# why stoc,

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 UNIVERSITY PRESS

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)

- Financial crises—History. 2. Stocks—Prices—History.
- Financial crises—United States—History.
- Stock exchanges—United States—History.
- 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. �

www.pupress.princeton.edu

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

Contents

xiii Preface

Chapter 1

financial crashes:

what, how, why,

and when?

3

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

26

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

Lunches”

45 Prices Are Unpredictable,

or Are They?

47 Risk–Return Trade-Off

Chapter 3

financial crashes

are “outliers”

49

49 What Are “Abnormal” Returns?

51 Drawdowns (Runs)

51 Definition of Drawdowns

54 Drawdowns and the Detection of

“Outliers”

56 Expected Distribution of “Normal”

Drawdowns

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

81

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

Psychology

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

Behavior

130 Complex Evolutionary Adaptive

Systems of Boundedly

Rational Agents

Chapter 5

modeling financial

bubbles and

market crashes

134

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

Models

Chapter 6

hierarchies,

complex fractal

dimensions, and

log-periodicity

171

173 Critical Phenomena by Imitation on

Hierarchical Networks

173 The Underlying Hierarchical

Structure of Social Networks

177 Critical Behavior in Hierarchical

Networks

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

Group

contents ix

198 Complex Fractal Dimensions and

Log-Periodicity

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

Singularity

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

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

Mid-1980

279 Synthesis: “Emergent” Behavior of the

Stock Market

Chapter 8

bubbles, crises,

and crashes in

emergent markets

281

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

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

Signatures?

334 A Hierarchy of Prediction Schemes

334 The Simple Power Law

335 The “Linear” Log-Periodic Formula

contents xi

336 The “Nonlinear” Log-Periodic

Formula

336 The Shank’s Transformation on a

Hierarchy of Characteristic Times

337 Application to the October 1929

Crash

338 Application to the October 1987

Crash

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

Predictions

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

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

Barriers

395 The Increasing Propensity to Emulate

the Stock Market Approach

397 References

419 Index

Preface

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

1983

35

Developing

countries

Other developed

United Kingdom

Japan

US

30

25

20

1986 1989 1992 1995

15

10

5

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

occur.

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

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chapter 1

financial crashes: what,

how, why, and when?

WHAT ARE CRASHES, AND

WHY DO WE CARE?

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

instabilities?

financial crashes: what, why, and when? 5

THE CRASH OF OCTOBER 1987

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. - 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. - 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 - 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. - 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

partner. - 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

crashes.

HISTORICAL CRASHES

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 - 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

participants.

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

models.

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

1910

20

10

9

8

7

30

r = 0.58

25

20

1915 1920 1925 1930

15

10

5

1935 1940

Books

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

crashed.

EXTREME EVENTS IN COMPLEX SYSTEMS

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

100

100

80

70

60

50

40

1895

1000

800

600

700

1900 1905 1910 1915

500

400

300

200

100

80

70

60

50

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

masses.

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

limitation.

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.

IS PREDICTION POSSIBLE? A WORKING HYPOTHESIS

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.

THE BASICS

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

1

10

100

1,000

10,000

1800 1840 1880 1920 1960 2000

Dow Jones Industrial Average Jan 1790–Sept 2000

Close Prices

Years

2.9%/y

6.8%/y

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

plots.

500

1,500

1,000

2,000

2,500

3,000

1980 1982 1984 1986 1988

Dow Jones Industrial Average 2 Jan 1980–31 Dec 1987

Close Prices

Years

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�

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0 100 200 300 400

June, 20th, 1995

One Minute Returns

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

-0.04

-0.02

0

0.02

0.04

1-1-80 31-12-81 1-1-84 31-12-85

Dow Jones Index Returns Jan. 2nd 1980–Dec.31st 1987

Daily Returns

Time

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

-0.4

-0.2

0

0.2

0.4

1800 1850 1900 1950 2000

Dow Jones Index Jan. 1790–Sept. 2000

Returns

Time

0.1

0.3

-0.3

-0.1

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.

-0.04

-0.02

0

0.02

0.04

0 200 400 600 1000

Gaussian White Noise

Time

0.01

0.03

-0.03

-0.01

800

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

1

10

100

1000

Distribution Function

return DJ>0

return DJ<0

return NAS>0

return NAS<0

|Returns|

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,

- 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

-0.2

0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8

S&P 500

Correlation

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

mt

≡ 1

B�t� t�

�

i<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

decrease).

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.

THE EFFICIENT MARKET HYPOTHESIS

AND THE RANDOM WALK

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

0.9

1.0

1.1

1.2

1.3

1.4

1.5

0 200 400 600 800 1000

Position

Time

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

1

0

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

traders.

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.

- 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. - 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. - 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):

pt

= 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

pt

= 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

1

� (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%).

RISK–RETURN TRADE-OFF

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.

WHAT ARE “ABNORMAL” RETURNS?

