Fundamentals in Risk Analysis

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Financial Market Risk

What is nancial market risk? How is it measured and analyzed? Is all nancialmarket risk dangerous? If not, which risk is hedgeable?

These questions, and more, are answered in this comprehensive book writtenby Cornelis A. Los. The text covers such issues as:

competing nancial market hypotheses; degree of persistence of nancial market risk; timefrequency and timescale analysis of nancial market risk; chaos and other nonunique equilibrium processes; consequences for term structure analysis.

This important book challenges the conventional statistical ergodicity paradigmof global nancial market risk analysis. As such it will be of great interest tostudents, academics and researchers involved in nancial economics, internationalnance and business. It will also appeal to professionals in international bankinginstitutions.

Cornelis A. Los is Associate Professor of Finance at Kent State University, USA.In the past he has been a Senior Economist of the Federal Reserve Bank ofNew York and Nomura Research Institute (America), Inc., and Chief Economistof ING Bank, New York. He has also been a Professor of Finance at NanyangTechnological University in Singapore and at Adelaide and Deakin Universities inAustralia. His PhD is from Columbia University in the City of New York.

Routledge International Studies in Money and Banking

1 Private Banking in EuropeLynn Bicker

2 Bank Deregulation andMonetary OrderGeorge Selgin

3 Money in IslamA study in Islamic politicaleconomyMasudul Alam Choudhury

4 The Future of EuropeanFinancial CentresKirsten Bindemann

5 Payment Systems in GlobalPerspectiveMaxwell J. Fry, Isaak Kilato,Sandra Roger, KrzysztofSenderowicz, David Sheppard,Francisco Soils and John Trundle

6 What is Money?John Smithin

7 FinanceA characteristics approachEdited by David Blake

8 Organisational Change andRetail FinanceAn ethnographic perspectiveRichard Harper, Dave Randall andMark Rounceeld

9 The History of theBundesbankLessons for the EuropeanCentral BankJakob de Haan

10 The EuroA challenge and opportunityfor nancial marketsPublished on behalf of SocitUniversitaire Europenne deRecherches Financires(SUERF)Edited by Michael Artis,Axel Weber and ElizabethHennessy

11 Central Banking in EasternEuropeNigel Healey

12 Money, Credit and PriceStabilityPaul Dalziel

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14 Adapting to FinancialGlobalisationPublished on behalf of SocitUniversitaire Europenne deRecherches Financires (SUERF)Edited by Morten Balling,Eduard H. Hochreiter andElizabeth Hennessy

15 Monetary MacroeconomicsA new approachAlvaro Cencini

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19 HRM and Occupational Healthand SafetyCarol Boyd

20 Central Banking SystemsComparedThe ECB, The pre-EuroBundesbank and the FederalReserve SystemEmmanuel Apel

21 A History of Monetary UnionsJohn Chown

22 DollarizationLessons from Europe for theAmericasEdited by Louis-Philippe Rochonand Mario Seccareccia

23 Islamic Economics and Finance:A Glossary, 2nd EditionMuhammad Akram Khan

24 Financial Market RiskMeasurement and analysisCornelis A. Los

Financial Market RiskMeasurement and analysis

Cornelis A. Los

First published 2003by Routledge11 New Fetter Lane, London EC4P 4EESimultaneously published in the USA and Canadaby Routledge29 West 35th Street, New York, NY 10001Routledge is an imprint of the Taylor & Francis Group

2003 Cornelis A. LosAll rights reserved. No part of this book may be reprinted or reproduced orutilized in any form or by any electronic, mechanical, or other means, nowknown or hereafter invented, including photocopying and recording, or inany information storage or retrieval system, without permission in writingfrom the publishers.British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Cataloging in Publication DataLos, Cornelis Albertus, 1951-

Financial market risk : measurement & analysis / Cornelis A. Los.p. cm. (Routledge international studies in money and banking ; 24)

Includes bibliographical references and index.1. Hedging (Finance) 2. Risk management. I. Title. II. Series.

HG6024.A3L67 2003332.015195dc21 2003040924

ISBN 041527866X

This edition published in the Taylor & Francis e-Library, 2005.

To purchase your own copy of this or any of Taylor & Francis or Routledgescollection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.

