 p.m., a massive sell-off began, eventually ripping
    300 more points off the Dow. At the closing bell, the Dow appeared
    to have suffered an amazing loss of about 400 points. However, heavy
    volume kept the NYSE’s computers running hours behind trading. Only
    about two hours later would investors realize that the day’s total loss
    exceeded 500 points. Reaction to the crash varied from sentiments that
    the market was due for a correction to feelings of outright despair.
    President Ronald Reagan sought to reassure investors, saying: “All the
    economic indicators are solid. There is nothing wrong with the economy.”
    And the day after the crash, Federal Reserve Chairman Alan
    Greenspan gave a lucid one-sentence statement indicating the Fed would
    provide sufficient funds to banks, allowing them to provide credit to
    securities firms. “The Federal Reserve, consistent with its responsibilities
    as the nation’s central bank, affirmed today its readiness to serve
    as a source of liquidity to support the economic and financial system,”
    230 chapter 7
    the statement said. The NYSE did end up opening for business as usual
    on October 20, and the Dow rose 102�27—its largest one-day gain ever
    up to that time—to close at 1,841.01. But making up the full extent of
    Black Monday’s losses would take longer. The Dow only returned to its
    pre-crash levels in January 1989, 15 months after Black Monday. The
    broader S&P 500 index took 21 months to fully recover.
    It is interesting to quantify the relative weight of various participants
    during these volatile times. Based on the Federal Reserve’s Flow of
    Funds Accounts of the US analyzed by Fung and Hsieh [146], the market
    value of U.S. corporate equities stood at U.S.$3,511 billion at the end
    of September 1987. The major owners were households (49%), private
    pension funds (21%), mutual funds (7%), state and local government
    retirement funds (6%), bank personal trusts and estates (6%), foreigners
    (6%), insurance companies (5%), and brokers and dealers (<1%). In
    the last quarter of 1987, households had been the largest sellers, with
    sells worth U.S.$19.6 billion, followed by the rest of the world, with
    sells worth U.S.$7.5 billion, brokers and dealers, U.S.$4.8 billion, and
    mutual funds, U.S.$3.0 billion. These sells were almost fully balanced
    by purchases of equities back from investors by U.S. corporations for
    the amount of U.S.$30.2 billion.
    The net sells thus amounted to less than 1% of the total value of
    U.S. corporate equities. Studies carried out by the Investment Company
    Institute (ICI) confirm the following specific findings about mutual fund
    shareholders and their reactions to market volatility:
    � The largest net outflow within a short period occurred during and
    immediately after the October 1987 stock market break and amounted
    to only 4.5% of total equity fund assets.
    � An estimated 95% of stock fund owners did not redeem shares immediately
    after the 1987 stock market break.
    � The responses of shareholders to other sharp drops in stock prices
    since 1945 were considerably more restrained than the reaction in

The Investment Company Institute [207] is the national association of
the American investment company industry. Founded in 1940, its membership
in 2000 included 8,414 mutual funds, 489 closed-end funds, and
8 sponsors of unit investment trusts. Its mutual fund members represent
more than 83 million individual shareholders and manage approximately
$7 trillion.
autopsy of major crashes 231
Precursory Pattern
In the sequel, time is often converted into decimal year units: for
nonleap years, 365 days = 1�00 year, which leads to 1 day = 0�00274
years. Thus 0�01 year = 3�65 days and 0�1 year = 36�5 days or 5 weeks.
For example, October 19, 1987 corresponds to 87�800.
Figure 7.1 shows the evolution of the NYSE index S&P 500 from July
1985 to the end of October 1987 after the crash. The plusses (+) represent
the best fit to an exponential growth obtained by assuming that the
market is given an average return of about 30% per year. This first representation
does not describe the apparent overall acceleration before the
crash, occurring more than a year in advance. This acceleration (cusp-like
85.5 86.5 87
Time (year)
86 87.5
Fig. 7.1. Evolution as a function of time of the NYSE S&P 500 index from July
1985 to the end of October 1987 (557 trading days). The + represent a constant
return increase of ≈30%/year and gives var�Fexp� ≈ 113 (see text for definition).
The best fit to the power law (14) gives A1
≈ 327, B1
≈ ?79, tc
≈ 87�65, m1
≈ 0�7,
and varpow
≈ 107. The best fit to expression (15) gives A2
≈ 412, B2
≈ ?165,
≈ 87�74, C ≈ 12, � ≈ 7�4, T = 2�0, m2
≈ 0�33, and varlp
≈ 36. One can observe
four well-defined oscillations fitted by the expression (15) before finite size effects
limit the theoretical divergence of the acceleration, at which point the bubble ends
in the crash. All the fits are carried over the whole time interval shown, up to 87�6.
The fit with (15) turns out to be very robust with respect to this upper bound, which
can be varied significantly. Reproduced from [401].
232 chapter 7
shape) is better represented by using power law functions that chapters 5
and 6 showed to be signatures of critical behavior of the market. The
monotonic line corresponds to the following power law parameterization:
Fpow�t� = A1

  • B1�tc
    ? t�m1 � (14)
    where tc denotes the time at which the power law fit of the S&P 500
    presents a (theoretically) diverging slope, announcing an imminent crash.
    In order to qualify and compare the fits, the variances (denoted var,
    equal to the mean of the squares of the errors between theory and data)
    or its square-root (called the root-mean-square [r.m.s.]) are calculated.
    The ratio of two variances corresponding to two different hypotheses is
    taken as a qualifying statistic. The ratio of the variance of the constant
    rate hypothesis to that of the power law is equal to varexp/varpow

    1�1, indicating only a slightly better performance of the power law in
    capturing the acceleration, the number of free variables being the same
    and equal to 2.
    However, to the naked eye, the most striking feature in this acceleration
    is the presence of systematic oscillatory-like deviations. Inspired
    by the insight given in chapter 5 and especially chapter 6, the oscillatory
    continuous line is obtained by fitting the data by the following
    mathematical expression:
    Flp�t� = A2
  • B2�tc
    ? t�m2 �1 + C cos�� log��tc
    ? t�/T ���� (15)
    This equation is the simplest example of a log-periodic correction to a
    pure power law for an observable exhibiting a singularity at the time tc
    at which the crash has the highest probability. The log-periodicity here
    stems from the cosine function of the logarithm of the distance tc
    ? t to
    the critical time tc. Due to log-periodicity, the evolution of the financial
    index becomes (discretely) scale invariant close to the critical point.
    As shown in chapter 6, the log-periodic correction to scaling implies
    the existence of a hierarchy of characteristic time intervals tc
    ? tn, given
    by expression (11) on page 215, with a preferred scaling ratio denoted
    g or �. For the October 1987 crash, we find � � 1�5 ? 1�7 (this value
    is remarkably universal and is found approximately the same for other
    crashes, as we shall see). We expect a cut-off at short time scales (i.e.,
    above n ~ a few units) and also at large time scales due to the existence
    of finite size effects. These time scales tc
    ? tn are not universal but
    depend upon the specific market. What is expected to be universal are the
    ratios tc
    = �. For details on the fitting procedure, we refer to [401].
    autopsy of major crashes 233
    It is possible to generalize the simple log-periodic power law formula
    used in Figure 7.1 by using a mathematical tool, called bifurcation
    theory, to obtain its generic nonlinear correction, which allows one to
    account quantitatively for the behavior of the Dow Jones and S&P 500
    indices up to eight years prior to October 1987 [397]. The result of
    this theory, presented in [397], is used to generate the new fit shown in
    Figure 7.2. One sees clearly that the new formula accounts remarkably
    well for almost eight years of market price behavior compared to only
    a little more than two years for the simple log-periodic formula shown
    in Figure 7.1. The nonlinear theory developed in [397] leads to “logfrequency
    modulation,” an effect first noticed empirically in [128]. The
    remarkable quality of the fits shown in Figures 7.1 and 7.2 have been
    assessed in [214].
    80 83 84 85 86 87 88
    81 82
    Fig. 7.2. Time dependence of the logarithm of the NYSE S&P 500 index from January
    1980 to September 1987 and best fit by the improved nonlinear log-periodic
    formula developed in [397] (dashed line). The exponent and log-periodic angular
    frequency are m2
    = 0�33 and �1987 = 7�4. The crash of October 19, 1987 corresponds
    to 1987�78 decimal years. The solid line is the fit by (15) on the subinterval
    from July 1985 to the end of 1987 and is represented on the full time interval starting
    in 1980. The comparison with the thin line allows one to visualize the frequency
    shift described by the nonlinear theory. Reproduced from [397].
    234 chapter 7
    In a recent reanalysis, J. A. Feigenbaum [127] examined the data in a
    new way by taking the first differences for the logarithm of the S&P 500
    from 1980 to 1987. The rationale for taking the price variation rather than
    the price itself is that the fluctuations, noises, or deviations are expected
    to be more random and thus more innocuous than for the price, which is
    a cumulative quantity. By rigorous hypothesis testing, Feigenbaum found
    that the log-periodic component cannot be rejected at the 95% confidence
    level: in plain words, this means that the probability that the log-periodic
    component results from chance is about or less than 0�05.
    D. S. Bates [34] has studied the transaction prices of the S&P 500
    futures options over 1985–87 and found evidence of expectations prior
    to October 1987 of an impending stock market crash in this data. These
    expectations are based on patterns of intermittent accelerating “fears,”
    possibly related to the evidence presented so far. S&P 500 futures options
    are contracts that derive from the underlying S&P 500 index and whose
    price depends on three main variables, (1) the so-called exercise or strike
    price of the option, (2) the interval of time between the present and
    the maturity date of the option, and (3) a measure of the perceived
    volatility of the underlying S&P 500 index. So-called “put” (respectively,
    “call”) options have increasing value the smaller (respectively, larger)
    is the expected future index price at the maturity date and the larger is
    the perceived volatility. Put options are thus direct probes of the sentiment
    of traders on the downside risk of the underlying market, that
    is, of the risk of a large drop of the market that would make these put
    options very valuable. Symmetrically, call options are direct probes of
    the sentiment of traders on the upside risk of the underlying market,
    i.e., of the possibility of a large rally of the market which would make
    these call options very valuable. Figure 7.3 summarizes how this idea
    can be used concretely for a quantification of the perceived asymmetry
    between large downside and upside risks. It shows the percentage
    deviation �C ? P�/P between call and put option prices (which Bates
    called a “skewness premium”). The curve at the bottom, called “at-themoney
    options,” quantifies the percentage deviation �C ? P�/P of put
    and call options, which take a significant value as soon as the price
    deviates from the present price (so-called at-the-money options). Since
    the at-the-money options are mostly sensitive to price variations around
    zero, they do not provide a good measure of the perceived risks of large
    moves. The curve at the top called, “4% OTM [out-of-money] options,”
    corresponds to put (respectively, call) options that become valuable only
    when the price has decreased (respectively, increased) by at least 4%.
    autopsy of major crashes 235
    850102 850506
    At-the-Money Options
    4% OTM Options
    850906 851224 860415 860801 861118 870312 870630 871016
    1 2 5 6 7 8
    October 1-16, 1987: Hourly Data
    9 12 13 14 15 16
    Fig. 7.3. Percentage deviation �C ? P�/P of call from put prices (skewness premium)
    for options at-the-money and 4% out-of-the-money, over 1985–87. The percentage
    deviation �C ? P�/P is a measure of the asymmetry between the perceived
    distribution of future large upward moves compared to large downward moves of the
    S&P 500 index. Deviations above (below) 0% indicate optimism (fear) for a bullish
    market (of large potential drops). The inset shows the same quantity �C ?P�/P calculated
    hourly during October 1987 prior to the crash: ironically, the market forgot
    its “fears” close to the crash. Reproduced from [34].
    Such put and call options thus sample the perceived tails of the potential
    distribution of price variations. Figure 7.3 shows that, most of the
    time in the 1985–87 period, call options were more expansive than put
    options, corresponding to an optimistic view of the market felt to be
    oriented positively, with small risks of a market drop. However, stronger
    and stronger bursts of “fear” can be observed, first at the end of 1985,
    then in November 1986, and finally in August 1987. These bursts of fear
    correspond to a very significant overpricing of the put options (negative
    spikes on Figure 7.3), quantifying a perceived risk of a probably significant
    drop of the market. Notice a contraction of the time intervals
    between the spikes of “fear,” reminiscent of the log-periodic acceleration
    236 chapter 7
    towards a critical point tc (see the section titled “Nonparametric Test of
    Log-Periodicity” later in this chapter and the section titled “The Shank’s
    Transformation on a Hierarchy of Characteristic Times” in chapter 9).
    Quantitatively, however, the contraction of the time intervals between the
    spikes is not sufficiently fast to converge to a date close to the crash time
    and overshoots it by about a year and a half. Bates noted that his results
    are fully consistent with the model of rational expectation bubbles (see
    chapter 5) with an explosive divergence away from the fundamentals
    which is sustained by an expected sudden drop [34].
    Aftershock Patterns
    If the concept of a crash as a kind of critical point has any value, we
    should be able to identify post-crash signatures of the underlying cooperativity.
    In fact, we should expect an at least qualitative symmetry between
    patterns before and after the crash. In other words, we should be able
    to document the existence of a critical exponent as well as log-periodic
    oscillations on relevant quantities after the crash. Such a signature in
    the volatility of the S&P 500 index, implied from the price of S&P 500
    options (which are derivative assets with price varying as a function of
    the price of the S&P 500), can indeed be seen in Figure 7.4.
    The term “implied volatility” has the following meaning. First, one
    must recall what an option is: this financial instrument is nothing but
    an insurance that can be bought or sold on the market to insure oneself
    against unpleasant price variations. The price of an option on the S&P
    500 index is therefore a function of the volatility of the S&P 500. The
    more volatile and the more risky is the S&P 500, the more expensive is
    the option. In other words, the price of an option on the market reflects
    the value of the variance of the stock as estimated by the market with
    its offer-and-demand rules. In practice, it is very difficult to have a good
    model for market price volatilities or even to measure it reliably. The
    standard procedure is then to see what the market forces decide for the
    option price and then determine the implied volatility by inversion of
    the Black and Scholes formula for option pricing [294]. Basically, the
    implied volatility is a measure of the market risks perceived by investors.
    Figure 7.4 presents the time evolution of the implied volatility of the
    S&P 500, taken from [84]. The perceived market risk is small prior to the
    crash, jumps up abruptly at the time of the crash, and then decays slowly
    over several months. This decay to “normal times” of perceived risks
    is compatible with a slow power law decay decorated by log-periodic
    autopsy of major crashes 237
    87.6 88.0 88.2 88.4
    σ2 (S&P 500)
    Time (year)
    87.8 88.6 88.8
    Fig. 7.4. Time evolution of the implied volatility of the S&P 500 index (in logarithmic
    scale) after the October 1987 crash, taken from [84]. The + represent an
    exponential decrease with var�Fexp� ≈ 15. The best fit to a power law, represented
    by the monotonic line, gives A1
    ≈ 3�9, B1
    ≈ 0�6, tc
    = 87�75, m1
    ≈ ?1�5, and
    ≈ 12. The best fit to expression (15) with tc
    ? t replaced by t ? tc gives
    ≈ 3�4, B2
    ≈ 0�9, tc
    ≈ 87�77, C ≈ 0�3, � ≈ 11, m2
    ≈ ?1�2, and varlp
    ≈ 7. One
    can observe six well-defined oscillations fitted by (15). Reproduced from [401].
    oscillations, which can be fitted by expression (15) on page 232 with
    ? t (before the crash) replaced by t ? tc (after the crash). Our analysis
    of expression (15) with tc
    ?t replaced by t ?tc again gives an estimation
    of the position of the critical time tc, which is found correctly within a
    few days. Note the long time scale covering a period of the order of a
    year involved in the relaxation of the volatility after the crash to a level
    comparable to the one before the crash. This implies the existence of
    a “memory effect”: market participants remain nervous for quite a long
    time after the crash, after being burned out by the dramatic event.
    It is also noteworthy that the S&P 500 index as well as other markets
    worldwide have remained close to the after-crash level for a long
    time. For instance, by February 29, 1988, the world index stood at 72�7
    (reference 100 on September 30, 1987). Thus, the price level established
    in the October crash seems to have been a virtually unbiased estimate of
    238 chapter 7
    S&P 500
    Time (year)
    87.80 87.82 87.84 87.86 87.88 87.90
    Fig. 7.5. Time evolution of the S&P 500 index over a time window of a few weeks
    after the October 19, 1987 crash. The fit with an exponentially decaying sinusoidal
    function shown in the continuous line suggests that a good model for the short-time
    response of the U.S. market is a single dissipative harmonic oscillator or damped
    pendulum. Reproduced from [401].
    the average price level over the subsequent months (see also Figure 7.5).
    Note also that the present value of the S&P 500 index is much larger
    than it was even before the October 1987 crash, showing again that nothing
    fundamental happened then. All this is in support of the idea of a
    critical point, according to which the event is an intrinsic signature of a
    self-organization of the markets worldwide.
    There is another striking signature of the cooperative behavior of the
    U.S. market, found by analyzing the time evolution of the S&P 500 index
    over a time window of a few weeks after the October 19, 1987 crash.
    A fit shown in Figure 7.5 with an exponentially decaying sinusoidal
    function suggests that the U.S. market behaved, for a few weeks after
    the crash, as a single dissipative harmonic oscillator, with a characteristic
    decay time of about one week equal to the period of the oscillations.
    In other words, the price followed the trajectory of a pendulum moving
    back and forth with damped oscillations around an equilibrium position.
    autopsy of major crashes 239
    This signature strengthens the view of a market as a cooperative selforganizing
    system. The basic story suggested by these figures is the
    following. Before the crash, imitation and speculation were rampant and
    led to a progressive “aggregation” of the multitude of agents into a large
    effective “superagent,” as illustrated in Figures 7.1 and 7.2; right after
    the crash, the market behaved as a single superagent, rapidly finding the
    equilibrium price through a return to equilibrium, as shown in Figure 7.5.
    On longer time scales, the superagent progressively was fragmented and
    diversity of behavior was rejuvenated, as seen in Figure 7.4.
    The crash of October 1929 is the other major historical market event of
    the twentieth century on the U.S. market. Notwithstanding the differences
    in technologies and the absence of computers and other modern means
    of information transfer, the October 1929 crash exhibits many similarities
    with the October 1987 crash—so much so, as shown in Figures 7.6 and
    7.7, that one wonders about the similitudes: what has remained unchanged
    over the history of mankind is the interplay between the human craving
    for exchanges and profits, and our fear of uncertainty and losses.
    27 27.5 28 28.5 29 29.5 30
    Dow Jones
    Fig. 7.6. The DJIA prior to the October 1929 crash on Wall Street. The fit shown
    as a continuous line is the equation (15) with A2
    ≈ 571� B2
    ≈ ?267� B2C ≈
    ≈ 0�45� tc
    ≈ 1930�22�� ≈ 7�9, and � ≈ 1�0. Reproduced from [212].
    240 chapter 7
    21 24 25 26 27 28 29 30
    Log(Dow Jones)
    22 23
    Fig. 7.7. Time dependence of the logarithm of the DJIA from June 1921 to September
    1929 and best fit by the improved nonlinear log-periodic formula developed in
    [397]. The crash of October 23, 1929 corresponds to 1929�81 decimal years. The
    parameters of the fit are: r.m.s.= 0�041, tc
    = 1929�84 year, m2
    = 0�63, � = 5�0,
    �� = ?70, �t = 14 years, A2
    = 61, B2
    = ?0�56, C = 0�08. �� and �t are two
    new parameters introduced in [397]. Reproduced from [397].
    The similarity between the situations in 1929 and 1987 was in fact noticed
    at a qualitative level in an article in the Wall Street Journal on October
    19, 1987, the very morning of the day of the stock market crash (with
    a plot of stock prices in the 1920s and the 1980s). See the discussion in
    The similarity between the two crashes can be made quantitative by
    comparing the fit of the Dow Jones index with formula (15) from June
    1927 until the maximum before the crash in October 1929, as shown in
    Figure 7.6, to the corresponding fit for the October 1987 crash shown
    in Figure 7.1. Notice the similar widths of the two time windows, the
    similar acceleration and oscillatory structures, quantified by similar exponents
    m2 and log-periodic angular frequency �: m1987
    = 0�33 compared
    to m1929
    = 0�45; �1987 = 7�4 compared to �1987 = 7�9. These numerical
    values are remarkably close and can be considered equal to within their
    autopsy of major crashes 241
    Figure 7.7 for the October 1929 crash is the analog of Figure 7.2
    for the October 1987 crash. It uses the improved nonlinear log-periodic
    formula developed in [397] over a much larger time window starting in
    June 1921. Also according to this improved theoretical formulation, the
    values of the exponent m2 and of the log-periodic angular frequency �
    for the two great crashes are quite close to each other: m1929
    = 0�63 and
    = 0�68. This is in agreement with the universality of the exponent
    m2 predicted from the renormalization group theory exposed in chapter 6.
    A similar universality is also expected for the log-frequency, albeit with
    a weaker strength, as it has been shown [356] that fluctuations and noise
    will modify � differently depending on their nature. The fits indicate that
    = 5�0 and �1987
    = 8�9. These values are not unexpected and fall
    within the range found for other crashes (see below). They correspond
    to a preferred scaling ratio equal, respectively, to �1929
    = 3�5 compared
    to �1987
    = 2�0.
    The October 1929 and October 1987 crashes thus exhibit two similar
    precursory patterns on the Dow Jones index, starting, respectively, 2.5
    and 8 years before them. It is thus a striking observation that essentially
    similar crashes have punctuated the twentieth century, notwithstanding
    tremendous changes in all imaginable ways of life and work. The only
    thing that has probably changed little are the ways humans think and
    behave. The concept that emerges here is that the organization of traders
    in financial markets leads intrinsically to “systemic instabilities,” which
    probably result in a very robust way from the fundamental nature of
    human beings, including our gregarious behavior, our greediness, our
    instinctive psychology during panics and crowd behavior, and our risk
    aversion. The global behavior of the market, with its log-periodic structures
    that emerge as a result of the cooperative behavior of traders, is
    reminiscent of the process of the emergence of intelligent behavior at a
    macroscopic scale that individuals at the microscopic scale cannot perceive.
    This process has been discussed in biology, for instance in animal
    populations such as ant colonies or in connection with the emergence of
    consciousness [8].
    There are, however, some differences between the two crashes. An
    important quantitative difference between the great crash of 1929 and the
    collapse of stock prices in October 1987 was that stock price variability
    in the year following the crash was much higher in 1929 than in 1987
    [351]. This has led economists to argue that the collapse of stock prices
    in October 1929 generated significant temporary increased uncertainty
    about future income that led consumers to forego purchases of durable
    goods. Forecasters were then much more uncertain about the course of
    242 chapter 7
    future income following the stock market crash than was typical even for
    unsettled times. Contemporary observers believed that consumer uncertainty
    was an important force depressing consumption, which may have
    been an important factor in the strengthening of the great depression. The
    increase of uncertainty after the October 1987 crash has led to a smaller
    effect, as no depression ensued. However, Figure 7.4 clearly quantifies
    an increased uncertainty and risk, lasting months after the crash.
    Actually, this phenomenon, known as the “leverage effect,” is a robust
    property of markets observed for losses that are not necessarily of a crash
    amplitude: after a drop of an equity’s value, the return volatility tends to
    increase more than after a gain. In other words, negative unanticipated
    returns result in an upward revision of conditional volatility, whereas
    positive unanticipated returns appear to result in a downward revision of
    the conditional volatility [242, 160, 86, 11].
    Naively, this property seems in contradiction with the risk-driven
    model described in chapter 5 in which the price goes up because the risk
    of a crash increases. If the price goes up, the volatility should go down
    according to the “leverage effect.” Since the volatility is usually taken
    as the measure of risks, this seems in contradiction with the increasing
    crash risk driving the underlying price increase in the risk-driven model.
    Actually, this contradiction is easily resolved by noting that the risk for
    a crash to occur is very different from the risk captured by the volatility.
    The former is sensitive to the most extreme possible yet unrealized
    price fluctuation, while the latter is an average estimation of small- and
    medium-size fluctuations of prices.
    The negative correlation, quantified by the leverage effect, between
    the volatility of equity’s rate of return and the value of equity reflects a
    larger perceived risk and uncertainty after a loss and might relate to a
    fundamental psychological trait of humans. Indeed, it is well documented
    that people perform better after initial success compared to initial failure.
    Failure or events perceived as unlucky undermine confidence in people’s
    abilities and in the future [125].
    OF 1987, 1994, AND 1997
    The Hong Kong Crashes
    Hong Kong has a strong free-market attitude, characterized by very
    few restrictions on either residents or nonresidents, private persons or
    autopsy of major crashes 243
    companies, to operate, borrow, and repatriate profit and capital. This
    continued even after Hong Kong reverted to Chinese sovereignty on
    July 1, 1997 as a Special Administrative Region (SAR) of the People’s
    Republic of China, as it was promised a “high degree of autonomy” for
    at least 50 years from that date according to the terms of the Sino-British
    Joint Declaration. The SAR is ruled according to a miniconstitution,
    the Basic Law of the Hong Kong SAR. Hong Kong has no exchange
    controls, and cross-border remittances are readily permitted. These rules
    have not changed since China took over sovereignty from the U.K. Capital
    can thus flow in and out of the Hong Kong stock market in a very
    fluid manner. There are no restrictions on the conversion and remittance
    of dividends and interest. Investors bring their capital into Hong Kong
    through the open exchange market and remit it the same way.
    Accordingly, we may expect speculative behavior and crowd effects to
    be free to express themselves in their full force. Indeed, the Hong Kong
    stock market provides perhaps the best textbook-like examples of speculative
    bubbles decorated by log-periodic power law accelerations followed
    by crashes. Over just the last fifteen years, one can identify
    three major bubbles and crashes. They are indicated as I, II, and III in
    Figure 7.8.
    80 82 84 86 88 90 92 94 96 98
    Hong-Kong III
    Fig. 7.8. The Hong Kong stock market index as a function of time. Three extended
    bubbles followed by large crashes can be identified. The approximate dates of the
    crashes are October 87 (I), January 94 (II), and October 97 (III). Reproduced from
    244 chapter 7
    84.5 85 85.5 86 86.5 87 87.5
    ’Hong-Kong I’
    Best fit
    Second best fit
    Fig. 7.9. Hong Kong stock market bubble ending with the crash of October 1987.
    On October 19, 1987, the Hang Seng index closed at 3362�4. On October 26, it
    closed at 2241�7, corresponding to a loss of 33�3%. See Table 7.1 for the parameter
    values of the fit with equation (15). Note that the two fits are almost indistinguishable
    except at the very end of the bubble. Reproduced from [218].
  1. The first bubble and crash are shown in Figure 7.9 and are synchronous
    to the worldwide October 1987 crash already discussed. On October
    19, 1987, the Hang Seng index closed at 3,362.4. On October 26, it
    closed at 2,241.7, corresponding to a cumulative loss of 33�3%.
  2. The second bubble ends in early 1994 and is shown in Figure 7.10.
    The bubble ends with what we could call a “slow crash”: on February
    4, 1994, the Hang Seng index topped at 12,157.6 and, a month later
    on March 3, 1994, it closed at 9,802, corresponding to a cumulative
    loss of 19�4%. It went even further down over the next two months,
    with a close at 8,421.7 on May 9, 1994, corresponding to a cumulative
    loss since the high on February 4 of 30�7%.
  3. The third bubble, shown in Figure 7.11, ended in mid-August 1997 by
    a slow and regular decay until October 17, 1997, followed by an abrupt
    crash: the drop from 13,601 on October 17 to 9,059.9 on October
    28 corresponds to a 33�4% loss. The worst daily plunge of 10% was
    the third biggest percentage fall following the 33�3% crash in October
    1987 and the 21�75% fall after the Tiananmen Square crackdown in
    June 1989.
    autopsy of major crashes 245
    92 92.5 93 93.5 94
    ’Hong-Kong II’
    Best fit
    Fig. 7.10. Hong Kong stock market bubble ending with the crash of early 94. On
    February 4, 1994, the Hang Seng index topped at 12,157.6. A month later, on
    March 3, 1994, it closed at 9,802, corresponding to a cumulative loss of 19�4%.
    It went even further down two months later, with a close at 8,421.7 on May, 9,
    1994, corresponding to a cumulative loss since the high on February 4 of 30�7%.
    See Table 7.1 for the parameter values of the fit with equation (15) shown as the
    continuous line. Reproduced from [218].
    Table 7.1 gives the parameters of the fits with equation (15) of the
    bubble phases of the three events I, II, and III shown in Figures 7.9–
    7.11. It is quite remarkable that the three bubbles on the Hong Kong
    stock market have essentially the same log-periodic angular frequency
    � within ±15%. These values are also quite similar to what has been
    found for bubbles on the U.S. market and for the FOREX (see below).
    In particular, for the October 1997 crash on the Hong Kong market, we
    have m1987
    = 0�33 < mHK1997
    = 0�34 < m1929
    = 0�45 and �1987 = 7�4 <
    �HK1997 = 7�5 < �1929 = 7�9; the exponent m2 and the log-periodic
    angular frequency � for the October 1997 crash on the Hong Kong
    Stock Exchange are perfectly bracketed by the two main crashes on Wall
    Street! Figure 7.12 demonstrates the “universality” of the log-periodic
    component of the signals in the three bubbles preceding the three crashes
    on the Hong Kong market.
    246 chapter 7
    95 95.5 96 96.5 97 97.5 98
    Hang Seng
    ’Hong-Kong III’
    Fig. 7.11. The Hang Seng index prior to the October 1997 crash on the Hong Kong
    Stock Exchange. The index topped at 16,460.5 on August 11, 1997. It then regularly
    decayed to 13,601 reached on October 17, 1997. It then crashed abruptly, reaching
    a close of 9,059.9 on October 28, 1997, with an intraday low of 8,775.9. The
    amplitude of the total cumulative loss since the high on August 11 is 45%. The
    amplitude of the crash from October 17 to October 28 is 33�4%. The fit, shown
    as the solid line, is equation (15) with A2
    ≈ 20077, B2
    ≈ ?8241, C ≈ ?397,
    ≈ 0�34, tc
    ≈ 1997�74, � ≈ 7�5, and � ≈ 0�78. Reproduced from [212] and
    The Crash of October 1997 and Its Resonance
    on the U.S. Market
    The Hong Kong market crash of October 1997 has been presented as a
    textbook example of contagion and speculation taking a course of their
    own. When Malaysian Prime Minister Dr. Mahathir Mohamad made his
    now-famous address to the World Bank International Monetary Fund
    seminar in Hong Kong in September 1997, many critics pooh-poohed his
    proposal to ban currency speculation as an attempt to hide the fact that
    Malaysia’s economic fundamentals were weak. They pointed to the fact
    that the currency turmoil had not affected Hong Kong, whose economy
    was basically sound. Thus, if Malaysia and other countries were affected,
    that’s because their economies were weak. At that time, it was easy to
    point out the deficits in the then-current accounts of Thailand, Malaysia,
    and Indonesia. In contrast, Hong Kong had a good current account
    situation and, moreover, had solid foreign reserves worth U.S.$88 bilTable
    Stock market A2 B2 B2C m2 tc � �
    Hong Kong I 5523� 4533 ?3247�?2304 171�?174 0�29� 0�39 87�84� 87�78 5�6� 5�2 ?1�6� 1�1
    Hong Kong II 21121 ?15113 ?429 0�12 94�02 6�3 ?0�6
    Hong Kong III 20077 ?8241 ?397 0�34 97�74 7�5 0�8
    Fit parameters of the three speculative bubbles on the Hong Kong stock market shown in Figures 7.9–7.11 leading to a large crash. Multiple entries correspond
    to the two best fits. Reproduced from [218].
    248 chapter 7
    0 1 2 3 4 5 6 7 8
    Spectral Weight
    Fig. 7.12. The Lomb spectral analysis of the three bubbles preceding the three
    crashes on the Hong Kong market shown in Figures 7.9–7.11. See the section titled
    “Nonparametric Test of Log-Periodicity” later in this chapter. All three bubbles are
    characterized by almost the same “universal” log-frequency f ≈ 1 corresponding to
    a preferred scaling ratio of the discrete scale invariance equal to � = exp�1/f � ≈
    2�7. Courtesy of A. Johansen.
    lion. This theory of the strong-won’t-be-affected had already suffered a
    setback when the Taiwan currency’s peg to the U.S. dollar had to be
    removed after Taiwanese authorities spent U.S.$5 billion to defend their
    currency from speculative attacks, and then gave up. The coup de grace
    came with the meltdown in Hong Kong in October 1997, which shocked
    analysts and the media, as this high-flying market was considered the
    safest haven in Asia. Notwithstanding the meltdown in Asia’s lesser
    markets, as country after country, led by Thailand in July 1997, succumbed
    to economic and currency problems, Hong Kong was supposed
    to be different. With its Western-style markets, the second largest in Asia
    after Japan, it was thought to be immune to the financial flu that had
    swept through the rest of the continent. It is clear from our analysis in
    chapters 4 and 5 and from the lessons of the two previous bubbles ending
    in October 1987 and in early 1994 that those assumptions naively overlooked
    the contagion, leading to overinvestments in the build-up period
    preceding the crash and resulting instability, which left the Hong Kong
    market vulnerable to so-called speculative attacks. Actually, hedge funds
    in particular are known to have taken positions consistent with a possible
    autopsy of major crashes 249
    crisis on the currency and on the stock market, by “shorting” (selling)
    the currency to drive it down, forcing the Hong Kong government to
    raise interest rates to defend it by increasing the currency liquidity, but
    as a consequence making equities suffer and making the stock market
    more unstable.
    As we have already emphasized, one should not confuse the “local”
    cause with the fundamental cause of the instability. As the late George
    Stigler—Nobel laureate economist from the University of Chicago—
    once put it, to blame “the markets” for an outcome we don’t like is like
    blaming restaurants waiters for obesity. Within the framework defended
    in this book, crashes occur as possible (but not necessary) outcomes of
    long preparation, which we term “herding,” which pushes the market
    into increasingly unstable regimes. When in this state, there are many
    possible “local” causes that may cause it to stumble. Pushing the argument
    to the extreme to make it crystal clear, let us compare this to laying
    responsibility for the collapse of the infamous Tacoma Narrows Bridge
    that once connected mainland Washington with the Olympic peninsula
    on strong wind. It is true that, on November 7, 1940, at approximately
    11:00 a.m., the bridge suddenly collapsed after developing a remarkably
    “ordered” sway in response to a strong wind [418] after it had been
    open to traffic for only a few months. However, the strong wind of that
    day was only the “local” cause, while there was a more fundamental
    cause: The bridge, like most objects, has a small number of characteristic
    vibration frequencies, and one day the wind was exactly the strength
    needed to excite one of them. The bridge responded by vibrating at
    this characteristic frequency so strongly, that is, by “resonating,” that it
    fractured the supports holding it together. The fundamental cause of the
    collapse of the Tacoma Narrows Bridge thus lies in an error of conception
    that enhanced the role of one specific mode of resonance. In sum,
    the collapse of the Tacoma Narrows Bridge as well as that of many stock
    markets during crashes, is the result of built-in or acquired instabilities.
    These instabilities are in turn revealed by “small” perturbations that lead
    directly to the collapse.
    The speculative attacks in periods of market instabilities are sometimes
    pointed to as possible causes of serious potential hazards for developing
    countries when allowing the global financial markets to have free
    play, especially when these countries come under pressure to open up
    their financial sectors to large foreign banks, insurance companies, stockbroking
    firms, and other institutions, under the World Trade Organization’s
    financial services negotiations. We argue that the problem comes
    in fact fundamentally from the overenthusiastic initial influx of capital as
    250 chapter 7
    a result of herding, which initially profits the country, but carries the risk
    of future instabilities: developing countries as well as investors “can’t
    have their cake and eat it too!” From an efficient market viewpoint, the
    speculative attacks are nothing but the revelation of the instability and
    the means by which markets are forced back to a more stable dynamical
    Interestingly, the October 1997 crash on the Hong Kong market caused
    important echos in other markets worldwide, and in particular in the
    U.S. markets. The story is often told as if a “wave of selling,” starting
    in Hong Kong, spread first to other southeast Asian markets based on
    negative sentiment—which served to reaffirm the deep financial problems
    of the Asian “tiger” nations—then to the European markets, and
    finally to the U.S. market. The shares that were hardest hit in Western
    markets were the multinational companies that receive part of their earnings
    from the southeast Asian region. The reason for their devaluation is
    that the region’s economic slowdown would lower corporate profits. It is
    estimated that the 25 companies that make up one-third of Wall Street’s
    S&P 500 index of market capitalization earn roughly half of their income
    from non-U.S. sources. Lower growth in southeast Asia heightened one
    of the biggest concerns of Wall Street investors. To carry on the thenpresent
    bull run, the market needed sustained corporate earnings; if they
    were not forthcoming, the cycle of rising share prices would whither into
    one of falling share prices. Concern over earnings might have proved to
    be the straw that broke Wall Street’s six-year bull run.
    Fingerprints of herding and of incoming instability were detected by
    several groups independently and announced publicly. According to our
    theory, the turmoil on the U.S. financial market in October 1997 should
    not be seen only as a passive reaction to the Hong Kong crash. The logperiodic
    power law signature observed on the U.S. market over several
    years before October 1997 (see Figure 7.13) indicates that a similar herding
    instability was also developing simultaneously. In fact, the detection
    of log-periodic structures and a prediction of a stock market correction
    or a crash at the end of October 1997 was formally issued jointly ex ante
    on September 17, 1997 by A. Johansen and the current author, to the
    French office for the protection of proprietary softwares and inventions,
    with registration number 94781. In addition, a trading strategy was been
    devised using put options in order to provide an experimental test of
    the theory. A 400% profit had been obtained in a two-week period covering
    the minicrash of October 28, 1997. The proof of this profit is
    available from a Merrill Lynch client cash management account released
    in November 1997. Using a variation [435] of our theory, which turns
    autopsy of major crashes 251
    out to be slightly less reliable (see comparative tests in [214]), a group
    of physicists and economists also made a public announcement published
    on September 18, 1997 in a Belgian journal [115] and communicated
    their methodology in a scientific publication afterwards [433]. Two
    other groups have also analyzed, after the fact, the possibility of having
    predicted this event. Feigenbaum and Freund analyzed the log-periodic
    oscillations in the S&P 500 and the NYSE in relation to the October 27
    “correction” seen on Wall Street [129]. Gluzman and Yukalov proposed a
    new approach based on the algebraic self-similar renormalization group
    to analyze the time series corresponding to the October 1929 and 1987
    crashes and the October 1997 correction of the NYSE [161].
    The prices of stocks and their convertible bonds also gave a clear signal
    of the market reversal and of the minimum range of the stock price
    change during the Hong Kong stock market bubble of 1997 and its subsequent
    crash [82]. Recall that convertible bonds are debt instruments
    that can be converted into equities at a certain price, which is called the
    conversion price. A convertible bond is essentially a bond plus a call
    (buy) option on the equity. Because of the call option on the equity, convertible
    bonds usually pay lower coupon than the straight bonds. When
    the share price trades below the conversion price, the call option has very
    little value and the convertible bond behaves mostly like a straight bond.
    When the share prices trade higher than the conversion price, the convertible
    bond behaves more and more like an equity because the possibility
    of conversion is very high. For most convertible bonds, the issuers can
    call back the bonds and force the conversion when the underlying stocks
    reach a certain price, which is called the call price. So a convertible
    bond is a hybrid of debt and equity. Since a convertible bond contains a
    call option on the equity and the value of an option is always positive, a
    convertible bond should always trade at a premium over the share price;
    that is, the price of the convertible bond should always be higher than
    the corresponding share price. If a convertible is traded at a discount, this
    usually indicates that either there are some restrictions on the convertible
    bonds that reduce their values or some additional information has been
    revealed by this pricing anomaly, which is the effect documented for the
    end of the Hong Kong bubble [82]. There is thus additional information
    to be found in the relationship between underlying stocks and their
    derivatives during market bubbles.
    The best fit of the logarithm of the S&P 500 index from January 1991
    until September 4, 1997 by the improved nonlinear log-periodic formula
    developed in [397], already used in Figures 7.2 and 7.7, is shown in
    Figure 7.13. This result and many other analyses led to the prediction
    252 chapter 7
    91 92 93 94 95 96 97 98
    Logarithm(S&P 500 Index)
    Time (years)
    Fig. 7.13. The best fit shown as the smooth continuous line of the logarithm of
    the S&P 500 index from January 1991 until September 4, 1997 (1997.678) by
    the improved nonlinear log-periodic formula developed in [397], already used in
    Figures 7.2 and 7.7. The exponent m2 and log-periodic angular frequency � are,
    respectively, m2
    = 0�73 (compared to 0�63 for October 1929 and 0�33 for October
  1. and � = 8�93 (compared to 5�0 for October 1929 and 7�4 for October 1987).
    The critical time predicted by this fit is tc
    = 1997�948, that is, mid-December 1997.
    Courtesy of A. Johansen.
    alluded to above, which will be further discussed in chapter 9. It turned
    out that the crash did not really occur. What happened was that the
    Dow plunged 554.26 points, finishing the day down 7.2%, and Nasdaq
    posted its biggest-ever (up to that time) one-day point loss. In accordance
    with a new rule passed after the October 1987 Black Monday, trading
    was halted on all major U.S. exchanges. Private communications from
    professional traders to the author indicate that many believed that a crash
    was coming, but this turns out to be incorrect. This sentiment must also
    be put into the perspective of the earlier sell-off at the beginning of the
    month triggered by Greenspan’s statement that the boom in the U.S.
    economy was unsustainable and that the current rate of gains in the stock
    market was unrealistic.
    autopsy of major crashes 253
    It is actually interesting that the critical time tc identified around this
    data (see chapter 9) indicated a change of regime rather than a real
    crash: after this turbulence, the U.S. market remained more or less flat,
    thus breaking the previous bullish regime, with large volatility until the
    end of January 1998, and then started a new bull phase that was later
    stopped in its course in August 1998, which we shall analyze below. The
    observation of a change of regime after tc is in full agreement with the
    rational expectation model of a bubble and crash described in chapter 5:
    the bubble expands, the market believes that a crash may be increasingly
    probable, the prices develop characteristic structures of speculation and
    herding, but the critical time passes without the crash happening. This
    can be interpreted as the nonzero probability scenario also predicted
    by the rational expectation model of a bubble and crash described in
    chapter 5, that it is possible that no crash occurs over the whole lifetime
    of the bubble including tc.
    What could be additional reasons for the abortion of the crash predicted
    in October 1997 on the U.S. market? One origin may be found in
    the behavior of household investors. U.S. households own the majority of
    the mutual fund industry, with an ownership of $2.626 trillion, or 74.2%
    of the $3.539 trillion of mutual fund assets (value at the end of 1996),
    while banks and individuals serving as trustees, guardians, or administrators
    and other institutional investors hold the remaining $913 billion, or
    25.8%. As shown in Figure 7.14, the purchase of equities by households
    has evolved over the last decade by being more and more concentrated
    on mutual funds. An analysis of the Investment Company Institute covering
    more than 50 years, including fourteen major market contractions
    and several sharp market sell-offs, found no historical evidence of mass
    redemptions from stock mutual funds during U.S. stock market contractions.
    Even the severe market break of October 19, 1987 failed to trigger
    substantial outflows from mutual funds. This analysis is consistent with
    evidence from shareholder surveys suggesting that mutual fund owners
    have a long-term investment horizon and basic understanding of risk.
    Thus a larger share of the market by these long-term horizon investors
    provides more stability and less reactivity to local turn-downs. The limited
    drop in October 1997 that stopped just short of cascading in a crash
    might be due to this stabilizing effect, which was stronger in 1997 than
    in 1987 as a result of the larger market share owned by households.
    The simultaneity of the critical times tc of the Hong Kong crash and
    of the end of the U.S. and European speculative bubble phases at the
    end of October 1997 are neither a lucky occurrence nor a signature of
    a causal impact of one market (Hong Kong) onto others, as has often
    254 chapter 7
    Purchases of Equities by Households
    (billions of dollars)
    1986 1987 1988
    Direct Purchases
    Purchases Made through Mutual Funds
    1985 1989 1990 1992 1993 1994 1995 1996
    Fig. 7.14. Reproduced from a report of the Investment Company Institute based on
    sources from the Federal Reserve Board, the Employee Benefit Research Institute,
    and the Investment Company Institute (http://www.ici.org/). The ICI is the national
    association of the American investment company industry. Founded in 1940, its 2001
    membership includes 8,414 mutual funds, 489 closed-end funds, and 8 sponsors
    of unit investment trusts. Its mutual fund members represent more than 83 million
    individual shareholders and manage approximately $7 trillion. The negative “Direct
    Purchases” correspond to sales.
    been discussed too naively. This simultaneity can actually be predicted
    in a model of rational expectation bubbles allowing the coupling and
    interactions between stock markets. For general interactions, if a critical
    time appears in one market, it should also be present in other markets as
    a result of the nonlinear interactions existing between the markets [219].
    This will be discussed further in chapter 10 in relation to the interaction
    between the world population, its global economic output, and global
    market indices.
    In sum, two lessons can be taken home from the Hong Kong October
    1997 crash: the trend-setting power of the “global village” and the
    might of the general investor sentiment forged by forces of imitation and
    Currencies can also develop bubbles and crashes. The bubble on the
    dollar starting in the early 1980s and ending in 1985 is a remarkable
    example, as shown in Figure 7.15.
    autopsy of major crashes 255
    83.0 83.5 84.0 84.5 85.0 85.5
    Exchange Rate
    Fig. 7.15. The U.S. dollar expressed in German Mark DEM (top curve) and in
    Swiss franc CHF (bottom curve) prior to its collapse on mid-1985. The fit to the
    DEM currency against the U.S. dollar with equation (15) is shown as the continuous
    and smooth line and gives A2
    ≈ 3�88, B2
    ≈ ?1�2, B2C ≈ 0�08, m2
    ≈ 0�28, tc