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

“outlier.”

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.

DRAWDOWNS (RUNS)

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

1600

1800

2000

2200

2400

2600

2800

1987.77 1987.79 1987.81 1987.83 1987.85

DJIA

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

varies).

any structure in the distribution of drawdowns absent in that of price

variations.

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

rule:

p�t� =

�t� +

�t ? 1�

�t ? 2�� (5)

where

�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

before!

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 ?

1��

�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

moves.

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

predictability.”

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

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0

Log[P(|u11|2/<|u11|2>)]

|u11|2/<|u11|2>

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

-15

-10

-5

0

Ln[P(Un

2)]

Un

2

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.

DRAWDOWN DISTRIBUTIONS OF STOCK

MARKET INDICES

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

1

2

3

4

5

6

7

8

9

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

0

-5

-10

-15

-20

-25

-30

1 10 100 1000

Loss (%)

Rank

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

0.001

0.01

0.1

1.0

-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

[217].

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

controlled.

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:

- We looked at financial data and found it apparently random.
- We formed the hypothesis that the time evolution of stock market

prices are random walks. - We used this hypothesis to make the prediction that the distribution of

drawdowns should be exponential. - 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

1

10

100

1000

10000

-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 PRESENCE OF OUTLIERS

IS A GENERAL PHENOMENON

The data sets that have been analyzed [220] comprise - 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); - currencies: U.S. dollar versus German mark (UD$/DM), U.S. dollar

versus Japanese yen (UD$/Yen), U.S. dollar versus Swiss franc

(UD$/CHF); - gold;
- 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

Ford).

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

1

2

3

4

5

6

7

8

0.3 0.25 0.2 0.15 0.1 0.05 0

Log(Cumulative Number)

Draw Down

Null Hypothesis

Draw Down

0

1

2

3

4

5

6

7

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

1

2

3

4

5

6

7

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

0

1

2

3

4

5

6

7

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

1

2

3

4

5

6

7

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

1

2

3

4

5

6

7

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

1

2

3

4

5

6

7

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

1

0

2

3

4

5

6

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

1

2

3

4

5

6

7

8

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

1

0

2

3

4

5

6 Log(Cumulative Number)

Draw Down

Null Hypothesis

Draw Down

Fig. 3.12. DAX (Germany) (left) and MIBTel (Italy) (right). Reproduced from

[220].

72 chapter 3

0

-2

-4

2

4

6

8

-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

1

0

2

3

4

5

7

6

8

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

2

3

1

4

5

6

7

8

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

1

0

2

3

4

5

7

6

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

[220].

74 chapter 3

0

2

3

1

4

5

6

7

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

1

0

2

3

4

5

7

6

Log(Cumulative Number)

Draw Down

Null Hypothesis

Draw Down

Fig. 3.15. Microsoft (left) and Cisco (right). Reproduced from [220].

0

2

3

1

4

5

6

7

8

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

1

0

2

3

4

5

7

6

Log(Cumulative Number)

Draw Down

Null Hypothesis

Draw Down

Fig. 3.16. General Electric (left) and Intel (right). Reproduced from [220].

0

2

3

1

4

5

6

7

8

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

1

0

2

3

4

5

7

6

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

Synthesis

We have found the following facts [211, 217, 220].

- 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.” - 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. - For companies, large drawups of more than 15% occur approximately

twice as often as large drawdowns of similar amplitudes. - 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.

SYMMETRY-BREAKING ON CRASH AND RALLY DAYS

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.

IMPLICATIONS FOR SAFETY REGULATIONS

OF STOCK MARKETS

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

100

104

a b c

100

104

d e f

100

104

g h i

PDF PDF PDF

-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

100

104

a b c

100

104

d e f

100

104

g h i

PDF PDF PDF

-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

killers.

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.

FEEDBACKS AND SELF-ORGANIZATION IN ECONOMICS

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

Phew!

Bears

enter

here

Enter

Crash here

Equity Markets Overview

Hibernate

instead?

B2C

Joe

Larry

Ralph

Abby

Bears bail

Rally!!!

Momentum

Weird

yield curve

Trade

deficit

e-broker

TV ads

W$W

elves

Hot

market

IPO

billionaires

DOW

36,000

New

Economy

Old

Economy

Margin

call

Gold

auctions

30yr bond

extinct

Oil

up

P/E’s of 2000

Small

volume

MSFT

breakup New Era

Soros

out

401k

inflows

“Bottom

is in”

16 year olds

beat market vets

Dollar goes every

which way

Mergers

Fed

tightens

Fed

loosens

Buy on dips

Flight to safety

Buy and

hold forever

Futures

up

Earnings

slowdown

Nothing

matters

Ignore

history

Optimistic

CNBC guest

P/Sales

new metric

tiny jobs

number

big jobs

number

Bad

breadth

Wealth

effect

Asia up Telecom Fuel Cell 4:1 splits Stewart

Daytraders

From “Hold”

to “Strong Buy”

CNBC

NASDAQ special

GDP at 7%

growth rate

Greenspan speaks

Greenspan

silent

Big

volume

Bubble?