ISBN 0-203-98763-2 Master e-book ISBN

(Print Edition)

ToJanie and Klaas Los, Saba and Leopold Haubenstock, and P. Khne

withGratitude for life, liberty and the pursuit of happiness

Contents

List of gures xiiiList of tables xixPreface xxiIntroduction xxvii

PART IFinancial risk processes 1

1 Risk asset class, horizon and time 31.1 Introduction 31.2 Uncertainty 71.3 Nonparametric and parametric distributions 171.4 Random processes and time series 311.5 Software 411.6 Exercises 41

2 Competing nancial market hypotheses 472.1 Introduction 472.2 EMH: martingale theory 472.3 FMH: fractal theory 532.4 Importance of identifying the degree of market efciency 652.5 Software 672.6 Exercises 67

3 Stable scaling distributions in nance 713.1 Introduction 713.2 Afne traces of speculative prices 723.3 Invariant properties: stationarity versus scaling 763.4 Invariances of (ParetoLvy) scaling distributions 773.5 Zolotarev parametrization of stable distributions 803.6 Examples of closed form stable distributions 92

x Contents

3.7 Stable parameter estimation and diagnostics 943.8 Software 963.9 Exercises 97

4 Persistence of nancial risk 1024.1 Introduction 1024.2 Serial dependence 1024.3 Global dependence 1054.4 (G)ARCH processes 1114.5 Fractional Brownian Motion 1174.6 Range/Scale analysis 1204.7 Critical color categorization of randomness 1224.8 Software 1284.9 Exercises 128

PART IIFinancial risk measurement 133

5 Frequency analysis of nancial risk 1355.1 Introduction 1355.2 Visualization of long-term nancial risks 1355.3 Correlation and time convolution 1365.4 Fourier analysis of stationary price innovations 1415.5 Software 1525.6 Exercises 152

6 Fourier timefrequency analysis of risk 1556.1 Introduction 1556.2 FT for aperiodic variables 1566.3 Hurst exponent identication from risk spectrum 1696.4 Heisenberg Uncertainty Principle 1716.5 Windowed FT for transient price innovations 1736.6 Software 1866.7 Exercises 186

7 Wavelet timescale analysis of risk 1907.1 Introduction 1907.2 Wavelet analysis of transient pricing 1927.3 Mallats MRA 2097.4 Wavelet Parseval Risk Decomposition Theorem 223

Contents xi

7.5 Software 2247.6 Exercises 224

8 Multiresolution analysis of local risk 2308.1 Introduction 2308.2 Measurement of local nancial market risk 2378.3 Homogeneous Hurst exponents of monofractal price series 2508.4 Multiresolution analysis of multifractal price series 2658.5 Software 2818.6 Exercises 281

PART IIITerm structure dynamics 287

9 Chaos nonunique equilibria processes 2899.1 Introduction 2899.2 Logistic parabola regimes 2929.3 General nonlinear dynamic systems 3179.4 Detecting attracting points and aperiodic orbits 3279.5 Summary of aperiodic cyclical steady-state equilibria 3289.6 Software 3309.7 Exercises 331

10 Measuring term structure dynamics 33710.1 Introduction 33710.2 Dynamic investment cash ow theory 34010.3 Nonlinear relationships in nance 34710.4 Liquidity and nancial turbulence 35810.5 Software 37210.6 Exercises 373

11 Simulation of nancial turbulence 38011.1 Introduction 38011.2 Theories of physical and nancial turbulence 38111.3 Measurement and simulation of turbulence 38811.4 Simulation of nancial cash ow turbulence 39611.5 Multiresolution analysis of nancial turbulence 39811.6 Wavelet solutions of nancial diffusion equations 40211.7 Software 41411.8 Exercises 414

xii Contents

PART IVFinancial risk management 423

12 Managing VaR and extreme values 42512.1 Introduction 42512.2 Global dependence of nancial returns 42512.3 VaR for stable distributions 42812.4 VaR for parametric distributions 43112.5 Extreme value theory 43712.6 VaR and fractal pricing processes 44012.7 Software 44512.8 Exercises 445

Appendix A: original scaling in nancial economics 449Appendix B: S&P500 daily closing prices for 1988 450

Index 453

Figures

1 Risk = Danger (Wei) + Opportunity (Ji) xxx1.1 Historical average annual returns and return volatility 41.2 Levels and returns of empirical nancial time series: AMEX

stock and oil indices and DEMUSD exchange rate 91.3 Simple and relatively inexpensive radiation monitor 111.4 The nth-order moments and cumulants for n = 1, 2, 3, 4

of the Laplace, Gaussian and Uniform p.d.fs 211.5 The nth-order moments and cumulants for n = 1, 2, 3, 4

of Exponential, Rayleigh and K-distribution p.d.fs 221.6 Construction of the histogram of a time series by binning 241.7 Raw and transformed daily returns of the DAX. The histograms

on the right show the relative frequencies of the returns in thesame scale 25

1.8 Empirical histogram of minute-by-minute log-increments of theJPY in June 1997 25

1.9 Volatility smile of foreign currrency options 261.10 Implied distribution and (log-) normal distribution of foreign

currency options 261.11 Skewed volatility smile of equities 271.12 Implied distribution and (log-) normal distribution 281.13 The empirical cumulative distributions for USD/DEM and USD

6 months cash interest rate, shown for different time horizons 301.14 Semi-annual cumulative distributions of THBFX increments,

JanuaryDecember 1997 361.15 Semi-annual cumulative distributions of DEMFX increments,