    1985�20, � ≈ 6�0, and � ≈ ?1�2. The fit to the Swiss franc against the U.S.
    dollar with equation (15) gives A2
    ≈ 3�1, B2
    ≈ ?0�86, B2C ≈ 0�05, m2
    ≈ 0�36,
    ≈ 1985�19, � ≈ 5�2, and � ≈ ?0�59. Note the small fluctuations in the value of
    the scaling ratio 2�2 ≤ � ≤ 2�7, which constitute one of the key tests of our “critical
    herding” theory. Reproduced from [212].
    To understand what happened, we need to retrace a piece of exchange
    rate history. In 1975, the U.S. Treasury Secretary informed the International
    Monetary Fund annual meeting that “We strongly believe that
    countries must be free to choose their own exchange rate system.” Both
    these developments were the successful culmination of “campaigns” led
    by the economist Milton Friedman during the previous quarter-century.
    Friedman’s case for flexible exchange rates was transformed from heresy
    to majority academic recommendation and from there (via two U.S. treasury
    secretaries) to become the cornerstone of the post-1973 international
    “monetary order” [261]. As flexible exchange rates were legitimized, several
    leading countries began to experiment with monetary targeting, with
    the idea that a flexible exchange rate is a precondition for independent
    256 chapter 7
    national monetary policy. This was the death of the previous 1944 Bretton
    Woods agreement, designed to provide postwar international stability
    to facilitate the approach towards both free trade and full employment.
    It turned out that fixed-exchange rates led to numerous crises and problems:
    indeed, the whole point of going from a world fixed-exchange rate
    to floating exchanges between local currencies was to give governments
    the ability to have independent monetary policies so they could fight their
    local recessions when necessary. The flexibility to develop an independent
    monetary policy thus gives a country an essential additional degree
    of freedom to stabilize its economy. However, a country cannot simultaneously
    print money to fight a recession and maintain the value of its
    currency on the foreign exchange market. A country can also improve
    its competitive position by devaluing. But hints that a devaluation might
    be looming can cause massive speculation against the vulnerable currency,
    as we shall discuss in chapter 8. See also [248] for an eye-opening
    description of the conundrums of monetary policies.
    With the end of Bretton Woods in the early 1970s, the market for foreign
    currency grew rapidly in both size and instability. The liberalization
    of capital flows that followed the adoption of floating-exchange rates
    brought vastly larger flows of capital between nations. The first naive
    presumption is that the exchange rate between two currencies, say the
    U.S. dollar and the European euro (since January 1999), would be determined
    by the needs of trade: by North Americans trading with Europeans
    for euros in order to buy European goods, and conversely. However,
    there is another important population, the investors: people who are buying
    and selling currencies in order to purchase stocks and bonds in the
    U.S. and/or the European markets. Since these investment demands are
    highly variable, including a fluctuating component of speculation, currency
    values prove volatile and prone to the same forces as described in
    chapters 4 and 5 for stocks and general financial markets. Such forces
    proved to be at the origin of the speculative bubble on the dollar in the
    first half of the 1980s [340].
    The role of monetary policy allowed by the floating-exchange rate
    was particularly clear in the context of the large deficit of the U.S. federal
    budget in the early 1980s, which led to fears that inflation would go
    sky-high. According to supporters of monetary policy, the key to controlling
    inflation was that the Federal Reserve did not pump up the money
    supply too much. Indeed, by allowing a strong dollar (which slows the
    U.S. economy) and restricting the money supply, the Federal Reserve
    chopped inflation from 13.3% in 1979 to 4.4% in 1987 to about 2% at
    the end of the twentieth century. Many even believed that the value of the
    autopsy of major crashes 257
    U.S. dollar has been high because of large U.S. budget deficits. Indeed,
    the large U.S. budget deficit of the early 1980s had to be financed in
    particular by foreign investors encouraged by a high interest rate to buy
    U.S. Treasury bonds and securities. A high interest rate automatically
    makes the dollar attractive and thus in strong demand, raising it up. Statistical
    tests over several periods of whether the dollar appreciates when
    the federal budget deficit increases showed results globally counter to the
    held belief [121]. The Economic Recovery Tax Act of 1981 constituted
    the origin of the megadeficits, as it was designed to increase savings
    and investment and thus increase real economic growth; that increased
    growth would in turn offset a tax cut. It turned out that the act has not
    produced the increase in revenues necessary to reduce the budget deficit,
    in turn augmenting the foreign trade deficit linked to the federal deficit
    by high U.S. interest rates, which encourage foreign investors to buy
    U.S. securities. This is why there is general belief in the importance of a
    gradual and steady reduction of the federal deficit as the best long-term
    solution to reducing the high interest rates and the trade deficit.
    In fact, the relationship between exchange rates and economic health
    is more complex due to other factors as well as the role of investor expectations
    and anticipations. As anything in the economic sphere, exchange
    rates are first determined by the interaction of supply and demand forces.
    For example, if the prices of products increase in the United States relative
    to those in France, the value of U.S. currency should decrease.
    Indeed, if initially, a bottle of wine, which is valued at $1 in the United
    States and 1 euro in France, later costs $2 in the United States and still
    1 euro in France, the effective exchange rate $1 = 1 euro based on the
    bottle of wine taken as a reference has become $1 = 2 euros. Due to
    travel costs and other “frictions,” the adjustment of the exchange rate
    does not, however, closely follow this relationship. If the exchange rate
    remains at $1 = 1 euro for other reasons, the price increase is also felt
    in France: 1 bottle of wine = $2 = 2 euros. The French will stop buying
    any wine from the United States, as it is twice as expensive as their
    homemade brands.
    Actually, a more significant determinant of exchange rate is the
    (inflation-adjusted) real interest rate. If real interest rates increase in a
    country, the value of its currency should increase, as investors will get a
    larger return by owning the currency with the largest real interest rate.
    This currency is thus in strong demand, driving its price up. But this is
    not always the case: short-term data on exchange rates and interest rates
    during the 1980s shows a negative correlation, which probably occurred
    258 chapter 7
    because most analysts anticipated higher inflation, even though interest
    rates were relatively high [35].
    The U.S. dollar experienced an unprecedented cumulative appreciation
    against the currencies of the major industrial countries starting
    around 1980, with several consequences: loss of competitiveness, with
    important implications for domestic industries, and increase of the U.S.
    merchandise trade deficit by as much as $45 billion by the end of 1983,
    with export sales about $35 billion lower and the import bill $10 billion
    higher. For instance, in 1982, it was already expected that, through its
    effects on export and import volume, the appreciation would reduce real
    gross national product by the end of 1983 to a level 1% to 1.5% lower
    than the third-quarter 1980 preappreciation level [130]. The appreciation
    of the U.S. dollar from 1980–84 was accompanied by substantial
    decline in prices for the majority of manufactured imports from Canada,
    Germany, and Japan. However, for a substantial minority of prices, the
    imported items’ dollar prices rose absolutely and in relation to the general
    U.S. price level. The median change was a price decline of 8% for
    imports from Canada and Japan and a decrease of 28% for goods from
    Germany [133]. As a positive effect, the impact on the U.S. inflation outlook
    was to improve it very significantly. There is also evidence that the
    strong dollar in the first half of the 1980s forced increased competition
    in U.S. product markets, especially vis a vis continental Europe [240].
    As we explained in chapter 5, according to the rational expectation
    theory of speculative bubbles, prices can be driven up by an underlying
    looming risk of a strong correction or crash. Such a possibility has been
    advocated as an explanation for the strong appreciation of the U.S. dollar
    from 1980 to early 1985 [230]. If the market believes that a discrete
    event may occur when the event does not materialize for some time,
    this may have two consequences: drive price up and lead to an apparent
    inefficient predictive performance of forward exchange rates. (Forward
    and future contracts are financial instruments that closely track “spot”
    prices, as they embody the best information on the expectation of market
    participants on near-term spot price in the future.) Indeed, from October
    1979 to February 1985, forward rates systematically underpredicted the
    strength of the U.S. dollar. Two discrete events could be identified as
    governing market expectations [230]: (1) change in monetary regime in
    October 1979 and the resulting private sector doubts about the Federal
    Reserve’s commitment to lower money growth and inflation; (2) private
    sector anticipation of the dollar’s depreciation beginning in March 1985,
    that is, anticipation of a strong correction, exactly as in the bubblecrash
    model of chapter 5. The corresponding characteristic power law
    autopsy of major crashes 259
    acceleration of bubbles decorated by log-periodic oscillations is shown
    in Figure 7.15.
    Expectations of future exchange rate have been shown to be excessive
    in the posterior period from 1985.2 to 1986.4, indicating bandwagon
    effects at work and the possibility of a rational speculative bubble [278].
    As usual before a strong correction or a crash, analysts were showing
    overconfidence, and there was much reassuring talk about the absence of
    significant danger of collapse of the dollar, which had risen to unprecedented
    heights against foreign currencies [199]. In the long term, however,
    it was clear that such a strong dollar was unsustainable, and there
    were indications that the dollar was overvalued, in particular because
    foreign exchange markets generally hold that a nation’s currency can
    remain strong over the longer term only if the nation’s current account is
    healthy. By constrast, for the first half of 1984, the U.S. current account
    suffered a seasonally adjusted deficit of around $44.1 billion.
    A similar but somewhat attenuated bubble of the U.S. dollar
    expressed, respectively, in Canadian dollar and Japanese Yen, extending
    over slightly less than a year and bursting in the summer of 1998, is
    shown in Figure 7.16. Paul Krugman, a professor of economics at the
    Massachusetts Institute of Technology, has suggested that this run-up
    on the Yen and the Canadian dollar, as well as the near collapse of
    U.S. financial markets at the end of the summer of 1998, which is
    discussed in the next section, are the unwanted “byproduct of a vast
    get-richer-quick scheme by a handful of shadowy financial operators”
    which backfired [246]. The remarkable quality of the fits of the data
    with our theory does indeed give credence to the role of speculation,
    imitation, and herding, be them spontaneous, self-organized, or manipulated
    in part. Actually, Frankel and Froot have found that, over the
    period 1981–85, the market shifted away from the fundamentalists and
    toward the chartists [139, 140].
    From its top in mid-June 1998 (1998�55) to its bottom in the first days
    of September 1998 (1998�67), the U.S. S&P 500 stock market lost 19%.
    This “slow” crash, and in particular the turbulent behavior of stock markets
    worldwide starting in mid-August, are widely associated with and
    even attributed to the plunge of the Russian financial markets, the devaluation
    of its currency, and the default of the government on its debt
    260 chapter 7
    96.8 97.0 97.2 97.4 97.6 97.8 98.0 98.2 98.4 98.6 98.8
    Price Ratio of Yen to US$
    Price Ratio of CAN$ to US$
    Fig. 7.16. The U.S. dollar expressed in Canadian dollars and Yen currencies prior to
    its drop starting in August 1998. The fit with equation (15) to the two exchange rates
    gives A2
    ≈ 1�62, B2
    ≈ ?0�22, B2C ≈ ?0�011, m2
    ≈ 0�26, tc
    ≈ 98�66, � ≈ ?0�79,
    � ≈ 8�2 and A2
    ≈ 207, B2
    ≈ ?85, B2C ≈ 2�8, m2
    ≈ 0�19, tc
    ≈ 98�78, � ≈ ?1�4,
    � ≈ 7�2, respectively. Reproduced from [221].
    obligations (see chapter 8 for information and analysis of other crises on
    the Russian market).
    The analysis presented in Figure 7.17 suggests a different story: the
    Russian event may have been the triggering factor, but not the fundamental
    cause! One can observe clear fingerprints of a kind of speculative
    herding, starting more than three years before, with its characteristic
    power law acceleration decorated by log-periodic oscillations. Table 7.2
    gives a summary of the parameters of the log-periodic power law fit
    to the main bubbles and crashes discussed until now. The crash of
    August 1998 is seen to fit nicely in the family of crashes with “herding”
    This indicates that the stock market was again developing an unstable
    bubble which would have culminated at some critical time tc
    ≈ 1998�72,
    close to the end of September 1998. According to the rational expectation
    bubble models of chapter 5, the probability for a strong correction or
    a crash was increasing as tc was approached, with a rising susceptibility
    to “external” perturbations, such as news or financial difficulties occurautopsy
    of major crashes 261
    95.0 95.5 96.0 96.5 97.0 97.5 98.0 98.5
    S&P 500
    Hang Seng
    Fig. 7.17. The Hang Seng index prior to the October 1997 crash on the Hong Kong
    Stock Exchange already shown in Figure 7.11 and the S&P 500 stock market index
    prior to the crash on Wall Street in August 1998. The fit to the S&P 500 index is
    equation (15) with A2
    ≈ 1321, B2
    ≈ ?402, B2C ≈ 19�7, m2
    ≈ 0�60, tc
    ≈ 98�72,
    � ≈ 0�75, and � ≈ 6�4. Reproduced from [221].
    ring somewhere in the “global village.” The Russian meltdown was just
    such a perturbation. What is remarkable is that the U.S. market somehow
    contained the information of an upcoming instability through its unsustainable
    accelerated growth and structures! The financial world being an
    extremely complex system of interacting components, it is not farfetched
    to imagine that Russia was led to take actions against its unsustainable
    debt policy at the time of a strongly increasing concern by many about
    risks on investments made in developing countries. This concept is further
    developed in the section on the Russian crashes in chapter 8.
    The strong correction starting in mid-August was not specific to the
    U.S. markets. Actually, it was much stronger in some other markets, such
    as the German market. Indeed, within the period of only nine months
    preceding July 1998, the German DAX index went up from about 3�700
    to almost 6�200 and then quickly declined over less than one month to
    below 4�000. Precursory log-periodic structures have been documented
    for this event over the nine months preceding July 1998 [111], with the
    addition that analogous log-periodic oscillations also occurred on smaller
    Table 7.2
    Crash tc tmax tmin drop m2 � � A2 B2 B2C Var
    1929 (WS) 30.22 29.65 29.87 47% 0.45 7.9 2.2 571 ?267 14�3 56
    1985 (DEM) 85.20 85.15 85.30 14% 0.28 6.0 2.8 3�88 ?1�16 0�08 0�0028
    1985 (CHF) 85.19 85.18 85.30 15% 0.36 5.2 3.4 3�10 ?0�86 +0�055 0�0012
    1987 (WS) 87.74 87.65 87.80 30% 0.33 7.4 2.3 411 ?165 12�2 36
    1997 (HK) 97.74 97.60 97.82 46% 0.34 7.5 2.3 20077 ?8241 ?397 190360
    1998 (WS) 98.72 98.55 98.67 19% 0.60 6.4 2.7 1321 ?402 19�7 375
    1998 (YEN) 98.78 98.61 98.77 21% 0.19 7.2 2.4 207 ?84�5 2�78 17
    1998 (CAN$) 98.66 98.66 98.71 5.1% 0.26 8.2 2.2 1�62 ?0�23 ?0�011 0�00024
    1999 (IBM) 99.56 99.53 99.81 34% 0.24 5.2 3.4
    2000 (P&G) 00.04 00.04 00.19 54% 0.35 6.6 2.6
    2000 (Nasdaq) 00.34 00.22 00.29 37% 0.27 7.0 2.4
    Summary of the parameters of the log-periodic power law fit to the main bubbles and crashes discussed in this chapter (see Figures 7.22, 7.23, and 7.24 for
    the April 2000 crash on the Nasdaq and the two crashes on IBM and on Procter & Gamble). tc is the critical time predicted from the fit of each financial time
    series to the equation (15) on page 232. The other parameters of the fit are also shown. � = exp� 2�
    � � is the preferred scaling ratio of the log-periodic oscillations.
    The error Var is the variance between the data and the fit and has units of price × prices. Each fit is performed up to the time tmax at which the market index
    achieved its highest maximum before the crash. tmin is the time of the lowest point of the market after the crash, disregarding smaller “plateaus.” The percentage
    drop is calculated as the total loss from tmax to tmin. Reproduced from [221].
    autopsy of major crashes 263
    time scales as precursors of smaller intermediate decreases, with similar
    preferred scaling ratio � at the various levels of resolution. However,
    the reliability of these observations at smaller time scales established
    by visual inspection in [111] remain to be established with rigorous
    statistical tests.
    Until now, the evidence presented in support of the “critical crash” concept
    is based on so-called parametric fits of financial prices with the
    formula of the power law decorated by log-periodic oscillations. Fitting
    data with sufficiently complex formulas with a rather large number
    of adjustable parameters is a delicate problem. In particular, one could
    question the explanatory power of a formula with too many parameters.
    The following sentence, often attributed to the famous Italian physicist
    Enrico Fermi, epitomizes (actually, exaggerates) the problem: “Give me
    five parameters and I will describe an elephant.” In order to address
    this possible criticism, we have emphasized the remarkable robustness
    and quasi-universality of the two key meaningful parameters across the
    10 crashes analyzed so far, the exponent m2 controlling the acceleration
    close to the critical time and the preferred scale ratio � quantifying the
    hierarchical organization in the time domain. If the log-periodic power
    law acceleration were the result of noise or luck, these parameters should
    vary wildly from one crash to the next.
    As we emphasized in chapter 6 and in the present chapter, the logperiodic
    component is a key signature of discrete scale invariance, taken
    as a crucial witness of the critical self-organization of financial markets.
    This suggests another non-parametric test, specifically aimed at detecting
    the log-periodic component of the financial signals. A first example is
    shown in Figure 7.18 for the October 1987 crash. A simple and robust
    method is used to quantify the amplitude of the deviation from the overall
    growth of the DJIA [434]. This deviation is then seen to be close to an
    oscillation accelerating with the approach to the critical crash time, in
    agreement with the log-periodic prediction.
    Another formulation of the same idea has been developed to quantify
    in addition the statistical significance of the putative log-periodicity
    [221]. As in Figure 7.18, the idea is first to detrend the financial time
    series to remove the trend and acceleration and keep only the noisy
    oscillatory residue shown in Figure 7.19. In the implementation shown
    264 chapter 7
    ymax - ymin
    82 83 84 85 86 87 88
    Fig. 7.18. Top panel: Evolution of the DJIA from January 1982 to August 1987;
    the continuous curve represents a fit with a pure power law with small exponent.
    Actually, Vandewalle et al. [434] use the limit of a vanishing exponent corresponding
    to a fit with a logarithmic acceleration ?ln�tc
    ? t�. This method provides inferior
    fits [214] but has the advantage of decreasing by one the number of adjustable
    parameters. The pattern in the bottom panel is a measure of the detrended oscillatory
    component. At any given time t, it is obtained by measuring the difference between
    the running maximum until time t and the running minimum from t to the end
    of the time series. The zeros of this difference correspond to new records since
    the maximum of the past is equal to the minimum of the future. The continuous
    oscillatory line is a pure log-periodic cosine cos�� ln�tc
    ? t��. Reproduced from
    [434] with permission from Elsevier Science.
    in the figures, the detrending is performed by substrating and normalizing
    by the pure power law fit. The residual is then analyzed by a
    spectral analysis as a function of the variable ln�tc
    ? t� (specifically
    here the so-called Lomb periodogram method adapted to nonequidistant
    autopsy of major crashes 265
    -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5
    log((tc - t)/tc)
    Fig. 7.19. The residual as defined by the transformation explained in the text as a
    function of log� tc
    � for the October 1987 crash. Reproduced from [221].
    sampled data points), which should give a pure angular frequency � if
    the log-periodicity was perfect. In Figure 7.20, a peak around the logfrequency
    f = �/2� ≈ 1�1 (corresponding to the angular frequency
    = 2�f ≈ 7) is obtained consistently for all eight cases reported in
    Table 7.2 and shown in previous figures (excluding the two companies
    and the Nasdaq index). This is in remarkable agreement with the results
    on � listed in Table 7.2 that were obtained by the parametric log-periodic
    power law fits. If the noise was the standard white Gaussian process,
    the confidence given by the Lomb periodograms would be well above
    99�99% for all cases shown [338]; that is, the probability that the logfrequency
    peaks observed in the bubble data could result from chance
    would be less than one in ten thousand (10?4) for each of the events;
    for ten events supposed to be independent, it would be �10?4�10 = 10?40
    or one in ten thousand billion billion billion billion! However, since the
    “noise” spectrum is unknown and very likely different for each crash,
    we cannot estimate precisely the confidence interval of the peak in the
    usual manner [338] and compare the results for the different crashes.
    Therefore, to be conservative, only the relative level of the peak for each
    separate periodogram can be taken as a measure of the significance of
    the oscillations, and the periodograms have hence been normalized. In
    all cases, the mean peak is well above the background and is consistent
    across the crashes. Therefore, this spectral analysis demonstrates that the
    266 chapter 7
    0 1 2 3 4 5 6
    Arbitrary Units
    Fig. 7.20. The Lomb periodogram for the 1929, 1987, and 1998 crashes on Wall
    Street, the 1997 crash on the Hong Kong Stock Exchange, the 1985 U.S. dollar
    currency crash in 1985 against the DM and CHF and in 1998 against the Yen and the
    5�1% correction against the Canadian dollar. For each periodogram, the significance
    of the peak should be estimated against the noise level. Reproduced from [221].
    observed log-periodic oscillations have a very strong power spectrum,
    much above noise level. It would be very difficult and much less parsimonious
    to account for these structures by another model.
    In order to investigate further the statistical significance of these results,
    fifty 400-week intervals in the period 1910 to 1996 of the Dow Jones
    average were picked at random and fitted with the log-periodic power
    law formula [209]. The approximate end-dates of the 50 data sets are
    1951, 1964, 1950, 1975, 1979, 1963, 1934, 1960, 1936, 1958, 1985,
    1884, 1967, 1943, 1991, 1982, 1972, 1928, 1932, 1946, 1934, 1963,
    1979, 1993, 1960, 1935, 1974, 1950, 1970, 1980, 1940, 1986, 1923,
    autopsy of major crashes 267
    1963, 1964, 1968, 1975, 1929, 1984, 1944, 1994, 1967, 1924, 1974,
    1954, 1956, 1959, 1926, 1947, and 1965.
    The motivation was to see whether the method has many false alarms,
    in other words, if the formula can find periods where it detects a speculative
    regime interpreted as a precursor of a critical time corresponding
    to a high probability for a strong correction or a crash, while in fact
    nothing of the sort happens. If a substantial fraction of the random intervals
    exhibit similar patterns as for the ten cases discussed above, there
    is no value in such a method with no discriminating power. On the other
    hand, if only the pre-crash periods are characterized by the log-periodic
    power law–like fits, we have a method to characterize, detect, identify,
    and maybe forecast critical times (more on this in chapters 9 and 10).
    The results reported in [209] were as follows. Out of the fifty time
    intervals, only eleven had a quality of fit comparable with that of the
    other crashes, and only six of them produced values for the exponent m2
    and log-periodic angular frequency � which were in the same range. This
    criterion embodies the expected “universality” of the critical cooperative
    regime underlying the critical point, as discussed in chapters 4–6. Among
    the six fits, five belonged to the periods prior to the crashes of 1929 and
  1. The sixth was identifying a speculative regime culminating in the
    spring of 1962, an event that we did not expect, as we were unaware
    of any crash during these times. The existence of a “crash” in 1962
    was unknown to us before these results, and the identification of this
    crash naturally strengthens the case. After this discovery, a little search
    in the history of economic booms and busts (see, for instance, [282])
    taught us that, indeed, the late 1950s and early 1960s had their “new
    industry” and “growth stocks,” with soaring stock prices ending with the
    slow 1962 crash. Growth stocks in the new electronic industry like Texas
    Instruments and Varian Associates, expected to exhibit a very fast rate
    of earning growth, were highly prized and far outdistanced the standard
    blue-chip stocks. Many companies associated with the esoteric high-tech
    of space travel and electronics sold in 1961 for over 200 times their
    previous year’s earning. Previously, the traditional rule had been that the
    price should be a multiple of 10 to 15 times their earnings. This is a
    story all too familiar! The “tronics boom,” as it was called, actually has
    remarkably similar features to the New Economy boom preceding the
    October 1929 crash or the New Economy boom of the late 1990s, ending
    in the April 2000 crash on the Nasdaq index.
    The best fit of the DJIA from 1954 to the end of 1961 by the logperiodic
    power law formula is shown in Figure 7.21. This period of
    time was followed by a “slow crash,” in the sense that the stock market
    268 chapter 7
    54 55 56 57 58 59 60 61 62 63
    Dow Jones
    Fig. 7.21. The DJIA prior to the 1962 slow “crash” on Wall Street. The fit (solid
    line) is equation (15) with A2
    ≈ 960, B2
    ≈ ?120, B2C ≈ ?14�9, m2
    ≈ 0�68,
    ≈ 1964�83, � ≈ 12�1, and � ≈ 4�1. Reproduced from [209].
    declined approximately 27% in three months, not in one or two weeks as
    for the other crashes. In terms of the rational expectation model presented
    in chapters 5 and 6, some external shock may have provoked this slow
    crash before the stock market was “ripe.” Indeed, within the rational
    expectation model, a bubble that starts to “inflate” with some theoretical
    critical time tc can be perturbed and not go to its culmination due to the
    influence of external shocks. Recall that the critical time tc of the power
    law is the time at which the crash is the most probable, but this does not
    prevent the bubble from crashing or stoping before a crash, albeit with
    a smaller probability. If this happens, as seems to have been the case in
    1962, this does not prevent the log-periodic structures from developing
    up to the time when the course of the bubble evolution is modified by
    these external shocks. These structures are the signatures of a strong
    speculative phase announcing a coming unstable phase.
    A recurring theme of this book is that bubbles and crashes result from
    speculation. The objects of speculation differ from boom to boom, as
    we have seen in the first chapters of this book, including metallic coins,
    tulips, selected companies, import commodities, country banks, foreign
    mines, building sites, agricultural and public lands, railroad shares, copper,
    silver, gold, real estate, derivatives, hedge-funds, and new industries
    [236]. The euphoria derived from the infatuation with new industries
    autopsy of major crashes 269
    especially marked the bubble preceding the great crash of October 1929
    as well as the “tronics boom” before the slow crash of 1962 or the
    Internet/Information technology (IT) boom before the Nasdaq crash of
    April 2000 discussed below. As the euphoria of a boom gives way to
    the pessimism of a bust, one ought to wonder what really happens to
    the buying plans and business projects of overextended consumers and
    In the last few years of the second millenium, there was a growing divergence
    in the stock market between New Economy and Old Economy
    stocks, between technology and almost everything else. Over 1998 and
    1999, stocks in the Standard & Poor’s technology sector rose nearly
    fourfold, while the S&P 500 index gained just 50%. And without technology,
    the benchmark would be flat. In January 2000 alone, 30% of
    net inflows into mutual funds went to science and technology funds,
    versus just 8.7% into S&P 500 index funds. As a consequence, the average
    price-over-earnings ratio (P/E) for Nasdaq companies was above 200
    (corresponding to a ridiculous earnings yield of 0�5%), a stellar value
    above anything that serious economic valuation theory would consider
    reasonable. It is worth recalling that the very same concept and wording
    of a so-called New Economy was hot in the minds and mouths of
    investors in the 1920s and in the early 1960s, as already mentioned. In
    the 1920s, the new technologies of the time were General Electric, ATT,
    and other electric and communication companies, and they also exhibited
    impressive price appreciations of the order of hundreds of percentage
    points in an eighteen-month time intervals before the 1929 crash.
    The Nasdaq composite index (see chapter 2 for definition) dropped
    precipitously, with a low of 3,227 on April 17, 2000, corresponding to a
    cumulative loss of 37% counted from its all-time high of 5,133 reached
    on March 10, 2000. The Nasdaq composite consists mainly of stock
    related to the New Economy, that is, the Internet, software, computer
    hardware, telecommunication, and so on. A main characteristic of these
    companies is that their P/Es, and even more so their price-over-dividend
    ratios, often came in three digits prior to the crash. Some companies,
    such as VA LINUX, actually had a negative earnings/share of ?1�68.
    Yet they were traded around $40 per share, which is close to the price
    of Ford in early March 2000. Opposed to this, so-called Old Economy
    companies, such as Ford, General Motors, and DaimlerChrysler, had
    270 chapter 7
    P/Es ≈ 10. The difference between Old Economy and New Economy
    stocks is thus the expectation of future earnings [395]: investors, who
    expect an enormous increase in, for example, the sale of Internet and
    computer-related products rather than in car sales, are hence more willing
    to invest in Cisco than in Ford notwithstanding the fact that the
    earning-per-share of the former is much smaller than for the latter. For a
    similar price per share (approximately $60 for Cisco and $55 for Ford),
    the earning per share is $0.37 for Cisco compared to $6.00 for Ford
    (Cisco had a total market capitalization of $395 billions [close of April,
    14, 2000] compared to $63 billion for Ford). In the standard fundamental
    valuation formula, in which the expected return of a company is the
    sum of the dividend return and of the growth rate, New Economy companies
    are supposed to compensate for their lack of present earnings
    by fantastic potential growth. In essence, this means that the bull market
    observed in the Nasdaq in 1997–2000 was fueled by expectations
    of increasing future earnings rather than economic fundamentals (and
    by the expectation that others will expect the same thing and will help
    increase the capital gains): the price-over-dividend ratio for a company
    such as Lucent Technologies with a capitalization of over $300 billion
    prior to its crash on January 5, 2000 was over 900, which means that
    you get a higher return on your checking account (!) unless the price of
    the stock increases. Opposed to this, an Old Economy company such as
    DaimlerChrysler gave a return that was more than thirty times higher.
    Nevertheless, the shares of Lucent Technologies rose by more than 40%
    during 1999, whereas the shares of DaimlerChrysler declined by more
    than 40% in the same period. The recent crashes of IBM, Lucent, and
    Procter & Gamble shown in chapter 1 correspond to a loss equivalent
    to the state budgets of many countries. This is usually attributed to a
    “business-as-usual” corporate statement of a slightly revised, smallerthan-
    expected earnings!
    These considerations make it clear that it is the expectation of future
    earnings and future capital gains rather than present economic reality
    that motivates the average investor, thus creating a speculative bubble. It
    has also been proposed [289] that better business models, the network
    effect, first-to-scale advantages, and real options effect could account for
    the apparent overvaluation, providing a sound justification for the high
    prices of dot.com and other New Economy companies. In a nutshell, the
    arguments are as follows.
  2. The better business models refer to the fact that dot.com companies
    such as Amazon require little capital investment compared to their
    autopsy of major crashes 271
    brick-and-mortar competitors. In addition, the reduced delay in receiving
    electronic payments from customers compared to sending payments to
    suppliers means that, as the business grows, it actually generates cash
    from working capital.
  3. Usually, positive feedback stems from economies of scale: the largest
    companies sustain the lowest unit costs. Economies of scale are driven
    by the “supply side,” and consequently, may run into natural limitations
    and wane at a point well below market dominance. In the Internet
    economy, in contrast, positive feedback is fueled by the network effect,
    whose fundamental principle is that a network becomes more valuable
    to each user as incremental users are added. More specifically, the value
    of the network grows exponentially as the number of members grows
    arithmetically. A network of users is very valuable and becomes more
    so as it grows over time, locking in the customer base and enhancing
    the sustainability of excess returns. As companies start to enjoy the virtuous
    cycle, their revenue growth often meaningfully outstrips their cost
  4. First-to-scale advantages describe those companies that establish user
    bases large enough to launch them into the previously described virtuous
    cycle. According to this concept, it may often make sense for companies
    to forego current profits in an effort to build their network of users. Being
    first in a given space is important, as it offers the opportunity to establish
    a brand, set industry standards, and increase switching costs.
  5. The real option effect refers to the concept that New Economy companies
    can use their already developed networks to grasp new opportunities
    as they unfold in the future. In other words, their customer network and
    their strong intellectual capital allow them to move rapidly in new markets,
    providing the potential for new gains. Their present structure thus
    gives them an “option” for the future, similar to a financial option: a
    financial option gives its owner the right, but not the obligation, to purchase
    or sell a security at a given price. Analogously, a company that
    owns a real option has the possibility, but not the obligation, to make a
    potentially value-accretive investment to enter a new market. The remarkable
    consequence is that, the larger the “volatility,” that is, the larger
    the uncertainty of future market developments, the greater is the value
    of this option, because volatility and uncertainty highlight the value of
    future opportunities. For example, Amazon’s e-commerce expertise and
    customer franchise in the book market gave it a “real option” to invest
    in the e-commerce markets for music, movies, and gifts.
    272 chapter 7
    These interesting views expounded in early 1999 were in synchrony
    with the bull market of 1999 and preceding years. They participated in
    the general optimistic view and added to the strength of the herd by a
    mechanism analogous to that exemplified in Figure 1.4. They seem less
    attractive in the context of the bearish phase of the Nasdaq market that
    has followed its crash in April 2000 and that is still running more than
    two years later. For instance, Koller and Zane [241] argued that the traditional
    triumvirate of earnings growth, inflation, and interest rates explains
    most of the growth and decay of U.S. indices (while not excluding the
    existence of a bubble of hugely capitalized new-technology companies).
    Indeed, as already emphasized in chapter 1, history provides many
    examples of bubbles, driven by unrealistic expectations of future earnings,
    followed by crashes [454, 236]. The same basic ingredients are
    found repeatedly: fueled by initially well-founded economic fundamentals,
    investors develop a self-fulfilling enthusiasm by an imitative process
    or crowd behavior that leads to an unsustainable accelerating overvaluation.
    The fundamental origin of the crashes on the U.S. markets in 1929,
    1962, 1987, 1998, and 2000 belongs to the same category, the difference
    being mainly in which sector the bubble was created: in 1929, it
    was utilities; in 1962, it was the electronic sector; in 1987, the bubble
    was supported by a general deregulation and new private investors with
    high expectations; in 1998, it was fueled by strong expectation regarding
    investment opportunities in Russia that ultimately collapsed; in 2000,
    it was powered by expectations regarding the Internet, telecommunication,
    and the rest of the New Economy sector. However, sooner or later,
    investment values always revert to a fundamental level based on real
    cash flows.
    This fact did not escape U.S. Federal Reserve chairman Alan Greenspan,
    who said: Is it possible that there is something fundamentally new about
    this current period that would warrant such complacency? Yes, it is possible.
    Markets may have become more efficient, competition is more global,
    and information technology has doubtless enhanced the stability of business
    operations. But, regrettably, history is strewn with visions of such
    “new eras” that, in the end, have proven to be a mirage. In short, history
    counsels caution [176].
    Figure 7.22 shows the logarithm of the Nasdaq composite fitted with
    the log-periodic power law equation (15) on page 232. The data interval
    to fit was identified using the same procedure as for the other crashes: the
    first point is the lowest value of the index prior to the onset of the bubble,
    and the last point is that of the all-time high of the index. There exists
    autopsy of major crashes 273
    97.5 98 98.5 99 99.5 00
    Log(Nasdaq Composite)
    Best fit
    Third best fit
    Fig. 7.22. Best (r.m.s. ≈ 0�061) and third best (r.m.s. ≈ 0�063) fits with equation
    (15) to the natural logarithm of the Nasdaq composite. The parameter values of the
    fits are A2
    ≈ 9�5, B2
    ≈ ?1�7, B2C ≈ 0�06, m2
    ≈ 0�27, tc
    ≈ 2000�33, � ≈ 7�0,
    � ≈ ?0�1 and A2
    ≈ 8�8, B2
    ≈ ?1�1, B2C ≈ 0�06, m2
    ≈ 0�39, tc
    ≈ 2000�25,
    � ≈ 6�5, � ≈ ?0�8, respectively. Reproduced from [217].
    some subtlety with respect to identifying the onset of the bubble, the
    end of the bubble being objectively defined as the date when the market
    reached its maximum. A bubble signifies an acceleration of the price. In
    the case of Nasdaq, it tripled from 1990 to 1997. However, the increase
    was a factor 4 in the three years preceding the current crash, thus defining
    an “inflection point” in the index. In general, the identification of such
    an “inflection point” is quite straightforward on the most liquid markets,
    whereas this is not always the case for the emergent markets that we
    shall discuss in chapter 8. With respect to details of the methodology of
    the fitting procedure, we refer the reader to [221].
    Undoubtedly, observers and analysts have forged post mortem stories
    linking the April 2000 crash with the effect of the crash of Microsoft Inc.,
    which resulted from the breaking off of its negotiations with the U.S.
    federal government on the antitrust issue during the weekend of April 1,
    as well as from many other factors. Here, we interpret the Nasdaq crash
    as the natural death of a speculative bubble, antitrust or not, the results
    presented here strongly suggesting that the bubble would have collapsed
    anyway. However, according to our analysis based on the probabilistic
    model of bubbles described in chapters 5 and 6, the exact timing of the
    death of the bubble is not fully deterministic and allows for stochastic
    274 chapter 7
    97.5 98.0 98.5 99.0 99.5
    Price of IBM Shares
    Fig. 7.23. Best (r.m.s. ≈ 3�7) fit, shown as solid line, with equation (15) to the price
    of IBM shares. The parameter values of the fits are A2
    ≈ 196� B2
    ≈ ?132� B2C ≈
    ≈ 0�24� tc
    ≈ 99�56�� ≈ 5�2, and � ≈ 0�1. Reproduced from [217].
    influences, but within the remarkably tight bound of about one month
    (except for the slow 1962 crash).
    Log-periodic critical signatures can also be detected on individual
    stocks, as shown in Figures 7.23 for IBM and 7.24 for Procter & Gamble.
    98.8 99.0 99.2 99.4 99.6 99.8 00 00.2
    Price of Procter & Gamble Shares
    Fig. 7.24. Best (r.m.s. ≈ 4�3) fit (solid line) with equation (15) to the price of
    Procter & Gamble shares. The parameter values of the fit are A2
    ≈ 124, B2
    ≈ ?38,
    B2C ≈ 4�8, m2
    ≈ 0�35, tc
    ≈ 2000�04, � ≈ 6�6, and � ≈ ?0�9. Reproduced from
    autopsy of major crashes 275
    These two figures extend Figures 1.7 and 1.9 of chapter 1 by offering a
    quantification of the precursory signals. The signals are more noisy than
    for large indices but are nevertheless clearly present. There is a weaker
    degree of generality for individual stocks as the valuation of a company
    is also a function of many other idiosyncratic factors associated with the
    specific course of the company. Dealing with broad market indices averages
    out all these specificities to mainly keep track of the overall market
    “sentiment” and direction. This is the main reason why the log-periodic
    power law precursors are stronger and more significant for aggregated
    financial series in comparison with individual assets. If speculation, imitation,
    and herding become at some time the strongest forces driving the
    price of an asset, we should then expect the log-periodic power law signatures
    to emerge again strongly above all the other idiosyncratic effects.
    We now summarize the evidence that imitation between traders and their
    herding behavior not only lead to speculative bubbles with accelerating
    overvaluations of financial markets possibly followed by crashes, but
    also to “antibubbles” with decelerating market devaluations following
    all-time highs [213]. There is thus a certain degree of symmetry between
    the speculative behavior of the “bull” and “bear” market regimes. This
    behavior is documented on the Japanese Nikkei stock index from January
    1, 1990 until December 31, 1998 and on gold future prices after
    1980, both after their all-time highs.
    The question we ask is whether the cooperative herding behavior of
    traders might also produce market evolutions that are symmetric to the
    accelerating speculative bubbles that often end in crashes. This symmetry
    is performed with respect to a time inversion around a critical time
    tc such that tc
    ? t for t < tc
    is changed into t ? tc for t > tc
    . This
    symmetry suggests looking at decelerating devaluations instead of accelerating
    valuations. A related observation has been reported in Figure 7.4
    in relation to the October 1987 crash showing that the implied volatility
    of traded options relaxed after the October 1987 crash to its long-term
    value, from a maximum at the time of the crash, according to a decaying
    power law with decelerating log-periodic oscillations. It is this type of
    behavior that we document now, but for real prices.
    The critical time tc then corresponds to the culmination of the market,
    with either a power law increase with accelerating log-periodic oscillations
    preceding it or a power law decrease with decelerating log-periodic
    276 chapter 7
    oscillations after it. In chapter 8 we shall show an example using the
    Russian market where both structures appear simultaneously for the same
    tc. This is, however, a rather rare occurrence, probably because accelerating
    markets with log-periodicity almost inevitably end up in a crash,
    a market rupture that thus breaks down the symmetry (tc
    ? t for t < tc
    into t ? tc for t > tc
    ). Herding behavior can occur and progressively
    weaken from a maximum in “bearish” (decreasing) market phases, even
    if the preceding “bullish” phase ending at tc was not characterized by
    a strengthening imitation. The symmetry is thus statistical or global in
    general and holds in the ensemble rather than for each single case individually.
    The “Bearish” Regime on the Nikkei Starting
    from January 1, 1990
    The most recent example of a genuine long-term depression comes from
    Japan, where the Nikkei decreased by more than 60% in the nine years
    following the all-time high of December 31, 1989. In Figure 7.25, we see
    (the logarithm of) the Nikkei from January 1, 1990 until December 31,
  6. The three fits, shown as the undulating lines, use three mathematical
    expressions of increasing sophistication: the dotted line is the simple
    log-periodic formula (15) on page 232; the continuous line is the
    improved nonlinear log-periodic formula developed in [397] and already
    used for the 1929 and 1987 crashes over eight years of data; the dashed
    line is an extension of the previous nonlinear log-periodic formula (19) on
    page 336 to the next order of description, which was developed in [213].
    This last most sophisticated mathematical formula (25) on page 339 predicts
    the transition from the log-frequency �1 close to tc to �1
  • �2 for
    T1 < � < T2 and to the log-frequency �1
  • �2
  • �3 for T2 < �, where
    T1 and T2 are characteristic time scales of the model. The correspondence
    of notations is
    = m, �t
    = T1, ��
    = T2, �w
    = w2, and ��w = w3.
    Using indices 1, 2, and 3, respectively, for the simplest to the most sophisticated
    formulas, the parameter values of the first fit of the Nikkei are
    ≈ 10�7� B1
    ≈ ?0�54� B1C1
    ≈ ?0�11�m1
    ≈ 0�47� tc
    ≈ 89�99��1