What bubble?

Wireless! Maria excited

B2B

Cash is

trash

Question basic

assumptions

Fund trade

deadlines

Back into

cave

8 week

bear market!?

Bull market

forever?

Better off

in cash?

worthless

puts

market

madness

Looking good!

Any gains lost

in next day rally

Arggh!

Pain

Disbelief

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”

[389].

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”

[391].

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

[333].

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

markets[66].

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

situations. - 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. - 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. - 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. - 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

research. - 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

experiments. - 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. - 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”

prices.

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

difficulties.

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

HEDGING DERIVATIVES, INSURANCE PORTFOLIOS,

AND RATIONAL PANICS

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

markets.

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

difficult.

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 - 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

“HERD” BEHAVIOR AND “CROWD” EFFECT

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.

Herding

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]. - 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. - 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]. - 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

sell. - 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,

N1→4

= 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

� =

5

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

[451].

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.

FORCES OF IMITATION

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

A

Signal

Result

B

X Y

C

D

Signal

Signal

Signal

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

N

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

�

N

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

�

K

�

j∈Ni

sj

- i

�

(6)

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

fB

= 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

programs.

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].

Rumors

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

society.

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.

“ANTI-IMITATION” AND SELF-ORGANIZATION

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

0

100

90

80

70

20 30 40 50

Time

Numbers Attending

60

50

40

30

20

10

0

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.

When

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

- 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. - 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

Price

0

1

Time

Generator Trend Line

Piece 1

Piece 2

Piece 3

0

1

Interpolated

Generator

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.

COOPERATIVE BEHAVIORS RESULTING

FROM IMITATION

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

cluster.

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

agent.

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

dynamics.

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: - a system of traders who are influenced by their “neighbors”;
- local imitation propagating spontaneously into global cooperation;
- global cooperation among noise traders causing collective behavior;

positive feedbacks 129 - prices related to the properties of this system;
- 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

message

Node B

Node E

Node C Node D

Node F

Node I

Node G Node H

Node A

Ancestor Structure

message

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

welfare.

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

strategies.

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

crashes

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.

WHAT IS A MODEL?

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.

STRATEGY FOR MODEL CONSTRUCTION IN FINANCE

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

markets?

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

organization.

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

bubbles.”

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

symmetry.

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

probability.

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

Cash

or payment

Consumer

Positive Prices

Undesired good

or service

Cash

or payment

Consumer

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�

d.

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 ?

r

� at a later time, pocketing 100

r

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�

d

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

p

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

consequence.

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

p

H

μ>μc

μ<μc

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

self-fulfilling.

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�

d

= 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.

THE RISK-DRIVEN MODEL

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

values.

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

Time

Price

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

interactions.

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

S*

K/Kc

0

10

9

8

7

0.2 0.4 0.6 0.8

6

5

4

3

2

1

1.0

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

K/Kc

0

0.7

0.6

0.2 0.4 0.6 0.8

0.5

0.4

0.3

0.2

0.1

0

1.0

Crash Hazard Rate

K/Kc

0

10

8

0.2 0.4 0.6 0.8

6

4

2

0

1.0

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

K

Kc

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

curve.

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

conclusions.

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

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 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

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

W(t)

0

1.0

0.5

500 1000 1500 2000

0

2500

B(t)

0

4.0

2.0

500 1000 1500 2000

0

2500

dB(t)

0

0.2

0

500 1000 1500 2000

-0.2

2500

dW(t)

0

0.1

0

500 1000 1500 2000

-0.1

2500

t

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�

diverges.

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

?W�t��

, 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.

RISK-DRIVEN VERSUS PRICE-DRIVEN MODELS

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.

CRITICAL PHENOMENA BY IMITATION

ON HIERARCHICAL NETWORKS

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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Tank, R.W.

Tsallis, C.

Watson, B.P.

Yeomans, J.M.

Young, A.P.

Allan, J.S.

Barahona, M.

Broner, F.

Colet, P.

Cuomo, K.M.

Czeisler, C.A.

Duwel, A.E.

Freitag, W.O.

Goldstein, G.

Heij, C.P.

Hohl, A.

Kow, Yeung

Kronauer, R.E.

Marcus, C.M.

Matthews, P.C.

Mirollo, R.E.

Nadim, A.

Oppenheim, A.V.

Orlando, T.P.

Rappell, W.-J.

Richardson, G.S.

Rios, C.D.

Ronda, J.M.

Roy, R.

Sanchez, R.

Stone, H.A.

Swift, J.W.

Trias, E.

vanderLinden, H.J.C.

vanderZant, H.S.J.

Vardy, S.J.K.