JanuaryDecember 1997 372.1 Annualized volatility of theoretical Random Walk model of

constant, normalized, asset return volatilities 552.2 Empirical annualized volatility of nancial market returns 562.3 Mandelbrots Julia set 582.4 Clear-air turbulence 592.5 Intraday return variance of the HSI 612.6 Intraday return variance of the HSIF 62

xiv List of gures2.7 One-month Eurodollar yield and time-varying (a) turbulent

volatility and (b) daily data 632.8 Five-year CMT yield and time-varying (a) turbulent volatility

and (b) daily data 643.1 Stable probabilistic schemes 783.2 Stable density in the Zolotarev S(Z, , , ; 0) =

S(Z, 0.8, 1, 0; 0) parametrization 863.3 Stable density in the Zolotarev S(Z, , , ; 1) =

S(Z, 0.8, 1, 0; 1) parametrization 873.4 Comparison of the t = 1 minute p.d.f. for high-frequency

S&P500 price changes with the Gaussian p.d.f. and with a Lvystable p.d.f. 90

3.5 Non-convergent moving variance of 253 daily rates of return(in 100 percent) of the S&P500 stock market index in 1998 90

3.6 Estimates of four parameters of the Zolotarev parametrizationof FX distributions 95

4.1 Successive increments of ARCH(1) simulations with the sameunconditional variance ( 2 = 1) 114

4.2 Probability density function of the successive increments shownin Figure 4.1. 115

4.3 Comparison of the scaling properties of the unconditional p.d.f.of a GARCH(1, 1) stochastic process 117

4.4 Autocorrelograms of equally-weighted CRSP daily andmonthly stock return indices 119

4.5 Sample of power spectra of white, pink and brown noise 1254.6 Relations between and constraints on d, H and Z 1275.1 Fourier series approximation of a square wave 1445.2 Heat diffusion analysis 1465.3 A sample signal constructed from sine functions representing

three pulsations 1505.4 The FT of the sampled signal s(t) 1515.5 Fourier series analysis of pure musical harmonics: dominant

frequencies of the clarinet, violin and bagpipe 1516.1 Granger and Morgensterns global risk spectrum of Standard

and Poor series, based on annual data, 18751952 1666.2 Semi-log plot of the autocorrelation function () for the

S&P500 index, sampled at a 1-minute time scale 1706.3 Spectral density of high-frequency data from the S&P500 index 1716.4 Gbors atom g0,i as a function of time for three frequencies:

(a) high 1, (b) middle 2 and (c) low 3 1766.5 Heisenberg boxes of two windowed Fourier atoms gu, and g, 1776.6 Timefrequency analysis by the Gbor Transform with

adapted to the time coherence of frequencies 1 and 3 1806.7 Signal, spectrograms and scalogram 181

List of gures xv6.8 Spectrogram PS(, ) of time series with two superimposed

time-varying frequencies 1826.9 Laughter data and their global histogram 1836.10 Spectrogram of laughter data with three dominant harmonics 1846.11 Changes in the daily level of the three-month Treasury yield 1846.12 Comparison of the modulated spectrogram of empirical

DEM/USD increments with the at spectogram of white noise 1857.1 A sine wave and a Daubechies wavelet D20 1937.2 Self-similarity of wavelets: translation (every fourth k) and

scaling of a wavelet 1947.3 Wavelet coefcients are correlation or resonance

coefcients. Here a wavelet is correlated with an irregularsignal. Different sections of the signal produce differentresonance coefcients 196

7.4 A scalogram: a plot of the magnitude of wavelet coefcients 1977.5 A 3D scalogram: a plot of the magnitude of the wavelet

coefcients in three dimensions 1977.6 Heisenberg boxes of two wavelets. Smaller scales decrease the

time dispersion, but increase the frequency support, which isshifted towards higher frequencies 198

7.7 Timefrequency resolution and basis functions of theWindowed FT and the Wavelet Transform 199

7.8 A scalogram with modulus |W(, a)| using a Morlet waveletfor 25 different scales 201

7.9 Normalized scalogram (/)PW(, a) 2027.10 Timescale tiling for a sinusoidal function with an isolated

singularity at t0 2037.11 Empirical 3D scalogram of Thai Baht increments in July 1997 2047.12 Time signal observations on f (x) 2117.13 Haar and triangle scaling functions and their respective MRA

equations 2167.14 Haar and triangle wavelets and their respective MRA equations 2208.1 USD/DEM exchange rate on a time scale of t = 20 minutes 2328.2 Scaling law behavior of the USD/DEM exchange rate in the

period 19931994, for various subperiods 2338.3 Trading transaction density based on daily and weekly averages

of tick-by-tick data 2348.4 Daily and weekly averaged volatility per hour of the USD/DEM

exchange rate 2358.5 The cone of inuence of an abscissa singularity 2458.6 Singularity cones of inuence of a Dirac pulse at t = t0 2458.7 Wavelet decomposition of a time series with singularities 2488.8 How to measure the degree of irregularity of local risk of a

series of price singularities x(t) 2498.9 Complete wavelet tiling 251

xvi List of gures8.10 Wavelet-based persistence analysis of heartbeat interarrival

times for a healthy patient with a Daubechies(5) wavelet 2548.11 Wavelet-based persistence analysis of weekly Dow Jones