    ?0�86, and �1
    ≈ 4�9 for equation (15). The parameter values of the second
    fit of the Nikkei are A2
    ≈ 10�8� B2
    ≈ ?0�70� B2C2
    ≈ ?0�11�m2

    0�41� tc
    ≈ 89�97��2
    ≈ 0�14��1
    ≈ 4�8� T1
    ≈ 9�5, �2
    ≈ 4�9. The third fit
    uses the entire time interval and is performed by adjusting only T1, T2,
    �2, and �3, while m3
    = m2, tc and �1 are fixed at the values obtained
    from the previous fit. The values obtained for these four parameters are
    ≈ 4�3, T2
    ≈ 7�8, �2
    ≈ ?3�1, and T2
    ≈ 23. In all these fits, T1 and T2
    autopsy of major crashes 277
    90 92 94 96 98 2000
    Fig. 7.25. Natural logarithm of the Nikkei stock market index after the start of the
    decline from January 1, 1990 until December 31, 1998. The dotted line is the simple
    log-periodic formula (15) on page 232 used to fit adequately the interval of ≈2�6
    years starting from January 1, 1990. The continuous line is the improved nonlinear
    log-periodic formula (19) on page 336 developed in [397] and already used for the
    1929 and 1987 crashes over 8 years of data. It is used to fit adequately the interval
    of ≈5�5 years starting from January 1, 1990. The dashed line is the extension (25)
    on page 339 of the previous nonlinear log-periodic formula to the next order of
    description, which was developed in [213] and is used to fit adequately the interval
    of ≈9 years starting from January 1, 1990. Reproduced from [213].
    are given in years unit. Note that the values obtained for the two time
    scales T1 and T2 confirms their ranking. This last fit predicts a change
    of regime and that the Nikkei should increase in 1999. The value of this
    prediction will be analyzed in detail in chapter 9.
    Not only do the first two equations agree remarkably well with respect
    to the parameter values produced by the fits, but they are also in good
    agreement with previous results obtained from stock market and Forex
    bubbles with respect to the values of exponent m2. What lends credibility
    to the fit with the most sophisticated formula is that, despite its
    complex form, we get values for the two crossover time scales T1, T2
    which correspond very nicely to what is expected from the ranking and
    278 chapter 7
    from the nine-year interval of the data. We refer to [213] for a detailed
    and rather technical discussion.
    The Gold Deflation Price Starting in Mid-1980
    Another example of log-periodic decay is that of the price of gold after
    the burst of the bubble in 1980, as shown in Figure 7.26. The bubble has
    an average power law acceleration, as shown in the figure, but without
    any visible log-periodic structure. A pure power law fit will, however,
    not lock in on the true date of the crash, but insists on an earlier date
    than the last data point. This suggests that the behavior of the price might
    be different in some sense in the last few weeks prior to the burst of the
    bubble. Again, we obtain a reasonable agreement with previous results
    for the exponent m2 with a good preferred scaling ratio � ≈ 1�9 for
    the anti-bubble. In this case, the strength of the log-periodic oscillations
    compared to the leading behavior is ≈ 10%. The parameter values of the
    fit to the anti-bubble after the peak are A2
    ≈ 6�7� B2
    ≈ ?0�69� B2C ≈
    ≈ 0�45� tc
    ≈ 80�69�� ≈ 1�4, and � ≈ 9�8. The line before the
    77 78 79 80 81 82
    Log(Gold Price)
    Fig. 7.26. Natural logarithm of the gold 100 Oz Future price in U.S. dollars after the
    decline of the price in the early 1980s. The dotted line before the peak is expression
    (15) fitted over an interval of almost 3 years. The continuous line after the peak is
    expression (15) with tc
    ? t changed into t ? tc, fitted over an interval of 2 years.
    Reproduced from [213].
    autopsy of major crashes 279
    peak is expression (15) fitted over an interval of ≈3 years. The parameter
    values of the fit to the bubble before the peak are A2
    ≈ 8�5� B2

    ?111� B2C ≈ ?110�m2
    ≈ 0�41� tc
    ≈ 80�08�� ≈ ?3�0�� ≈ 0�05.
    In this chapter, we have shown that large stock market crashes are analogous
    to so-called critical points studied in the statistical physics community
    in relation to magnetism, melting, and similar phenomena. Our
    main assumption is the existence of cooperative behavior among traders
    imitating each other, as described in chapters 4–6. A general result of
    the theory is the existence of log-periodic structures decorating the time
    evolution of the system. The main point is that the market anticipates the
    crash in a subtle self-organized and cooperative fashion, hence releasing
    precursory “fingerprints” observable in the stock market prices. In other
    words, this implies that market prices contain information on impending
    crashes. If the traders were to learn how to decipher and use this
    information, they would act on it and on the knowledge that others act
    on it; nevertheless, the crashes would still probably happen. Our results
    suggest a weaker form of the “weak efficient market hypothesis” [122],
    according to which the market prices contain, in addition to the information
    generally available to all, subtle information formed by the global
    market that most or all individual traders have not yet learned to decipher
    and use. Instead of the usual interpretation of the efficient market
    hypothesis in which traders extract and consciously incorporate (by their
    action) all information contained in the market prices, we propose that
    the market as a whole can exhibit “emergent” behavior not shared by
    any of its constituents. In other words, we have in mind the process
    of the emergence of intelligent behaviors at a macroscopic scale that
    individuals at the microscopic scale cannot perceive. This process has
    been discussed in biology, for instance in animal populations such as ant
    colonies or in connection with the emergence of conciousness [8, 198]
    Let us mention another realization of this concept, which is found in
    the information contained in options prices on the fluctuations of their
    underlying assets. Despite the fact that the prices do not follow geometrical
    Brownian motion, whose existence is a prerequisite for most
    options pricing models, traders have apparently adapted to empirically
    incorporating subtle information in the correlation of price distributions
    with fat tails [337]. In this case and in contrast to the crashes, the traders
    280 chapter 7
    have had time to adapt. The reason is probably that traders have been
    exposed for decades to options trading in which the characteristic time
    scale for option lifetime is in the range of month to years at most. This
    is sufficient for an extensive learning process to occur. In contrast, only
    a few great crashes occur typically during a lifetime and this is certainly
    not enough to teach traders how to adapt to them. The situation may be
    compared to the ecology of biological species, which constantly strive
    to adapt. By the forces of evolution, they generally succeed in surviving
    by adaptation under slowly varying constraints. In constrast, life may
    exhibit successions of massive extinctions and booms probably associated
    with dramatically fast-occuring events, such as meteorite impacts
    and massive volcanic eruptions. The response of a complex system to
    such extreme events is a problem of outstanding importance that is just
    beginning to be studied [89].
    Most previous models proposed for crashes have pondered the possible
    mechanisms for explaining the collapse of the price at very short
    time scales. Here, in contrast, we propose that the underlying cause of
    the crash must be searched years before it in the progressive accelerating
    ascent of the market price, reflecting an increasing build-up of the
    market cooperativity. From that point of view, the specific manner by
    which prices collapsed is not of real importance since, according to the
    concept of the critical point, any small disturbance or process may have
    triggered the instability, once ripe. The intrinsic divergence of the sensitivity
    and the growing instability of the market close to a critical point
    might explain why attempts to unravel the local origin of the crash have
    been so diverse. Essentially all would work once the system was ripe.
    Our view is that the crash has an endogenous origin and that exogenous
    shocks only serve as triggering factors. We propose that the origin of the
    crash is much more subtle and is constructed progressively by the market
    as a whole. In this sense, this could be termed a systemic instability.
    chapter 8
    bubbles, crises, and
    crashes in emergent
    In periods of optimistic consensus, emerging
    markets have the favor of investors looking for opportunities to leverage
    their returns. Bubbles may ensue, and their demise is often associated
    with large swings and extreme corrections leading to financial crises
    Holdings of foreign stock by U.S. residents reached 10% of all equity
    holdings, or $876 billion by the end of 1996. More than one-third of that,
    $336 billion, was held through U.S. mutual funds specializing in international
    (non-U.S.) and global markets. Global and international mutual
    funds now represent 12.1% of net assets in long-term equity and bond
    funds. In addition, public and corporate U.S. pension funds report that
    on average they hold 10% and 9%, respectively, of their portfolios in
    non-U.S. assets. Trading in non-U.S. stocks on U.S. markets exceeded
    $1 trillion in 1996. Foreign investors are also increasingly actively in the
    U.S., with a trading volume of $1.2 trillion in 1996. Worldwide international
    trading of equities amounted to $5.9 trillion in 1996 according to
    NYSE estimates [422].
    282 chapter 8
    The record flows of capital towards emerging markets (essentially
    those in Asia and Latin America) in the 1990s were stimulated by
    three factors [136]. First, there was the search for higher yields leading
    to a strong increase in the demand for high-yield sovereign and
    corporate bonds issued by emerging market countries. Second, the continuing
    drive by institutional managers to increase their exposure to
    emerging markets and to achieve greater diversification of portfolios provided
    an important stimulus for flows to emerging markets. In November
    1997, institutional investors (pension funds, insurance companies, and
    mutual funds in the Organisation for Economic Co-operation and Development
    [OECD] countries) had under management over $20 trillion in
    assets, only a small portion of which was invested in emerging markets.
    If institutional investors had reallocated just 1% of total assets under
    management toward the emerging markets, this shift would have constituted
    a capital flow of $200 billion. Third, the resurgence of capital flows
    also reflected the clear recognition by investors that the economic fundamentals
    in most emerging markets in the 1990s had vastly improved
    over those that prevailed in the late 1970s.
    Since 1987, both the direct barriers, such as capital controls, and the
    indirect barriers, such as difficulties in evaluating corporate information,
    that prevented the free flow of capital had gradually been reduced. As
    capital controls were gradually lifted, global investors with more diversified
    portfolios began to influence stock prices, particularly in emerging
    markets [422]. This trend of opening up financial markets meant that
    firms from emerging markets were able to raise capital, both domestically
    and internationally, at a lower cost. In fact, firms from emerging
    markets were able to raise long-term equity and debt capital in global
    markets at unprecedented rates [422]. The capital infusion from foreign
    investors made it possible for emerging market firms to capitalize on
    their growth opportunities in a way that would have been impossible had
    they been restricted to raising funds in domestic markets. Moreover, previously
    state-owned assets were successfully sold off to both domestic
    and foreign investors, raising much-needed revenue for governments in
    both developed and emerging markets. The world financial markets are
    now on the road to becoming internationally integrated, but are still far
    from being there [422].
    The story of financial bubbles and crashes has repeated itself over the
    centuries and in many different locations since the famous tulip bubble
    of 1636 in Amsterdam, almost without any alteration in its main global
    characteristics [152].
    bubbles and crashes in emergent markets 283
  1. The bubble starts smoothly with some increasing production and sales
    (or demand for some commodity) in an otherwise relatively optimistic
  2. The attraction to investments with good potential gains then leads
    to increasing investments, possibly with leverage coming from novel
    sources, often from international investors. This leads to price appreciation.
  3. This in turn attracts less sophisticated investors and, in addition, leveraging
    is further developed with small downpayment (small margins),
    which leads to the demand for stock rising faster than the rate at which
    real money is put in the market.
  4. At this stage, the behavior of the market becomes weakly coupled or
    practically uncoupled from real wealth (industrial and service) production.
  5. As the price skyrockets, the number of new investors entering the speculative
    market decreases and the market enters a phase of larger nervousness,
    until a point when the instability is revealed and the market
    This scenario applies essentially to all market crashes, including old ones
    such as October 1929 on the U.S. market, for which the U.S. market was
    considered to be at that time an interesting “emerging” market with good
    investment potentialities for national as well as international investors.
    In addition, the concept of a New Economy was used profusely in the
    medias of the time, reminiscent of several other New Economy phases
    in more recent times, including the recent crash of the Internet bubble
    documented in chapter 7. The robustness of this scenario is presumably
    deeply rooted in investor psychology and involves a combination of imitative/
    herding behavior and greediness (for the development of the speculative
    bubble) and overreaction to bad news in periods of instabilities.
    There is also a simple mechanical effect which tends to sustain bubbles
    and then make them crash abruptly, stemming from the so-called
    buying on margin, that is, buying stocks on borrowed money. If there
    are huge amounts of borrowed money in the market, then it is no longer
    possible to slow things down. Prices must constantly increase, faster and
    faster. If they don’t, the interest payments on all the borrowed money
    invested in the market will not be get paid. Money will be withdrawn
    to settle debts, leading to lower prices, leading to more money getting
    withdrawn, and so on in a vicious circle. This may lead to total market
    collapse and bank failure. This mechanism was active during the bubble
    284 chapter 8
    preceding the Nasdaq crash of April 2000 discussed in chapter 7. Indeed,
    the economist, Kurt Richebacher [344], warned:
    There is something unique and unprecedented about the recent U.S. bubble:
    the phenomenal magnitude of the credit excesses. Credit creation is
    completely out of control in relation to economic activity and domestic
    savings. For each dollar added to gross domestic product (at current
    prices), there have been 4.5 of additional debt in 1999. By this measure,
    the U.S. stock market’s bull run does not only rank as a bubble, but as
    the biggest and the worst of its kind in history. Bubbles and booms often
    continue much longer than anyone thinks possible. Nevertheless, all bubbles
    eventually burst with a vengeance, and the current one will not be
    any exception.
    In addition, the constant effort to set a price with the available information,
    and in particular with the rational valuation formula, drives
    corporate managers to adopt policies aimed at producing a return for
    shareholders. There is thus an asymmetry in stock market variation:
    many more profit from an increase than from a decrease, in contrast
    with the symmetry between buyers and sellers of commodities traded
    nonspeculatively. The same mechanism is discussed by Franklin Allen
    and Douglas Gale [3, 4], who documented that bubbles may be caused
    by relationships between investors and banks: investors use money borrowed
    from banks to invest in risky assets, which are relatively attractive
    because investors can avoid losses in low-payoff states by defaulting on
    the loan. This risk shifting leads investors to bid up the asset prices. Such
    risk can originate in both the real and the financial sectors. Financial
    fragility occurs when the positive credit expansion eventually becomes
    insufficient to prevent a crisis.
    The purpose of this chapter, based in large part on [218], is to extend
    the empirical basis for the observations presented in chapter 7 on major
    financial markets by analyzing a wide range of emerging markets. Eighteen
    significant bubbles followed by large crashes or severe corrections
    in Latin-American and Asian stock markets are identified. With very few
    exceptions, these speculative bubbles can be quantitatively described by
    the rational expectation model of bubbles presented in chapters 5 and 6,
    which predicts a specific power law acceleration as well as log-periodic
    geometric patterns. This study indicates that such large downward movements
    in the markets are nothing but depletions of the preceding bubble,
    thus bringing the market back towards a state closer to “rational” pricing
    bubbles and crashes in emergent markets 285
    via a relaxation process following the crash that may take hours, days,
    weeks, or longer (see, for instance, Figure 7.5).
    The methodology presented here follows the one previously used for
    major financial markets in chapter 7, which consists of a combination of
    parametric fits using the log-periodic formula (15) on page 232 as well
    as a so-called spectral analysis in the variable log�tc
    ? t�/tc aiming at
    quantifying the oscillating part of the market prices (see the section on
    “Nonparametric Test of Log-Periodicity” in chapter 7). For such a “spectral”
    analysis, a so-called Lomb periodogram was used, which consists
    in a local fit of an oscillatory cosine function (with a phase) using some
    user-chosen range of frequencies. The relative level of the peak for each
    separate “periodogram” can be taken as a measure of the significance of
    the oscillations.
    The use of the same methodology allows us to test the hypothesis
    that emerging markets exhibit bubbles and crashes with similar logperiodic
    signatures as in the major financial markets. It is very important
    for confirmation of the theory that no or little parameter tuning
    be done, as the danger of overfitting is always looming in this kind of
    Identifying a speculative bubble is very difficult because there are
    several conceptual problems that obscure the economic interpretation of
    bubbles, starting with the absence of a general definition: bubbles are
    model specific and generally defined from a rather restrictive framework
    [1]. It is therefore difficult to avoid a subjective bias, especially since
    the very existence of bubbles is still hotly debated [459, 411, 229, 144,
    120, 342, 108, 187, 109, 359, 453, 185, 320]. A major problem with
    arguments in favor of bubbles is also that apparent evidence for bubbles
    can be reinterpreted in terms of market fundamentals that are unobserved
    by the researcher [120, 135, 185].
    We have thus taken a pragmatic and very straightforward approach
    consisting in selecting “bubbles” based on the following three
    � the existence of a sharp peak in the spirit of [348],
    � the existence of a preceding period of increasing price that extends
    over at least six months and that should preferably be comparable with
    those of the larger crashes discussed in chapter 7,
    286 chapter 8
    � the existence of a fast price decrease following the peak over a time
    interval much shorter than the accelerating period.
    A bubble is defined as a period of time going from a pronounced minimum
    to a large maximum by a prolonged price acceleration, followed by
    a crash or a large decrease. As for the major financial markets, such a
    bubble is defined unambiguously by identifying its end with the date tmax,
    where the highest value of the index is reached prior to the crash/decrease.
    For the bubbles prior to the largest crashes on the major financial markets,
    the beginning of a bubble is clearly identified as coinciding always with
    the date of the lowest value of the index prior to the change in trend. However,
    this identification is not as straightforward for some of the emerging
    markets discussed below. In approximately half the cases, the date of the
    first data point used in defining the beginning of the bubble had to be
    moved forward in order to obtain fits with nonpathological values for the
    exponent and of the angular log-frequency. This may well be an artifact
    stemming from the restrictions in the fitting imposed by using a single
    cosine as the periodic function in expression (15). In order to filter out fits,
    the exponent m2 has been chosen consistently between zero and one and
    it should be not too close to either zero or one: too small an m2 implies
    a flat bubble with a very sudden acceleration at the end. Too large an m2
    corresponds to a nonaccelerating bubble. The angular frequency � of the
    log-periodic oscillations must also not be too small or too large. If it is
    too small, less than one oscillation occurs over the whole interval and the
    log-periodic oscillation has little meaning. If it is too large, the oscillations
    are too numerous and they start to fit the high-frequency noise.
    We refer to [218] for more details of the procedures.
    In Figures 8.1–8.6, the evolution of six Latin-American stock market
    indices (Argentina, Brazil, Chile, Mexico, Peru, and Venezuela) is shown
    as a function of time in the 1990s.
    For these six Latin American stock market indices, four Argentinian
    bubbles, one Brazilian bubble, two Chilean bubbles, two Mexican
    bubbles, two Peruvian bubbles, and a single Venezuelan bubble were
    identified [218], with a subsequent large crash/decrease, as shown in
    Figures 8.1 to 8.6.
    bubbles and crashes in emergent markets 287
    90 91 92 93 94 95 96 97 98 99
    Fig. 8.1. The Argentinian stock market index as a function of time. Four bubbles
    with a subsequent very large drawdown can be identified. The approximate dates
    are in chronological order: mid-91 (I), early 93 (II), early 94 (III), and late 97 (IV).
    Reproduced from [218].
    94 95 96 97 98 99
    ’Brazil’ I
    Fig. 8.2. The Brazilian stock market index as a function of time. One bubble with a
    subsequent very large drawdown can be identified. The approximate date is mid-97
    (I). Reproduced from [218].
    288 chapter 8
    91 92 93 94 95 96 97 98 99
    Fig. 8.3. The Chilean stock market index as a function of time. Two bubbles with
    a subsequent very large drawdown can be identified. The approximate dates are in
    chronological order: mid-91 (I) and early 94 (II). Reproduced from [218].
    91 92 93 94 95 96 97 98 99
    Mexico II
    Fig. 8.4. The Mexican stock market index as a function of time. Two bubbles with
    a subsequent very large drawdown can be identified. The approximate dates are in
    chronological order: early 94 (I) and mid-97 (II). Reproduced from [218].
    bubbles and crashes in emergent markets 289
    93 94 95 96 97 98 99
    Fig. 8.5. The Peruvian stock market index as a function of time. Two bubbles with
    a subsequent very large drawdown can be identified. The approximate dates are in
    chronological order: late 93 (I) and mid-97 (II). Reproduced from [218].
    94 95 96 97 98 99
    Venezuela I
    Fig. 8.6. The Venezuelan stock market index as a function of time. One bubble
    with a subsequent very large drawdown can be identified. The approximate date is
    mid-97 (I). Reproduced from [218].
    290 chapter 8
    91.1 91.9
    91.3 91.5 91.7 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Argentina I’
    Best fit
    Second best fit
    ’Argentina I’
    Fig. 8.7. Left panel: The Argentinian stock market bubble of 1991. See Table 8.1
    for the main parameter values of the fits with equation (15). Right panel: Only the
    best fit is used in the Lomb periodogram. Reproduced from [218].
    Figures 8.7–8.20 show the fits of the bubbles indicated in Figures 8.1–
    8.6 as well as the spectral Lomb periodogram of the difference between
    the indices and the pure power law, which quantifies the strength of the
    log-periodic component. The overall quality of these fits is good, and
    both the acceleration and the accelerating oscillations are rather well
    captured by the log-periodic power law formula. However, these fits
    do not have the same excellent quality as for those obtained for the
    major financial markets reported in chapter 7 as well as for the Russian
    stock market [221] (see below). A plausible interpretation is that these
    ’Argentina II’
    Best fit
    Second best fit
    Third best fit
    Best fit antibubble
    92.2 92.4 92.6 92.8 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Argentina II: Bubble’
    ’Argentina II: Anti-bubble’
    Fig. 8.8. Left panel: The Argentinian stock market bubble and antibubble of 1992.
    See Table 8.1. Right panel: Only the best fit is used in the Lomb periodograms.
    Reproduced from [218].
    bubbles and crashes in emergent markets 291
    93.7 93.8 93.9 94 94.1 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Argentina III’
    Best fit
    Second best fit
    ’Argentina III’
    Fig. 8.9. Left panel: The Argentinian stock market bubble ending in 1994. See
    Table 8.1 for the main parameter values of the fit. Right panel: Only the best fit is
    used in the Lomb periodogram. Reproduced from [218].
    are relatively small markets in terms of capitalization and number of
    investors, for which finite size effects, in the technical sense given in
    statistical physics [70], are expected and thus may blur out the signal
    with systematic distortions and unwanted fluctuations. See chapters 5
    and 6 for a discussion.
    In Table 8.1, the main parameters of the fits are given as well
    as the beginning and ending dates of the bubble and the size of the
    95.5 96 96.5 97 97.5 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Argentina IV’
    Best fit
    ’Argentina IV’
    Fig. 8.10. Left panel: The Argentinian stock market bubble ending in 1997. See
    Table 8.1 for the main parameter values of the fit. Right panel: Only the best fit is
    used in the Lomb periodogram. Reproduced from [218].
    292 chapter 8
    96.4 96.6 96.8 97 97.2 97.4 0 1 2 3 4 5 6 7 8
    Spectral Power
    Best fit
    Fig. 8.11. Left panel: The Brazilian stock market bubble ending in 1997. See
    Table 8.1 for the main parameter values of the fit. Right panel: Only the best fit is
    used in the Lomb periodogram. Reproduced from [218].
    crash/correction, defined as
    drop % = I�tmax� ? I�tmin�
    � (16)
    Here, tmin is defined as the date after the crash/correction where the
    index I�t� achieves its lowest value before a clear novel market regime
    is observed. The duration tmax
    ? tmin of the crash/correction is found to
    90.8 91 91.2 91.4 91.6 91.8 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Chile I’
    Best fit
    Second best fit
    Third best fit
    ’Chile I’
    Fig. 8.12. Left panel: The Chilean bubble ending in 1991. See Table 8.1 for the
    main parameter values of the fit. Right panel: Only the best fit is used in the Lomb
    periodogram. Reproduced from [218].
    bubbles and crashes in emergent markets 293
    93.4 93.5 93.6 93.7 93.8 93.9 94 94.1 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Chile II’
    Best fit
    ’Chile II’
    Fig. 8.13. Left panel: The Chilean bubble of 1993. See Table 8.1 for the main
    parameter values of the fit. Right panel: Lomb periodogram of the oscillatory component
    of the market price shown in the left panel. Reproduced from [218].
    range from a few days (a crash) to a few months (a less abrupt change
    of regime).
    Table 8.1 shows that the fluctuations in the parameter values m2 and
    � obtained for the eleven Latin-American crashes are considerable. The
    lower and upper values for the exponent m2 are 0.12 and 0.62, respectively.
    For �, the lower and upper values are 2.9 and 11.4, corresponding
    to a range of �’s in the interval 1.8–8.8. Removing the two largest values
    for � reduces the fluctuations to 2�8 ± 1�1, which is still a much larger
    95.5 95.6 95.7 95.8 95.9 96 96.1 96.2 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Chile Anti-bubble’
    Best fit
    ’Chile Anti-bubble’
    Fig. 8.14. Left panel: The Chilean antibubble beginning in 1995 fitted by the logperiodic
    power law with m2
    = 0�36, tc
    = 1�995�51, and � = 9�7. Right panel:
    Lomb periodogram of the oscillatory component of the market price shown in the
    left panel. Reproduced from [218].
    294 chapter 8
    93.5 93.6 93.7 93.8 93.9 94 94.1 0 1 2 3 4 5 6 7 8
    Spectral Power
    Mexico I
    Best fit ’Mexico I’
    Fig. 8.15. Left panel: The Mexican bubble ending in 1994. See Table 8.1 for the
    main parameter values of the fit. Right panel: Lomb periodogram of the oscillatory
    component of the market price shown in the left panel. Reproduced from [218].
    interval than the 2�5 ± 0�3 previously seen on major financial markets
    discussed in chapter 7. Three cases of antibubbles could be identified for
    the Latin-American markets analyzed here; see Figures 8.8, 8.14, and
    8.20. Quite remarkably, the first and the last are preceded by a bubble,
    thus exhibiting a qualitative symmetry around comparable critical times
    tc. Similar behavior will be shown later in this chapter for the Russian
    market in 1996–97 [221].
    96 96.4 96.8 97.2 97.6 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Mexico II’
    Best fit
    Second best fit
    ’Mexico II’
    Fig. 8.16. Left panel: The Mexican bubble ending in 1997. See Table 8.1 for the
    main parameter values of the fit. Right panel: Only the best fit is used in the Lomb
    periodogram. Reproduced from [218].
    bubbles and crashes in emergent markets 295
    ’Peru I’
    Best fit
    Second best fit
    Third best fit
    93.2 93.3 93.4 93.5 93.6 93.7 93.8 93.9 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Peru I’
    Fig. 8.17. Left panel: The Peruvian bubble of 1993. See Table 8.1 for the main
    parameter values of the fit. Right panel: Lomb periodogram of the oscillatory component
    of the market price shown in the left panel. Reproduced from [218].
    In Figures 8.21–8.25, the evolution of five Asian stock market indices
    (Indonesia, Korea, Malaysia, Philippines, and Thailand) is shown as
    a function of time from 1990 to February 1999. Two bubbles on the
    Indonesian stock market and one each on the Korean, the Malaysian,
    the Philippine, and the Thai markets are detected with subsequent
    crashes/decreases, as indicated in Figures 8.21–8.25.
    97.1 97.2 97.3 97.4 97.5 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Peru II’
    Fig. 8.18. Left panel: The Peruvian bubble ending in 1997. See Table 8.1 for the
    main parameter values of the fit. Right panel: Lomb periodogram of the oscillatory
    component of the market price shown in the left panel. Reproduced from [218].
    296 chapter 8
    96.5 97 97.5 98 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Venezuela Bubble’
    Best fit
    ’Venezuela Bubble’
    Fig. 8.19. Left panel: The Venezuelan bubble ending in 1997. See Table 8.1 for the
    main parameter values of the fit. Right panel: Lomb periodogram of the oscillatory
    component of the market price shown in the left panel. Reproduced from [218].
    In Figures 8.26–8.31, the fits of the bubbles indicated in Figures 8.21–
    8.25 are given, as well as the spectral Lomb periodogram of the difference
    between the indices and the pure power law. Similarly to the
    Latin-American markets, somewhat larger fluctuations in the values for
    the exponent m2 and the angular log-frequency � can be observed compared
    to the major financial markets.
    97.6 97.8 98 98.2 98.4 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Venezuela Anti-bubble’
    Best fit
    Second best fit
    Third best fit
    ’Venezuela Anti-bubble’
    Fig. 8.20. Left panel: The Venezuelan antibubble starting in 1997 fitted by the logperiodic
    power law with m2
    = 0�58� 0�35, tc
    = 97�75� 97�75 and � = 6�7� 3�9 (two
    best fits). Right panel: Only the best fit is used in the Lomb periodogram. Reproduced
    from [218].
    bubbles and crashes in emergent markets 297
    Table 8.1
    Crash and fit characteristics of the various speculative bubbles on the Latin-American
    market leading to a large drawdown in the 1990s
    Stock market tc tmax tmin % drop m2 � �
    Argentina I 91�80 91�80 91�90 26% 0�37 4�8 3�7
    Argentina II 92�43 92�42 92�90 59% 0�22 11�4 1�7
    Argentina III 94�13 94�13 94�30 30% 0�19 7�2 2�4
    Argentina IV 97�89 97�81 97�87 27% 0�20 10�1 1�9
    Brazil 97�58 97�52 97�55 18% 0�49 5�7 3�0
    Chile I 91�77 91�75 91�94 22% 0�50 7�2 2�4
    Chile II 94�10 94�09 94�26 20% 0�30 2�9 8�8
    Mexico I 94�10 94�09 94�30 32% 0�12 4�6 3�9
    Mexico II 97�93 97�80 97�82 21% 0�50 6�1 2�8
    Peru I 93�84 93�83 93�88 22% 0�62 11�2 1�8
    Peru II 97�43 97�42 98�15 30% 0�14 14�0 1�6
    Venezuela 97�75 97�73 98�07 42% 0�35 3�9 5�0
    tc is the critical time predicted from the fit of the market index to equation (15) on page 232.
    When multiple fits exist, the fit with the smallest difference between tc and tmax is chosen. Typically,
    this will be the best fit, but occasionally it is the second best fit. The other parameters m2, �, and
    � of the fit are also shown. The fit is performed up to the time tmax, at which the market index
    achieved its highest maximum before the crash. The percentage drop is calculated from the total
    loss from tmax to tmin, where the market index achieved its lowest value as a consequence of the
    91 92 93 94 95 96 97 98 99
    Indonesia II
    Fig. 8.21. The Indonesian stock market index as a function of time. Two bubbles
    with a subsequent very large drawdown can be identified. The approximate dates
    for the drawdowns are early 94 (I) and mid-97 (II). Reproduced from [218].
    298 chapter 8
    91 92 93 94 95 96 97 98 99
    ’Korea’ I
    Fig. 8.22. The Korean stock market index as a function of time. One bubble with a
    subsequent very large drawdown can be identified culminating at the end of 1994.
    Reproduced from [218].
    91 92 93 94 95 96 97 98 99
    ’Malaysia’ I
    Fig. 8.23. The Malaysian stock market index as a function of time. One extended
    bubble with a subsequent very large drawdown occurring early in 1994 can be
    identified. Reproduced from [218].
    bubbles and crashes in emergent markets 299
    91 92 93 94 95 96 97 98 99
    Philippines I
    Fig. 8.24. The Philippines stock market index as a function of time. One bubble
    with a subsequent very large drawdown occurring early in 1994 can be identified.
    Reproduced from [218].
    91 92 93 94 95 96 97 98 99
    ’Thailand’ I
    Fig. 8.25. The Thai stock market index as a function of time. One bubble with a
    subsequent very large drawdown occurring early in 1994 can be identified. Reproduced
    from [218].
    300 chapter 8
    93.2 93.4 93.6 93.8 94 94.2 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Indonesia I’
    Best fit
    ’Indonesia I’
    Fig. 8.26. Left panel: Indonesian stock market bubble ending in January 1994 with
    log-periodic power law fit with parameters m2
    = 0�44� tc
    = 1994�09, and � = 15�6.
    Right panel: Lomb periodogram of the log-periodic oscillatory component of the
    price shown in the left panel. The abscissa is the log-frequency f defined as f =
    �/2�. Reproduced from [218].
    95.5 96 96.5 97 97.5 98 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Indonesia II’
    Fig. 8.27. Left panel: Indonesian stock market bubble ending in 1997 with logperiodic
    power law fit with parameters m2
    = 0�23� tc
    = 1998�05, and � = 10�1.
    Right panel: Lomb periodogram of the log-periodic oscillatory component of the
    price shown in the left panel. Reproduced from [218].
    bubbles and crashes in emergent markets 301
    93 93.5 94 94.5 95 0 1 2 3 4 5 6 7 8
    Spectral Power
    ’Korea I’
    Fig. 8.28. Left panel: Korean stock market bubble ending in 1994 fitted by the logperiodic
    power law formula with main parameters m2
    = 1�05� tc
    = 1�994�87, and
    � = 8�15. Right panel: Lomb periodogram of the log-periodic oscillatory component
    of the price shown in the left panel. Reproduced from [218].
    A crisis is not always preceded by the log-periodic power law pattern.
    For instance, the log-periodic precursory behavior of the crisis of 1997 is
    clearly visible in the Asian markets only for Hong Kong and Indonesia.
    However, it is strongly present in the Argentinian, Brazilian, Mexican,
    Peruvian, and Venezuelan stock markets, as described in the previous
    Log (Index)
    92.8 93 93.2 93.4 93.6 93.8 94 0 1 2 3 4 5 6 7 8
    Spectral Power
    Best fit
    Fig. 8.29. Left panel: Malaysian stock market bubble ending with the crash of
    January 1994 fitted by the log-periodic power law formula with main parameters
    = 0�24� tc
    = 1�994�02, and � = 10�9. Right panel: Lomb periodogram of the
    log-periodic oscillatory component of the price shown in the left panel. Reproduced
    from [218].
    302 chapter 8
    92 92.5 93 93.5 94 0 1 2 3 4 5 6 7 8
    Spectral Power
    Best fit
    Fig. 8.30. Left panel: Philippine stock market bubble ending with the crash of
    January 1994 fitted by the log-periodic power law formula with main parameters
    = 0�16� tc
    = 1�994�02, and � = 8�2. Right panel: Lomb periodogram of the
    log-periodic oscillatory component of the price shown in the left panel. Reproduced
    from [218].
    section. The reason lies in the highly interconnected economic and market
    dynamics of these multiple countries. As a consequence, an adequate
    modeling requires a multidimensional approach. It can be shown that
    such multidimensional bubbles, which are extensions of the models of
    chapter 5, exhibit both synchronized and asynchronous crashes.
    93.4 93.5 93.6 93.7 93.8 93.9 94 94.1 0 1 2 3 4 5 6 7 8
    Spectral Power
    Best fit
    Fig. 8.31. Left panel: Thai stock market bubble ending with the crash of January
    1994 fitted by the log-periodic power law formula with main parameters m2
    0�48� tc
    = 1�994�07, and � = 6�1. Right panel: Lomb periodogram of the logperiodic
    oscillatory component of the price shown in the left panel. Reproduced
    from [218].
    bubbles and crashes in emergent markets 303
    Recall that the 1997 crisis began in July in Southeast Asia when foreign
    bankers, investors, currency speculators, and market analysts lost
    confidence in Thailand’s ability to cope with a deteriorating economic
    situation, including a rising trade deficit and a growing international debt
    that had reached 50% of gross domestic product. In the face of falling
    profits and mounting bankruptcies on the part of companies and financial
    institutions in Southeast Asia and South Korea, foreign investors
    dumped regional stocks and foreign lenders stopped rolling over their
    short-term loans. After depleting its hard currency reserves to counter
    speculative attacks on the baht, which had been pegged to the U.S. dollar,
    the Thai government had little choice but to adopt a managed float
    of its currency. The resultant plunge in the baht led to a series of forced
    currency devaluations that soon swept through Indonesia, Malaysia and
    the Philippines, and then spread to South Korea and, to a lesser extent,
    Singapore, Taiwan, and Japan.
    The main causes of the 1997 Asian crisis involved the following ingredients:
    excessive reliance on foreign borrowing by business enterprises
    and banks and, especially, overdependence on short-term debt; overinvestment
    in real estate and excess manufacturing capacity; inadequate
    supervision of financial institutions and politically influenced allocations
    of credit to unsound companies; overly expansive fiscal and macroeconomic
    policies in several countries; and declining terms of trade for
    countries whose currencies have been pegged closely to the U.S. dollar,
    which had been strengthening against the Japanese yen.
    Companies, banks, and governments unwisely piled up short-term
    debt on the unfounded assumption of never-ending growth. Since the
    mid-1990s these excesses were substantially facilitated by the easy
    availability of low-cost foreign capital, often at lower interest rates than
    domestically available credit. Such a credit distortion is an important
    mechanism helping the development of bubbles and their ensuing crises.
    In the period of the bubble growth, Asian government agencies have
    implicitly or explicitly guaranteed the credit risk of foreign loans, thus
    leading to lower interest rates compared to domestic loans (a smaller
    remuneration is required on foreign loans which bear less risk) [83]. As
    a consequence, there is an incentive to borrow from foreign sources on a
    great scale, even if savings are large: these foreign loans can be used in
    many different ways domestically to provide remunerations larger than
    the cost of the loan. Thus, heavy foreign borrowing and overinvestment
    in real estate are rational consequences when a particular currency is
    overvalued and cheap credit is available. This fuels lending to poorly
    managed local banks, a real estate boom while the economy has slowed
    304 chapter 8
    down considerably, and consistent domestic real exchange appreciation
    while the currency becomes weaker.
    Many of the excesses on the part of Asian governments, banks, and
    corporations would not have been possible except for the comparatively
    recent globalization of capital markets, including the relaxation of previous
    controls in Asian countries on international borrowing by banks
    and private corporations. By one estimate, 90% of international transactions
    were accounted for by trade before 1970, and only 10% by capital
    flows. Today, despite a vast increase in global trade, that ratio has been
    reversed, with 90% of transactions accounted for by financial flows not
    directly related to trade in goods and services [96]. Most of these capital
    flows are accounted for by highly volatile portfolio investment and
    short-term loans.
    Vast outflows of investor funds from the strong U.S. economy in
    search of higher returns and the recycling of huge Japanese trade surpluses
    (Japanese lending to Asia alone rose from $40 billion in 1994 to
    $265 billion in 1997, i.e., 40% of their total foreign lending) contributed
    to a dangerous build-up of debt and excessive property development
    and manufacturing capacity. In testimony before the House Banking
    and Financial Services Committee, Federal Reserve Chairman Alan
    Greenspan noted with understatement that “In retrospect, it is clear that
    more investment monies flowed into these economies than could be
    profitably employed at modest risk” [96].
    After the collapse of the Soviet Union in December 1991, following the
    highly symbolic destruction of the Berlin wall in 1990, the Russian stock
    market developed as an emerging market open to foreign investments.
    It is thus interesting to analyze whether the same patterns observed
    for essentially all emerging markets are also found there. As can be
    expected from the universal behavior of investors, the answer is positive.
    The post-crisis special 1999 report of the St. Petersburg Times
    is particularly instructive on the interplay between hypes of fast gains,
    for instance in Russian telecoms and other state-owned industries, and
    the psychology of political and financial risks permeating this chaotic
    period [412]. Indeed, in mid-1997, Russia benefitted from billions of
    dollars of International Monetary Fund (IMF), World Bank, and bilateral
    aid that initially permitted the Russian Central Bank to accumulate
    reserves at a pace of $1.5 billion a month. The Russian stock market
    bubbles and crashes in emergent markets 305
    became the world’s leading developing country stock market, as speculators
    chased stratospheric investment returns. This hid many problems
    [408]: the early corruption of the nonmarket “privatization” to insiders;
    the spread of organized crime; the impending complete collapse of the
    Russian economy in 1998; the rise of weapons proliferation as a means
    of generating hard currency; and the increasing estrangement of Russia
    from the United States, essentially reversing the trends that existed in
  6. Russia’s total economic collapse in 1998, following the bubble,
    inflicted pain, suffering, and disruption on millions of Russians.
    Due to the difficulty in getting a reliable measure of the Russian stock
    market, it is useful to analyze four Russian stock market indices: The
    Russian Trading System Interfax Index (IRTS), The Agence Skate Press
    Moscow Times Index (ASPMT), The Agence Skate Press General Index
    (ASPGEN), and The Credit Suisse First Boston Russia Index (ROSI).
    The ROSI is generally considered the best of the four. As the Russian
    stock market is highly volatile, companies go in and out of the indices
    and it is difficult to maintain a representative stock market index. Using
    four different indices mitigates this problem if the results turn out to be
    In Figure 8.32, we see the ROSI fitted with equation (15) on page
    232 in the interval �96�21 � 97�61�. The interval is chosen by identifying
    the start of the bubble and the end represented by the date of the highest
    value of the index before the crash, similarly to the major market crashes
    96.2 96.4 96.6 96.8 97 97.2 97.4 97.6
    ROSI Index
    Fig. 8.32. The ROSI Index fitted with equation (15). The parameter values of the
    fit with equation (15) are A2
    ≈ 4254� B2
    ≈ ?3166� B2C ≈ 246�m2
    ≈ 0�40� tc