Watanabe, S.

Weisenfeld, J.C.

Westervelt, R.M.

Wiesenfeld, K.

Winfree, A.T.

Yeung, M.K.S.

Adamchik, V.S.

Aharony, A.

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.

i-G

Nienhu

Prakash

Raghavan

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

Greenlaw

Guyer

Hallo

Kir

le

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

vertices.

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–

4.10.

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

K/Kc

0

0.7

0.6

0.2 0.4 0.6 0.8

0.5

0.4

0.3

0.2

0.1

0

1.0

Crash Hazard Rate

K/Kc

0

10

8

0.2 0.4 0.6 0.8

6

4

2

0

1.0

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.

�Kc

? 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

1.0

0.01

0.1 0.01

0.1

1.0

1.0 0.1 0.01 0.001 0.001

Crash Hazard Rate

(Kc - K)/Kc

1

10

0.1

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

�Kc

? 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

rate.

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

system.

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

modified.

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

Time

Position

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.

ORIGIN OF LOG-PERIODICITY

IN HIERARCHICAL SYSTEMS

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

world.

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

L�

� 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:

2

=

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

- For very corrugated coastlines exhibiting roughness at all

length scales, as the ruler becomes very small, the length grows without

190 chapter 6

ε2

ε1

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

L�

� changes with the ruler length

(which we also call “resolution”

as details smaller than

are not seen by definition). By construction,

L�

� 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�

� =

L�

�/

elements. The number of elements resolved with a ruler

is

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

L�

� ～

1?d becomes independent of the resolution

since

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

performed.

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

space.

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�

��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

criticality.

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.

- 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. - 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. - 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

agents.

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

t

0.8

3.5

0.85 0.9

2.5

1.5

2.0

3.0

1.0

3.4

3.0

2.6

2.4

2.8

3.2

2.2

0 0.2 0.4 0.6 0.8 1.0 0.95 1.0

Weierstrass Function

t

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

exponent

= 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

t

0

3.5

3.0

0.2 0.4 0.6 0.8

2.5

2.0

1.5

1.0

1.0

0.95

0.99

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

3i

2i

i

-i

-2i

-3i

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

1

zw

w

z

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

Time

Time

-1.2

1.2

-0.8 -0.4

0.4

-0.4

0

0.8

-0.8

-1.2

0.4

-0.4

0

-0.8

-1.2

1.2

0.8

0 4 8 12 16 0 0.4 0.9 1.2

y

y

-1.2 -0.8 -0.4 0 0.4 0.8 1.2

0

4

12

16

8

x

x

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

z5

z4

z3

z2

z

1

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

oscillations.

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

0.30

0

-0.15

0.15

-0.30

-10 -8 -6 -4 -2 0

Log2 C( l) - d Log2 l

0.30

0

-0.15

0.15

-0.30

-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

0

-4

-6

-8

-2

-10

-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,

existed.

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.

Prediction.

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

RU(ω)

ω [v = -1, k = 3, λ = 2, (211 points)]

2

0

-1

1

-2

-10 -8 -6 -4 -2 0 2 4 6 8 10

Ru(t)

t

0.2

0

-0.1

0.1

-0.2

-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

fundamentals.

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

principle.

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

n

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

digit.

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 - R = 19, N1

= 11, and F1

= 11/19 = 0�5789, while for R = 99, N1

=

11, and F1

= 11/99 = 1/9 = 0�1111; - R = 199, N1

= 111, and F1

= 111/199 = 0�5578, while for R = 999,

N1

= 111 and F1

= 111/999 = 1/9 = 0�1111; - 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

system!

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�: - R = 89, N9

= 1, and F9

= 1/89 = 0�0112, while for R = 99,

N1

= 11 and F9

= 11/99 = 1/9 = 0�1111; - R = 899, N9

= 11, and F9

= 11/899 = 0�0122 while for R = 999,

N1

= 111 and F9

= 111/999 = 1/9 = 0�1111;

hierarchies and log-periodicity 213 - 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

1/9.

Averaging these frequencies Fn�R� over a log-period (i.e., a factor of

- 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

ln

Tc

? T

of the time T to some critical time Tc. Periodicity of some

observable in the variable ln

Tc

? 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

Tn

= 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

Tc

= T 2

n+1

? Tn+2Tn

2Tn+1

? 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

Tn+3

= T 2

n+1

- T 2

n+2

? TnTn+2

? Tn+1Tn+2

Tn+1

? 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.

NONLINEAR TREND-FOLLOWING VERSUS NONLINEAR

FUNDAMENTAL ANALYSIS DYNAMICS

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=4

n=0.2

n=1 n=10

0

1

3

4

5

0 0.5 1 1.5

Net Order Size

Observed Price - Fundamental Price

2

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

otherwise.