Industrial Index data with Daubechies(5) wavelet 2568.12 Wavelet MRA by Morlet(6) wavelet of the various exchange

rate regimes of the Mexican Peso/USD and the variousBrazilian nancial market crises in the 1990s 257

8.13 Wavelet MRA of the various exchange rate regimes of the rst(log) differences of the Mexican Peso/USD in the 1990s 260

8.14 Wavelet MRA, based on daily data, of Chilean stock index rateof returns in the 1990s 261

8.15 The rst four monthly moments of the distributions of theminute-by-minute quotations of nine currency rates inMayAugust 1997 (USD is the numraire) 264

8.16 Development of Kochs snowake with Hausdorff dimensionD = 1.2619 267

8.17 Schematic convex multifractal singularity spectrum D(L),with various Gibbs exponent regimes 271

8.18 Computation of singularity spectrum of the devils staircase, itspartition function Z(q, a) scaling exponents (q) and itstheoretical spectrum D(L) 274

8.19 Wavelet MRA of Fractional Brownian Motion 2758.20 Time-warped Geometric Brownian Motion 2768.21 Multifractal spectrum analysis of time-warped GBM 2778.22 Mandelbrots early multifractal turbulence trace modeling in a

laboratory experiment 2788.23 Multifractal spectrum of physical (windtunnel generated)

turbulence 2798.24 Turbulent correlation between the S&P500 spot and futures

market, 19821991 2809.1 Nonunique dynamic equilibria of the logistic parabola 2969.2 The Hurst exponent may not be the best measure of the global

dependence of intermittent and chaotic processes 2979.3 The various stability regimes of the logistic process are

determined by the value of the scaling parameter 2989.4 When a dynamic process is chaotic, its later values are directly

dependent on the precision of its initial condition 2999.5 After a xed attraction point turns unstable, an orbit of period

length p = 2 emerges 3019.6 Period-doubling appears rst at a scaling parameter value just

above = 3 3029.7 The relationship between the parabolic map f (x) for an orbit of

period length p = 2 and the 1 iterated map f (2)(x) 3039.8 Oscillation of the logistic process between two steady-state

equilibria at x = 0.5 and x = 0.809 304

List of gures xvii9.9 The 1 iterated map f (2)(x) for period length p = 4, with

2 2 stable steady-state equilibria and one unstable equilibrium 3059.10 Oscillation of the logistic process between four steady-state

equilibria at x = 0.5, 0.875, 0.383 and 0.827, respectively 3069.11 The 2 iterated parabolic map for the scaling parameter

= 1 + 8 + 103 3069.12 A sample window of 100 observations of an undened orbit, or

frequency, of innite period length with scaling parameter = 3.6 309

9.13 Intermittency in time series is characterized by periods ofstability alternating with periods of chaos 310

9.14 The reappearance of a period of apparent stability 3109.15 Another period of apparent stability with periodicity with six

steady-state equilibria 3119.16 Another instance of intermittency in the time series of the

logistic process, after the birth of period length 3 3119.17 Complete chaos is dened by the coexistence of an innite

number of deterministic unstable equilibrium orbits 3139.18 Complete logistic chaos consists of innitely many coexisting

steady-state dynamic equilibria and is not white noise 3159.19 Complete chaos exhibits innitely many aperiodic oscillations

with each oscillation having its own amplitude 3159.20 Complete chaos exhibits innitely many aperiodic oscillations

with each oscillation having its own amplitude 3169.21 Wavelet scalogram and scalegram of the completely chaotic

logistic parabola process with scaling parameter = 4.0 3179.22 The trajectory of a billiard ball depends on the shape of the

elastic boundary that constraints it 3189.23 The state space trajectory of a chaotic system shows aperiodic

cyclicity with non-overlapping orbits 3199.24 First 10 observations of the state space trajectory of the chaotic

logistic process x(t) for = 4.0 3209.25 First 10 observations of the steady-state equilibrium points

where the trajectory touches the parabolic constraint of thechaotic logistic process x(t) for = 4.0 320