    97�61�� ≈ 0�44, and � ≈ 7�7. Reproduced from [221].
    306 chapter 8
    discussed previously. For all four indices, the same start-day and end-day
    can be identified to within a day.
    As can be seen from Table 8.2, the nondimensional parameters m2��,
    and � as well as the predicted time of the crash tc for the fit to the
    different indices agree very well except for the exponent m2 obtained
    from the ASPGEN index. In fact, the value obtained for the preferred
    scaling ratio � is fluctuating by no more than 5% for the four fits showing
    good numerical stability.
    The origin of this bubble is well known. In 1996, large international
    investors (U.S., German, and Japanese) began to invest heavily
    in the Russian markets believing that the financial situation of Russia
    had finally stabilized. Nothing was further from the truth [206, 281], but
    the belief and hope in a new investment haven with large returns led
    to herding and bubble development. This means that the same herding
    that created the log-periodic bubbles on Wall Street (1929, 1987, 1998),
    Hong Kong (1997), and the Forex (1985, 1998), entered an emerging
    market and brought along the same log-periodic power law pattern characterizing
    the global markets. The fact that the consistent values of �
    obtained for the four indices of the Russian market are comparable to
    that of the Wall Street, Hong Kong, and Forex crashes supports this
    interpretation. Furthermore, it supports the idea of the stock market as
    a self-organizing complex system of surprising robustness in one of its
    most dramatic behaviors.
    Inspired by this clear evidence of log-periodic oscillations decorating
    the power law acceleration signaling a bubble in the Russian stock
    market, it is natural to search for possible log-periodic signatures in the
    antibubble that followed the log-periodic bubble described above.
    As described in chapter 7, the decay of the Japanese Nikkei index
    starting January 1, 1990 and lasting until the present can be excellently
    modeled by a log-periodically decorated power law. In Figure 8.33, the
    ROSI index for the antibubble is fitted with equation (15), where tc and
    t have been interchanged. The “symmetry” around tc is rather striking.
    It may seem odd to argue for the log-periodic power law precursory
    patterns while one can forcefully argue that the market is largely reflecting
    the vagaries of the Russian political institutions. For instance, in
    the antibubble case, February–April 1998 was a revival period for the
    market characterized by the returning of Western investors after the postcrash
    calm-down. This can be followed by studying the dynamics of
    the Russian external reserves. The timing of the return can be argued to
    be dictated by the risk policies of larger investors more than anything
    else. The next large drop of the Russian index in April 1998 originated
    Table 8.2
    Bubble tc tmax tmin drop m2 � � A2 B2 B2C Var
    ASPMT 97�61 97�61 97�67 17% 0�37 7�5 2�3 1280 ?1025 59�5 907
    IRTS 97�61 97�61 97�67 17% 0�39 7�6 2�3 633 ?483 38�8 310
    ROSI 97�61 97�61 97�67 20% 0�40 7�7 2�3 4254 ?3166 246 12437
    ASPGEN 97�62 97�60 97�67 8.9% 0�25 8�0 2�2 2715 ?2321 72�1 1940
    Anti-bubble tc tmax tmin drop m2 � � A B C Var
    ROSI 97�72 97�77 98�52 74% 0�32 7�9 2�2 4922 ?3449 472 59891
    Nikkei (15) 89�99 90�00 92�63 63% 0�47 4�9 3�6 10�7 ?0�54 ?0�11 0�0029
    Nikkei (Nonlinear log-periodic eq.) 89�97 90�00 95�51 63% 0�41 4�8 3�7 10�8 ?0�70 ?0�11 0�0600
    tc is the predicted time of the crash from the fit of the market index to equation (15). The other parameters of the fits to the preceding bubble are also given.
    The error Var is the variance between the data and the fit and has units price2 except for the Nikkei, where the units are �log�price��2. The fit to the bubble is
    performed up to the time at which the market index achieved its highest maximum before the crash. The parameters tc , m2, �, and � correspond to the fit with
    equation (15), where tc and t have been interchanged. Here tmax and tmin represent the endpoints of the interval fitted.
    308 chapter 8
    96.5 97 97.5 98 98.5
    ROSI Index
    Fig. 8.33. Symmetric “bubble” and “antibubble”: in addition to the ascending part
    of the ROSI Index, which is reproduced from Figure 8.32 with the same fit, we
    show the deflating part fitted with equation (15) by changing tc
    ? t into t ? tc.
    The parameter values are A2
    ≈ 4922� B2
    ≈ ?3449� B2C ≈ 472�m2
    ≈ 0�32� tc