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

n=5

n=3

n=1

n=15

-5

0

10

15

20

0 0.4 0.6

y

Time

5

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� =

√50

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

-300

-100

100

200

300

3.2 3.4 3.5

y1

Time

α = 1 γ = 10 m = 1.3 n = 3 D(0) = 1

-200

0

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,

10

100

0.2 0.3 0.4

|y1|

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

y1�t�

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

α=10

α=1000

-1.0

-0.5

0

1.0

2.0

2.5

2 4 5

y1

Time

1.5

0.5

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

-1

1

1

-1

Δb-(3,2)

Δb+(3,2)

Δb-(2,1)

Δb+(1,0)

ΔB+(1,0)

ΔB-(2,1)

b- Bp-

2

p+1

p-0

Δb+(2,1)

Δb-(1,0)

ΔB-(1,0)

ΔB+(2,1)

Δe(-1,+0)

Δe(+2,-1)

B+ b+

p+2

p+0

p-1

Δe(-2,+1)

Δe(+1,-0)

y2

y1

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

log-periodicity

THE CRASH OF OCTOBER 1987

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

280

320

340

85.5 86.5 87

S&P

Time (year)

300

260

240

220

200

180

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,

tc

≈ 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

?tn+1

tc

?tn

= �. 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].

5.4

5.6

5.8

6.0

80 83 84 85 86 87 88

Log(S&P)

Year

5.2

5.0

4.8

4.6

4.4

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

(C-P)/P

Date

40%

30%

20%

10%

4%

0%

4%

-4%

0%

-10%

-20%

12%

1 2 5 6 7 8

October 1-16, 1987: Hourly Data

9 12 13 14 15 16

8%

4%

0%

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

60

70

90

80

87.6 88.0 88.2 88.4

σ2 (S&P 500)

Time (year)

50

40

30

20

10

0

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

varpow

≈ 12. The best fit to expression (15) with tc

? t replaced by t ? tc gives

A2

≈ 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

tc

? 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

280

300

320

S&P 500

Time (year)

260

240

220

200

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

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.

150

200

250

300

350

400

27 27.5 28 28.5 29 29.5 30

Dow Jones

Date

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 ≈

14�3�m2

≈ 0�45� tc

≈ 1930�22�� ≈ 7�9, and � ≈ 1�0. Reproduced from [212].

240 chapter 7

5.4

5.6

5.8

6.0

21 24 25 26 27 28 29 30

Log(Dow Jones)

Year

5.2

5.0

4.8

4.6

4.4

4.2

4.0

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

[374].

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

2

= 0�33 compared

to m1929

2

= 0�45; �1987 = 7�4 compared to �1987 = 7�9. These numerical

values are remarkably close and can be considered equal to within their

uncertainties.

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

2

= 0�63 and

m1987

2

= 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

�1929

= 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].

THE THREE HONG KONG CRASHES

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.

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

80 82 84 86 88 90 92 94 96 98

Index

Date

I

II

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

[218].

244 chapter 7

1000

1500

2000

2500

3000

3500

4000

84.5 85 85.5 86 86.5 87 87.5

Index

Date

’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].

- 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%. - 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%. - 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

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

13000

14000

92 92.5 93 93.5 94

Index

Date

’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

2

= 0�33 < mHK1997

2

= 0�34 < m1929

2

= 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

6000

8000

10000

12000

14000

16000

18000

20000

95 95.5 96 96.5 97 97.5 98

Hang Seng

Date

’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,

m2

≈ 0�34, tc

≈ 1997�74, � ≈ 7�5, and � ≈ 0�78. Reproduced from [212] and

[218].

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

7.1

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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 6 7 8

Spectral Weight

Log-frequency

1987

1994

1997

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

state.

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

4.6

5.8

6.0

6.2

91 92 93 94 95 96 97 98

Logarithm(S&P 500 Index)

Time (years)

5.6

5.4

5.2

5.0

4.8

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

- 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)

1984

300

200

100

0

1986 1987 1988

Direct Purchases

Purchases Made through Mutual Funds

1991

-100

-200

-300

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

herding.

CURRENCY CRASHES

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

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

83.0 83.5 84.0 84.5 85.0 85.5

Exchange Rate

Date

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,

tc

≈ 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].

THE CRASH OF AUGUST 1998

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

110

120

130

140

150

160

170

180

190

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$

Yen

1.30

1.35

1.40

1.45

1.50

1.55

1.60

Price Ratio of CAN$ to US$

Date

CAN$

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”

signatures.

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

400

500

600

700

800

900

1000

1100

1200

1300

1400

95.0 95.5 96.0 96.5 97.0 97.5 98.0 98.5

S&P 500

Date

HK

WS

6000

8000

10000

12000

14000

16000

18000

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.

NONPARAMETRIC TEST OF LOG-PERIODICITY

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

2000

2500

3000

DJIA

Date

ymax - ymin

1500

1000

250

200

150

100

500

0

500

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

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

-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

?t

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

�1

= 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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 6

Arbitrary Units

Frequency

’CAN$-98’

’YEN-98’

’WS-29’

’WS-87’

’HK-97’

’WS-98’

’DM-85’

’CHF-85’

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.