9.26 First 20 observations of the state space trajectory of the chaoticlogistic process x(t) for = 4.0 321

9.27 First 20 steady-state equilibria points on the attractor set of thechaotic logistic process x(t) for = 4.0 321


9.29 First 50 steady-state equilibria points on the attractor set of thechaotic logistic process x(t) for = 4.0 322


xviii List of gures9.31 First 90 steady-state equilibria points on the attractor set of the

chaotic logistic process x(t) for = 4.0 3239.32 The physical or institutional resource constraint of the chaotic

process determines its global, long-term predictability 3249.33 A close return or recurrence plot of the BelousovZhabotinsky

chemical reaction 3279.34 Close return histograms of (a) a chaotic time series with

aperiodic cyclicity and (b) Geometric Brownian Motion 32810.1 Nelson and Siegel curve tted to UK government bond (gilt)

rates derived from nine bonds for t0 = September 4, 1996 35810.2 Vertical ocean shear at various depths 35910.3 A 6-scale wavelet coecient sequence decomposition of ocean

shear series 36010.4 More than 9,300 minute-by-minute quotations on the Philippine

pesos collected in real time for the month of July 1997 36110.5 Three-scale wavelet resonance coefcient series of the

minute-by-minute quotations on the Philippine pesos 36210.6 The term structure gradient, TSG(t) 36411.1 An indictment of global (average) statistical analysis by two

time series with the same global risk spectrum P() 38211.2 Simulated evolution, t = 10, 20, 40, of a 2D vortex spiral,

based on a pseudo-wavelet computation of 2D NavierStokesequations 384

11.3 Theoretical and empirical representations of a shock wave 38611.4 Data in the time domain from nine different turbulent ows 39111.5 Gibbs phenomenon 40511.6 Approximation of the Heaviside function f (x) in panel of

Figure 11.5, using a wavelet basis 40612.1 Typical time dependence of nancial price volatility, log 2 42612.2 Empirical distribution of daily revenues of JP Morgan in 1994 43112.3 Potential large-scale catastrophic ow risk: the Yangtze River 43412.4 Drawing of the completed Three Gorges Dam: the spillway to

release water and control ooding is in the center 43512.5 Emergence of large-scale dynamic catastrophic ow risk

management 436A.1 Mandelbrots original evidence for scaling in economic pricing

processes 449B.1 S&P500 daily closing prices taken from table 2.7 in Sherry

(1992), pp. 2932 451

Tables

1.1 Ellsberg Paradox payoffs: Options 1 and 2 161.2 Ellsberg Paradox payoffs: Options 3 and 4 161.3 First four moments of FX returns: USD/DEM and USD/JPY 281.4 First four moments of FX returns by time interval 302.1 Volatility matrix of European option prices for various strike

prices and expiration dates 603.1 Comparison of tail P(X > c) probabilities 934.1 ARCH(1) limit kurtosis 1144.2 ACFs of long- and short-memory series 1184.3 Equivalence of various critical irregularity exponents 1246.1 Risk spectrum of FBM increments 1688.1 Degree of Lipschitz irregularity of Daubechies wavelets 2428.2 Heterogeneous Hurst exponents of subsequent exchange rate

regimes in Mexico in the 1990s 2598.3 Measured homogeneous Hurst exponents of Latin American

stock and foreign exchange markets 2628.4 Values of homogeneous Hurst exponents for nine currencies 2659.1 Levels of short- and long-term predictability 2909.2 Steady-state equilibrium regimes of the logistic process 329

Preface

In the spring semester of 2000, I was asked to teach a new course on RiskTheory III for 15 third- and fourth-year undergraduate students and 2 postgraduateMasters students at the School of Economics of Adelaide University, in Adelaide,South Australia.1 I could have chosen an existing textbook on Risk Theory foractuarialists,2 and that would have saved me countless hours of research andwriting, but, instead, I decided to be courageous and develop a new course fromscratch and to focus on (1) the measurement, and (2) the analysis of nancial mar-ket risk, and, perhaps, to discuss some of the implications for nancial portfoliomanagement.

Previous professional experiences had widened my perception of nancialmarket risk, both of nancial crises and of nancial turbulence, when Iwas a SeniorEconomist for Nomura Research Institute (America), Inc. between the years of1987 and 1990. An example of this being the following event: on November 19,1987, theDow Jones Industrial Average plunged 508.32 points, losing 22.6 percentof its total value. That drawdown far surpassed the one-day loss of 12.9 percent thatbegan the great stock market crash of 1929 and foreshadowed the Great Depres-sion. The Dows 1987 plunge also triggered panic selling and similar drops instock markets worldwide. But the US stock market recovered, after the Fed rstpumped in a massive amount of liquidity and then drained it two weeks later ina classic monetary action, that prevented an already persistent, and now rapidlybecoming illiquid, stockmarket to grind to a screeching halt. Although this was amassive market failure, it was a nancial crisis without many consequences thanksto the rapid successful monetary engineering action by the Fed (in contrast to theFeds bungling in the 1930s!).