    97�72�� ≈ 1�4, and � ≈ 7�9. Reproduced from [221].
    by the decision of Mr. Yeltsin to sack Mr. Chernomyrdin’s government,
    which destabilized the political situation and created uncertainty. Further
    political disturbance was introduced twice by the Duma when it rejected
    Mr. Yeltsin’s candidates for the prime minister’s office and put itself on
    the brink of dissolution.
    The August 1998 crash, which had such a large effect on the markets
    of the rest of the world (see chapter 7), was often attributed to a devaluation
    of the ruble and to events on the Russian political scene. While we
    do not underestimate the effect of “news,” we observe that markets are
    constantly bombarded by news, and it will always be possible to attribute
    the crash to a specific one, after the fact. In contrast, we view markets’
    reactions more often than not as reflecting their underlying stability (or
    instability). In the case of the August 1998 crash, the market was ripe for
    a major crisis and the “news” made it occur. If nothing had occurred on
    the Russian scene, other news would probably have triggered the event
    anyway [221], within a time scale of about a month, which seems to be
    the relevant lifetime of a market instability associated with the burst of
    a bubble.
    We emphasize again that one must not mistake a systematically unstable
    situation for the specific historical action that triggered the instability.
    bubbles and crashes in emergent markets 309
    Consider a ruler put vertically on a table. Being in an unstable position,
    the stick will fall in some direction and the specific air current or slight
    initial imperfection in the initial condition are of no real importance.
    What is important is the intrinsically unstable initial state of the stick.
    We argue that a similar situation applies for crashes. They occur because
    the market has reached a state of global instability. Of course, there will
    always be specific events that may be identified as triggers of market
    motions, but they simply reveal the instability rather than being its deep
    sources. Furthermore, political events must also be considered as indicators
    of the state of the dynamical system which includes the market.
    There is, in principle, no decoupling between the different events. Specifically,
    the 1997 Russian crash may have been triggered by the Asian
    crises, but it was to a large extent fueled by the collapse of a banking
    system, which in the course of the bubble had created an outstanding
    debt of $19.2 billion [281].
    It is well known that the October 1987 crash was an international event,
    occurring within a few days in all major stock markets worldwide [30].
    It is also often noted that smaller western European stock markets as
    well as other markets around the world are influenced by dominating
    trends on the U.S. market.
    There are counterexamples. An instance of a pronounced synchronization
    unrelated to a U.S. event is the rash of crashes/corrections on
    most emerging stock markets in early 1994. These crises occurred from
    January to June 1994 and concerned the currency markets (Mexico,
    South Africa, Turkey, Venezuela) and the stock markets (Chile, Hungary,
    India, Indonesia, Malaysia, Philippines, Poland, South Africa, Turkey,
    Venezuela, Germany, Hong Kong, Singapore, U.K.) [271]. In terms of
    the bubbles discussed above, the corresponding maxima of the stock
    markets occurred on 1994.13 (Argentina III), 1994.09 (Chile II), 1994.09
    (Mexico I), Peru (1993.83), 1994.01 (Hong Kong II), 1994.01 (Indonesia
    I), 1994.01 (Malaysia), 1994.01 (Philippines), and 1994.01 (Thailand).
    The crises were particularly severe in Latin-American countries, the
    worst of which was felt in Mexico. The United States, helped by Canada
    and Europe, came to its rescue twice, first in April 1994 and then in
    early 1995 with a massive rescue fund of $50 billion [236].
    310 chapter 8
    Similarly, another rash of several crises evolved from the troubles in
    Thailand, which spilled over worldwide. This set of crises can be seen
    already contained in the death of the bubbles previously analyzed. The
    maximum of the bubbles was 1997.81 (Argentina IV), 1997.51 (Brazil),
    1997.80 (Mexico II), 1997.42 (Peru II), 1997.73 (Venezuela), 1997.60
    (Hong Kong III), and 1997.52 (Indonesia II). These maxima are followed
    by sharp corrections triggered by and following the abandonment by
    Thailand of the fixed-exchange rate system after strong attacks on its
    currency. When the Thailand domino fell, three other Asian countries
    immediately got caught up in the turmoil: the Philippines, Indonesia,
    and Malaysia. None had situations as bad as Thailand, but they all had
    currencies pegged to a strong dollar, so they were hit hard.
    Such financial contagion is based on the same mechanisms as that
    leading to speculative bubbles. Investors’ and lenders’ moods follow
    regime shifts: when times are good, they think less about risk and focus
    on potential gain. When something bad happens, they start worrying
    about risk again, and the whole structure of hope and greed that had
    driven the market up collapses. Such sudden shifts in market psychology
    are nowadays amplified by the internationalization of investments:
    the same fund managers and bankers who got burned in Thailand also
    had money in Malaysia, Indonesia, and other emerging markets. In addition,
    they share much of the same information from similar channels.
    As a consequence, they often collectively reevaluate the risks they faced
    all over the globe particularly where economies and financial systems
    resemble Thailand’s. In particular, the real economic adversity, the fundamentals,
    surfaces again with its interconnection across national borders
    by real economic ties.
    There are also simple technical reasons for such cascades. The main
    players in emerging markets are the hedge funds and mutual funds. The
    former borrow money from banks to leverage their investments. If the
    value of these investments drops far enough, the bank calls in the loan
    and the hedge fund has to sell other securities to pay back the loan.
    The same mechanism can then operate on those securities that have been
    sold, as they also drop from the wave of selling. Mutual funds do not use
    leverage, but they have to keep a cushion of cash in case retail investors
    want their money back. They do it by selling securities from countries
    that have not been hit by the crisis yet.
    The causes of currency crises such as the 1997–98 Asian currency
    crises, can probably be traced back to the interplay of countries’
    structural imbalances and weak policies with shifts in market expectations,
    both amplifying each other to provide the principal source of
    bubbles and crashes in emergent markets 311
    instability [332]. In other words, the crisis resulted from the interaction
    of structural weaknesses and volatile international capital markets, as
    well as inadequate supervision of the banking and financial sectors and
    the rapid transmission of the crisis across countries linked by trade
    and common credit sources. Using a panel of annual data for over
    100 developing countries from 1971 through 1992, it is found that
    currency crashes tend to occur when output growth is low, the growth
    of domestic credit is high, and the level of foreign interest rates are
    high [141].
    In the theoretical framework developed in chapter 5, it is possible to
    incorporate a feedback loop whereby prices affect the probability of a
    crash and vice versa. The higher the price, the higher the hazard rate
    or the increase rate of the crash probability. This process reflects the
    phenomenon of a self-fulfilling crisis, a concept that has recently gained
    wide attention, in particular with respect to the crises that occurred in
    seven countries (Mexico, Argentina, Thailand, South Korea, Indonesia,
    Malaysia, and Hong Kong) [245]. They have all experienced severe economic
    recessions, worse than anything the United States had seen since
    the 1930s. It is believed that this is due to the feedback process associated
    with the gain and loss of confidence from market investors. Playing the
    confidence game forced these countries into macroeconomic policies that
    exacerbated slumps instead of relieving them [245]. For instance, when
    the Asian crisis struck, countries were told to raise interest rates, not to
    cut them, in order to persuade some foreign investors to keep their money
    in place and thereby limit the exchange-rate plunge. In effect, countries
    were told to forget about macroeconomic policy; instead of trying to prevent
    or even alleviate the looming slumps in their economies, they were
    told to follow policies that would actually deepen those slumps, all this
    for fear of speculators. Thus, it is possible that a loss of confidence in a
    country can produce an economic crisis that justifies that loss of confidence:
    countries may be vulnerable to what economists call self-fulfilling
    speculative attacks. If investors believe that a crisis may occur in the
    absence of certain actions, they are surely right, because they themselves
    will generate that crisis. In other words, because their growth has been
    predicated on access to foreign capital, the Asian countries faced a kind
    of Hobson’s choice between economic policies that reassure the financial
    markets and policies that might produce better results in the domestic
    economy and create less stress on social stability. One side is concerned
    with creating the right response in the financial markets. The other is
    more concerned with the impact of the IMF’s reforms on the domestic
    economies and political stability of the affected countries.
    312 chapter 8
    In the same spirit, the Joint Economic Committee of the Congress of
    the United States recently released a new study finding a combination
    of perverse incentives as a key contributing factor to recent financial
    crises in Asian emerging economies [362]. The report, entitled “Financial
    Crises in Emerging Markets: Incentives and the IMF,” finds that
    incentives to overextend credit (created by a combination of government
    guarantees, risky lending opportunities, and low levels of ownercontributed
    equity capital) often produce conditions resulting in financial
    crises. The impact of these incentives is similar to what troubled U.S.
    savings and loan and banking industries in the 1980s and early 1990s.
    The study demonstrates that recent IMF lending and prospects for its
    future lending serve to reinforce existing counterproductive incentives
    and create an additional layer of risk subsidies at the international level.
    The corresponding moral hazard problem is that investors take unreasonable
    risks because they know that the IMF will act as a lender of
    last resort. The imitation and herding mechanisms are thus unleashed
    without much restraint.
    Another example of a pronounced synchronization between western
    European stock markets, unrelated to a U.S. event, comes from the period
    following the crashes/corrections on most emerging stock markets in
    early 1994. This time period is associated with sharply rising U.S. interest
    rates. Whereas the S&P 500 dipped less than 10% and recovered
    within a few months, the effect of the emergent market crisis was much
    more profound on smaller Western stock markets worldwide. The toll
    on a range of Western countries resembled that of a minirecession, with
    drops between 18% (London) and 31% (Hong Kong) over a period from
    about five months (London) to about thirteen months (Madrid), as summarized
    in Table 8.3. For each stock market, the decline in the logarithm
    of the index has been fitted with the log-periodic power law equation. In
    Figures 8.34–8.37, the decreases in all the stock markets analyzed can
    be quantified as log-periodic antibubbles.
    From Table 8.3, we observe that the value of the preferred scaling
    ratio � = e2�/� is very consistent with � ≈ 2�0 ± 0�3. This is remarkable
    considering that these stock markets belong to three very different
    geographical regions of the world (Europe, Asia, and the Pacific).
    With respect to the value of the exponent m2, the fluctuations are, as
    usual, much larger. However, excluding New Zealand and Hong Kong,
    we obtain m2
    ≈ 0�4 ± 0�1, which again is quite reasonable compared
    to the major financial markets [209]. The amplitudes of the log-periodic
    oscillations are remarkably similar with B2C ≈ 0�03 ? 0�04, except for
    London (≈ 0�02) and Milan (≈ 0�05).
    bubbles and crashes in emergent markets 313
    Table 8.3
    Characteristics of the fits of the 1994 antibubble on the Western financial markets plus
    Hong Kong following the emerging markets collapse in early 1994
    Stock market tc tmax tmin % drop m2 � �
    United Kingdom 94�08 94�09 94�48 18% 0�25 7�6 2�3
    Hong Kong 94�09 94�09 94�53 31% 0�03 11 1�8
    Australia 94�08 94�09 95�11 22% 0�46 8�0 2�2
    New Zealand 94�08 94�09 94�95 23% 0�09 7�7 2�3
    France 94�06 94�09 95�20 27% 0�51 12 1�7
    Spain 94�08 94�09 95�23 27% 0�28 13 1�6
    Italy 94�36 94�36 95�21 28% 0�35 9�2 2�0
    Switzerland 94�08 94�08 94�54 22% 0�45 12 1�7
    tc is the critical time predicted from the fit of the market index to equation (15). When multiple
    fits exist, the fit with the smallest difference between tc and tmax is chosen. Typically, this will be
    the best fit, but occasionally it is the second best fit. The other parameters m2, �, and � of the fit
    are also shown. The fit is performed from the time tmax, at which the market index achieved its
    highest maximum before the decrease, to the time tmin, which is the time of the lowest point of the
    market before a shift in the trend. The percentage drop is calculated from the total loss from tmax
    to tmin. Reproduced from [218].
    94.1 94.2 94.3 94.4 94.5 94.1 94.2 94.3 94.4 94.5
    ’FTSE (London)’
    Best fit
    Second best fit
    Best fit
    ‘Hong Kong’
    Fig. 8.34. Left panel: FTSE (London). The two lines are the best and the second
    best fit with equation (15). Right panel: Hong Kong. Note the small value for the
    exponent m2 given in Table 8.3. This is presumably due to the undersampling of
    the data in the very first part of the data set. Reproduced from [218].
    314 chapter 8
    94.2 94.4 94.6 94.8 95 94 94.2 94.4 94.6 94.8 95 95.2
    Best fit
    New Zeeland
    Best fit
    ’CAC (Paris)’
    Best fit
    Second best fit
    Fig. 8.35. Left panel: The Australian and New Zealand stock market indices. Right
    panel: The French CAC40. The two lines are the best and the second best fit with
    equation (15). Reproduced from [218].
    Several notable economists, J. E. Stiglitz and, recently, P. Krugman in
    particular as well as financier George Soros, have argued that markets
    should not be left completely alone. The mantra of the free-market
    purists requiring that markets should be totally free may not always be
    the best solution, because it overlooks two key problems: (1) the tendency
    of investors to develop strategies that may destabilize markets in
    a fundamental way and (2) the noninstantaneous adjustment of possible
    imbalance between countries. Soros has argued that real-world interna-
    94.1 94.2 94.3 94.4 94.5 94.4 94.6 94.8 95 95.2
    Best fit
    Second best fit
    Best fit
    Fig. 8.36. Left panel: The Swiss stock market index. The lines are the two best fits
    with equation (15). Right panel: Italian stock market index. Reproduced from [218].
    bubbles and crashes in emergent markets 315
    Angular Frequency w
    Distribution P(w)
    94.2 94.4 94.6 94.8 95 95.2 2 4 6 8 10 12 14 16 18
    Best fit
    Fig. 8.37. Left panel: The Spanish (Madrid) stock market index. Right panel: Distribution
    of the log-periodic angular frequency � for the fits of the previous antibubbles
    using the price (dashed line) and the logarithm of the price (continuous line).
    Reproduced from [218].
    tional financial markets are inherently volatile and unstable since “market
    participants are trying to discount a future that is itself shaped by market
    expectations.” This question is of course at the center of the debate
    on whether local and global markets are able to stabilize on their own
    after a crisis such as the Asian crisis, which started in 1997. In this
    example, to justify the intervention of the IMF, U.S. Treasury Secretary
    Rubin warned in January 1998 that global markets would not be able to
    stabilize in Asia on their own and that a strong role on the part of the
    IMF and other international institutions and governments was necessary,
    lest the crisis spread to other emerging markets in Latin America and
    Eastern Europe.
    The following analogy with forest fires is useful to illustrate the nature
    of the problem: In many areas around the world, the dry season sees
    numerous large wildfires, sometimes with deaths of firefighters and other
    people and the destruction of many structures and large forests. It is
    widely accepted that livestock grazing, timber harvesting, and fire suppression
    over the past century have led to unnatural conditions, such
    as excessive biomass (too many trees without sufficient biodiversity and
    dead woody material) and altered species mix, in the pine forests of the
    western United States, in the Mediterranean countries, and elsewhere.
    These conditions make the forests more susceptible to drought, insect and
    disease epidemics, and other forest-wide catastrophes, and in particular
    large wildfires [167]. Interest in fuel management to reduce fire control
    costs and damage has been renewed due to the numerous, destructive
    316 chapter 8
    wildfires that have spread across the western United States. The most
    frequently used technique of fuel management is fire suppression. Recent
    reviews comparing southern California on the one hand, where management
    has been active since 1900, and Baja California (northern Mexico)
    on the other hand, where management is essentially absent (a “let-burn”
    strategy) highlight a remarkable fact [301, 308]: only small and relatively
    moderate patches of fires occur in Baja California, compared to a wide
    distribution of fire sizes in southern California, including huge destructive
    fires. The selective elimination of small fires (those that can be
    controlled) in normal weather in southern California restricts large fires
    to extreme weather episodes, a process that encourages broad-scale high
    spread rates and intensities. It is found that the danger of fire suppression
    is the inevitable development of coarse-scale bush fuel patchiness and
    large-instance fires, in contradistinction with the natural self-organization
    of small patchiness in let-burn areas. Taken at face value, the let-burn
    theory seems paradoxically the correct strategy for maximizing the protection
    of property and of resources, at minimal cost.
    This conclusion seems to be correct when the fuel is left on its own
    to self-organize in a way consistent with the dynamics of fires. In other
    words, the fuel-fire constitutes a complex nonlinear system with negative
    and positive feedbacks that may be close to optimal: more fuel favors
    fire; fires decrease the instantaneous level of fuel but may accelerate its
    future production; many small fires create natural barriers for the development
    and extension of large fires; fires produce rich nutrients in the
    soil; fires have other benefits, for instance, a few species, notably lodgepole
    pine and jack pine, are serotinous: their cones will only open and
    spread their seeds when they have been exposed to the heat of a wildfire.
    The possibility for complex nonlinear systems to find the “optimal” or
    to be close to the optimal solution has been stressed before in several
    contexts [97, 300, 404]. Let us mention, for instance, a model of fault
    networks interacting through the elastic deformation of the crust and
    rupturing during earthquakes, which finds that faults are the optimal geometrical
    structures accommodating the tectonic deformation: they result
    from a global mathematical optimization problem that the dynamics of
    the system solves in an analog computation, that is, by following its
    self-organizing dynamics (as opposed to digital computation performed
    by digital computers). One of the notable levels of organization is called
    self-organized criticality [26, 394] and has been applied in particular to
    explain forest fire distributions [280].
    Baja California could be a representative of this self-organized regime
    of the fuel-fire complex left to itself, leading to many small fires and
    bubbles and crashes in emergent markets 317
    few big ones. Southern California could illustrate the situation where
    interference both in the production of fuel and also in its combustion
    by fires (by trying to stop fires) leads to a very broad distribution, with
    many small and moderate controlled fires and too many uncontrollable
    very large ones.
    Where do stock markets stand in this picture? The proponents of
    the “let-alone” approach could get ammunition from the Baja/southern
    California comparison, but they would forget an essential element: stock
    markets and economies are more like southern California than Baja
    California. They are not isolated. Even if no government or regulation
    interferes, they are “forced” by many external economic, political, and
    climatic influences that impact them and on which they may also have
    some impact. If the example of the wildland fires has something to teach
    us, it is that we must incorporate into our understanding both the selforganizing
    dynamics of the fuel-fire complex as well as the different
    exogenous sources of randomness (weather and wind regimes, natural
    lightning strike distribution, etc.).
    The question of whether some regulation could be useful is translated
    into whether southern California fires would be better left alone. Since
    the management approach fails to function fully satisfactorily, one may
    wonder whether the let-burn scenario would not be better. This has in
    fact been implemented in Yellowstone Park as the “let-burn” policy but
    was abandoned following the huge Yellowstone fires of 1988. Even the
    let-burn strategy may turn out to be unrealistic from a societal point-ofview
    because allowing a specific fire to burn down may lead to socially
    unbearable risks or emotional sensitivity, often discounted over a very
    short time horizon (as opposed to the long-term view of land management
    implicit in the let-burn strategy).
    We suggest that the most momentous events in stock markets, the large
    financial crashes, can indeed be seen as the response of a self-organized
    system forced by a multitude of external factors in the presence of regulations.
    The external forcing is an essential element to consider and it
    modifies the perspective on the “let-alone” scenario. For instance, during
    the recent Asian crises, the IMF and the U.S. government considered that
    controls on the international flow of capital were counterproductive or
    impractical. J. E. Stiglitz, the chief economist of the IMF until 2000, has
    argued that in some cases it was justified to restrict short-term flows of
    money in and out of a developing economy and that industrialized countries
    sometimes pushed developing nations too fast to deregulate their
    financial systems. The challenge remains, as always, to encourage and
    318 chapter 8
    work with countries that are ready and able to implement strong corrective
    actions and to cooperate toward finding the financial solutions best
    suited to the needs of the individual case and the broader functioning of
    the global financial system when difficulties arise [81].
    In 1987, economist and Nobel prize winner James Tobin suggested
    two possible routes for reform of the international monetary system in
    order to control the world’s speculative financial system and the “casino
    economy” [439]:
  7. The first route consists in making currency transactions more costly to
    reduce capital mobility and speculative exchange rate pressures. This
    approach, which has become known as the “Tobin tax,” has become
    most popular among many new economists in the form of an internationally
    uniform tax on all spot conversions of one currency into
    another, proportional to the size of the transaction. Conventional anti–
    Tobin tax arguments include that it will dry up liquidity, be impossible
    to collect, and invite offshore forex operations.
  8. The second route consists in a greater world economic integration,
    implying eventual monetary union and a World Central Bank. This
    could take the form of an International Currency Unit administered
    by a World Central Bank and based on an equivalent “basket” of
    goods in each country. The value of these “baskets” in domestic currency
    would determine relative exchange rates, which would therefore
    depend on real domestic economic conditions rather than short-term
    currency movements [439]. Another form of integration could take
    the form of coordination of interest-rate policies among the United
    States, the European Union, and Japan, thereby enabling countries to
    pursue their own interest-rate objectives without destabilization from
    competing interest rate policies induced by foreign exchange rate transmissions
    and speculation. Another proposition is to establish a notfor-
    profit global foreign exchange facility (FXE) to perform foreign
    currency exchange transactions. It could be set up as a public utility,
    possibly franchised by a group of governments and the United Nations
    (UN) to offer a little competition to private forex banks, and in partnership
    with the UN, IMF, and BIS (Bank for International Settlements)
    [439]. Currency market “circuit-breaker fees,” analogous to a similar
    fee on Wall Street, could also be used in conjunction with halts in
    trading (common on all stock exchanges) if a currency came under
    speculative attack. This would represent an important social innovation
    because it offers national governments and central banks a new domestic
    macromanagement tool to insulate their currencies and economies
    bubbles and crashes in emergent markets 319
    from attack without having to raise interest rates and subject their citizens
    and businesses to a recession [439].
    At present, it seems that the proposals and measures taken for combating
    risks inherent in the global financial system have no real chance for
    chapter 9
    prediction of bubbles,
    crashes, and
    The time arrow is inexorably projecting us
    towards the undetermined future. Predicting the future captures the
    imagination of all and is perhaps the greatest challenge. “Prophets” have
    historically terrified or inspired the masses by their visions of the future.
    Until recently, science has mostly avoided this question by focusing
    on another kind of prediction, that of novel phenomena such as the
    prediction by Einstein of the deviation of light by the sun’s gravitation
    field, the prediction of the elusive particles called neutrinos by Pauli, and
    the prediction of the intermediate bosons within the electroweak theory
    by Weinberg and Salam, to cite just a few examples. Scientifically based
    predictions of the future, typically using computerized mathematical
    models, is a more recent phenomenon which is becoming pervasive in a
    modern society trying to control its environment and mitigate risks. In
    the real world, efforts to predict are frustrated because scientists have
    not nailed down all the physical workings and because a substantial
    measure of uncertainty remains in the characterization of the system,
    present and future. The result is a considerable range of uncertainty.
    Therefore, while mathematical modeling and computer simulations made
    prediction of crashes and of antibubbles 321
    reasonable predictions possible, they are always uncertain; results are,
    by definition, a model of reality, not reality itself.
    Predictions of trend-reversals, changes of regime, or “ruptures” is
    extraordinarily difficult and unreliable in essentially all real-life domains
    of applications, such as economics, finance, weather, and climate. It
    is possibly the most difficult challenge and arguably the most interesting
    and useful. The two known strategies for modeling, namely analytical
    theories and brute-force numerical simulations of resulting large
    algebraic systems, are both unable to offer effective solutions for most
    concrete problems. Simulation studies of ruptures suffer from numerous
    sources of error, including model mispecifications and inaccurate
    numerical representation of the mathematical models, which are especially
    important for rare extreme events [232].
    The following example borrowed from the field of climatology illustrates
    the point. In view of the growing concensus on global warming,
    it is instructive to remember that in the 1970s there was growing concern
    among scientists that the earth was cooling down and might enter
    a new ice-age similar to the previous little 1400–1800 ice age or even
    worse [61, 368, 429, 155]! Now that global warming is almost universally
    recognized, we can appreciate in hindsight how short-sighted was
    this “prediction.” The situation is essentially the same nowadays: estimations
    of future modest changes of economic growth rates are rather good,
    but predictions of strong recessions and of crashes are utterly unreliable
    most of the time. For instance, the almost overwhelming concensus on
    the reality and magnitude of global warming is based on a clear trend
    over the twentieth century that has finally emerged above the uncertainty
    level. We stress that this concensus is not based on the prediction of a
    reversal or regime switch. In other words, scientists are good at recognizing
    a trend once already deeply immersed in it: we needed a century
    of data to extract a clear signal of a trend on global warming. In contrast,
    the techniques presently available to scientists are bad at predicting most
    changes of regime.
    In economics and finance, the situation may be even worse, as the people’s
    expectation of the future, their greediness, and their fear intertwine
    to construct the indeterminate future. On this question of prediction, Federal
    Reserve Chairman Alan Greenspan [177] said: “Learn everything
    you can, collect all the data, crunch all the numbers before making a
    prediction or a financial forecast. Even then, accept and understand that
    nobody can predict the future when people are involved. Human behavior
    hasn’t changed; people are unpredictable. If you’re wrong, correct
    your mistake and move on.” The fuzziness resulting from the role of
    322 chapter 9
    the expectation and discount of the future on present investor decisions
    may be captured by another famous quote from Greenspan before the
    Senate Banking Committee, June 20, 1995: “If I say something which
    you understand fully in this regard, I probably made a mistake.”
    Uncertainty in predictions is inherent in the complexity of the task.
    Nevertheless, predictions are useful. For instance, weather forecasts
    retain a large degree of uncertainty. Nevertheless, they are useful because
    they are better than pure chance once users know their shortcomings
    and take those into consideration. Predictions can be compared with
    observations and corrected for new improved predictions, a process
    called assimilation of data into the forecast. It is thus essential to use
    “error bars” and quantify uncertainties associated with any given prediction:
    hard numbers on predictions are misleading; only their probability
    distribution of success carries the relevant information. The flood of
    Grand Forks, ND, by the Red River of the North is a case in point.
    When it was rising to record levels in the spring of 1997, citizens and
    officials relied on scientists’ predictions about how high the water would
    rise. A 49-foot forecast lulled the town into a false sense of security,
    because more precision was assigned to the forecast than was warranted.
    Actually, there was a wider range of probabilities; the river ultimately
    crested at 54 feet, forcing 50,000 people to abandon their homes fast.
    Had the full range of scenarios and probabilities been appreciated by
    the citizens, countermeasures could probably have been taken, allowing
    more people to preserve their possessions. The important message here
    is that the 49-foot forecast was not necessarily wrong. The possible
    deviations from this best guess were sorely missing. A probabilistic
    forecast allowing for at least two scenarios would have been much more
    instructive. It could have been phrased, for instance, as “there is a 50%
    probability that the river will crest at a level no larger than 49 foot and a
    90% probability that the river will crest at a level no larger than 52 foot.”
    Note that the first part of this statement carries the same information
    about the best guess (in the median sense) of the crest, while the second
    part provides a quantification of the uncertainty. With that, it is then
    possible in principle to weigh the cost of mitigation measures to respond
    to any given fluctuations from the best guess. The message here is to
    keep in mind the coexistence of several possible scenarios (and not of a
    best one or an average one) with their associated estimated likelihood.
    The importance of working with several scenarios is illustrated in
    Figure 9.1, which represents the evolution of an ensemble of trajectories
    obeying a set of equations (now called the Lorenz system) proposed by
    the meteorologist Lorenz [270] as a parody of atmospheric dynamics.
    prediction of crashes and of antibubbles 323
    Fig. 9.1. Evolution of the probability density function represented in perspective for
    the variable v in a perfect ensemble under the Lorenz equations, which provide a
    simplified model of atmospheric dynamics. The variable v is plotted along the horizontal
    axis such that the center of symmetry is the initial condition. Time t is plotted
    along the vertical. As time increases (upwards), the initially sharp distribution at
    t = 0 decays and widens, but then shows true return of skill (at t = 0�4) by growing
    and sharpening. Later, the distribution bifurcates in two branches: the variable v is
    either largely above or below the initial value, while an average prediction predicts
    a value at the center, which in reality is almost never observed. This illustrates the
    fundamental limits of forecasts based on one representative value. Reproduced from
    Note that the study of this system was instrumental in the development
    of the theory of chaos in the 1970s and 1980s. The horizontal axis represents
    the proxy for a meteorological variable, say wind velocity v. The
    vertical axis is time, which goes from 0 to 5 in this plot. For each time,
    324 chapter 9
    the third dimension in perspective shows the probability distribution of
    the wind velocity v: the maximum of the initial bell-shape distribution
    corresponds to the best initial guess of what is the present state of the
    system. The width of the bell-shape curve quantifies the initial uncertainty
    of our observations: we perform an initial measurement of the
    wind velocity and we know that any measure has some uncertainty, here
    quantified by the probability that the true initial condition deviates from
    the best estimate corresponding to the peak. To evolve this distribution,
    each of 4,096 initial conditions chosen at random is evolved according to
    the Lorenz equations of motion. Each of the 4,096 initial conditions thus
    defines a possible trajectory. At each time of interest, the value of v for
    each trajectory is measured and the aggregation of the 4,096 measures
    provides the statistics to construct the distribution of v. At early times,
    the distribution spreads out: notice that the peak decreases in amplitude
    and the distribution widens. This reflects an increasing uncertainty
    in the value of v after some time and thus a loss of prediction skill.
    Up to time t = 1�5, we observe alternative deterioration and improvement
    of prediction skill, as the distribution function widens and sharpens
    again periodically. This is the first rather nonintuitive lesson: regions of
    decreasing uncertainty may exist in a chaotic dynamics [388]. Increasing
    the forecast horizon does not always lead to a degradation of the
    prediction, in contrast to standard views on chaotic dynamics. Beyond
    t = 1�5, the distribution function bifurcates into two separate branches.
    At t = 2�5, it is clear that the velocity will have either a large positive or
    large negative deviation from the initial value, yet the optimal prediction
    made by averaging over all possible trajectories is close to the initial
    value. This is a fundamental shortcoming of such standard forecasting
    techniques for nonlinear systems [388]. It underscores the importance of
    thinking in terms of distributions or ensembles of scenarios, as opposed
    to a mean, an average, a median, or a representative forecast. At time
    t = 2�5, no single trajectory is a reliable representative of the complexity
    of the dynamics. Because of the structure of the dynamics in this
    example, as least two leading scenarios must be envisaged.
    Thinking of predictions as intrinsically linked to their associated
    uncertainty is even more important when taking into account the combination
    of observational uncertainty and model error. Model error refers
    to the fact that, in general, we do not know the exact equations of the
    dynamics of the system we are interested in forecasting. We have only
    an approximate understanding of its complexity, and the models used
    for prediction by force capture only a part of all ingredients. This model
    error obviously places severe limits on what we can say about the future
    prediction of crashes and of antibubbles 325
    of a system. Working with an ensemble of trajectories for each model
    belonging to an ensemble of models is advocated as one way to mitigate
    these fundamental limitations [386].
    We describe below how these ideas can be put into concrete form
    for the prediction of financial crashes. The different models will correspond
    to different implementations of the theory of critical points with
    log-periodic power laws. Different scenarios will be generated for each
    model by the different solutions obtained by the fitting procedure.
    Before studying the issue of prediction, the question of a possible selection
    bias of the fitted financial time series presented in chapters 7 and
    8 must be addressed. By selecting time windows on the basis of the
    existence of (1) a change of regime and acceleration of the market price
    and of (2) a crash or large correction at their end, we may have pruned
    the data so that, by chance alone, the fits with the log-periodic power
    law formula may have been qualified. This issue has to be raised each
    time a pattern is proposed as an indicator with some predictive skill.
    There is a fundamental mathematical reason for this: the English mathematician
    F. P. Ramsey proved that complete disorder is an impossibility
    [173, 172]. Every large set of numbers, such as an ensemble of financial
    price series or points or objects, necessarily contains highly regular
    patterns. For instance, the night sky appears to be filled with constellations
    in the shape of straight lines, rectangles and pentagons, which
    bear suggestive names such as the lion, the bull, or the scorpion, given
    by ancient astronomers. Could it be that such geometric patterns arise
    from unknown forces in the cosmos? In 1928, Ramsey proved that such
    patterns are implicit in any large structure. Given enough stars, one can
    always find a group that very nearly forms a particular pattern. Given a
    sufficiently long series of numbers, you will find any pattern in it, such
    as your birthdate or any other number of special interest to you. Intuitively,
    the argument underlying this theorem is that if it was not the case
    that any pattern could be approximately found in a random set, this set
    would not be really random. Randomness is such that any pattern can
    The relevant question is, then, to figure out just how many stars, numbers,
    or figures are required to guarantee a certain desired pattern. In
    other words, how probable is it to observe a desired substructure in a
    326 chapter 9
    given set? Answering this question is the domain of statistics and its
    economic application, econometrics. If one can show that the number of
    stars needed to obtain a particular pattern is not much larger than the
    observed number, we can ask with reason whether this particular pattern
    may not result from chance alone in this particular set. This is the
    essence of the method of statistical hypothesis testing which constructs
    so-called “statistical confidence levels”: if the confidence level of a phenomenon
    is, say, 99%, this means that there is only a remote probability
    of 1 in 100 that the phenomenon in question is due to chance.
    In the present context, we first refer to the computer experiment summarized
    in the section titled “The Slow Crash of 1962 Ending the ‘Tronics’
    Boom” of chapter 7, in which fifty 400-week intervals in the period
    1910–1996 of the DJIA were chosen at random [209]. This experiment
    shows that fits, which in terms of the fitting parameters correspond to
    the three crashes of 1929, 1962, and 1987, are not likely to occur “accidentally.”
    Feigenbaum and Freund have also looked at randomly selected
    time widows in the real data and generally found no evidence of logperiodicity
    in these windows unless they were looking at a time period
    in which a crash was imminent [128]. More recently, Feigenbaum has
    examined the first differences for the logarithm of the S&P 500 from
    1980 to 1987 and finds that he cannot reject the log-periodic component
    at the 95% confidence level [127]: in plain words, this means that the
    probability that the log-periodic component results from chance is about
    or less than one in twenty.
    To test furthermore the solidity of the advanced log-periodic hypothesis,
    Johansen, Ledoit, and I [209] tested whether the null hypothesis that
    a standard statistical model of financial markets, called the GARCH(1,1)
    model with Student-distributed noise, could “explain” the presence of
    log-periodicity. In the 1,000 surrogate data sets of length 400 weeks generated
    using this GARCH(1,1) model with Student-distributed noise and
    analyzed as for the real crashes, only two 400-week windows qualified.
    This result corresponds to a confidence level of 99�8% for rejecting the
    hypothesis that GARCH(1,1) with Student-distributed noise can generate
    meaningful log-periodicity. There is no reference to a crash; the question
    is solely to test if log-periodicity of the strength observed before
    the 1929 and 1987 crashes can be generated by one of the standard
    benchmarks of financial time series used intensively by both academics
    and practitioners. If in addition, we add that the two spells of significant
    log-periodicity generated in the simulations using GARCH(1,1) with
    Student-distributed noise were not followed by crashes, then the case is
    even stronger for concluding that real markets exhibit behaviors that are
    prediction of crashes and of antibubbles 327
    dramatically different from the one predicted by one of the most fundamental
    benchmarks of the industry. Indeed, the frequency of crashes
    in the Monte Carlo simulations was much smaller than the frequency
    of crashes in real data: if one of the most frequently used benchmarks
    of the industry is incapable of reproducing the observed frequency of
    crashes, this indeed means that there is something to explain that may
    require new concepts and methods.
    We should stress, however, that no truth is ever demonstrated in science;
    the only thing that can be done is to construct models and reject
    them at a given level of statistical significance. Those models that are not
    rejected when pitted against more and more data progressively acquire
    the status of theory (think, for instance, of quantum mechanics, which
    is repeatedly put to tests). In the present context, it is clear that, in a
    purist sense, we shall never be able to “prove” the existence of a logperiodicity
    genuinely associated with specific market mechanisms. The
    next best thing we can do is to take one by one the best benchmarks of
    the industry and test them to see if they can generate the same structures
    as we document. It would, of course, be interesting to test more
    sophisticated models in the same way as for the GARCH(1,1) model
    with Student-distributed noise. However, we caution that rejecting one
    model after another will never prove that log-periodicity exists. This is
    outside the realm of statistical and econometric analysis. If more and
    more models are unable to “explain” the observed log-periodicity, this
    means, however, that log-periodicity is an important fact that needs to
    be understood.
    Another worry is that integrated processes, like a random walk which
    sums up random innovations over time, can generate log-periodic patterns
    from pure chance. Actually, Huang et al. [203] specifically tested
    the following problem: Under what circumstances can an integrated process
    produce spurious log-periodicity? The answer obtained after lengthy
    and thorough Monte Carlo tests is twofold. (1) For approximately regularly
    sampled time series, as in the case of the financial time series,
    taking the integral of a noisy log-periodic function destroys the logperiodic
    signal! (2) Only when sampling rates increase exponentially or
    as a power law of tc
    ? t can spurious log-periodicity in integrated processes
    be observed. The name “Monte Carlo” refers to the notion that
    random (as in a casino) series with prescribed properties are used to test
    the probability that a given pattern can occur by chance: if this probability
    is very small, the corresponding pattern is probably not due to
    chance. The consequence is that it may result from a causal set of effects
    that can be understood and used.
    328 chapter 9
    Ultimately, only forward predictions can demonstrate the usefulness of
    a theory (see the section below titled “Forward Predictions”), thus only
    time will tell. However, as we have suggested by the many examples
    reported in chapters 7 and 8 and from the discussion offered below, the
    analysis points to an interesting predictive potential. However, a fundamental
    question concerns the use of a reliable crash prediction scheme,
    if any. Assume that a crash prediction is issued stating that a crash of
    an amplitude between 20% and 30% will occur between one and two
    months from now. At least three different scenarios are possible [217]:
    � Nobody believes the prediction, which was then futile, and, assuming
    that the prediction was correct, the market crashes. One may consider
    this as a victory for the “predictors” but as we have experienced in
    relation to our quantitative prediction of the change in regime of the
    Nikkei index [213, 216], this would only be considered by some critics
    just another “lucky one” without any statistical significance (see the
    section below entitled “Estimation of the Statistical Significance of
    the Forward Predictions” [216] and below for an alternative Bayesian
    � Everybody believes the warning, which causes panic, and the market
    crashes as consequence. The prediction hence seems self-fulfilling and
    the success is attributed more to the panic effect than to real predictive
    � Sufficiently many investors believe that the prediction may be correct,
    investors make reasonable adjustments, and the steam goes off the
    bubble. The prediction hence disproves itself.
    None of these scenarios is attractive. In the first two, the crash is
    not avoided, and in the last scenario the prediction disproves itself and
    as a consequence the theory looks unreliable. This seems to be the
    inescapable lot of scientific investigations of systems with learning and
    reflective abilities, in contrast with the usual inanimate and unchanging
    physical laws of nature. Furthermore, this touches upon the key problem
    of scientific responsibility. Naturally, scientists have a responsibility to
    publish their findings. However, when it comes to the practical implementation
    of those findings in society, the question becomes considerably
    more complex, as history has taught us. We believe, however, that
    increased awareness of the potential for market instabilities, offered in
    particular by our approach, will help in constructing a more stable and
    efficient stock market.
    prediction of crashes and of antibubbles 329
    Time is converted into decimal year units: for nonleap years, 365
    days = 1�00 year, which leads to 1 day = 0�00274 years. Thus 0�01
    year = 3�65 days and 0�1 year = 36�5 days or 5 weeks. For example,
    October 19, 1987 corresponds to 87�800.
    What Is the Predictive Power of Equation (15) on Page 232?
    Table 9.1 presents a summary of equation (15)’s predictive power for the
    1929, 1987, and 1998 crashes on Wall Street and the 1987, 1994, and
    1997 crashes on the Hong Kong stock exchange as well as the collapse
    of the U.S. dollar in 1985 and the crash on the Nasdaq in April 2000,
    all cases previously discussed in chapter 7.
    We see that, in all nine cases, the market crash started at a time
    between the date of the last point and the predicted tc. And with the
    exception of the October 1929 crash, in all cases the market ended its
    decline less than approximately one month after the predicted tc. These
    results suggest that predictions of crashes with equation (15) is indeed
    Table 9.1
    Crash tc tmax tmin % drop m2 � �
    1929 (DJ) 30�22 29�65 29�87 47% 0�45 7�9 2�2
    1985 (DM) 85�20 85�15 85�30 14% 0�28 6�0 2�8
    1985 (CHF) 85�19 85�18 85�30 15% 0�36 5�2 3�4
    1987 (S&P) 87�74 87�65 87�80 30% 0�33 7�4 2�3
    1987 (HK) 87�84 87�75 87�85 50% 0�29 5�6 3�1
    1994 (HK) 94�02 94�01 94�04 17% 0�12 6�3 2�7
    1997 (HK) 97�74 97�60 97�82 42% 0�34 7�5 2�3
    1998 (S&P) 98�72 98�55 98�67 19�4% 0�60 6�4 2�7
    1999 (IBM) 99�56 99�53 99�81 34% 0�24 5�2 3�4
    2000 (P&G) 00�04 00�04 00�19 54% 0�35 6�6 2�6
    2000 (Nasdaq) 00�34 00�22 00�29 37% 0�27 7�0 2�4
    tc is the critical time predicted from the fit of the financial time series to the equation (15). The
    other parameters m2��, and � of the fit are also shown. The fit is performed up to the time tmax
    at which the market index achieved its highest maximum before the crash. tmin is the time of the
    lowest point of the market before rebound. The percentage drop is calculated from the total loss
    from tmax to tmin. Several of these crashes have also been listed in Table 7.2. Reproduced from
    330 chapter 9
    How Long Prior to a Crash Can One Identify
    the Log-Periodic Signatures?
    Not only would one like to predict future crashes, but it is important
    to further test how robust the results are. Obviously, if the log-periodic
    structure of the data is purely accidental, then the parameter values
    obtained should depend heavily on the size of the time interval used in
    the fitting. The systematic testing procedure reported in [209] using a
    second-order expansion of the crash hazard rate [397] and a time interval
    of eight years prior to the two crashes of 1929 and 1987 consists in the
    For each of these two crashes, the time interval used in the fitting has
    been truncated by removing points and relaunching the fitting procedure
    for each truncated data set. Specifically, the logarithm of the S&P 500
    was truncated down to an end-date of approximately 1985 and fitted.
    Then, 0�16 years was added consecutively and the fitting was relaunched
    until the full time interval was recovered. Table 9.2 reports the number
    of minima obtained for the different time intervals. This number is to
    some extent rather arbitrary since it naturally depends on the number of
    points used in the preliminary scan as well as the size of the time interval
    used for tc. Specifically, 40�000 points were used and the search on tc
    was chosen in the interval from 0�1 years from the last data point used
    to 3 years forward. What is more interesting is the number of solutions
    of these fits (each “solution” corresponds to a minimum of the error
    between the data and the theoretical function) with reasonable parameters
    referred to as “physical” especially for the values of tc�m2��� and �t ,
    where �t is an additional time parameter quantifying the size of the
    critical region. The general picture to be extracted from this Table 9.2 is
    that a year or more before the crash, the data is not sufficient to give any
    conclusive results at all. This point corresponds to the end of the fourth
    oscillation. Approximately a year before the crash, the fit begins to lock
    in on the date of the crash with increasing precision. In fact, in four of
    the last five time intervals, a fit exists with a tc, which differs from the
    true date of the crash by only a few weeks.
    In order to better investigate this, Table 9.3 shows the corresponding
    parameter values for the other three pertinent variables m2��, and �t .
    The scenario resembles that for tc. This suggests that the fitting procedure
    of [397] is rather robust up to approximately one year prior to the crash.
    However, if one wants to actually predict the time of the crash, a major
    obstacle is the fact that the fitting procedure produces several possible
    dates for the date of the crash, even for the last data set.
    prediction of crashes and of antibubbles 331
    Table 9.2
    Number of minima obtained by fitting different truncated versions of the S&P 500
    time series shown in Figure 7.2 to predict the October 1987 crash using the procedure
    described in the text. Note that the predicted times for the crash are progressively
    postponed as the end-date increases. However, the correct time is identified early (in
    hindsight) and is recurrent in the set of solutions as the end-date increases. Reproduced
    from [397]
    Total “Physical”
    End-date # minima minima tc of “physical” minima
    85�00 33 1 86�52
    85�14 25 4 4 in �86�7 � 86�8�
    85�30 26 7 5 in �86�5 � 87�0�, 2 in �87�4 � 87�6�
    85�46 29 8 7 in �86�6 � 86�9�,1 with 87�22
    85�62 26 13 12 in �86�8 � 87�1�,1 with 87�65
    85�78 23 7 87�48, 5 in �87�0 ? 87�25�, 87�68
    85�93 17 4 87�25� 87�01� 87�34� 86�80
    86�09 18 4 87�29� 87�01� 86� 98� 87�23
    86�26 28 7 5 in �87�2 � 87�4�, 86�93� 86�91�
    86�41 24 4 87�26� 87�36� 87�87� 87�48
    86�57 20 2 87�67� 87�34
    86�73 28 7 4 in �86�8 � 87�0�, 87�37� 87�79� 87�89
    86�88 22 1 87�79
    87�04 18 2 87�68� 88�35
    87�20 15 2 87�79� 88�03
    87�36 15 2 88�19� 88�30
    87�52 14 3 88�49� 87�92� 88�10
    87�65 15 3 87�81� 88�08� 88�04
    Table 9.3
    End-date tc m2 � �t
    86�88 87�79 0�66 5�4 7�8
    87�04 87�68� 88�35 0�61� 0�77 4�1� 13�6 12�3� 10�2
    87�20 87�79� 88�03 0�76� 0�77 9�4� 11�0 10�0� 9�6
    87�36 88�19� 88�30 0�66� 0�79 7�3� 12�2 7�9� 8�1
    87�52 88�49� 87�92� 88�10 0�51� 0�71� 0�65 12�3� 9�6� 10�3 10�2� 9�8� 9�8
    87�65 87�81� 88�08� 88�04 0�68� 0�69� 0�67 8�9� 10�4� 10�1 10�8� 9�7� 10�2
    For the last five time intervals shown in Table 9.2, the corresponding parameter values for the
    other three variables m2����t are shown. Reproduced from [397].
    332 chapter 9
    Table 9.4
    The average of the values listed in Table 9.3. Reproduced from [397].
    End-date tc m2 � �t
    86�88 87�79 0�66 5�4 7�8
    87�04 88�02 0�69 8�6 11�3
    87�20 87�91 0�77 10�20 9�8
    87�36 88�25 0�73 9�6 8�0
    87�52 88�17 0�62 10�7 9�9
    87�65 87�98 0�68 9�8 10�2
    As a naive solution to this problem, Table 9.4 shows the average of
    the different minima for tc�m2��, and �t . The values for m2��, and
    �t are within 20% of those for the best prediction, but the prediction
    for tc has not improved significantly. The reason for this is that the fit
    in general “overshoots” the true day of the crash. This overshooting is
    consistent with the rational expectation models of a bubble and crash
    described in chapter 5. Indeed, the critical time tc is not the time of the
    crash but only its most probable value, that is, the time for which the
    asymmetric distribution of the possible times of the crash peaks. The
    occurrence of the crash is a biased random phenomenon which occurs
    with a probability that increases as time approaches tc. Thus, we expect
    that fits will give values of tc which are in general close to but systematically
    later than the real time of the crash: the critical time tc is included
    in the log-periodic power law structure of the bubble, whereas the crash
    is randomly triggered with a biased probability increasing strongly close
    to tc.
    The same procedure was used on the logarithm of the Dow Jones
    index prior to the crash of 1929 shown in Figure 7.7 and the results are
    shown in Tables 9.5, 9.6, and 9.7. One has to wait until approximately
    four months before the crash before the fit locks in on the date of the
    crash, but from that point the picture is the same as for the crash in
  9. The reason for the fact that the fit “locks-in” at a later time for
    the 1929 is obviously the difference in the transition time �t for the two
    crashes which means that the index prior to the crash of 1929 exhibits
    fewer distinct oscillations.
    Feigenbaum [127] has recently confirmed that, for the October 1987
    crash, “excluding the last year of data, the log-periodic component is
    no longer statistically significant.” This should not be a surprise for specialists
    of critical phenomena, and it is naive to expect otherwise. The
    prediction of crashes and of antibubbles 333
    Table 9.5
    Same as Table 9.2 for the October 1929 crash. Reproduced from [397].
    End-date Total # minima “Physical” minima tc of “physical” minima
    27�37 12 1 31�08
    27�56 14 2 30�44� 30�85
    27�75 24 1 30�34
    27�94 21 1 31�37
    28�13 21 4 29�85� 30�75� 30�72� 30�50
    28�35 23 4 30�29� 30�47� 30�50� 36�50
    28�52 18 1 31�3
    28�70 18 1 31�02
    28�90 16 4 30�40� 30�72� 31�07� 30�94
    29�09 19 2 30�52� 30�35
    29�28 33 1 30�61
    29�47 24 3 29�91� 30�1� 29�82
    29�67 23 1 29�87
    Table 9.6
    Same as Table 9.3 for the October 1929 crash. Reproduced from [397].
    End-date tc m2 � �t
    28�90 30�40� 30�72, 0�60� 0�70, 7�0� 7�6, 12�3� 9�5,
    31�07� 30�94 0�70� 0�53 10�2� 13�7 9�0� 11�6
    29�09 30�52� 30�35 0�54� 0�62 11�0� 7�8 12�6� 10�2
    29�28 30�61 0�63 9�5 9�5
    29�47 29�91� 30�1� 29�82 0�60� 0�67� 0�69 5�8� 6�2� 4�5 15�9� 11�0� 10�9
    29�67 29�87 0�61 5�4 15�0
    Table 9.7
    The average of the values listed in Table 9.6. Reproduced from [397].
    End-date tc m2 � �t
    28�90 30�78 0�63 9�6 10�6
    29�09 30�44 0�58 9�4 11�4
    29�28 30�61 0�63 9�5 9�5
    29�47 29�94 0�65 5�5 12�6
    29�67 29�87 0�61 5�4 15�0
    334 chapter 9
    determination of a power law B2�tc
    ? t�m2 is indeed very sensitive to
    noise and to the distance from tc of the data used in the estimation. This
    is well known by experimentalists and numerical scientists working on
    critical phenomena who have invested considerable efforts in developing
    reliable experiments that could probe the system as closely as possible
    to the critical point tc, in order to get reliable estimations of tc and m2.
    A typical rule of thumb is that an error of less than 1% in the determination
    of tc can lead to tenth-of-a-percent errors in the estimation of
    the critical exponent m2. While the situation is improved by the addition
    of the log-periodic component because the fit can lock in on the
    oscillations, the problem remains qualitatively the same. Being one year
    before the critical time corresponds to the situation in which a worker
    on critical phenomena would be trying to get a reliable estimation of tc
    and m2 by trashing the last 15% of the data, which are of course the
    most relevant—an almost impossible task in general.
    We thus caution the reader that jumping into the prediction game
    may be hazardous and misleading: one deals with a delicate optimization
    problem that requires extensive backward and forward testing. Furthermore,
    the formulas discussed here are only “first-order” approximations,
    and novel improved methods have been developed that are not published.
    Finally, one must never forget that the crash has to remain in part a random
    event in order to exist! This is according to the rational expectation
    models described in chapter 5.
    The Simple Power Law
    The concept that a crash is associated with a critical point suggests fitting
    a simple power law
    log�p�t�� = A + B�tc
    ? t�� (17)
    to the price or the logarithm of the price. A fit of the logarithm of the
    S&P 500 index before the October 87 crash gives tc
    = 87�65, � = 0�72,
    �2 = 107, A = 327, and B = ?79 when using data from 1985.7 to
    1987.65. Note that the value of tc obtained from the fit is completely
    dominated by the last values used in the fit. The reason is that the
    information on tc is contained essentially in or is dominated by the
    prediction of crashes and of antibubbles 335
    acceleration in the last points. In contrast, the log-periodic structures contain
    the information on tc in their oscillations that develop much before
    All attempts to use this formula (17) for prediction have been unsuccessful,
    because it is virtually impossible to distinguish this law (17)
    from a noncritical exponential growth when data are noisy. A smooth
    increase like (17) is well known to constrain very poorly the time tc in
    noisy time series. This is why all our empirical efforts were focused on
    the log-periodic formulas.
    The “Linear” Log-Periodic Formula
    We rewrite here the equation (15) that was used before for fitting the
    price of a financial time series in terms of the logarithm of the price:
    log�p�t�� = A + B�tc
    ? t�� �1 + C cos�� log�tc
    ? t� + ��� � (18)
    It turns out that, for time scales of about two years or less, the amplitude
    of the price variation in such periods is not large enough for detecting a
    significant difference between the goodness-of-fit of the fits of p�t� and
    of log�p�t��.
    For practical implementation of the fit of such a formula to a financial
    time series, it is important to stress that the variables A, B, and C enter
    linearly once the other four variables tc, �, �, and � are fixed. The
    best procedure is to determine them analytically through so-called least
    squares minimization and to plug them into the objective function to
    derive a concentrated objective function that depends only on tc, �, �,
    and �.
    Due to the noisy nature of the data and the fact that we are performing
    a highly nonlinear four-parameter fit, there are several local minima. The
    best strategy is to perform a first grid search and then start an optimizer
    (for instance the Levenberg–Marquardt) from all the local optima of the
    grid. The best of the resulting convergence points is taken as the global
    A priori restrictions are imposed on parameter values, which ensure
    that they are plausible. The exponent � needs to be between 0 and 1
    for the price to accelerate and to remain finite. The more stringent criterion
    0�2 < � < 0�8 has been found useful to avoid the pathologies
    associated with the endpoints 0 and 1 of the interval. Recall that the
    angular log-periodic frequency � determines the ratio � of successive
    336 chapter 9
    time intervals between local maxima through the following relationship:
    � = e2�/�. Experience across many disciplines and some theoretical
    arguments [392] suggest that this ratio � should typically lie in the range
    2–3. In practice, we have often used the constraint 5 < � < 15, which
    corresponds to 1�5 < � < 3�5. Obviously, tc must be greater than the last
    date in the sample data being fitted. The phase � cannot be meaningfully
    The “Nonlinear” Log-Periodic Formula
    The nonlinear log-periodic formula used to fit the longest financial time
    series discussed in chapters 7 and 8 is [397]
    log�p�t�� = A + B
    ? t��

    1 +

    ? t

1 + C cos

? t�

  • 2�

    1 +

    ? t
    � (19)
    Using the same least-squares method as for the linear log-periodic formula
    allows one to concentrate away the linear variables A, B, and C and
    form an objective function depending only on tc, �, �, and � as before,
    with the addition of two parameters �t and ��. Since �t is a transition
    time between two regimes, this transition should be observed in the data
    set, and therefore we require it to be between 1 and 20 years. As before,
    the nonlinearity of the objective function creates multiple local minima,
    and the preliminary grid search is used to find starting points for the
    The Shank’s Transformation on a Hierarchy
    of Characteristic Times
    The fundamental idea behind the appearance of log-periodicity is the
    existence of a hierarchy of characteristic scales. Reciprocally, any logperiodic
    pattern implies the existence of a hierarchy of characteristic time
    scales. This hierarchy of time scales is determined by the local positive
    maxima of the function such as log�p�t��. They are given by
    ? tn
    = ��
    � (20)
    prediction of crashes and of antibubbles 337
    � ∝ exp