THE SLOW CRASH OF 1962 ENDING

THE “TRONICS” BOOM

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

- 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

250

300

350

400

450

500

550

600

650

700

750

54 55 56 57 58 59 60 61 62 63

Dow Jones

Date

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,

tc

≈ 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

businesspeople.

THE NASDAQ CRASH OF APRIL 2000

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. - 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. - 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

increases. - 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. - 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

7.0

7.2

7.4

7.6

7.8

8.0

8.2

8.4

8.6

8.8

9.0

97.5 98 98.5 99 99.5 00

Log(Nasdaq Composite)

Date

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

20

40

60

80

100

120

140

160

180

97.5 98.0 98.5 99.0 99.5

Price of IBM Shares

Date

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 ≈

?6�1�m2

≈ 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.

50

60

70

80

90

100

110

120

98.8 99.0 99.2 99.4 99.6 99.8 00 00.2

Price of Procter & Gamble Shares

Date

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

[217].

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.

“ANTIBUBBLES”

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, - 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, ��

t

= 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

A1

≈ 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

T1

≈ 4�3, T2

≈ 7�8, �2

≈ ?3�1, and T2

≈ 23. In all these fits, T1 and T2

autopsy of major crashes 277

9.4

9.6

9.8

10.2

10.4

10.6

90 92 94 96 98 2000

Log(Nikkei)

Date

eq.(19)

eq.(15)

eq.(25)

10.0

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�06�m2

≈ 0�45� tc

≈ 80�69�� ≈ 1�4, and � ≈ 9�8. The line before the

4.8

5.0

5.2

5.4

5.6

5.8

6.0

6.2

6.4

6.6

6.8

77 78 79 80 81 82

Log(Gold Price)

Date

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.

SYNTHESIS: “EMERGENT” BEHAVIOR

OF THE STOCK MARKET

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

markets

SPECULATIVE BUBBLES

IN EMERGING MARKETS

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

[271].

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

- The bubble starts smoothly with some increasing production and sales

(or demand for some commodity) in an otherwise relatively optimistic

market. - 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. - 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. - At this stage, the behavior of the market becomes weakly coupled or

practically uncoupled from real wealth (industrial and service) production. - 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

collapses.

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).

METHODOLOGY

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

analysis.

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

criteria:

� 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.

LATIN-AMERICAN MARKETS

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

0

5000

10000

15000

20000

25000

30000

90 91 92 93 94 95 96 97 98 99

Index

I

II III IV

Date

’Argentina’

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].

0

2000

4000

6000

8000

10000

12000

14000

94 95 96 97 98 99

Index

Date

’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

1000

2000

3000

4000

5000

6000

91 92 93 94 95 96 97 98 99

Index

I

Date

II

Chile

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].

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

91 92 93 94 95 96 97 98 99

Index

Date

I

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

0

500

1000

1500

2000

2500

93 94 95 96 97 98 99

Index

Date

I

II

Peru

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].

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

94 95 96 97 98 99

Index

Date

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

0

5000

10000

15000

20000

91.1 91.9

Index

Date

91.3 91.5 91.7 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

10000

18000

16000

14000

12000

22000

20000

24000

26000

92

Index

Date

92.2 92.4 92.6 92.8 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

18000

16000

14000

12000

22000

20000

24000

26000

93.6

Index

Date

93.7 93.8 93.9 94 94.1 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

12000

18000

16000

14000

22000

20000

24000

26000

Index

Date

95.5 96 96.5 97 97.5 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

8000

6000

10000

12000

14000

96.2

Index

Date

96.4 96.6 96.8 97 97.2 97.4 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’Brazil’

Best fit

’Brazil’

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�

I�tmax�

� (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

1000

1500

2500

2000

3000

Index

Date

90.8 91 91.2 91.4 91.6 91.8 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

4000

3500

3000

4500

5000

Index

Date

93.4 93.5 93.6 93.7 93.8 93.9 94 94.1 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

5200

5600

5400

6000

5800

6200

6400

Index

Date

95.5 95.6 95.7 95.8 95.9 96 96.1 96.2 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

2200

1800

1600

2000

2600

2400

2800

3000

Index

Date

93.5 93.6 93.7 93.8 93.9 94 94.1 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

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].

2000

3500

3000

2500

4500

4000

5000

Index

Date

96 96.4 96.8 97.2 97.6 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

300

700

600

500

400

1000

800

900

1100

1200

93.1

Index

Date

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

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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].

ASIAN MARKETS

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.

1200

1600

1400

2000

1800

2200

2400

97

Index

Date

97.1 97.2 97.3 97.4 97.5 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

4000

2000

8000

6000

10000

12000

96

Index

Date

96.5 97 97.5 98 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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.