On Friday January 20, 1990, as a Senior Economist of Nomura ResearchInstitute (America), Inc., the research arm of the Japanese global securities rmNomura Securities, Inc., I predicted on CNBC TV, in response to a question bythe Chief Economist of Business Week, that the Tokyo stock market would declineby 40 + percent. I stated that, because the Japanese stock market was an admin-istered market, it would happen in carefully controlled phases, in the rst- andthird quarter of that year. This would happen in response to a slight tightening ofthe money supply by the Bank of Japan in December 1989, which attempted todeate Japans asset bubble. The Japanese stock market actually lost 69 percent

xxii Prefaceof its value in 1990, in the respective predicted quarters. Until now, it never fullyrecovered from that nancial crisis, because Japans banks were determined notto restructure and to write off any book value of their nancial institutions. Thesenancial institutions obstructed the free working of the market system and becamezombies, or livingdead,which continued to destroyglobal capital formore thana decade thereafter. For more than ten years, Japans nancial system operated asa black hole for capital ows, retarding global economic development. Even theavailability of free money (= money available at zero interest) did not induce anydomestic activity in what had become a pure Keynesian liquidity gap. The Bank ofJapan was pushing on a string. Why the difference between the US stock marketand the Japanese stock market?

In 19911993, as Chief Economist of ING Bank, Inc., later ING Capital, Inc.,in New York City, I became familiar with the trading of distressed debt of theLatin American and domestic US markets, resulting from the collapses of creditworthiness and the increases in the respective country and regional risk premia. Atthat time I wrote sales revenue generating country risk reports on Latin Americanemerging markets, e.g., my report on Mexico, in July 1993, generated $21 millionin sales of Mexican distressed debt for ING Bank with one week.3 In 19921993,I also monitored the European Financial Crisis, when the European MonetarySystem became undone and accelerated the rush into the overvalued Euro, whichsubsequently after January 1, 1999 depreciated by more than 15 percent in valueversus the US dollar. Its only recently that the Euro is returning to par with theUS dollar.

Finally, in 19951999, when I was an Associate Professor in Banking andFinance at the Nanyang Business School of the Nanyang Technological Univer-sity (NTU), from my vantage point in Singapore, I closely monitored the AsianFinancial Crisis in 1997, i.e., the collapse of the Asian bank loan, stock and cur-rency markets, closely followed by the Russian Financial Crisis of 1998, i.e., thedefault of Russian government debt, which led, via the implosion of the Germanbond market to the $500 billion collapse and subsequent bail-out by the Fed of Long Term Capital Management, Inc.4

Primarily in reaction to the Asian Financial Crisis of 19971998, I designedand supervised a Masters thesis research project on the Wavelet MultiresolutionAnalysis (MRA) of Asian foreign exchange markets (Karuppiah and Los, 2000),to demonstrate that most of the Asian currency markets were antipersistent andhad continued their regular trading. There was a sharp discontinuity in the Thaibaht on July 2, 1997, the day after the handover of Hong Kong back to the PeoplesRepublic of China, followed by considerable nancial turbulence. But therewas nocollapse of the other Asian FX markets, which continued to function properly, asshown by our analysis.5 In fact, for the rst time we could measure the differencesin the degrees of persistence of the various Asian FX markets and show that somemarkets operated faster and more efciently than others.

I had become deeply involved in that fascinating Asian FX project and, with theassistance of Dr Er Meng Hwas Center for Signal Processing at NTU, I becameconvinced that nancial market risk should not only be measured in terms of

Preface xxiiiits frequency distributions, as the conventional MarkowitzJorion Value-at-Riskapproach suggested, but that it should be completely analyzed in terms of itslong-term time dependencies, preferably in the all-encompassing timefrequencydomain.

The fundamental characteristic of nancial time series, such as FX rates orasset returns, is that they are nonstationary (stationarity is a precondition for clas-sical correlation, spectral and harmonic analysis) and singular (non-singularityis still a precondition for Gbors Windowed Fourier analysis). This insight wasreinforced when I learned from signal processing engineers about the technologi-cal advance made by Mallats wavelet multiresolution analysis (MRA) in 1989. Itdemonstrated that timefrequency visualization and analysis of non-differentiable,singular, nonstationary and non-ergodic nancial time series is possible bywaveletMRA. Simultaneous analysis of many frequency and time dependencies is madepossible because the newwaveletMRA operates as a gigantic data microscope thatcan be ne-tuned to any level of analytic resolution one wants to use for research.

I obtained further inspiration in Adelaide from the provocative 1994 bookFractal Market Analysis of Edgar Peters, Manager of PanAgora Management,who substitutes his FractalMarketHypothesis (FMH) for the 1970EfcientMarketHypothesis (EMH) of Eugene Fama. A foray into the rapidly expanding eld ofparametric stable distributions brought me in contact with Nolans clear expla-nations of their Zolotarev parametrization. In October 2000, at an internationalconference at the University of Konstanz, I noted that some of my European col-leagues had also made progress in that direction, in the context of the emergingExtreme Value Theory.

All these efforts helped me to sort out the confusing array of critical expo-nents in Chapter 4. Additional reading of Benoit Mandelbrots awe-inspiring 1982monograph, The Fractal Geometry of Nature, and the compilations of his articlesin his recent books, on Fractals and Scaling in Finance and Multifractals and 1/fNoise, immeasurably inuenced the direction ofmy research. They also stimulatedmany of the computer graphics and other illustrations of this book.