    ?log �
    tan?1 2�
    � log �

    � = e
    � � (22)
    The spacing between successive values of tn approaches zero as n
    becomes large and tn converges to tc. This hierarchy of scales tc
    ? tn
    is not universal but depends upon the specification of the system. What
    is expected to be universal are the ratios tc
    = �
    2 . From three
    successive observed values of tn, say tn, tn+1, and tn+2, we have
    = t2
    ? tn+2tn
    ? tn
    ? tn+2
    � (23)
    This relation applies the so-called Shanks transformation to accelerate
    the convergence of series. In the case of an exact geometrical series, three
    terms are enough to converge exactly to the asymptotic value tc. Notice
    that this relation is invariant with respect to an arbitrary translation in
    time. In addition, the next time tn+3 is predicted from the first three ones
    = t2
  • t2
    ? tntn+2
    ? tn+1tn+2
    ? tn
    � (24)
    The weakness of this method lies in the identification of the characteristic
    times tn’s which may be quite subjective.
    Application to the October 1929 Crash.
    Looking at the Dow Jones index by eye, we can try to identify the “characteristic”
    times as those of the successive “coarse-grained” local maxima
    that form a geometrical series. We propose t1
    = 1926�3, t2
    = 1928�2,
    and t3
    = 1929�1. Inserting into (23), we get a prediction tc
    = 1929�91.
    This prediction is less than a month off the true date. Notwithstanding
    this positive result, this method is rather unstable, as a change of one
    month or about 0�1 of one of these dates t1� t2� t3 may move the predicted
    date by one month or more. This is why this can be no more
    than an indication that must be taken with a grain of salt. Compared to
    338 chapter 9
    fitting with a complete mathematical formula, this method focuses only
    on specific times and thus loses what may be an important part of the
    Application to the October 1987 Crash.
    Looking at the S&P 500 index, we identify the characteristic times as
    those of the successive coarse-grained local maxima that form a geometrical
    series. We find t1
    = 1986�5, t2
    = 1987�2, and t3
    = 1987�5 or 1987�55.
    Inserting into (23), we get the prediction tc
    = 87�725 and 87�900, respectively.
    These predictions correctly bracket the true date, 87.800.
    As we said earlier, only forward predictions provide a reliable test, avoiding
    the many traps of statistical bias and data snooping.
    We are now going to narrate the forward predictions that A. Johansen
    and the present author have made in the last few years. Specifically, we
    have examined a few of the major indices in real time, essentially continuously
    since 1996, and have tried to apply the methodology described
    before in order to predict a crash, a severe correction, or even a depression
    (called an antibubble in chapter 7). Here, the word “prediction” is
    taken with its full meaning, since the future was unknown at the time
    when each prediction was performed. The term “forward” prediction
    stresses this fact. In contrast, “post-dictions” or retroactive predictions
    are performed by artificially cutting a part of the most recent past in
    recorded time series to perform a prediction of this hidden past. Such
    post-dictions, which were described in the previous section, are very useful
    for testing the skills of a forecasting system by providing much faster
    a larger testing set than would be otherwise available by waiting for the
    future to confirm or disprove a given prediction. However, they never
    completely reproduce the real-time and real-life situations of forward
    We report all cases, successes, and failures, so that the reader can
    judge for himself or herself. We report three successes (U.S. market
    August 1998, Nikkei Japanese market 1999, Nasdaq April 2000), two
    failures (U.S. market December 1997, Nasdaq October 1999), and one
    “semifailure” (U.S. market October 1997).
    Of the successes, only the Nikkei 1999 prediction was publicly
    announced in advance and published. The two others (U.S. market
    prediction of crashes and of antibubbles 339
    August 1998 and Nasdaq April 2000) were made about one month in
    advance but were not published.
    The semifailure concerns the prediction of a crash on the U.S. market
    in October 1997 which was registered by an official agency about
    a month in advance. As we showed in chapter 7, this prediction may
    actually be counted as a semifailure or semisuccess depending on taste,
    since something did happen as far as investors and market commentators
    are concerned (the main U.S. market indices dropped about 7% in
    a day), as can be seen from the many available reports on this event, but
    it was not of sufficient magnitude to qualify as a crash, as the market
    quickly recovered. Other groups have also analyzed this event [129] and
    predicted it [433] with a similar log-periodic analysis.
    Successful Prediction of the Nikkei 1999 Antibubble
    Following the general guidelines described above (see also [214]),
    a prediction was made public on January 25, 1999 by posting a
    preprint on the Los Alamos Internet server; see http://xxx.lanl.gov/abs/
    cond-mat/9901268. The preprint was later published as [213]. The
    prediction stated that the Nikkei index should recover from its 14-year
    low (13�232�74 on January 5, 1999) and reach ≈20�500 a year later,
    corresponding to an increase in the index of ≈50%. This prediction
    was mentioned in a wide-circulation journal in physical sciences which
    appeared in May 1999 [413].
    Specifically, based on a third-order “Landau” expansion generalizing
    the nonlinear log-periodic formula (19), the following formula was established:
    log�p�t�� ≈ A� + �

1 +




B� + C� cos

�log � + ��


1 +


  • 4


1 +


  • � (25)
    describing the time evolution of the Nikkei index p�t�, where � ≡ t ?tc,
    and tc
    = December 31, 1989 is the time of the all-time high of the
    Nikkei index. Equation (25) was then fitted to the Nikkei index in the
    340 chapter 9
    time interval from the beginning of 1990 to the end of 1998, that is,
    a total of nine years. Extending the curve beyond 1998 thus provided
    us with a quantitative prediction for the future evolution of the Nikkei.
    The original figure published in [213], which formed the basis for the
    prediction, is Figure 7.25 of chapter 7.
    In Figure 9.2, the actual and predicted evolution of the Nikkei over
    1999 and later are compared [216]. Not only did the Nikkei experience
    a trend reversal as predicted, but it has also followed the quantitative
    prediction with rather impressive precision. In particular, the prediction
    of the 50% increase at the end of 1999 is validated accurately. The
    prediction of another trend reversal is also accurately predicted, with the
    correct time for the reversal occuring at the beginning of 2000: the predicted
    maximum and the observed maximum match closely. It is impor-
    90 92 94 96 98 100
    Fig. 9.2. Natural logarithm of the Nikkei stock market index after the start of the
    decline from January 1, 1990 until February 2001. The continuous smooth line is
    the extended nonlinear log-periodic formula (25), which was developed in [213] and
    is used to fit adequately the interval of ≈9 years starting from January 1, 1990. The
    Nikkei data is separated into two parts. The dots show the data used to perform the
    fit with formula (25) (dotted line) and to issue the prediction in January 1999 (see
    Figure 7.25). Its continuation as a continuous line gives the behavior of the Nikkei
    index after the prediction has been made. Reproduced from [216].
    prediction of crashes and of antibubbles 341
    tant to note that the error between the curve and the data has not grown
    after the last point used in the fit over 1999. This tells us that the prediction
    has performed well for more than a year. Furthermore, since the
    relative error between the fit and the data is within ±2% over a time
    period of ten years, not only has the prediction performed well, but so
    has the underlying model.
    The fulfillment of this prediction is even more remarkable than the
    comparison between the curve and the data indicates, because it included
    a change of trend: at the time when the prediction was issued, the market
    was declining and showed no tendency to increase. Many economists
    were at that time very pessimistic and could not envision when Japan
    and its market would rebound. For instance, the well-known economist
    P. Krugman wrote on July 14, 1998, at the time of the banking scandal:
    “The central problem with Japan right now is that there just is not enough
    demand to go around—that consumers and corporations are saving too
    much and borrowing too little. � � � So seizing these banks and putting
    them under more responsible management is, if anything, going to further
    reduce spending; it certainly will not in and of itself stimulate the economy.
    � � � But at best this will get the economy back to where it was a year
    or two ago—that is, depressed, but not actually plunging [247].
    Then, on January 20, 1999, Krugman wrote: “The story is starting to look
    like a tragedy. A great economy, which does not deserve or need to be in
    a slump at all, is heading for the edge of the cliff—and its drivers refuse
    to turn the wheel” [249]. In an October 1998 poll of thirty economists
    performed by Reuters (one of the major news and finance data providers
    in the world), only two economists predicted growth for the fiscal year
    of 1998–99. For the year 1999–2000 the prediction was a meager 0.1%
    growth. This majority of economists said that “a vicious cycle in the
    economy was unlikely to disappear any time soon as they expected little
    help from the government’s economic stimulus measures. � � � Economists
    blamed moribund domestic demand, falling prices, weak capital spending
    and problems in the bad-loan laden banking sector for dragging down
    the economy” [315].
    It is in this context that we predicted an approximately 50% increase
    of the market in the 12 months following January 1999, assuming that
    the Nikkei would stay within the error-bars of the fit. Predictions of
    trend reversals is notably difficult and unreliable, especially in the linear
    framework of autoregressive models used in standard economic analyses.
    The present nonlinear framework is well adapted to the forecasting of
    changes of trends, which constitutes by far the most difficult challenge
    342 chapter 9
    posed to forecasters. Here, we refer to our prediction of a trend reversal
    within the strict confines of equation (25): trends are limited periods of
    time when the oscillatory behavior shown in Figure 9.2 is monotonous.
    A change of trend thus corresponds to crossing a local maximum or
    minimum of the oscillations. Our formula seems to have predicted two
    changes of trends, bearish to bullish at the beginning of 1999 and bullish
    to bearish at the beginning of 2000.
    Successful Prediction of the Nasdaq Crash of April 2000
    This prediction was performed by using equation (18). The last point
    used in the fitted data interval was March 10, 2000. The predicted time
    of the crash was May 2, 2000 for the best fit and March 31, 2000 for the
    third best fit. The second best fit had a rather small value for � ≈ 0�08
    and was not considered. Except for slight gains on March 31 and April
    5, 6, and 7, the closing of the Nasdaq composite had been in continuous
    decline since March 24 and lost over 25% in the week ending on Friday,
    April 14. Consequently, the crash occured approximately in between the
    predicted date of the two fits. The best fit is shown in Figure 7.22 in the
    section entitled “The Nasdaq Crash of April 2000” of chapter 7.
    Table 7.2 reports the main characteristics of the fit of the Nasdaq index
    as well as ten other cases. Observe that, in all cases, the market crash
    started at a time between the date of the last point and the predicted tc.
    And with the exception of the October 1929 crash and using the third
    best fit of the Nasdaq crash (this fit had �/2� ≈ 1), in all cases, the
    market ended its decline less than approximately one month after the
    predicted tc.
    The U.S. Market, December 1997 False Alarm
    The shock at the end of October 1997 on the U.S. markets might be
    considered as an aborted crash. Private polling of professional investors
    indeed suggests that many traders were actually afraid that a crash was
    coming at the end of October 1997. After this event, we continued to
    monitor the market closely to detect a possible resurgent instability. An
    analysis using data up to Friday, November 21, 1997 using the three
    methods based on the log-periodic formula (18), its nonlinear extension
    (19), and the Shank formula (23) suggested a prediction for a decrease of
    the price approximately mid-December 1997, with an error bar of about
    two weeks.
    prediction of crashes and of antibubbles 343
    Table 9.8 shows an attempt at predicting a critical time tc with the linear
    log-periodic formula using data ending at the “last date” given in the
    first column, in order to test for robustness. The last “last date” 97�8904
    corresponds to Friday, November 21, 1997 and the data includes the
    close of this Friday. In the fit on the data going up to Friday, November
    21, 1997, ten solutions are found. The first eight all give 25 ≤ � ≤ 38,
    which is quite large. Since large values of � correspond to fast oscillations,
    there is the danger of fitting “noise,” that is, extracting information
    where there is none. It was thus considered safe to reject these solutions,
    which were all proposing tc
    = 98�6 ± 0�1. The last two solutions are
    those that are reported in Table 9.8. Their square error �2 is only 7%
    above the very best fast oscillating solution. Thus the �2 is not a parameter
    that allows one to qualify an acceptable or nonacceptable solution.
    Taking the two predictions obtained for the past dates of 97.7191 and
    of 97�6781 (the superscript 1 means that we select here the best fit of
    the formula to the data) as the values that should bracket the true tc,
    this suggested 97�922 ≤ tc
    ≤ 97�985, which corresponds to December 3,
    1997 ≤ tc
    ≤ December 25, 1997.
    In hindsight and knowing what happened in August 1998 (see the
    section titled “The Crash of August 1998” in chapter 7), the best eight
    Table 9.8
    Prediction of the next crash by the linear log-periodic formula (18) on the S&P 500
    index using time intervals from 1994�9 to “last date.”
    Last date tc � � �2 A B C
    97�89041 98�06 0�28 6�4 8�884 998 ?858 0�105
    97�89042 98�04 0�25 6�1 8�886 989 ?793 0�103
    97�7191 97�985 0�23 8�3 113�4 897 ?622 0�026
    97�6781 97�922 0�24 7�9 108�8 838 ?573 0�028
    97�6331 97�678 0�42 6�3 103�9 514 ?280 0�054
    97�6332 97�845 0�27 7�3 105�0 753 ?499 0�032
    97�5881 97�796 0�30 7�0 102 670 ?422 0�038
    97�5431 97�756 0�36 6�8 95�2 579 ?337 0�046
    97�4981 97�702 0�44 6�4 90�3 501 ?265 0�056
    97�4531 97�676 0�50 6�3 88�2 461 ?227 0�061
    97�4081 97�674 0�52 6�2 88�7 452 ?218 0�062
    97�4082 97�864 0�74 16 160 414 ?154 0�031
    97�3631 97�734 0�53 6�6 88�7 458 ?217 0�059
    The superscripts 1 and 2 refer, respectively, to the best and second best fits of the formula to the
    data. Unpublished results obtained in collaboration with A. Johansen.
    344 chapter 9
    solutions may have actually been relevant as indicating possible scenarios
    for the future further ahead. Here is a lesson to learn in connection
    with Figure 9.1: it may be that several scenarios are possible for the
    future evolution. The stock market dynamics will select one, but another
    branch may have occurred with some slight modifications of the different
    perturbations acting on the system. This brings us back to the beginning
    of this chapter, where we emphasized the importance of a view
    of prediction involving multiple scenarios. As used in a different context
    in [6], predictive patterns and their associated forecasts should be
    defined in probabilistic terms, allowing for multiple scenarios evolving
    from the same past evolution. Deeply imbedded in this approach is the
    view of the future as a set of potentially acceptable trajectories that can
    branch and bifurcate at special times. At certain times, only one main
    trajectory extrapolates with high probability from the past making the
    future depend almost deterministically (albeit possibly in a nonlinear and
    chaotic manner) on the past. At other times, the future is much less certain,
    with multiple almost equivalent choices. In this case, we return to
    an almost random walk picture. The existence of a unique future must
    not be taken as the signature of a single dynamical system but as the
    collapse of the large distribution of probabilities. This is the concept
    learned, for instance, from the famous Polya urn problem discussed in
    chapter 4 in which the historical trajectory appears to converge to a certain
    outcome, which is, however, solely controlled by the accumulation
    of purely random choices; a different outcome might have been selected
    by history with equal probability [20]. It is fundamental to view any
    forecasting program as essentially a quantification of probabilities for
    possible competing scenerios. This view has been vividly emphasized by
    Asimov in his famous science fiction Foundation series [23, 22].
    Table 9.9 shows an attempt at predicting a critical time tc with the
    nonlinear log-periodic formula (19) using data going from “first date”
    (first column) to “last date” (second column), in order to test for robustness.
    There seems to be a clear prediction toward 1997�94 ± 0�01. Some
    of the fits give a value for tc very close to the “last date” and must thus
    be rejected. This is the case for the rows marked by an asterisk ?. The
    preferred prediction date was tc
    ≈ December 12, 1997.
    The last attempt consisted in using the Shank formula (23). The difficulty
    here is to identify the “characteristic” times tn. For this, we used
    the successive “coarse-grained” local maxima. A rough estimation by
    eye gave t1
    = 94�05, t2
    = 96�15, t3
    = 97�1. Inserting in (23) provided a
    prediction t�1�
    = 97�884. Expression (24) predicts t4
    = 97�53, while we
    observed t4
    = 97�55, providing a rather good confirmation. Using t2, t3,
    Table 9.9
    First date Last date tc � � �� �t �2 A B C
    1991 97�89041 98�07 0�82 10�5 ?11�2 5�2 0�02483 6�33 ?0�338 ?0�085
    1991 97�6782 98�13 0�52 12�3 ?58�6 29 0�02576 6�51 ?0�510 ?0�076
    1991 97�6781 97�948 0�73 8�9 ?14�1 9�3 0�03916 6�26 ?0�505 ?0�091
    1991 97�6782 97�942 0�61 9�1 ?33�8 23�0 0�03930 6�33 ?0�584 ?0�089
    1991 97�6061 97�709 0�69 0�039 6�15
    1991 97�5161 97�819 0�80 0�039 6�13
    1991 97�4441 97�780 0�885 0�039 6�06
    1988 97�3921 97�982 0�94 0�076 6�17
    1992.4 97�3921 97�990 0�48 0�102 9�83
    1995? 97�3921 97�481 0�247 0�019 7�14
    1991 97�3721 97�818 0�94 0�0393 6�05
    1987.9 97�3041 98�788 0�86 9�9 ?6�6 7�1 0�102 596 ?134 0�066
    1988 97�2861 99�479 0�135 6�6 ?14�9 10�4 0�088 17�0 ?24�8 0�488
    1992.2 97�2681 98�962 0�39 8�5 ?76�7 16�4 0�100 11�5 ?4�63 0�034
    1991 97�2421 97�966 0�62 10�4 ?20�2 9�5 0�016 6�73 ?0�367 0�074
    1988? 97�2421 98�280 0�84 12�6 ?35�9 9�5 0�026 6�63 ?0�212 0�113
    1988 97�2422 97�361 0�79 7�0 ?34�1 13�9 0�026 6�46 ?0�196 0�158
    1991 97�2291 97�894 0�925 0�03915 6�10
    1988? 97�2151 98�229 0�88 12�6 ?10�5 3�3 11�3 786 ?173 0�055
    1991 97�1571 97�851 0�927 0�03935 6�08
    1991 97�0851 98�412 0�43 0�0405 7�40
    1988 97�0551 97�760 0�47 10�1 ?15�9 7�5 232 6�26 ?0�505 ?0�091
    Attempt to predict a crash by the nonlinear log-periodic formula using time intervals from the “first date” to the “last date” to fit the logarithm of the S&P
    500 index. The exponents i indicate the order of the minima found for the same “last date.” Unpublished results obtained in collaboration with A. Johansen.
    346 chapter 9
    and t4 in (23) gives the prediction t�2�
    = 97�955. This was taken as the
    preferred value because it uses the log-periodicity in the last two years
    for which the log-frequency shift described by the nonlinear log-periodic
    formula is not present. This again predicted tc
    ≈ December 15, 1997, in
    agreement with the two other methods.
    The U.S. Market, October 1999 False Alarm
    Following a similar methodology, we also closely monitored several U.S.
    markets and found that significant log-periodic power behavior could be
    detected in September 1999, suggesting the end of a bubble in October
  1. The world markets were actually sent into turmoil by a speech by
    Alan Greenspan, and the Dow Jones for the first time since April 8, 1999
    dipped below 10.000 on October 15 and 18, 1999. However, the market
    did not crash and instead quickly recovered and later started a renewed
    and strengthened bullish phase. In hindsight, we see that, similarly to
    October 1997, this may have been an aborted event, which turned into a
    precursor of the large crash in April 2000, which we correctly detected.
    We have just discussed in some detail two of the three successful predictions
    (U.S. market August 1998, Nikkei Japanese market 1999, Nasdaq
    April 2000) and the two false alarms (U.S. market December 1997, Nasdaq
    October 1999).
    Several remarks are in order.
    The Finite Probability That No Crash Will Occur
    during a Bubble
    We stress again that the fact that markets are approximately efficient
    and that investors try to arbitrage away gain opportunities lead to the
    fundamental constraint that crashs are stochastic events. The rational
    expectation model described in chapter 5 provides a benchmark for such
    behavior. It tells us that we should not expect all speculative bubbles
    to end in a crash: the crux of the theory is that there is always a finite
    probability that the bubble will deflate smoothly without a crash. Hence,
    according to this theory, the two false alarms may just correspond to
    prediction of crashes and of antibubbles 347
    this scenario of a smooth death of the bubble. The sample is not large,
    but without better knowledge, the existence of these two false alarms
    interpreted in this context suggests that the total probability for a crash to
    occur conditioned on the existence of a bubble is approximately 3/5 =
    60%. Thus, there is a probability of 40% for living through a speculative
    bubble safely without crash.
    In other words, the two cases of bubbles landing more or less smoothly
    are completely consistent with the theory of rational bubbles and crashes
    developed in [221] and reported in chapter 5. This also illustrates the
    difficulties involved in developing a crash-prediction scheme based on
    the critical point theory. According to the rational expectation model,
    the critical time tc is not necessarily the time of the crash, only its most
    probable time.
    Estimation of the Statistical Significance
    of the Forward Predictions
    Statistical Confidence of the Crash “Roulette.”
    Let us now be conservative and consider that the two false alarms are
    real failures. How can we quantify the statistical significance of the predictions?
    Let us formulate the problem precisely. First, we divide time
    into monthly intervals and ask what the probability is that a crash will
    occur in a given month interval. Let us consider N month intervals. The
    recent out-of-sample period over which we carried out our analysis goes
    from January 1996 to December 2000, corresponding to N = 60 months.
    In these N = 60 months, nc
    = 3 crashes occurred while N ? nc
    = 57
    monthly periods were without crash. Over this 5-year time interval, we
    made r = 5 predictions, and k = 3 of them where successful while
    r ? k = 2 were false alarms. What is the probability Pk of achieving
    such success by chance?
    This question has a clear mathematical answer and reduces to a wellknown
    combinatorial problem leading to the so-called hypergeometric
    As explained in the book of W. Feller [131], this problem is the same
    as the following game. In a population of N balls, nc are red and N ? nc
    are black. A group of r balls is chosen at random. What is the probability
    pk that the group so chosen will contain exactly k red balls?
    To make progress, we need to define a quantity called C�n� m�, which
    is the number of distinct ways to choose m elements among n elements,
    independently of the order with which we choose the m elements. The
    348 chapter 9
    combinatorial factor C�n� m� has a simple mathematical expression
    C�n� m� = n!/m!�n ? m�! where m!, called the factorial of m, is defined
    by m! = m × �m ? 1� × �m ? 2� × · · · × 3 × 2 × 1. C�n = 52�m =
    13� = 635�013�559�600 gives, for instance, the number of possible different
    hands at the game of bridge, and C�n = 52�m = 5� = 2�598�960
    gives the number of possible different hands at the game of poker.
    We can now use C�n� m� to estimate the probability pk. If, among the
    r chosen balls, there are k red ones, then there are r ? k black ones.
    There are thus C�nc� k� different ways of choosing the red balls and
    C�N ? nc� r ? k� different ways of choosing the black balls. The total
    number of ways of choosing r balls among N is C�N� r�. Therefore, the
    probability pk that the group of r balls so chosen will contain exactly k
    red balls is the product C�nc� k� × C�N ? nc� r ? k� of the number of
    ways corresponding to the draw of exactly k red balls among r divided
    by the total possible number C�N� r� of ways to draw the r ball (here
    we simply use the so-called “frequentist” definition of the probability of
    an event as the ratio of the number of states corresponding to that event
    divided by the total number of events):
    = C�nc� k� × C�N ? nc� r ? k�
    C�N� r�
    � (26)
    pk is the so-called hypergeometric function. In order to quantify a statistical
    confidence, we must ask a slightly different question: what is the
    probability Pk that, out of the r balls, there are at least k? red balls?
    Clearly, the result is obtained by summing pk over all possible values of
    k’s from k? up to the maximum of nc and r; indeed, the number of red
    balls among r cannot be greater than r, and it cannot be greater than the
    total number nc of available red balls.
    In the case of interest here, the number of monthly periods is N =
    60, the number nc of real crashes is equal to the number k of correct
    predictions nc
    = k = 3, N ? nc
    = 57, the total number of issued prediction
    is r = 5, and the number of false alarms is r ? k = 2. Since
    = k, Pk=3
    = pk=3
    = C�3�3�×C�57�2�
    = 0�03%: the probability that this
    result is due to chance is a very small value 0�03%, corresponding to an
    exceedingly strong statistical significance of 99�97%. We conclude that
    our track record, while containing only few cases, is highly suggestive
    of real significance.
    To obtain a feeling for the sensitivity of this estimation on the reported
    number of successes and failures, let us assume that instead of correctly
    predicting k = 3 crashes, we had predicted only two out of the five
    prediction of crashes and of antibubbles 349
    alarms that we declared. This corresponds to N = 60, nc
    = 3, N ? nc
    57, r = 5, k = 2, and r ?k = 3. The probability that this result is due to
    chance is Pk=2
    = p2
    = C�3�2�×C�57�3�
  • C�3�3�×C�57�2�
    = 1�9%+0�03%;
    the probability that this result would be due to chance is still very small
    and approximately equal to 2%, corresponding to a still strong statistical
    significance of 98%. While less overwhelming, two correct predictions
    and three false alarms is still strongly significant. We conclude that the
    statistical confidence level of our track record is robust.
    What will happen if we issue a sixth prediction in the following year,
    which turns out to be incorrect? The track record would then be such
    that N = 72, nc
    = 3, N ? nc
    = 69, r = 6, k = 3, and r ? k = 3.
    The probability that this result is due to chance is Pk=3
    = C�3�3�×C�69�3�
    0�033%, which gives a small degradation of the statistical significance:
    three correct predictions and three failures in a set of seventy-two targets
    remains highly nonrandom. We are thus justified in claiming that these
    results are nonrandom with high significance.
    We should stress that this contrasts with the view that three successes
    and two failures, or vice versa, would correspond to approximately one
    chance in two of being right, giving the impression that the prediction
    skill is no better than deciding that a crash will occur by random coin
    tosses. This conclusion would be very naive because it forgets an essential
    element of the forecasting approach, which is to identify a (short)
    time window (one month) in which a crash is probable: the main difficulty
    in making a prediction is indeed to identify the few monthly
    periods among the sixty in which there is the risk of a crash.
    Statistical Significance of a Single Successful Prediction
    via Bayes’s Theorem.
    Consider our prediction in January 1999 of the trend reversal of the
    Nikkei index in its antibubble regime. This is a single case of a prediction
    of an antibubble regime. In the standard “frequentist” approach to
    probability [224] and to the establishment of statistical confidence, this
    bears essentially no weight and should be discarded as storytelling. However,
    the “frequentist” approach is unsuitable for assessing the quality of
    such a unique experiment of the prediction of a global financial indicator.
    The correct framework is Bayesian. Within the Bayesian framework, the
    probability that the hypothesis is correct given the data can be estimated,
    whereas this is excluded by construction in the standard “frequentist”
    formulation, in which one can only calculate the probability that the
    null-hypothesis is wrong, not that the alternative hypothesis is correct
    (see also [279, 98] for recent introductory discussions). We now present
    350 chapter 9
    a simple application of Bayes’s theorem to quantify the impact of our
    prediction [216].
    Bayes’s view of the prediction skill given one successful prediction:
    We can approach the problem of the significance of a single successful
    prediction by using a fundamental result in probability theory, known as
    Bayes’ theorem. This theorem states that

D� = P�D
Hi� × � P�Hi�
j P�D
Hj�P�Hj �
� (27)
where the sum in the denominator runs over all the different conflicting
hypotheses. In words, equation (27) estimates that the probability that
hypothesis Hi is correct given the data D is proportional to the probability
Hi� of the data given the hypothesis Hi multiplied with the prior
belief P�Hi� in the hypothesis Hi divided by the probability of the data.
Consider our prediction in January 1999 of the trend reversal of the Nikkei
index. Translated in this context, we use only the two hypotheses H1 and
H2 that our model of a trend reversal is correct or that it is wrong. For the
data, we take the change of trend from bearish to bullish. We now want to
estimate whether the fulfillment of our prediction was a “lucky one.” We
quantify the general atmosphere of disbelief that Japan would recover by
the value P�D
H2� = 5% to the probability that the Nikkei will change
trend while disbelieving our model. We estimate the classical confidence
level of P�D
H1� = 95% to the probability that the Nikkei will change
trend while believing our model.
Let us consider a skeptical Bayesian with prior probability (or belief)
P�H1� = 10?n, n ≥ 1, that our model is correct. From (27), we get

D� = 0�95 × 10?n
0�95 · 10?n + 0�05 × �1 ? 10?n�
� (28)
For n = 1, we see that her posterior belief in our model has been amplified
compared to her prior belief by a factor ≈ 7 corresponding to P�H1

D� ≈
70%. For n = 2, the amplification factor is ≈ 16 and hence P�H1

D� ≈
16%. For large n (very skeptical Bayesian), we see that her posterior belief
in our model has been amplified compared to her prior belief by a factor
0�95/0�05 = 19.
Alternatively, consider a neutral Bayesian with prior belief P�H1� =
1/2; that is, a priori she considers it equally likely that our model is correct
prediction of crashes and of antibubbles 351
or incorrect. In this case, her prior belief is changed into the posterior
belief equal to

D� =
0�95 · 1
0�95 · 1

  • 0�05 · 1
    = 95%� (29)
    This means that this single case is enough to convince the neutral
    We stress that this specific application of Bayes’s theorem only deals
    with a small part of the model; the trend-reversal. It does not establish
    the significance of the quantitative description of ten years of data (of
    which the last one was unknown at the time of the prediction) by the
    proposed model within a relative error of ≈ ±2%.
    The Error Diagram and the Decision Process.
    In evaluating predictions and their impact on (investment) decisions, one
    must weigh the relative cost of false alarms with respect to the gain
    resulting from correct predictions. The Neyman-Pearson diagram, also
    called the decision quality diagram, is used in optimizing decision strategies
    with a single test statistic. The assumption is that samples of events
    or probability density functions are available both for correct signals
    (the crashes) and for the background noise (false alarms); a suitable test
    statistic is then sought which optimally distinguishes between the two.
    Using a given test statistic (or discriminant function), one can introduce
    a cut that separates an acceptance region (dominated by correct
    predictions) from a rejection region (dominated by false alarms). The
    Neyman-Pearson diagram plots contamination (misclassified events, that
    is, classified as predictions, which are in fact false alarms) against losses
    (misclassified signal events, that is, classified as background or failureto-
    predict), both as fractions of the total sample. An ideal test statistic
    corresponds to a diagram where the “acceptance of prediction” is plotted
    as a function of the “acceptance of false alarm” in which the acceptance
    is close to 1 for the real signals and close to 0 for the false alarms.
    Different strategies are possible: a “liberal” strategy favors minimal loss
    (i.e., high acceptance of signal, i.e., almost no failure to catch the real
    events but many false alarms), a “conservative” one favors minimal contamination
    (i.e., high purity of signal and almost no false alarms but
    many possible misses of true events).
    Molchan has shown that the task of predicting an event in continuous
    time can be mapped onto the Neyman-Pearson procedures. He has
    introduced “error diagram” which plots the rate of failure-to-predict (the
    352 chapter 9
    number of missed events divided by the total number of events in the
    total time interval) as a function of the rate of time alarms (the total
    time of alarms divided by the total time, in other words the fraction of
    time we declare that a crash is looming) [303, 304]. The best predictor
    corresponds to a point close to the origin in this diagram, with almost
    no failure-to-predict and with a small fraction of time declared as dangerous:
    in other words, this ideal strategy misses no event and does not
    declare false alarms! These considerations teach us that making a prediction
    is one thing, but using it is another, which corresponds to solving
    a control optimization problem [303, 304].
    Decision theory provides useful guidelines. Let c1 represent the cost
    of mispredicting a crash as a noncrash and c2 the cost of mispredicting
    a normal time as a crash. Let us assume that, conditioned on past data
    X, our model provides the probability � =Pr�Y = 1
    X� for a crash to
    occur (Y = 1). If a crash occurs, the average cost is C1
    = c2�1 ? ��,
    which represents the possibility that we may have mispredicted it. If the
    crash does not occur, the average cost is C2
    = c1�, which represents the
    possibility that we may have predicted a crash anyway. By comparing
    these two costs, it is clear that C1 > C2 if � < 1/�1 + �c1/c2�� and
    ≤ C2 if � ≥ 1/�1 + �c1/c2��. Thus, the optimal prediction (in the
    sense of minimizing total expected cost) is “crash” (Y = 1) when Pr�Y =
    X� > 1/�1 + �c1/c2�� and “no-crash” (Y = 0) otherwise (see also
    [345, pp. 19, 58]). Hence, if the two possible mispredictions are equally
    costly, c1/c2
    = 1, we would predict that a crash will occur when Pr�Y =
    X� > 0�5. However, if mispredicting a crash is, say, twice as costly as
    mispredicting a no-crash, c1/c2
    = 2, an optimal decision process would
    predict a crash whenever Pr�Y = 1
    X� > 1/3. By applying decision
    theory like this, we can compare model outputs to the data and judge
    our success in prediction. The key, however, is that the value of c1/c2
    must be decided independently of the data and of the development of
    the model. The model should also be able to provide a prediction in a
    probabilistic language. There is thus much to do in future research.
    Practical Implications on Different Trading Strategies
    A significant fraction of professional investors and managers, and in
    particular hedge fund managers, use a variety of strategies in order to
    prediction of crashes and of antibubbles 353
    improve their performance. It is clear that the two broad classes of
    strategies, trend following and market timing, would profit from ex ante
    detections of impending crashes.
    Fung and Hsieh [147] have recently provided a useful and simple classification
    scheme for strategies which we borrow here. They considered
    so-called buy-and-hold, market-timing, and trend-following strategies.
    Both market-timers and trend-followers attempt to profit from price
    movements. Roughly speaking, a market-timer forecasts the direction of
    an asset, buying to capture a price increase, and selling to capture a price
    decrease. A trend-follower attempts to capture trends, that is, serial correlations
    in price changes that make prices move persistently, mainly in one
    direction over a given time interval (for positive price correlations).
    A simple model of such strategies is as follows. Let pi� pf � pmax, and
    pmin be the initial asset price, the final price, the maximum price, and
    the minimum price achieved over a given time interval. Let us consider
    strategies that complete a single trade over the given time interval.
    The buy-and-hold strategy consists in buying at the beginning at the
    price pi and selling at the end at the price pf , pocketing or losing pf
    ?pi .
    In this example, the market-timing strategy attempts to capture the price
    movement between pi and pf . If pf is expected to be higher (lower) than
    pi , the trader buys (sells) an asset. The trade is reversed at the end of the
    period, to exit the market. Thus, the optimal payout of the market-timing
    strategy is pf
    ? pi if pf > pi, or pi
    ? pf if pf < pi , which can be noted