6000

5000

4000

8000

9000

7000

10000

11000

Index

Date

97.6 97.8 98 98.2 98.4 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

crash.

200

300

400

500

600

700

800

91 92 93 94 95 96 97 98 99

Index

Date

I

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

200

300

400

500

600

700

800

900

1000

1100

1200

91 92 93 94 95 96 97 98 99

Index

Date

’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].

200

400

600

800

1000

1200

1400

91 92 93 94 95 96 97 98 99

Index

Date

’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

500

1000

1500

2000

2500

3000

3500

91 92 93 94 95 96 97 98 99

Index

Date

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].

200

400

600

800

1000

1200

1400

1600

1800

91 92 93 94 95 96 97 98 99

Index

Date

’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

5.8

5.6

6.0

6.2

6.4

6.6

Log(Index)

Date

93.2 93.4 93.6 93.8 94 94.2 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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].

6.2

6.0

5.8

6.4

6.6

6.8

7.0

Log(Index)

Date

95.5 96 96.5 97 97.5 98 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

6.2

6.3

6.1

6.4

6.5

6.6

6.7

6.8

7.0

6.9

7.1

Log(Index)

Date

93 93.5 94 94.5 95 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’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

6.4

6.6

6.8

7.0

7.2

Log (Index)

Date

92.8 93 93.2 93.4 93.6 93.8 94 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’Malaysia’

Best fit

’Malaysia’

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

m2

= 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

7.4

7.2

7.0

6.8

7.6

7.8

8.0

8.2

Log(Index)

Date

92 92.5 93 93.5 94 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’Philippine’

’Philippines’

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

m2

= 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.

7.1

6.9

7.0

6.8

6.7

7.2

7.3

7.4

7.5

7.6

Log(Index)

Date

93.4 93.5 93.6 93.7 93.8 93.9 94 94.1 0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1.0

Spectral Power

Frequency

’Thailand’

Best fit

’Thailand’

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].

THE RUSSIAN STOCK MARKET

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 - 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

robust.

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

0

1000

2000

3000

4000

5000

96.2 96.4 96.6 96.8 97 97.2 97.4 97.6

ROSI Index

Date

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

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

96.5 97 97.5 98 98.5

ROSI Index

Date

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].

CORRELATIONS ACROSS MARKETS: ECONOMIC

CONTAGION AND SYNCHRONIZATION

OF BUBBLE COLLAPSE

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].

8.05

8.00

7.95

8.10

8.15

8.20

8.25

Log(Index)

Date

94.1 94.2 94.3 94.4 94.5 94.1 94.2 94.3 94.4 94.5

9

9.25

9.20

9.15

9.10

9.05

9.30

9.35

9.40

9.45

Log(Index)

Date

’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

7.65

7.60

7.55

7.50

7.70

7.75

7.80

7.85

Log(Index)

Date

94.2 94.4 94.6 94.8 95 94 94.2 94.4 94.6 94.8 95 95.2

7.45

7.65

7.60

7.55

7.50

7.70

7.75

7.80

Log(Index)

Date

Australia

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].

IMPLICATIONS FOR MITIGATIONS OF CRISES

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-

7.90

7.85

7.80

7.95

8.00

8.05

8.10

Log(Index)

Date

94.1 94.2 94.3 94.4 94.5 94.4 94.6 94.8 95 95.2

9.7

9.6

9.5

9.8

9.9

Log(Index)

Date

’Zurich’

Best fit

Second best fit

’Milano’

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)

8.1

8.0

8.2

8.3

Log(Index)

Date

94.2 94.4 94.6 94.8 95 95.2 2 4 6 8 10 12 14 16 18

0.08

0.06

0.04

0.02

0

0.12

0.10

0.14

0.16

’Madrid’

Best fit

’Log(Price)’

’Price’

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]: - 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. - 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

success.

chapter 9

prediction of bubbles,

crashes, and

antibubbles

THE NATURE OF PREDICTIONS

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

5.00

4.50

4.00

3.50

3.00

2.50

2.00

1.50

1.00

0.50

1.00

0.00

0.50

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

[388].

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.

HOW TO DEVELOP AND INTERPRET STATISTICAL

TESTS OF LOG-PERIODICITY

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

occur.

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

approach).

� 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

power.

� 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

FIRST GUIDELINES FOR PREDICTION

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

possible.

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

[218].

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

following.

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 - 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.

A HIERARCHY OF PREDICTION SCHEMES

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

tc.

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

optimum.

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

restricted.

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

�tc

? t��

�

1 +

�

tc

? t

�t

�2�

1 + C cos

�log�tc

? t�

- �

2�

log

�

1 +

�

tc

? t

�t

�2�

��

� (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

optimizer.