Finally, a relearning, and drastic upgrading, of my rudimentary secondaryschool knowledge of uid dynamics at Adelaide University, supported my originalhydrological or meteorological interpretation of global cash ow dynamicsand the measurement andmodeling of nancial turbulence and of nancial crises.6Of course, there exists the well-known historical precedent of an economist apply-ing hydrological concepts to Economics. In the 1950s, in the basement of theLondonSchool of Economics (LSE), theNewZealand economistA.W. Phillips, ofelusive Phillips curve fame, engineered an actual water ow model of the NationalIncomeAccounts of an economy.7 Many introductory textbooks of Economics stillrefer to this model by way of National Income ow diagrams. Also in Finance, thedynamic cash ow theory nds some resonance. For example, James van Horne ofStanford University has a (5th) edition introductory nance textbook on FinancialMarket Rates and Flows.

However, following Mandelbrot, my book applies hydrological concepts toFinance, in particular to the identication from inexact data of models for stable

xxiv Prefacenancial risk, nancial turbulence and nancial crises, that can quantitativelyassist in the timefrequency analysis and optimal management of such nancialmarket risks.

I thank Dean Colin Rogers and Professors Jonathan Pincus, Kym Andersonand Richard Pomfret in the School of Economics of Adelaide University in SouthAustralia, for providing me with a rustic, but productive research environmentin the year 2000 to develop a series of three new courses in Finance: (1) Com-putational Finance: A Scientic Perspective, which resulted in my eponymousbook published in 2001 by the World Scientic Publishing Co. in Singapore; (2)Financial Risk: Measurement and Management, which forms the basis for thisbook published by Routledge; and (3) Dynamic Valuation and Hedging, whichis used for parts of courses in the new Master of Science in Financial Engineer-ing program at Kent State University and which may develop further into a thirdbook. MyHonors students at AdelaideUniversity provided the necessary stimulus,raised lots of questions, produced self-correcting feedback and did helpful libraryresearch and computations during my series of lectures.

In the southern hemisphere Fall Semester of 2001, Dean Garry Carnegie andProfessors Jonathan Batten and Stewart Jones of Deakin University in Burwood,a suburb of Melbourne, Victoria, Australia, provided me a second place to workon this book, while, as a Visiting Associate Professor of Finance, I taught a courseon Business Finance Decision Making and tutored students on the ner points ofFinance and Financial Markets.

My employment as a tenure-track Associate Professor of Finance by Kent StateUniversity in Ohio allowed me to nish this book and to ready it for production. Inparticular, I would like to express my deepest thanks to Dean George Stevens andAssociate Dean Rick Schroath, who both have been unwavering in their supportof my tenuous position in Kent States Business School. At Kent State, I interactedwith and enjoyed the excellent services of the library and I received the highly desir-able computational assistance and error corrections of my postgraduate studentsKyaw Nyonyo from Burma, Joanna Lipka from Poland, Kang Zhixing (Richard),Zong Sijing and Chen Xiaoying from the Peoples Republic of China, RossitsaYalamova from Bulgaria, who found new relevant articles and checked a largenumber of bibliographic references, and Sutthisit Jamdee from Thailand, whoworked with me to produce a movie of a colorized dynamic scalogram for real-time high frequency nancial data. Doctoral candidates Kyaw and Zong preparedthe scalograms and scalegrams of the Financial Crises in Mexico, Brazil and Chilein Chapter 8, using interactive software available on the web. All these studentsactively participated in my doctoral seminars on Research in Finance in the Fallterms of 2001 and 2002.

I also enjoyed the exchanges about my research with Mohammed Kazim Khanand Chuck Gartland, both Professors of Mathematics, Richard Kent, Professorof Economics, Jim Boyd, Associate Professor of Finance, who hired me, and mycolleagueMarkHolder, Assistant Professor of Finance andDirector ofKent Statesnew Master of Science in Financial Engineering program, which we together

Preface xxvhelped to give birth to. This program is one of 38 such programs in the world andthe only one devoted to derivatives, in particular, energy and weather derivatives.

Its also high time that I acknowledge a lifelong debt of intellectual trust andencouragement to the lateMr P.Khne, theHead of theNicolaasBeets School, a nolonger existent elementary school inHeiloo, North-Holland, TheNetherlands, whoin the early 1960s, conducted a unique socio-economic experiment with four boysand four girls. MrKhne selected twoboys and twogirls frompoor to lowmoderatesocio-economic background (I was one of them) and two boys and two girls fromhigh-moderate to high socio-economic background and gave them a dedicatedpreparation for the entrance exam to theGymnasium, the formerDutch prep schoolfor university level education, to prove that meritorious education and not socio-economic background mattered for individual success. He proved to be right: alleight boys and girls passed their entrance exam and successfully completed theirGymnasium education. Later on, all eight students received university degreesand became very successful in their respective professions, unfortunately, afterMr Khne had already passed away.