? pi

, where the vertical bars correspond to taking the absolute value.
In other words, such an ideal market-timing strategy works like an electric
rectifier, changing negative price changes into positive gains.
In this example, the perfect trend-following strategy attempts to capture
the largest price movement during the time interval. Therefore, the optimal
payout is pmax
? pmin. It is clear that this strategy would profit the most
from a crash prediction.
Let us in addition mention investment strategies using financial derivatives,
such as “put” and “call” options. A put option is the natural tool
for leveraging a prediction on an incoming crash. Recall that a put (also
called sell) option gives the right (but not the obligation) obtained from a
counterparty (say a bank) to sell a stock at a prechosen price, called the
exercise price, during a given time period. When the real price becomes
much smaller than the exercise price, the put option becomes very valuable
because the investor can buy at a low price on the market and sell
at the high exercise price to the bank, thus pocketing the difference. The
leveraging embedded in the put option stems from the fact that its initial
354 chapter 9
price may be very small if the exercise price is initially chosen “out-ofthe-
money,” that is, much below current price, since the trader does not
get much from the possibility of selling at a price below market price. If
a crash occurs before the option comes to maturity and as a consequence
the price plunges close to or below the exercise price, the initially almost
valueless option suddenly acquires a large value. Its price may jump by
factors of up to hundreds for large crashes, corresponding to potential
gains of tens of thousands of percentage points! But this is more easily
said than done, as precise timing is of the essence.
Understandably, traders regard their trading systems to be proprietary
and are reluctant to disclose them. We are no exception: while we have
taken an open view by describing our underlying theory in great detail
and by providing explicit examples of some past implementations, key
recent progress has not been divulged yet. Recent theoretical studies
indeed suggest that new strategies coevolving with older ones may surpass
them if used only by a limited number of players.
chapter 10
2050: the end of the
growth era?
How will the stock markets of the world behave
in the months, years, or even decades ahead of us? This question underlies
much of our economic future and well-being. As discussed in previous
chapters, countries around the world are relying increasingly on
the stock market for the retirement of their elders, for quantifying the
value of companies, and for characterizing the health of the economy
in general. In addition, the stock market has become a powerful engine
of both developed and emerging economies as the principal source of
liquidity and capital for investment.
At the end of the twentieth century, several authors, emboldened by
the seemingly endless bull market of the time, proposed that the Dow
Jones index will climb to 36,000 [158], 40,000 [118], or even 100,000
[225] in the next two or three decades from the flat range 10,000–11,000
in which it has hovered from mid-1999 to the time of writing (mid-
2001). Are these predictions realistic or overblown? More generally, what
possible scenarios are ahead of us?
To address these questions, we generalize our approach by analyzing
financial as well as economic and population times series over the
longest time scales for which reliable data is available. The rationale for
this multivariate approach is that the future of the stock market cannot
356 chapter 10
be decoupled from that of the economy, which itself is linked to the
productivity of the labor, and hence to the dynamics of the population.
This leads us naturally to ask broader questions, such as whether the
present pace of human population growth and of its associated economic
development can continue along its accelerating path in the indefinite
future. Or, as a growing number of scholars threaten, are they bound to
stop catastrophically if mankind is not able to soon achieve a regime of
long-term sustainable development?
Indeed, contrary to common belief, both the global human population
as well as its economic output have grown faster than exponentially for
most of known history, and most strikingly in the last two centuries.
Recall that an exponential growth corresponds to a constant growth rate,
such as the interest rate one gets on a CD account or from a government
bond. A faster-than-exponential growth thus means that the growth rate
is itself growing with time (see the “Intuitive Explanation of the Creation
of a Finite-Time Singularity at tc” in chapter 5). We shall show
below that this observed accelerating growth rate is consistent with a
spontaneous apparent divergence at the same critical time around 2050,
with the same self-similar log-periodic patterns in three data sets: human
population, gross domestic product, and financial indices. This result can
be explained by the interplay between the dynamics of the growth of
population, of capital, and of technology, producing an “explosion” in
the economic output, even when the individual isolated dynamics do
not have strong enough positive feedbacks to do the same by their single
action. Interestingly, in the 1950s, two famous mathematicians and
computer scientists, S. Ulam and J. von Neumann (the father of modern
computing as well as game theory in economics) were aware of
this possibility. Indeed, in [428], Ulam recalled a conversation with von
Neumann: “One conversation centered on the ever accelerating progress
of technology and changes in the mode of human life, which gives the
appearance of approaching some essential singularity in the history of
the race beyond which human affairs, as we know them, could not continue.”
The tremendous pace of accelerated growth observed until now has
led to increasing worries about its sustainability. It has also led to rising
concerns that the human culture as a result may cause severe and
irreversible damage to ecosystems, global weather systems, and so on.
On the other hand, optimists expect that the innovative spirit of mankind
will be able to solve such problems and the economic development of
the world will continue as a succession of revolutions, for example, the
Internet, bio-technological, and other yet unknown major innovations
2050: the end of the growth era? 357
replacing the agricultural, industrial, medical, and information revolutions
of the past. The observed acceleration of economic development
seems to support the optimistic point of view.
However, the spontaneous apparent divergence around 2050, which
we shall document below, has the surprising consequence that even the
optimistic view needs to be revised, since an acceleration of the growth
rate contains endogenously its own limit in the form of a singularity. The
singularity is a mathematical idealization of a transition to a qualitatively
new behavior. The degree of abruptness of the transition to the new
regime can be inferred from the fact that the maximum of the world
population growth rate was reached in 1970, about 80 years before the
predicted singular time, corresponding to approximately 4% of the 2,000
years over which the acceleration is documented below. This roundingoff
of the finite-time singularity is probably due to a combination of
well-known finite-size effects and drag effects that are bound to become
dominant close to the singularity. It suggests that we have already entered
the transition region to a new regime, as we shall discuss in more detail
in this chapter.
As a bonus, we also offer the prediction that the U.S. market is in a
period of consolidation, or stagnation, which may last up to a full decade.
This period will be followed by renewed accelerated growth. We attempt
to unearth the origins of this behavior on the basis of macroeconomic
The rapid growth of the world population is a quite recent phenomenon
compared to the total history of modern homo sapiens. It is estimated
that 2,000 years ago the population of the world was approximately
300 million. It took more than 1,600 years for the world population
to double to 600 million, and since then the growth has accelerated. It
reached 1 billion in 1804, 2 billion in 1927 (123 years later), 3 billion in
1960 (33 years later), 4 billion in 1974 (14 years later), 5 billion in 1987
(13 years later), and 6 billion in 1999 (12 years later) (see Table 10.1).
Representatives of national academies of sciences from throughout the
world met in New Delhi in October 1993 at a “Science Summit” on
world population. The participants issued a statement, signed by representatives
of 58 academies on population issues, related to development,
notably on the determinants of fertility and the effect of demographic
358 chapter 10
Table 10.1
Year Population (billions) Source
0 0�30 Durand
1000 0�31 Durand
1250 0�40 Durand
1500 0�50 Durand
1750 0�79 D & C
1800 0�98 D & C
1850 1�26 D & C
1900 1�65 D & C
1910 1�75 Interp.
1920 1�86 WPP63
1920 1�86 WPP63
1930 2�07 WPP63
1940 2�30 WPP63
1950 2�52 WPP94
1960 3�02 WPP94
1970 3�70 WPP94
1980 4�45 WPP94
1990 5�30 WPP94
1994 5�63 WPP94
1999 6�00 WPP94
2001 6�14 WPP01
Data from the United Nations Population Division, Department
of Economic and Social Information and Policy Analysis.
Durand: J.D. Durand, 1974. Historical Estimates of World
Population: An Evaluation (University of Pennsylvania, Population
Studies Center, Philadelphia), mimeo. D & C: United
Nations, 1973. The Determinants and Consequences of Population
Trends, Vol. 1 (United Nations, New York). WPP63:
United Nations, 1966. World Population Prospects as Assessed
in 1963 (United Nations, New York). WPP94: United Nations,

  1. World Population Prospects: The 1994 Revision (United
    Nations, New York). Interp: Estimate interpolated from adjacent
    population estimates.
    growth on the environment and the quality of life. The statement asserted
    that “continuing population growth poses a great risk to humanity,” and
    proposed a demographic goal: “In our judgment, humanity’s ability to
    deal successfully with its social, economic, and environmental problems
    will require the achievement of zero population growth within the
    2050: the end of the growth era? 359
    lifetime of our children,” and “Humanity is approaching a crisis point
    with respect to the interlocking issues of population, environment and
    development because the Earth is finite” [366]. Accordingly, “Excessive
    peopling of the world is contributing to major environmental trauma,
    including famine, rain forest destruction, global warming, acid rain, pollution
    of air, water, overflow and even to the AIDS epidemic” [366].
    There are many documented cases of irreversible damage to ecosystems,
    global weather system perturbations, as well as increasing concerns
    about a severe shortage of water. Extrapolating present trends, it is
    estimated that, by 2025, two-thirds of the world population will live in
    water-stressed conditions [119]. These problems all have one common
    root: the fast-increasing human population and its associated economic
    development. The worry about human population size and growth is
    shared by many natural scientists, including the Union of Concerned
    Scientists (comprising 99 Nobel Prize winners), which asks nations to
    “stabilize population.”
    At what may be considered the other extreme, optimists expect that the
    innovative spirit of mankind will be able to solve the problems associated
    with a continuing increase in the growth rate [441, 380, 306]. Specifically,
    as we said above, they believe that world economic development
    will continue as a successive unfolding of revolutions, for example, the
    Internet, bio-technological, and other yet unknown innovations replacing
    the prior agricultural, industrial, medical, and information revolutions of
    the past.
    Indeed, by 1990, most of the economics profession has turned almost
    completely away from the previous view that population growth is a
    negative factor in economic development (see, however, [94, 145, 99]).
    In fact, they now consider it to be a positive factor: more people implies
    greater wealth, more resources, and a healthier environment. The argument
    goes: “Additional persons produce more than they consume in the
    long run, and natural resources are not an exception” [380, 306]. “Without
    exception, the relevant data, i.e., the long-run economic trends, and
    the appropriate measures of scarcity, i.e., the costs of natural resources
    in human labor and their prices relative to wages and to other goods, all
    suggest that natural resources have been becoming less scarce over the
    long run, right up to the present” [380]. On essentially all accounts, the
    360 chapter 10
    optimists thus argue that the situation has improved compared to past
    decades and will continue to improve in the coming decades [380, 306]:
  2. Pollution: Pollution has always been a problem since the beginning of
    time, but we now live in a more healthy and less dirty environment than
    in earlier centuries. Life expectancy, which is the best overall index of
    the pollution level, has improved markedly as the world’s population
    has grown.
  3. Food: Food production per capita has been increasing over the halfcentury
    since World War II. Famine has progressively diminished for
    at least the past century (quantified in relative values, as the fraction of
    the total population). There is compelling reason to believe that human
    nutrition will continue to improve into the indefinite future, even with
    continued population growth.
  4. Land: The amount of agricultural land has been increasing substantially,
    and it is likely to continue to increase where needed. For rich
    countries (United States, for instance), the quantity of land under cultivation
    has been decreasing. The amount of land used for forests, recreation,
    and wildlife has been increasing rapidly in the United States!
  5. Natural resources: Our supplies are not finite in any economic sense,
    nor does past experience give reason to expect natural resources to
    become more scarce. Natural resources will progressively become less
    costly, hence less scarce, and will constitute a smaller proportion of our
    expenses in future years. Population growth is likely to have a long-run
    beneficial impact on the natural-resource situation.
  6. Energy: The long-run future of our energy supply is at least as bright
    as that of other natural resources, though government intervention can
    temporarily boost prices from time to time. Finiteness is no problem
    here either. And the long-run impact of additional people is likely to
    speed the development of cheap energy supplies that are almost inexhaustible.
  7. The standard of living: In the short run, additional children imply
    additional costs, as all parents know. In the longer run, per capita
    income is likely to be higher with a growing population than with a
    stationary one, both in more-developed and less-developed countries.
  8. Human fertility: The contention that poor and uneducated people breed
    without constraint is demonstrably wrong, even for the poorest and
    most “primitive” societies [380, 306]. Well-off people who believe that
    2050: the end of the growth era? 361
    the poor do not weigh the consequences of having more children are
    simply arrogant, or ignorant, or both.
  9. Future population growth: Present trends suggest that even though
    total population for the world is increasing, the density of population
    on most of the world’s surface will decrease. This is already happening
    in the developed countries. Though the total populations of developed
    countries increased from 1950 to 1990, the rate of urbanization was
    sufficiently great that population density on most of their land areas
    (say, 97% of the land area of the United States) has been decreasing. As
    the poor countries become richer, they will surely experience the same
    trends, leaving most of the world’s surface progressively less populated,
    astonishing as this may seem.
    Let us start from Malthus’s exponential growth model, which assumes
    that the size of a population increases by a fixed proportion over a given
    period of time independently of the size of the population, and thus gives
    an exponential growth. Take, for instance, the proportion of 2.1% per
    year or 23.1% per decade corresponding to the all-time peak of the population
    growth rate reached in 1970. This leads to a population doubling
    time of forty-eight years. Starting from a population of, say 1,000, the
    population is 1.231 times 1,000 = 1,231 after one decade, 1.231 times
    1.231 times 1,000 = 1,515 after two decades, and so on. As we see,
    such an exponential growth corresponds to the multiplication of the population
    by a constant factor, here 1.231, for each additional unit of time,
    here ten years. It is thus convenient to visualize it by presenting the population
    on a scale such that successive values of the multiplication by a
    constant factor are equally spaced, which defines the so-called “logarithmic
    scale” already encountered several times in this book; we will use
    this scale for all figures presented below.
    In the Malthusian exponential model, the logarithm of the population
    should thus increase proportionally to, or linearly with, time. Figure 10.1
    shows the estimated (logarithm of the) world population (obtained from
    the United Nations Population Division, Department of Economic and
    Social Affairs) as a function of time. In contrast to the expected Malthusian
    straight line, we clearly observe a strong upward curvature characterizing
    “superexponential” behavior. Similar faster-than-exponential
    362 chapter 10
    0 500 1000 1500 2000
    World Population (millions)
    World GDP (billions 1990 US$)
    Year (AD)
    ’World population’
    ’World GDP’
    Fig. 10.1. World population and world GDP (gross domestic production) over 2,000
    years from 0 to the present in logarithmic scale as a function of time (linear scale),
    such that a straight line would qualify as exponential growth. The upward curvature
    of both time series shows that their growth cannot be accounted for by the
    exponential model and is “superexponential.”
    growth is also observed in the estimated GDP (gross domestic product)
    of the world estimated by DeLong at the Department of Economics at
    U.C. Berkeley [105], for the year 0 up to 2000.
    Over a shorter time period, a faster-than-exponential growth is also
    shown in Figure 10.2 for a number of financial indicators, such as the
    DJIA since 1790 obtained from the Foundation of the Study of Cycles
    (www.cycles.org/cycles.htm), the S&P 500 index since 1871, and a number
    of regional and global financial indices since 1920, including the
    Latin American index, the European index, the EAFE index, and the
    World index. The last five financial indices are obtained from Global
    Financial Data, Los Angeles (www.globalfindata.com). They are shown
    as their logarithm as a function of time, such that an exponential growth
    should be qualified by a linear increase.
    Source of data: The several data sets analyzed here express the development
    of mankind on Earth in terms of size and economic impact. They
    are as follows.
    � The human population data from 0 to 1998 was retrieved from
    the website of The United Nations Population Division, Depart2050:
    the end of the growth era? 363
    1800 1850 1900 1950 2000
    Real power law
    Complex power law
    Dow Jones
    Standard & Poor
    Latin America
    Fig. 10.2. Financial indices in logarithmic scale as a function of time (linear scale).
    The two largest time series, the Dow Jones extrapolated back to 1790 and the S&P
    (500) index from 1871, are fitted by a power law A�tc
    ? t�m shown as continuous
    lines. The log-periodic law (corresponding to a complex exponent of the power law)
    is shown only for the Dow Jones time series as a dashed line. A sophisticated power
    law analysis suggests an abrupt transition at around 2050 [219]. EAFE is the composite
    index regrouping Europe, Australia, and Far Eastern countries. Note again the
    upward curvature, which excludes exponential growth in favor of superexponential
    ment of Economic and Social Affairs (http://www.popin.org/
    � The GDP of the world from 0 to 1998, estimated by J. Bradford
    DeLong at the Department of Economics, U.C. Berkeley [105],
    was given to us by R. Hanson [186].
    � The financial data series include the DJIA from 1790 to 2000,
    the S&P index from 1871 to 2000, as well as a number
    of regional and global indices since 1920. The DJIA was
    constructed by The Foundation for the Study of Cycles
    (http://www.cycles.org/cycles.htm). It is the DJIA back to 1896,
    which has been extrapolated back to 1790 and further. The other
    364 chapter 10
    indices are from Global Financial Data [159]. These indices
    are constructed as follows. For the S&P, the data from 1871 to
    1918 are from the Cowles commission, which back-calculated
    the data using the Commercial and Financial Chronicle. From
    1918, the data is the Standard and Poor’s composite index
    (S&P) of stocks. The other indices use Global Financial Data’s
    indices from 1919 through 1969 and Morgan Stanley Capital
    International’s indices from 1970 through 2000. The EAFE
    index includes Europe, Australia, and the Far East. The Latin
    America index includes Argentina, Brazil, Chile, Colombia,
    Mexico, Peru, and Venezuela.
    Demographers usually construct population projections in a disaggregated
    manner, filtering the data by age, stage of development, region, and
    so on. Disaggregating and controlling for such variables is thought to be
    crucial for demographic development and for any reliable population prediction.
    Here, we propose a different strategy based on aggregated data,
    which is justified by the following concept: in order to get a meaningful
    prediction at an aggregate level, it is often more relevant to study aggregate
    variables than “local” variables, which can miss the whole picture in favor
    of special idiosyncrasies. To take an example from material sciences, the
    prediction of the failure of heterogeneous materials subjected to stress can
    be performed according to two methodologies. Material scientists often
    analyze in exquisite details the wave forms of the acoustic emissions or
    other signatures of damage resulting from microcracking within the material.
    However, this is of very little help in predicting the overall failure,
    which is often a cooperative global phenomenon [193] resulting from the
    interactions and interplay between the many different microcracks nucleating,
    growing, and fusing within the materials. In this example, it has
    indeed been shown that aggregating all the acoustic emissions in a single
    aggregated variable is much better for prediction purpose [215]. Similarly,
    the economic and financial development of the United States and Europe
    and of other parts of the world are interdependent due to the existence
    of several coupling mechanisms (exchanges of goods, services, transfer of
    research and development, immigration, etc.)
    The faster-than-exponential growths observed in Figures 10.1 and 10.2
    correspond to nonconstant growth rates, which increase with population
    or with the size of economic factors.
    Suppose, for instance, that the growth rate of the population doubles
    when the population doubles. For simplicity, we consider discrete time
    2050: the end of the growth era? 365
    intervals as follows. Starting with a population of 1,000, we assume it
    grows at a constant rate of 1% per year until it doubles. We estimate the
    doubling time as proportional to the inverse of the growth rate, that is,
    approximately 1/1% = 1/0.01 = 100 years. Actually, there is a multiplicative
    correction term equal to ln 2 = 0�69 such that the doubling time
    is ln 2/1% = 69 years. But we drop this proportionality factor ln 2 =
    0�69 for the sake of pedagogy and simplicity. Including it just multiplies
    all time intervals below by 0�69 without changing the conclusions. Thus,
    with this approximation, the first doubling time is one century.
    When the population turns 2,000, we assume that the growth rate doubles
    to 2% and stays fixed until the population doubles again to reach
    4,000. This takes only fifty years at this 2% growth rate. When the population
    reaches 4,000, the growth rate is doubled to 4%. The doubling
    time of the population is therefore approximately halved to twenty-five
    years and the scenario continues with a doubling of the growth rate every
    time the population doubles. Since the doubling time is approximately
    halved at each step, we have the following sequence (time = 0, population
    = 1,000, growth rate = 1%), (time = 100, population = 2,000,
    growth rate = 2%), (time = 150, population = 4,000, growth rate =
    4%), (time = 175, population = 8,000, growth rate = 8%), and so on.
    We observe that the time interval needed for the population to double
    is shrinking very rapidly by a factor of 2 at each step. In the same
    way that 1/2 + 1/4 + 1/8 + 1/16 + · · · = 1, which was immortalized
    by the ancient Greeks as Zeno’s paradox, the infinite sequence of doubling
    thus takes a finite time and the population reaches infinity at a
    finite “critical time” approximately equal to 100 + 50 + 25+· · · = 200
    (a rigorous mathematical treatment requires a continuous-time formulation,
    which does not change the qualitative content of the example).
    A spontaneous singularity has been created by the increasing growth
    This process is quite general and applies as soon as the growth rate
    possesses the property of being multiplied by some factor larger than 1
    when the population is multiplied by some constant larger than 1. Such
    spontaneous singularities are quite common in mathematical descriptions
    of natural and social phenomena, even if they are often looked at as
    monstrosities. They are found in many physical and natural systems.
    Examples are flows of fluids, the formation of black holes, the rupture
    of structures, and material failure in models of large earthquakes
    and of stock market crashes, as we have seen in previous chapters. The
    mathematics of singularities is applied routinely in the physics of phase
    366 chapter 10
    transitions to describe the transformations from ice to water or from a
    magnet to a demagnetized state when raising the temperature.
    The empirical test of the existence of singularities in the dynamics of
    the population or the economic indices rest on the way they increase up
    to the critical time. It turns out that they do so in a self-similar or fractal
    manner: for a given fixed contraction of the distance in time from the
    singularity, the population is multiplied by a fixed given factor. Repeating
    the contraction to approach closer to the singularity leads to the
    same magnification of the population by the same factor. These properties
    are captured by the mathematical law called a power law, already
    discussed in previous chapters. Power laws describe the self-similar geometrical
    structures of fractals. As we have seen in chapter 6, fractals
    are geometrical objects with structures at all scales that describe many
    complex systems, such as the delicately corrugated coast of Brittany or
    of Norway, the irregular surface of clouds, or the branched structure of
    river networks. The exponent of the power law is the so-called fractal
    dimension and, in the present context, quantifies the regular multiplicative
    structure appearing on the population, on financial indices, and on
    the distance in time to the singularity.
    Plotting the logarithm of the population as a function of the logarithm
    of the time from the singularity, a power law will appear as a
    straight line. This is shown in Figures 10.3 and 10.4 for the world population,
    the world GDP, and the financial indices shown in Figures 10.1
    and 10.2. Since the power laws characterizing the population and economic
    growth are expressed as a function of the time to the singularity,
    a value has to be chosen for this critical time. In Figure 10.3, the year
    2050 is used, which is close to the value obtained from a more sophisticated
    statistical analysis discussed later (see also [219]). For the financial
    indices, removing an average inflation of 4% or similar amounts
    does not change the results qualitatively, but the corresponding results
    are not quantitatively reliable as the inflation has varied significantly
    over history with quantitative impacts that are difficult to estimate. Correcting
    for inflation amounts to subtracting a linear term in the panel
    where the logarithm of the price is represented as a function of time.
    This will thus have no impact on the existence of the documented nonlinear
    upward curvature, qualified as an accelerated superexponential
    The issue of detrending by inflation to get constant-value dollars and
    indices: For the United States, it is generally agreed that the inflation
    factor converting U.S. dollars at the end of the nineteenth century to the
    2050: the end of the growth era? 367
    Fig. 10.3. World population and world GDP (with a logarithmic scale) as a function
    of the time to the critical time tc
    = 2050 (with a logarithmic scale) such that time
    flows from right to left. The straight lines are the best fit of the data to power laws
    (see text) and suggest an abrupt transition at 2050.
    end of the twentieth century is about 15: $1 in 1870 is equal approximately
    to $15 in 1995. This is small compared to France, for instance, where
    the conversion factor is already as large as 20 to convert 1959 francs into
    1995 francs. An example of a detrending to account for inflation of the
    DJIA since 1885 can be found in [378]. The conversion is performed by
    using the CPI (consumer price index). The problem is that the definition
    and way of calculation of the CPI has evolved a lot since its creation. At
    its origin, it was the wholesale price index, for its ease of measurement.
    Another way to measure inflation is to use the value of gold in U.S. dollars
    (about $300 per ounce at present, compared to about 20$ at the end of
    the nineteenth century, retrieving the factor 15 discussed above). There
    are many detrending techniques; they all have advantages and problems,
    which we have chosen to avoid.
    Inflation in the United States has undergone several phases:
  10. Before 1914, inflation was essentially zero on average, except during
    the civil war (famous “greenbacks”).
  11. From 1914 to 1921, there was high inflation followed by deflation in
    1921, and then during the depression of 1929–1932, which brought
    the CPI back to its pre-1914 level.
    368 chapter 10
    Fig. 10.4. Logarithm of financial indices as a function of the logarithm of the time
    to the critical time tc
    = 2050, such that time flows from right to left. The straight
    lines are the best fits, which qualify as power law behavior, as explained in the text,
    and suggest an abrupt transition at 2050.
  12. From 1933 to the present, there were some strong inflationary periods
    associated with World War II, the Cold War, the Korean war,
    the Vietnam war, as well as the oil shocks of the seventies.
    The factor 15 thus corresponds approximately to an average annual inflation
    rate of 4% since 1933. We present in Figure 10.5, the long-term time
    evolution of the debt of the U.S. federal government. There seems to be
    a relationship (a factor 2, approximately) between the growth of this debt
    and inflation rates. This relationship is especially strong in times of war,
    when inflation is galloping and the debt is accumulating at a fast rate. This
    is expected since inflation is a simple way for government to leverage
    taxes, in effect to finance expenses. Due to the complexity in accounting
    for these intermittent inflationary periods, we have not corrected our data
    for inflation.
    2050: the end of the growth era? 369
    1750 1800 1850 1900 1950 2000 2050
    Public Debt (US$)
    Debt from War
    of Independence
    War of
    Seminole War
    Civil War
    Spanish War
    WW I
    WW II
    Start of Cold War
    Average yearly growth rate = 8.6% End of Cold War
    Fig. 10.5. The debt of the U.S. federal government since the war of independence
    in logarithmic scale as a function of time (linear scale). The notation 1e + 09 corresponds
    to $1 billion and 1e +12 corresponds to $1 trillion. In 2000, the U.S. federal
    goverment debt was about $5.6 trillion. The straight line corresponds to an average
    exponential law with constant growth rate of 8.6% per year. Notice that the U.S. wars
    can be seen to punctuate the growth of the debt at many scales. U.S. wars seem to be
    the main large-scale features explaining the growth of the debt. The data is from the
    Bureau of the Public Debt (http://www.publicdebt.treas.gov/opd/opd.htm#history).
    Figure researched and prepared by A. Johansen.
    Complex Power Law Singularities
    The message to be extracted from the analysis of the previous section is
    that the world population, as well as the major economic indices, have
    on average grown at an accelerating growth rate which is compatible
    with a singular behavior occurring within a finite time horizon.
    Singularities and infinities were anathema for a long time until it was
    realized that they are often good mathematical idealizations of many natural
    phenomena. They are not fully present in reality; only the precursory
    acceleration can be observed and may announce an important transition.
    In the present context, they must be interpreted as a kind of “critical
    370 chapter 10
    point” signaling a fundamental change of regime. At this point in the
    analysis, there is still a relatively large uncertainty in the determination
    of the critical time tc. As can be seen from the figures, an important reason
    lies in the existence of large fluctuations around the average power
    law behavior.
    The mathematical theory of power laws, summarized in chapter 6,
    suggests an efficient way of taking these fluctuations into account by
    generalizing the concept of a real exponent into a complex exponent.
    As we have seen, this leads to so-called log-periodic oscillations, which
    decorate the overall power law acceleration. Fundamentally, this corresponds
    to replacing the continuous self-similar symmetry by a discrete
    self-similar symmetry. For instance, in the previous example, the population
    had a doubling growth rate each time it doubled. In this case,
    the dynamics is self-similar only under a change of times scales and a
    change of growth rate performed with a multiplication by a power of
    two. This leads to discreteness in the acceleration of the population such
    that the power law is modulated by steps in its slope occurring at each
    magnification by a factor of 2, that is, steps that are regularly spaced
    in the logarithmic representation. In reality, other factors than 2 can be
    selected by the dynamics. In addition, there are many other effects not
    taken into account in the analysis, which introduce some blurring of the
    steps and which then become smooth log-periodic oscillations as shown
    in Figure 10.2 in dashed lines for the DIJA. A nonparametric test of
    log-periodicity is shown in Figure 10.6, using the same approach as in
    chapters 7 and 8. One can observe a reliable log-periodic signal.
    There are fundamental reasons for introducing log-periodic corrections
    and complex exponents, deriving from the very structure of the theories
    describing fundamental particles at the smallest level on one hand and the
    organization of complex systems on the other hand. Again, examples are
    fluid flows, formation of black holes, material failure, and stock market
    crashes, as we have shown in chapters 7–9. The presence of log-periodic
    oscillations derived from general theoretical considerations may provide
    a first step to account for the ubiquitous observation of cycles at many
    scales in population growth and in the economy. Sensitivity analysis of
    the power law fits shown in Figures 10.3 and 10.4 and of the log-periodic
    power law fit shown for the Dow Jones in Figure 10.2 as well as tests of
    the statistical significance all give a large improvement on the position
    of the critical time tc. It is found to lie in the range 2042–2062, with
    70% probability [219].
    The best fit of equation (19) on page 336 to the 210 years of monthly
    quotes is shown in Figure 10.7, and its parameter values are given in
    2050: the end of the growth era? 371
    -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
    Spectral Power
    ’Data set 5’
    ’Data set 3’
    ’Data set 5’
    ’Data set 3’
    Fig. 10.6. Left panel: Residue between the best simple power law fit and the population
    data from 1250 to 1998 (called data set 3) and from 1500 to 1998 (called data
    set 5), performed to (i) check the sensitivity to the part of the demographic data in
    the past that is the most unreliable and (ii) detect the presence of log-periodicity.
    Right panel: Spectrum of the residues using a Lomb periodogram technique. For
    the population data from 1500 to 1998, the position of the peak corresponds to an
    angular log-frequency � ≈ 5�8, which should be compared with � ≈ 6�5 for the
    fit with the log-periodic power law formula. For the population data from 1250 to
    1998, the peak corresponds to � ≈ 6�1, which should be compared with � ≈ 6�5
    for the fit with the log-periodic power law formula. Reproduced from [219] .
    the caption. Note the close agreement between the value of the angular
    log-frequency � ≈ 6�5 compared to � ≈ 6�3 for the world population, as
    well as the value for the position of the singularity tc
    ≈ 2�053 compared
    to tc
    ≈ 2�056 for the population. Furthermore, the crossover time scale
    between the two log-frequencies �t ≈ 171 years is perfectly compatible
    with the total time window of 210 years.
    Prediction for the Coming Decade
    Figure 10.8 shows the extrapolation of the nonlinear log-periodic fit for
    the DJIA shown in Figure 10.7 up to the critical time tc
    ≈ 2053. Note
    that the trajectory of the DJIA since the last point (December 1999)
    used in the fit is following our prediction remarkably well up to the time
    of proof reading (mid-2002): the log-periodic fit predicts a plateau or
    a slowdown that may last for about a decade; and since mid-1999, the
    DJIA has indeed been stagnant.
    372 chapter 10
    1800 1850 1900 1950 2000
    Fig. 10.7. The corrugated line is the (natural) logarithm of the monthly quotes of the
    DJIA index from December 1790 to December 1999, already shown in Figure 10.2.
    The upward trending dashed line is the best fit with the simple power law equation
    giving � ≈ +0�27 (see definition with expression (18) on page 335) and tc
    ≈ 2068.
    The oscillating solid line is the best fit with the nonlinear log-periodic power law
    formula (19) giving an exponent � ≈ 0�39, tc
    ≈ 2053, � ≈ 6�5, and �t ≈ 171.
    Reproduced from [219].
    In Figure 10.8, five other periods of stagnation of the DJIA can be
    observed. They fall into two classes: (1) weak fluctuations around an
    approximately constant level (1790–1810, 1880–1900, and 1970–1980),
    and (2) strong acceleration followed by a crash/depression followed by
    a recovery (1830–1850 and 1920–1945). Note that the crash of October
    1987 belongs to an acceleration regime in this large-scale coarse-grained
    classification. Will the 2001–2010 decade be in the first or second class?
    This prediction of a period of consolidation is in line with the analysis
    of W. Godley [162], a scholar at the Levy Institute and professor
    emeritus of applied economics at Cambridge University, England. Godley
    examined the origin of the remarkable growth of the U.S. economy
    in the last decade of the twentieth century, based on an analysis of fiscal
    policy, foreign trade, and private income expenditure and borrowing, and
    found that it is unsustainable over the following decade.
    2050: the end of the growth era? 373
    1800 1850 1900 1950 2000 2050
    Fig. 10.8. Extrapolation of the nonlinear log-periodic fit for the DJIA shown in
    Figure 10.7 up to the critical time tc
    ≈ 2�053. The vertical axis is the (natural)
    logarithm of the DJIA. Note that the trajectory of the DJIA since the last point
    (December 1999) used in the fit is following our prediction remarkably well: the logperiodic
    fit predicts a plateau or a slowdown that may last for about a decade; from
    mid-1999 to mid-2002 (the time of proof-reading), the DJIA fluctuated between
    approximately 10,000 and 11,500 with no clear upward trend. Reproduced from
    To understand Godley’s arguments, let us recall a few basic principles
    of wealth conservation and flux. We are all aware of these principles even
    unconsciously when we try to balance our expenses by income. From
    the point of view of the private sector in a given country (consumers and
    companies), we become instantaneously richer at the aggregate level if
    � the government spends more as its spending translates into increasing
    income for companies and people, and
    � exports to foreign countries increase,
    as these two processes directly pump funds into the economy. Of course,
    an instantaneous measure of government spending, counted as a positive
    flow of funds for the private sector and for households on the short time
    scale, has to be funded by public borrowing (if deficit arises) whose
    374 chapter 10
    interests are paid from taxes which are part of the process that makes us
    poorer at the aggregate level. Thus, we are poorer if
    � taxes increase, and
    � imports increase for which we need to pay foreigners,
    as these two processes siphon funds out of the economy.
    If there is not growth of productivity, in the long run, the growth of
    the economy measured, for instance, by the GDP should thus follow
    one-to-one the growth of the difference between the amounts pumped in
    (government spending and exports) and siphoned out (taxes and imports).
    Godley shows that between 1961 and 1992 the GDP of the United States
    did indeed track this net balance of influx funds, within minor fluctuations.
    From the beginning of 1992 until 1999, GDP rose 3.3% per annum,
    while the net balance of influx funds rose only 0.6% per annum. The net
    spending from the government and net exports since 1992, which had
    been much weaker than in any other period since 1960, cannot be the
    cause of the large growth of the GDP.
    Godley [162] suggested that the GDP growth was fueled by an
    increasing private financial deficit, that is, excess of personal consumption
    and housing investment over personal disposable income, which
    became much larger than ever before. This increase of private deficit
    can be derived from two sets of evidence. First, the deficit of private
    households can be inferred from the fact that it must mechanically be
    equal to the government surplus plus the balance-of-payment deficit.
    Conversely, a positive private balance is equal to the government deficit
    plus positive export-minus-import balance. The intuition is that public
    deficits and balance-of-payment surpluses create income and financial
    assets for the private sector, whereas budget surpluses and balance-ofpayment
    deficits withdraw income and destroy financial assets. As the
    budget balance between 1992 and 1999 has changed by a larger amount
    than ever before (at least since the early 1950s) and has reached a record
    surplus (2.2% of GDP in the first quarter of 1999), and as the current
    balance of payments has deteriorated rapidly, the consequence is that
    the private sector balance has reached a record deficit (5.2% of GDP
    in the first quarter of 1999) [162]. The increase of private deficit can
    also be directly measured by comparing private income and expenditure:
    since 1993, the rise in private expenditure has been increasing much
    faster than the rise of income [162]. Data shows that most of the fall
    in private balance and the entire private deficit has taken place in the
    household sector, rather than by businesses, which financed most of
    their investment by internally generated funds.
    2050: the end of the growth era? 375
    Thus, the private sector as a whole has become a net borrower of
    money (or a net seller of financial assets) on a record and growing scale.
    The annual rate of net lending rose from about 1% of disposable income
    ($40 billion) at the end of 1991 to 15% (over $1 trillion) in the first
    quarter of 1999. The private financial deficit measures the extent to which
    the flow of payments into the private sector arising from the production
    and sale of goods and services exceeds private outlays on goods and
    services and taxes, which have to be made in money.
    Capital gains on the stock markets have probably been fueled by
    increasing borrowing invested in it and may also have fueled increasing
    consumption. In order to have a continuing influence, the stock market
    has to continue rising at an accelerating pace faster than exponential.
    Only a faster-than-exponential stock market growth makes private
    investors feel richer. They can sell a fraction of their stock without feeling
    poorer since the accelerating stock market compensates for the reduction
    in capital, providing a still rising capital. For instance, if investors
    are used to a stock market growth of 10% per year, they expect their
    capital to appreciate from $100 to $110 in a year. If during the following
    year, the growth rate rises to 20%, their capital rises to $120
    instead of the expected $110. They can thus spend $10 without having
    the impression of eating their capital, a psychological process associated
    with mental accounting [423, 373] (see the section titled “Behavioral
    Economics” in chapter 4). On the other hand, if there is not acceleration
    of stock market prices, capital gain only makes a one-time addition to
    the stock of wealth without changing the future flow of income. If the
    market is not accelerating, capital gains have only a transitory effect on
    expenditure. But even a faster-than-exponential accelerating market is
    unsustainable, as we have seen in preceding chapters. It may take years
    for the effect of a large rise in the stock market to burn itself out, but
    over a strategic time period, say 5 to 10 years, it is bound to do so [162].
    To summarize, the growth of the GDP and its associated stock market
    bubble can be associated with several unsustainable processes in the
    United States [162]: (1) the fall in private savings into ever deeper negative
    territory, (2) the rise in the flow of net lending to the private sector,
    (3) the rise in the growth rate of the real money stock, (4) the rise in
    asset prices at a rate that far exceeds the growth of profits (or of GDP),
    (5) the rise in the budget surplus, (6) the rise in the current account
    deficit, (7) the increase in the United States’ net foreign indebtedness
    relative to GDP.
    Godley concluded [162] that if spending were to stop rising relative to
    income without there being either a fiscal relaxation or a sharp recovery
    376 chapter 10
    in net exports, the impetus that has driven the expansion so far would
    evaporate and output would not grow fast enough to stop unemployment
    from rising. If, as seems likely, private expenditure at some stage reverts
    to its normal relationship with income, there will be, given present fiscal
    plans, a severe and unusually protracted recession with a large rise in
    unemployment. Because its momentum has become so dependent on
    rising private borrowing and rising capital gains, the real economy of
    the United States is at the mercy of the stock market to an unusual
    extent. A crash would probably have a much larger effect on output and
    employment now than in the past [162].
    However, there is one key ingredient that has been left out of this analysis:
    productivity gains. Recall that labor productivity is defined as real
    output per hour of work. Similarly, total factor productivity is defined as
    real output per unit of all inputs. Total factor productivity reflects, in part,
    the overall efficiency with which inputs are transformed into outputs. It
    is often associated with technology, but it also reflects the impact of a
    host of other factors, like economies of scale, any unaccounted inputs,
    resource reallocations, and so on. When productivity grows, the growth
    of the economy (GDP) can be larger than the growth of the difference
    between the amounts pumped in (government spending and exports) and
    siphoned out (taxes and imports), because more output per input creates
    internally new wealth at the aggregate level. As a consequence, it seems
    that Godley’s arguments do not apply directly.
    According to the official U.S. productivity statistics prepared by the
    U.S. Bureau of Labor Statistics, the average annual growth of total factor
    productivity was 2.7% between 1995 and 1999 (such a large growth rate
    implies that productivity would be 70% higher after 20 years). Clearly,
    the rate of productivity growth can have an enormous effect on real output
    and living standards. Productivity growth is a fundamental measure
    of economic health, and all of the major measures of aggregate labor and
    total factor productivity have recently shown improvements after long
    spells of sluggishness. If this improved performance continues, strong
    overall performance of real growth and low inflation may be sustained,
    although the short-run linkage of productivity to real income (and to output,
    after the very shortest period) is not as tight as some might expect
    [415]. Examination of the sources of productivity growth suggests that a
    major source of the better aggregate performance has been the remarkable
    surge of the high-technology sector (the New Economy argument!).
    A recent study of the link between information technology and the U.S.
    productivity revival in the late 1990s indeed shows that virtually all of
    the aggregate productivity acceleration can be traced to the industries
    2050: the end of the growth era? 377
    that either produced information technology or used it most intensively,
    with essentially no contribution from the remaining industries that were
    less involved in the IT revolution [416]. Faster productivity growth in this
    rapidly growing sector has directly added to aggregate growth, and the
    massive wave of investment in high-technology capital by other sectors
    has been equally important.
    This optimistic view should be tempered, however, by the fact that
    U.S. productivity growth shows a major cyclical component. In what
    amounts to a return to Godley’s argument [162], it has been shown
    recently that much of the rebound in productivity growth in the late
    1990s is a reflection of the strengthening of aggregate demand, rather
    than a fundamental improvement in the medium- or longer term productivity
    trend [165, 166]. The crash of the Nasdaq in April 2000, which
    reflects the collapse of the New Economy bubble, makes concrete that
    many New Economy industries have been far from delivering the enormous
    future incomes that were expected.
    The Aging “Baby Boomers.”
    Summarizing the world demographic structure and its financial assets
    by single statistics, as we have done so far, is restrictive and may miss
    important dimensions of the problem. In particular, understanding the
    economic consequences of the demographic development of the world
    over future decades probably requires us to distinguish between different
    segments of the populations, typically the young segment, the
    “asset accumulating” population (typically in the 40–65 range), and the
    65+ segment of the population. These population classes have different
    impacts in wealth creation, different consumption levels, they exert
    very different weights on society, and they have very different investment
    behaviors and needs.
    In particular, there is a concern that the aging “Baby Boomers,” the
    generation born in the two decades following the end of World War
    II, not only will exert an enormous strain on society due to retirement
    benefits but will correlatively cause a market meltdown as they start, in
    the decade of 2010, to sell their assets to finance their retirement. To
    appreciate why this may be of importance, let us recall that public and
    private pensions control almost a quarter of the United States’ tangible
    wealth, which is roughly equivalent to all of the country’s residential real
    estate. They account for most current savings in the country, are a crucial
    component of household retirement resources, and have significant
    effects on labor market mobility and efficiency. Collectively, they hold
    378 chapter 10
    a tremendous proportion of all common stock. Similar figures hold for
    most developed industrial countries in Europe and Japan.
    When the baby boomers retire, it is already clear that the Social Security
    system will require drastic changes to remain solvent. Will the stock
    market experience a similar meltdown as baby boomers withdraw their
    assets from pension plans [365]? The concern is that when the pension
    system begins to be a net seller of assets, roughly in the third decade of
    the century, this could depress stock prices.
    J. Poterba [336] of the Massachussets Institute of Technology argued
    that this simple logic is flawed because it neglects several important
    dimensions of the problem. First, lower demand for financial assets can
    lower price only if supply remains relatively unaffected. It is unrealistic
    to assume that the supply of stocks and bonds will remain fixed.
    For instance, a more balanced budget will lead goverments to issue
    fewer bonds. Second and more importantly, large demographic changes
    can have a substantial effect on economic performance and productivity
    growth, which in turn will impact asset returns. As we have argued before
    in this chapter, the magnitude of such indirect effects can be very significant
    and actually drive the accelerated growth of the economy. This can
    thus overwhelm any direct effect that population structure may have on
    asset returns. Third, the possible dependence between asset returns and
    demographic structure may be weakened by the increasing integration
    of world capital markets. For open economies with significant foreign
    investments, it is the global demographic structure that should matter.
    Finally, empirical data suggests that assets are sold much more slowly
    during retirement years than they are accumulated during working years.
    While not leading to a systematic meltdown, the stability of the markets
    and their susceptibility to external shocks may be significantly modified
    by the retirement of Baby Boomers. The impact of these Baby
    Boomers may also be one of the ingredients in the transition, by the
    middle of the twenty-first century, to another regime, which is discussed
    Related Works and Evidence
    Other authors have also documented a superexponential acceleration of
    human activity. Kapitza [231] recently analyzed the dynamical evolution
    of the human population, both aggregated and regionally, and also
    documented an overall acceleration until recent times, consistent with
    a power law singularity. He introduces an arbitrary saturation effect to
    2050: the end of the growth era? 379
    limit the blow-up on the basis that an infinity is impossible. Note that
    we, in contrast, prefer not to add any parameter and we interpret the
    approaching singularity as the signature of a transition. Using data from
    the Cambridge Encyclopedia, Kapitza also argued that epochs of characteristic
    evolutions or changes shrink as a geometrical series. In other
    words, the epoch sizes are approximately equidistant in the logarithm of
    the time to present, in agreement with our own findings [219].
    In a study of an important related human activity associated with
    research and development, A. van Raan [436] found that scientific production
    since the sixteenth century in Europe has accelerated much faster
    than exponentially [436]. Faster-than-exponential growth also occurs in
    computing power, as measured by the evolution of the number of MIPS
    per $1,000 of computer from 1900 to 1997 (see Figure 10.9). Thus, the
    so-called Moore’s law is incorrect, since it implies only an exponential
    growth. This faster-than-exponential acceleration has been argued to lead
    to a transition to a new era, around 2030, corresponding to the epoch
    when we will have the technological means to create superhuman intelligence
    From a more standard viewpoint, macroeconomic models have also
    been developed that predict the possibility of accelerated growth [352].
    Maybe the simplest model is that of M. Kremer [243], who noted that,
    over almost all of human history, technological progress has led mainly
    to an increase in population rather than an increase in output per person.
    Kremer developed a simple model in which the economic output per person
    is equal to a constant factor times the subsistence level, and is thus
    assumed fixed. The total output is supposed to increase with technology
    and knowledge and labor (proportional to population), for instance as
    proportional to their square root such that a multiplication of knowledge
    or of labor by 4 leads to a multiplication of output by only 2. The growth
    rate of knowledge and technology is taken proportional to population
    and to knowledge, embodying the concept that a larger population offers
    more opportunities for finding exceptionally talented people who will
    make important innovations and that new knowledge is obtained by leveraging
    existing knowledge. The resulting equation for the total population
    exhibits a growth rate, which is proportional to the population. Since the
    population growth rate grows as a positive power of population, this gives
    a finite-time singularity due to the positive feedback effects between
    population/labor, technology/knowledge, and output. Kremer tested this
    prediction by using population estimates extending back to 1 million
    B.C., constructed by archaeologists and anthropologists: he showed that
    the population growth rate is approximately linearly increasing with the
    380 chapter 10
    1995 Trend
    1920 1940 1960 1980 2000 2020 Year
    MIPS per $1000
    (1997 Dollars)
    Brain Power Equivalent
    per $1000 of Computer
    Evolution of Computer Power/Cost
    1985 Trend
    1975 Trend
    1965 Trend
    Gateway G5-200
    PowerMac 8100/50
    Mac II
    Macintosh 128k
    Commodore 64
    IBM PC
    DG Eclipse
    Apple II
    Power Tower 150e
    AT&T Globalyst 500
    IBM PS/2 90
    Mac IIfx
    Vax 11/750
    DEC Vax 11/780
    DG Nova
    SDS 920
    IBM 350/75
    IBM 7040
    Burroughs 5000
    IBM 1620
    IBM 650
    ASCC (Mark7)
    IBM Tabulator
    Burroughs Class 16
    IBM 704
    IBM 7090 IBM 1130
    DEC PDP-10
    CDC 7600
    Fig. 10.9.
    2050: the end of the growth era? 381
    population [243], in agreement with his prediction. This theory also predicts,
    in agreement with historical facts, that in the historical times when
    regions were separated, technological progress was faster in regions
    with larger populations, thus explaining the differences between Eurasia-
    Africa, the Americas, Australia, and Tasmania. Our results extend and
    refine his by showing the consistency of the determination of the critical
    time, not only for the population but also for the world GDP and for
    major financial indices.
    We have also generalized Kremer’s economic model by combining
    labor, capital, technology/innovation, and output/production to show that
    the finite-time singularities can be created from the interplay of these
    simultaneously growing variables, even if the individual quantities do not
    carry such singularities [219]. This interplay also explains the observation
    that the population and the financial indices have the same approximate
    critical time around 2050. The key point of these models is that the
    long-run growth is created endogenously rather than by random exogenous
    technical progress. Thus, rather than suffering from diminishing
    returns and dependence on exogenous innovations, the growth view provides
    an endogenous mechanism for long-run growth, either by avoiding
    diminishing returns to capital or by explaining technological progress
    A complementary and very simple approach is to incorporate a feedback
    between the population and the increasing “carrying capacity” of
    Fig. 10.9. Faster than exponential growth in computing power illustrated by the
    evolution of the number of MIPS (million of instructions per second) per $1,000 of
    computer from 1900 to 1997. Steady improvements in mechanical and electromechanical
    calculators before World War II had increased the speed of calculation a
    thousandfold over manual methods from 1900 to 1940. The pace quickened with
    the appearance of electronic computers during the war, and 1940 to 1980 saw a
    millionfold increase. Since then, the pace has been even quicker, a pace that would
    make humanlike robots possible before the middle of the twenty-first century. The
    vertical scale is logarithmic; the major divisions represent thousandfold increases in
    computer performance. Exponential growth would show as a straight line, and the
    upward curve indicates faster than exponential growth, an accelerating rate of innovation.
    The superexponential growth is also seen from the fact that the estimated
    exponential trends, represented as the straight lines, increase continuously from 1965
    to 1995. The reduced spread of the data in the 1990s is probably the result of intensified
    competition: underperforming machines are more rapidly squeezed out. The
    animals listed on the right provide a scale of reference of their effective calculation
    power. Figure reproduced from [307].
    382 chapter 10
    the Earth within Malthus’s model. Such feedback comes from technological
    progress such as the use of tools and fire, the development of
    agriculture, the use of fossil fuels, and fertilizers, as well the expansion
    into new habitats and the removal of limiting factors by the development
    of vaccines, pesticides, antibiotics, and so on. If the carrying
    capacity increases sufficiently fast, a finite-time singularity is obtained
    in the equations. In reality, the singularity will be smoothed out because
    the Earth is not infinite.
    The logistic equation of population growth and positive feedback on
    the earth’s carrying capacity: As a standard model of population growth,
    Malthus’s model assumes that the size of a population increases by a fixed
    proportion r over a given period of time independently of the size of the
    population and thus gives an exponential growth. The logistic equation
    attempts to correct for the resulting unbounded exponential growth by
    assuming a finite carrying capacity K such that the population instead
    evolves according to
    = rp�t��K ? p�t��� (30)
    The carrying capacity K is not fixed and has no simple relation with other
    variables as it depends on the structure of production and consumption.
    It is contingent on the changing interactions between the physical and
    biotic environment. While a single number for human carrying capacity is
    certainly reductionist because of the difficulties in knowing human innovations
    and biological evolutions, Vitousek et al. [440] have provided a
    general index of the current intensity of the impact of humans on the
    biosphere: the total net terrestrial primary production of the biosphere currently
    appropriated for human consumption is around 40%. This puts the
    scale of the human presence on the planet in perspective [15].
    Cohen and others (see [87] and references therein) have put forward
    idealized models taking into account interaction between the human population
    p�t� and the corresponding carrying capacity K�t� by assuming
    that K�t� increases with p�t� due to technological progress, as explained
    above. If dK�t�/dt is sufficiently larger than dp�t�/dt for all times, for
    instance if K ∝ p with > 1, then p�t� explodes to infinity after a finite
    time, creating a singularity. Indeed, in this case, the limiting factor ?p�t�
    can be dropped out and (30) becomes
    = r�p�t��1+ � (31)
    2050: the end of the growth era? 383
    where the growth rate accelerates with time according to r�p�t�� . The
    generic consequence of a power law acceleration in the growth rate is the
    appearance of singularities in finite time:
    p�t� ∝ �tc
    ? t�z� with z = ?1