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

tc

? tn

= ��

n2

� (20)

prediction of crashes and of antibubbles 337

where

� ∝ exp

�

?log �

2�

tan?1 2�

� log �

�

(21)

with

� = e

2�

� � (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

?tn+1

tc

?tn

= �

1

2 . From three

successive observed values of tn, say tn, tn+1, and tn+2, we have

tc

= t2

n+1

? tn+2tn

2tn+1

? 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

by

tn+3

= t2

n+1 - t2

n+2

? tntn+2

? tn+1tn+2

tn+1

? 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

information.

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.

FORWARD PREDICTIONS

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

predictions.

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 +

�

�

�t

�2

�

�

�

�

t

�4

×

B� + C� cos

�log � + ��

2

log

�

1 +

�

�

�t

�2

�

- �

�

4

log

�

1 +

�

�

��

t

�4

�

- �

�

� (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-

9.4

9.6

9.8

10.0

10.2

10.4

10.6

10.8

90 92 94 96 98 100

Log(Nikkei)

Date

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�

c

= 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�

c

= 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

- 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.

PRESENT STATUS OF FORWARD PREDICTIONS

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

distribution.

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):

pk

= 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

nc

= k, Pk=3

= pk=3

= C�3�3�×C�57�2�

C�60�5�

= 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

+p3

= C�3�2�×C�57�3�

C�60�5�

- C�3�3�×C�57�2�

C�60�5�

= 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�

C�72�6�

=

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

P�Hi

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

P�D

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

P�H1

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

P�H1

D� =

0�95 · 1

2

0�95 · 1

2

- 0�05 · 1

2

= 95%� (29)

This means that this single case is enough to convince the neutral

Bayesian.

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

C1

≤ 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 =

1

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 =

1

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

pf

? 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?

STOCK MARKETS, ECONOMICS,

AND POPULATION

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

reasoning.

THE PESSIMISTIC VIEWPOINT

OF “NATURAL” SCIENTISTS

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,

- 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.”

THE OPTIMISTIC VIEWPOINT

OF “SOCIAL” SCIENTISTS

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]: - 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. - 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. - 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! - 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. - 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. - 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. - 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. - 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.

ANALYSIS OF THE FASTER-THAN-EXPONENTIAL

GROWTH OF POPULATION, GDP,

AND FINANCIAL INDICES

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

1000

7000

100

1000

10000

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

10

100

1000

10000

1800 1850 1900 1950 2000

Index

Date

Real power law

Complex power law

Dow Jones

Standard & Poor

EAFE

Europe

Latin America

World

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

acceleration.

ment of Economic and Social Affairs (http://www.popin.org/

pop1998/).

� 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

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 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

process.

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: - Before 1914, inflation was essentially zero on average, except during

the civil war (famous “greenbacks”). - 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. - 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

10000

100000

1e+06

1e+07

1e+08

1e+09

1e+10

1e+11

1e+12

1e+13

1750 1800 1850 1900 1950 2000 2050

Public Debt (US$)

Date

Debt from War

of Independence

War of

1812

Seminole War

Mexican

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.

REFINEMENTS OF THE ANALYSIS

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

Log((tc-t)/tc))

-1

-2

-3

-4

-5

1

0

2

3

4

-4.0

Residue

-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

0

2

1

3

4

5

6

Spectral Power

Frequency

’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

1

2

3

4

5

6

7

8

9

1800 1850 1900 1950 2000

Log(DJ)

Date

Log(DJ)

eq.(8)

eq.(5)

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

0

2

4

6

8

10

12

14

16

18

20

1800 1850 1900 1950 2000 2050

Log(DJ)

Date

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

[219].

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

below.

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

[438].

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

1900

1

billion

1920 1940 1960 1980 2000 2020 Year

1

million

1

1000

1

1000

1,000,000

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

Gateway-455DX2/66

Mac II

Macintosh 128k

Commodore 64

IBM PC

Sun-2

DG Eclipse

Apple II

Power Tower 150e

AT&T Globalyst 500

IBM PS/2 90

Mac IIfx

Sun-3

Vax 11/750

DEC Vax 11/780

DEC-KL-10

DG Nova

SDS 920

IBM 350/75

IBM 7040

Burroughs 5000

IBM 1620

IBM 650

ASCC (Mark7)

Zuse-1

Monroe

Calculator

IBM Tabulator

Burroughs Class 16

Colossus

ENIAC

UNIVAC I

IBM 704

Whairlwind

IBM 7090 IBM 1130

DEC PDP-10

CDC 7600

Human

Monkey

Mouse

Lizard

Spider

Nematode

Worm

Bacterium

Manual

Calculation

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

internally.

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

dp

dt

= 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

dp

dt

= 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

tc

= 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.

SCENARIOS FOR THE “SINGULARITY”

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

Collapse

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

womb.

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

viewpoint.

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

future.

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

consciousness.

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 INCREASING PROPENSITY TO EMULATE

THE STOCK MARKET APPROACH

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

precursors!

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