and t close to tc� (32)
Equation (31) is said to have a “spontaneous” or “movable” singularity
at the critical time tc [37], the critical time tc being determined by the
constant of integration, that is, the initial condition p�t = 0�.
Nottale (an astrophysicist), Chaline (a paleontologist), and Grou (an
economist) [317, 318] have recently independently applied a log-periodic
analysis to the main crises of different civilizations. They first noticed
that historical events seem to accelerate. This was actually anticipated by
Meyer, who used a primitive form of log-periodic acceleration analysis
[295, 296]. Grou [181] has demonstrated that the economic evolution
since the neolithic can be described in terms of various dominating poles,
which are subjected to an accelerating crisis/no-crisis pattern.
The quantitative analysis of Nottale, Chaline, and Grou on the median
dates of the main periods of economic crisis in the history of Western civilization
(as listed in [181, 52, 156]) are as follows (the dominating pole
and the date are given in years with respect to Jesus Christ): Neolithic:
?6500, Egypt: ?3000, Egypt: ?900, Greece: ?100, Rome: +400,
Byzantium: +800, Arab expansion: +1100, Southern Europe: +1400,
Netherlands: +1650, Great Britain: +1775, Great Britain: +1830, Great
Britain: +1880, Great Britain: +1935, United States: +1975. A logperiodic
acceleration with scale factor � = 1�32 ± 0�018 occurs towards
= 2080 ± 30. Agreement between the data and the log-periodic law
is statistically significant (tstudent
= 145; the probability that this results
from chance is much less than 0.01%). It is striking that this independent
analysis based on a different data set gives a critical time that is
compatible with our own estimate, 2050 ± 10.
What could be the possible scenarios for mankind close to and beyond
the critical time? As seems fitting for the apex of this essay, this last part
is highly speculative in nature.
384 chapter 10
Contemporary thinkers foresee collapse from such catastrophes as
nuclear war, resource depletion, economic decline, ecological crises, or
sociopolitical disintegration (see [419] and references therein).
In such a gloomy scenario, humankind will enter a severe recession
fed by the slow death of its host (the Earth). W. Hern [192], from the University
of Colorado at Boulder, and other scientists have gone as far as
comparing the human species with cancer: the sum of human activities,
viewed over the past tens of thousand of years, exhibits all four major
characteristics of a malignant process: rapid uncontrolled growth, invasion
and destruction of adjacent tissues (ecosystems), metastasis (colonization
and urbanization), and dedifferentiation (loss of distinctiveness
in individual components as well as communities throughout the planet).
This worry about human population size and growth is shared by
many scientists, as we summarized at the beginning of this chapter.
Associated with predicted crises of overpopulation, possible scenarios
involve a systematic development of terrorism and the segregation of
mankind into at least two groups, a minority of wealthy communities
hiding behind fortresses from the crowd of “have-nots” roaming outside,
as discussed in a recent seminar of the National Academy of Sciences
of the United States. This could occur both within developed countries
as well as between them and developing countries.
In this respect, history tells us that civilizations are fragile, impermanent
things. Our present civilization is a relative newborn, succeeding
many others that have died. The fall of the Roman Empire is, in the West,
the most widely known instance of collapse. Yet it is only one case of a
common process. Collapse is a recurrent feature of human societies. The
archeological and historical record is indeed replete with evidence for
prehistoric, ancient, and premodern societal collapses. These collapses
occurred quite suddenly and frequently involved regional abandonment,
replacement of one subsistence base by another (such as agriculture by
pastoralism), or conversion to a lower energy sociopolitical organization
(such as local state from interregional empire).
Human history as a whole has been characterized by a seemingly
inexorable trend toward higher levels of complexity, specialization,
and sociopolitical control, processing of greater quantities of energy
and information, formation of ever larger settlements, and development
of more complex and capable technologies [419]. There is a growing
body of research suggesting that the complexity caused by high
technology could be humankind’s undoing. For instance, the Maya
2050: the end of the growth era? 385
of the southern Peten lowlands dominated Central America up to the
ninth century. They built elaborate irrigation systems to support their
booming population, which was concentrated in cities growing in size
and power, with temples and palaces built and decorated, the arts
flourishing, and the landscape being modified and claimed for planting.
Overpopulation and the overreliance on irrigation was a major factor
in making the Maya vulnerable to failure: the trigger event of their
collapse appears to have been a long drought beginning about 840 A.D.
(communication of V. Scarborough, an archaelogist from the University
of Cincinnati [90]). Among many factors, such as war and plagues, that
contributed to many of the collapses of ancient societies, there seem
to be two main causes: too many people and too little fresh water.
As a consequence, the civilization became vulnerable to environmental
stress, for instance, a prolonged drought or a change in climate [90].
The societies themselves appear to have contributed to their own demise
by encouraging growth of their population to levels that carried the
seeds of their own decline through overexploitation of the land (communication
of C. Scarre, an archaelogist from the Cambridge University
in England [90]). Similarly, the Akkadian empire in Mesopotamia,
the Old Kingdom of Egypt, the Indus Valley civilization in India,
and early societies in Palestine, Greece, and Crete all collapsed in a
catastrophic drought and cooling of the atmosphere between 2300 and
2200 B.C.
The accumulation of high-resolution paleoclimatic data that provide
an independent measure of the timing, amplitude, and duration of past
climate events shows that the climate during the past 11,000 years
has been punctuated by many climatic instabilities [449]: multidecadal
to multicentury-length droughts started abruptly, were unprecedented
in the experience of the existing societies, and were highly disruptive
to their agricultural foundations because social and technological
innovations were not available to counter the rapidity, amplitude, and
duration of changing climatic conditions. These climatic events were
abrupt, involved new conditions that were unfamiliar to the inhabitants
of the time, and persisted for decades to centuries. They were therefore
highly disruptive, leading to societal collapse—an adaptive response to
otherwise insurmountable stresses [449].
It is tempting to believe that modern civilization, with its scientific
and technological capacity, its energy resources, and its knowledge of
economics and history, should be able to survive whatever crises ancient
and simpler societies found insurmountable. But how firm is this belief
in view of the fact that our modern civilization has achieved the highest
386 chapter 10
level of complexity known to humanity? This complexity comes with a
high differentiation of human activities, a strong interdependence, and
a reliance on environmental resources to feed concentrated populations.
These ingredients seem to have been the roots of collapse of many previous
civilizations. Tainter [420] suggested that the diminishing returns to
problem solving due to increased complexity limited the ability of historical
societies to resolve their challenges. To allow contemporary societies
to address global change, he proposes encouraging and financing problem
solving in the context of a system of evolving complexity. This view
seems the opposite of our suggestion of a coming crisis announced by the
acceleration of population growth fed by its associated economic growth,
both relying on the unfolding of scientific and technological revolutions.
The acceleration of innovations is the solution that Tainter requires to
avoid the dead-ends confronted by previous civilizations. In contrast, we
suggest that this acceleration carries the roots of its own collapse in its
How can these two viewpoints be reconciled? To answer, we have
to draw on recent research in optimization/remediation of complex
systems, with applications in epidemiology, aeronautical and automotive
design, forestry and environmental studies, the Internet, traffic, and
power systems, which suggest that complex systems develop somewhat
paradoxically a remarkable robustness as well as a fragility [71, 394].
Indeed, there is a tendency for interconnected systems to gain robustness
against uncertainties in one area by becoming more sensitive in other
areas. A system might attain robustness against common uncertainties
and yet be hypersensitive to design flaws or rare events. For example,
organisms and ecosystems exhibit remarkable robustness to large variations
in temperature, moisture, nutrients, and predation, but can be
catastrophically sensitive to tiny perturbations of a different kind, such
as a genetic mutation, an exotic species, or a novel virus.
As an illustration, consider a forest in which spontaneous ignition
(sparks and lightning) occurs preferentially in some part of the forest;
in other words, the spatial distribution of sparks is not homogeneous.
The management problem is to conceive an optimal array of firewalls
that provides the highest possible yield of the forest, while taking into
account the cost of building and keeping firewalls in good working order.
To a given geometrical structure of firewalls corresponds a specific size
and a specific spatial distribution of protected domains or tree clusters.
When a spark falls on a tree within a cluster, the whole connected cluster
of trees delimited by the firewalls bounding it is supposed to burn
2050: the end of the growth era? 387
entirely. In other words, the fires are supposed to stop only at the firewalls.
We can thus reformulate the optimal management of the forest so
that it consists of building firewalls that maximize the yield after fires,
that is, that minimize the average destructive impact of fires, given the
cost of building and keeping firewalls in good working order.
In the presence of a heterogeneous spatial probability density � of
sparks, it is clear that the density r of firewalls should not be spatially
uniform: more firewalls are needed in sensitive regions where the sparks
are numerous. The density r of firewalls will thus not be constant according
to the optimization process but will adjust to the predefined distribution
� of sparks. This spatial distribution � of sparks determines the
probability pi that a spark ignites a fire in a given domain or cluster i
bounded by the fire walls: pi is the sum of � over the cluster. In the presence
of a nonuniform distribution of sparks, it can be shown [71, 394]
that the optimization of the yield, that is, the minimization of the average
fire size, given the cost of firewalls, leads to a power distribution of
domains delimited by firewalls. The optimization process provides robust
performance despite the uncertainties quantified by the probabilities pi.
In the forest fire example, the optimal distribution of spatial firewalls is
the result of the interplay between our a priori knowledge of the uncertainty
in the distribution of sparks and the cost resulting from fires. The
solutions are robust with respect to the existence of uncertainties, that is,
to the fact that we do not know deterministically where sparks are going
to ignite; we only know their probability distribution.
However, the optimal spatial geometry of firewalls is fragile with
respect to an error in the quantification of the probabilities pi, that is, to
model errors, to use the terminology of chapter 9. It is not the uncertainty
that is dangerous, but errors in quantifying this uncertainty: a different
set of pi would lead to a very different spatial distribution of firewalls.
Thus, an optimized system of firewalls will be fragile, that is, poorly
adapted to even a modest but long-term spatial redistribution of spark
ignitions [71, 394].
Following this concept, we can rephrase the problem and say that
the robustness of our modern society is derived from its adaptation to
a model of growth relying on a succession of technological revolutions
and its applications. However, our society may be fragile with respect
to a global change that may require a different dynamical regime. The
concept of a critical singularity suggests in addition that this fragility or
susceptibility to global changes will rise as the optimization of society
and its complexity increase. Following Tainter [420], we probably need
to develop solutions for qualitatively different regimes. These solutions
388 chapter 10
(a) (b)
(c) (d)
Fig. 10.10. Unoccupied sites are black, and occupied sites (trees) are white in a
system of N = 64 by N = 64 sites. The goal is to optimize the yield of the
model forest, that is, to optimize the number of trees minus the losses due to fires.
Sparks are assumed to be more probable in the top-left corner. The optimal tree
configurations of four different forest management strategies are compared in the
different panels. In panel (a), trees are grown at random step by step at previously
empty sites. The optimal tree configuration corresponds to the so-called percolation
critical density. This is the “laissez-faire” strategy. In panels (b)–(d), an optimization
is performed by calculating for each choice of an additional tree what would be the
resulting average yield, thus weighting the possible future impact of random sparks.
An increasing degree of sophistication is used from panel (b) to (d) according to
the “design parameter” D. D measures the number of tree configurations that are
considered upon the addition of a new tree in the calculation of the optimal tree
planting strategy. Panel (b) corresponds to D = 2; that is, only two tree positions
are examined and the best one is chosen. Panel (c) corresponds to D = N = 64
and panel (d) corresponds to D = N2 = 4096; that is, all possible positions for the
2050: the end of the growth era? 389
may not emerge spontaneously from the accelerated innovation process
and ensuing growth which feed on themselves while preventing exploration
of other dynamical modes.
A disruption that is particularly predicted is that future climatic change
will involve both natural and anthropogenic forces and will be increasingly
dominated by the latter. Current estimates show that we can expect
them to be large and rapid. Global temperature will rise and atmospheric
circulation will change, leading to a redistribution of rainfall that is difficult
to predict. These changes will affect a world population expected to
increase from about 6 billion people today to about 10 billion by 2050. In
spite of technological changes, most of the world’s people will continue
to be subsistence or small-scale market agriculturalists, who are similarly
as vulnerable to climatic fluctuations as the late prehistoric/early historic
societies. Furthermore, in an increasingly crowded world, habitat tracking
as an adaptive response will not be an option. We do, however, have
distinct advantages over societies in the past because we can anticipate
the future somewhat. We must use this information to design strategies
that minimize the impact of climate change on societies that are at greatest
risk. This will require substantial international cooperation, without
which the twenty-first century will likely witness unprecedented social
disruptions [449].
Transition to Sustainability
A more optimistic perspective is that “ecological” actions will grow in
future decades, leading to a smooth transition towards an ecologically
integrated industry and humanity. There are some signs that we are on
this path: during the 1990s, the use of wind power grew at a rate of 26%
a year, and solar photovoltaic power at 17%, compared to the growth
Fig. 10.10 continued. additional tree are studied with respect to their consequence
on the danger of fires. This is reminiscent of playing chess, in which D is the
number of combinations that the player examines. Note that, as the sophistication D
of the optimization process increases, the optimal forest becomes denser and denser,
with only a few empty sites remaining that are organized so as to form effective
firewalls. These firewalls have been optimized to disconnect the forest in an optimal
set of tree clusters, given the known distribution of dangerous sparks. Note that, if
the sparks were suddenly to become more numerous in the lower right corner of the
square, the optimal solution (d) would behave catastrophically, illustrating also the
fragile nature of this optimization. Figure reproduced from [72].
390 chapter 10
in coal and oil at under 2%; governments have ratified more than 170
international environmental treaties, on everything from fishing to desertification.
However, there is serious resistance, in particular because there is no
consensus on the seriousness of the situation, as described in the section
on “The Optimistic Viewpoint of Social Scientists.” The problem is not
that this optimistic view is wrong. By economic accounting, the optimistic
view is mostly right. The issues raised by the analysis presented
here [219] and by others is that the approach to a finite-time singularity
can be surprisingly fast in the last few decades preceding it. As a
result, linear extrapolations will be grossly misleading, with catastrophic
consequences. What our analysis shows is that the “optimistic viewpoint”
contains endogenously its own death, in the form of a predicted
singularity, precisely created by the acceleration feeding the optimistic
The transition to sustainability consists in the evolution from a
growth regime to a balanced symbiosis with nature and with the Earth’s
resources. This would require the transition to a knowledge-based society,
in which knowledge, intellectual, artistic, and humanistic values
replace the quest for material wealth. Indeed, the main economic difference
is that knowledge is “nonrival” [350]: the use of an idea or of
a piece of knowledge in one place does not prevent it from being used
elsewhere. In contrast, say an item of clothing used by an individual
precludes its simultaneous use by someone else. Only the emphasis
on nonrival goods will ultimately limit the plunder of the planet. The
incentives that people need to work and to find meaning in their lives
should be found beyond material wealth and power. Some so-called
“primitive” societies seem to have been able to evolve into such a state.
Many researchers and environmental groups advocate a transition from
our present energy systems, dominated by use of oil, gas, and coal,
which are not sustainable, to a more direct use of solar energy in the
form of radiation, wind, ocean motions, and biomass production (see,
for instance, [148, 149, 151] and references therein). The sustainable
production of food and biomass depends on a number of critical components,
which include soil quality, water quality with adequate quantity,
climate, air quality, agriculture technology, fertilizer technology, biotechnology,
and biodiversity. Novel advances in plant biotechnology must be
deployed for the benefits of the rising population of developing countries
as the gains in food production provided by the “green” revolution have
reached their ceilings while world population continues to rise [91].
2050: the end of the growth era? 391
There is also a global problem of soil erosion, as almost 1% of the
world’s topsoil is lost annually [151] (at this rate, half the soil will be
lost in less than 70 years). Soil erosion can be prevented by intelligent
use of water and of vegetation. The quality of soil is also a crucial
issue: soil is a very complex material formed by the action of the atmosphere,
the hydrosphere, and the biosphere on rocky materials, collectively
called “weathering.” To reform a soil from its parent rock once
the soil is removed takes many decades to millenia. There is a need for
total soil chemistry with the development of a new agriculture based
on diversity and integration of techniques for a multiplicity of fields.
Water and soil are closely associated. The management of water supplies
requires integration of knowledge from almost all sciences and engineering
with major input coming from sustainable sociology and economics
[148, 149, 151].
Extraction of ore and purification of minerals produce enormous
amounts of toxic elements and pollution like arsenic, halogens (fluorine,
chlorine, and bromine), mercury, lead, sulphur, and selenium. We need
new engineering technologies to collect materials with minimal disturbance
to the environment. As 75% of the population of industrial nations
live in cities, there is a vast problem of waste management, including
technologies leading to massive air and water pollution. We need good
quality control at the source and recycling technology. To produce truly
sustainable systems, all people must be educated and must understand
our life support system [150].
Last but not least, we need the will to act rather than lip service
[264]. The triumphalism around economic growth has left no time to
spare for concern about the environment. For the major multinationals
in the resource, energy, chemicals, and agriculture industries to work
really concretely towards sustainability, the market forces do not seem
sufficient [145] as long as the service really offered by the environment
is not adequately priced and inserted into the accounting balance.
Ecosystems are capital assets: when properly managed, they yield a
flow of vital goods and services [99]. The value of nature includes the
production of goods (such as seafood and timber), life support processes
(such as pollination, air and water purification, climate stabilization, mitigation
of floods and droughts, pest control, generation of fertile soils),
and life-fulfilling conditions (such as recreation, beauty, and serenity).
Moreover, ecosystems have value in terms of the conservation of options
(such as genetic diversity for future use). To take another example, the
economic value of part of the Amazon rainforest is not limited to its
financial value as a repository of future pharmaceutical products or as a
392 chapter 10
location for ecotourism. That “use” value may only be a small part of the
properly defined economic valuation. For decades, economists have recognized
the importance of the “non-use” value of environmental amenities
such as wilderness areas or endangered species. The public nature
of these goods makes it particularly difficult to quantify these values
empirically, as market prices do not exist [145]. Indeed, relative to other
forms of capital, ecosystems are poorly understood, scarcely monitored,
and (in many cases) undergoing rapid degradation and depletion.
It has been argued that the process of economic valuation could
improve stewardship [99]. Individuals and societies already assess the
value of nature implicitly in their collective decision making, too often
treating ecosystem services as “free.” Until recently, this was generally
safe to do: relatively speaking, ecosystem capital was abundant, and
the impacts of economic activity were minimal. Ecosystem capital is
becoming ever scarcer, however, so that it is now critical to understand
both how to value ecosystems and the limitations of such valuations
[145]. R. Costanza of the University of Maryland and twelve coauthors
have made one of the most controversial recent attempts to integrate
economics and ecology to obtain the total monetary value for the
world’s “ecosystem services and natural capital” [94]: they obtained the
figure of $33 trillion per year, which exceeds the sum of the world’s
gross national products. Costanza et al. described the $33 trillion per
year as “a minimum estimate” for the “current economic value” of 17
ecosystem services (from atmospheric gas regulation to the provision of
“cultural value”) summed over 16 types of ecosystems (from the open
ocean to urban centers). This work has raised much criticisms, from “a
serious underestimate of infinity” by M. Toman of Resources for the
Future, to “non-applicable as neoclassical economics measures value
in the context of a specific exchange.” In the neoclassical economics
view, it is nonsensical to ask what the value of the world’s ecosystem
services is. A related requirement is that one can evaluate only small
(or “marginal”) changes from current conditions. However, what is
important in our view is that this order of magnitude study corrects the
result of 1% of GNP or less for the value of ecosystem services that
many would have guessed. Having this number is better than no number
at all, as it can foster the integration of environmental sustainability into
industrial and economics approaches.
2050: the end of the growth era? 393
Resuming Accelerating Growth by Overpassing
Fundamental Barriers
The new regime announced by the finite-time singularity could be a
renewed race for growth, an even stronger acceleration enhanced by new
discoveries enabling mankind to fully exploit the vast resources of the
oceans (mostly untapped yet) and even that of other planets, especially
beyond our solar system. The conditions for this are rather drastic. For
the planets, novel modes of much faster propulsions are required as well
as revolutions in our control of the adverse biological effects of space on
humans with its zero gravity and high radiation. New drugs and genetic
engineering could prepare humans for the hardship of space, leading to a
new era of enhanced accelerated growth after a period of consolidation,
culminating in a new finite-time singularity, probably centuries in the
The growth rate of computer power (see Figure 10.9) followed more
recently by the advent of the large-scale use of the Internet makes more
probable a major evolution of human interactions with computers and
networks than with any other machines. V. Vinge, [438] emeritus professor
at the Department of Mathematical Sciences at San Diego State
University and an author of science fiction books, proposes that the acceleration
of technological progress will cause the creation by technology
of entities with greater than human intelligence before 2030. He explored
several routes by which science may achieve this breakthrough:
� There may be developed computers that are “awake” and superhumanly
intelligent. (To date, there has been much controversy as to
whether we can create human equivalence in a machine. But if the
answer is “yes, we can,” then there is little doubt that beings more
intelligent can be constructed shortly thereafter.)
� Large computer networks (and their associated users) may “wake up”
as a superhumanly intelligent entity.
� Computer–human interfaces may become so intimate that users may
reasonably be considered superhumanly intelligent.
� Biological science may provide means to improve natural human intellect.
Vinge used the word “singularity” quite adequately in the present context
to describe the point where our old models must be discarded and
a new reality rules as a result of this transition to a superhuman intelligence.
If or when greater-than-human intelligence will drive progress,
394 chapter 10
this progress will be much more rapid and will probably involve the
creation of still more intelligent entities, on a still-shorter time scale. In
the evolutionary past, animals adapted to problems and made inventions,
the world acting as its own simulator in the case of natural selection
over time scales of millions of years. Superhuman intelligence can lead
to a drastic acceleration of natural evolution by executing simulations
at much higher speeds. Developments that before were thought to be
possible in “a million years” (if ever) may happen in this or in the next
century [438]. This accelerated evolution may have disturbing consequences.
Superhumanly intelligent machines would not be humankind’s
“tools,” any more than humans are the tools of rabbits or robins or chimpanzees.
Will they treat us more kindly than we have treated animals?
There are several arguments opposing the possibility of human intelligence
and consciousness, not to speak of superhuman. R. Penrose, professor
of physics and mathematics at the University of Oxford and at
Penn State University, develops an argument based on G?del’s incompleteness
theorem that the mechanism for consciousness involves quantum
gravitational phenomena, acting through microtubules in neurons
[331]. J. Searle, professor of philosophy at U.C. Berkeley, holds that
the syntactic manipulation of formal symbols by computers does not by
itself constitute a semantics [367]. Computers are mindless manipulators
of symbols, and they don’t understand what they are “saying.” It should
be noted that Searle’s biological naturalism does not entail that brains
and only brains can cause consciousness. Searle is careful to point out
that while it appears to be the case that certain brain functions are sufficient
for producing conscious states, our current state of neurobiological
knowledge prevents us from concluding that they are necessary for producing
There is also the possibility that the computational competence of
single neurons may be far higher than generally believed. If so, our
present computer hardware might be as much as ten orders of magnitude
short of the equipment we carry around in our heads. If this is true (or for
that matter, if the Penrose or Searle critique is valid), we might never see
the singularity [438]. But if the technological singularity can happen, it
will. Vinge argues that we cannot prevent the singularity, that its coming
is an inevitable consequence of humans’ natural competitiveness and the
possibilities inherent in technology.
Within this scenario, a central feature of strongly superhuman entities
will likely be their ability to communicate at variable bandwidths,
including ones far higher than speech or written messages. What happens
when pieces of ego can be copied and merged, when the size
2050: the end of the growth era? 395
of a self-awareness can grow or shrink to fit the nature of the problems
under consideration [438]? These are probably essential features
of strong superhumanity, with time accelerated so much that it becomes
unending and with the ability to truly know one another and to understand
the deepest mysteries.
The immersion of our analysis of stock markets into this general demographic,
environmental, and economic framework was a necessary step
because, at long time scales, their future and, in particular, the occurrence
of financial crashes cannot be decoupled from the many other
components of the world in which they “live.”
We would like to conclude this essay by pointing out that, reciprocally,
the whole economy is progressively emulating the behavior of
stock markets. In his testimony on monetary policy on the last Wednesday
February 2001, Alan Greenspan, the chairman of the U.S. Federal
Reserve, made the following argument: “The same forces that have been
boosting growth in structural productivity seem also to have accelerated
the pace of cyclical adjustment.” In other words, the recent plunge in
manufacturing is just a matter of nimble firms, reflexes speeded up by
information technology, moving quickly to get rid of excess inventories
[250]. This faster adjustment contains a caveat: firms’ investment
decisions are starting to emulate the hair-trigger behavior of financial
investors. This was summarized in Greenspan’s testimony as follows:
“The hastening of the adjustment to emerging imbalances is generally
beneficial � � � But the faster adjustment process does raise some warning
flags � � � flags appear to be acting in far closer alignment with one
another than in decades past.”
This implies that a growing part of the economy may be starting to act
like a financial market, with all that implies, like the potential for bubbles
and panics. Indeed, Krugman has argued that, far from making the
economy more stable, the rapid responses of today’s corporations make
their investment in equipment and software vulnerable to the kind of selffulfilling
pessimism that used to be possible only for investment in paper
assets like stocks [250]. A typical behavior is that businesses are abruptly
scaling back their investment plans, not because they are already hurt
but because a developing climate of fear has convinced managers that
396 chapter 10
it would be “prudent to be prudent.” And since one company’s investment
is another company’s sales, such retrenchment can bring on the
very slump that managers fear [250]. We argue that, symmetrically, optimistic
views of the future can progressively transform into self-fulfilling
bubbles which define corporations strategies, and their investment and
recruitment objectives. If the bubbles inflate too much or for too long,
they may collapse in “crashes.”
Thus, far from being a thing of the past, it is probable that the speculative
and self-fulfilling bubble and antibubble behaviors are going to
inhabit a larger and larger portion of economic and human activities. The
phenomena and underlying mechanisms discussed in this book may thus
become even more relevant to a larger and larger portion of human activity.
Understand their origin, and be prepared for subtle but significant

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