I dedicate this book to Mr Khne, and I dedicate it also to my parents andmy mother-in-law and (now late) father-in-law for maintaining their faith in methroughout my life, but in particular during the past critical six years, when I wasan Associate Professor of Banking and Finance in the Australasian region. Thecoming years may be just as turbulent, but, hopefully, not as catastrophic as theAsian Financial Crisis of 1997, which originally inspired this book.

Finally, I want to acknowledgemy debt to Robert Langham, Editor Economicsof Routledge, who invited me to publish this book in Routledges InternationalStudies in Money and Banking; to Terry Clague, his Editorial Assistant, who keptme on track when we moved back to the States and when my wife underwent herlung cancer operation; to Moira Eminton, Editor for Taylor & Francis Books, whokept this and many other projects on track during a merger by her publisher, andwho had to move from London to New York; and to Vincent Antony, project man-ager of Newgen Imaging Systems (P) Ltd., in India, who was able to accomodatemy math and gures in a beautiful typesetting.

As always, Im very grateful to my beautiful and very dapper wife Rosie, whoprevents me from making the most serious grammatical errors and who, despitemajor setbacks, continues to brighten my days with love and laughter.

Cornelis A. LosKent State University

Notes1 That is, in the spring semester of 2000 in the northern hemisphere, which was the actual

Fall semester for Australia in the southern hemisphere.2 Such as Bhlmann, Hans (1970)MathematicalMethods in Risk Theory, Springer-Verlag,

New York, NY.3 That was before the Mexican Financial Crisis of 1994!

xxvi Preface4 Cf. Jorion (1999) How Long-Term Lost Its Capital, RISK, September, and Dunbar, N.

(2000) Inventing Money: The Story of Long-Term Capital Management and the LegendsBehind It, John Wiley & Sons, Chichester, UK.

5 The Asian Financial Crisis did not originate in the antipersistent Asian FX markets, butin the non-transparent and persistent Asian bank loan markets and in the governmentcontrolled, illiquid and persistent Asian stock markets.

6 Contained in my Dutch Doctorandus (= MPhil) thesis of 1976.7 I was a Research Student at the University of London in 19751976, when I rst

learned about system identication and control theory and about Phillips interestinghydrodynamic contraption.

Introduction

Probabilitatem Non Esse Delendam, Esse Deducendam1

This book covers the latest theories and empirical ndings of nancial marketrisk, its measurement, analysis and management, and its applications in nance,e.g., for dynamic asset valuation, derivatives pricing and for hedging and portfoliomanagement. A special and rather unique part of this book is devoted to measuringwhen nancial turbulence can occur and when nancial catastrophes are probable.

To gain a basic understanding of nancial market risk, we must ask at least fourfundamental questions:

(1) What is nancial market risk?(2) How do we measure nancial market risk? For example, which frequency and

timing distributions of nancial market risk do we actually measure?(3) Is all nancial market risk dangerous or can we distinguish between safe

nancial market risk and dangerous nancial market risk? For example,which nancial market risk is diversiable, which is hedgeable and which isnon-diversiable and non-hedgeable?

(4) How can we manage nancial market risk to our advantage? For example,how much nancial market risk is hedgeable?

These four questions will be answered, or at least discussed in technical detail,in the four consecutive Parts of this book.

In Part I on Risk Processes, we discuss the four different concepts of measuringrisk, such as uncertainty, randomness, irregularity and probability. We discuss riskinvariants, in particular, against time and frequency, called self-similarity, or, moreprecisely, self-afnity. We highlight the statistical invariants of stationarity andtimefrequency scaling and provide various descriptors of serial time dependence,of discontinuity and of concentration. Our objective is to determine the periodicity,aperiodic cyclicity, turbulence, intermittence and arrhythmias of the nancial timeseries currently produced in great abundance by the global nancial markets.

This detailed analysis of nancial time series helps us to determine what is thebest way of measuring nancial market risk. In this book, we nd that the best wayof measuring risk is as a residual, unexplained irregularity. For that purpose we

xxviii Introduction

analyze the fractality, or self-afnity, of speculative and cash market pricing, andpropose various forms ofmeasurement and visualization of long-term dependence,in particular, of market persistence or antipersistence, using wavelet Multiresolu-tion Analysis (MRA). As YvesMeyer comments in his interesting series of lecturenotes: The study of profound problems is often inuences by the available instru-ments and techniques.2 We put these concepts of nancial risk within the contextof two major market hypotheses: the Efcient Market Hypothesis of Eugene Famaand the Fractal Market Hypothesis of Benoit Mandelbrot and Edgar Peters.

In Part II on RiskMeasurement we discuss the various ways of measuring nan-cial market risk in both its time and frequency, c.q., scale, dimensions. The basictool for such inexact model identication of nancial market risk is correlationor, slightly more specic, convolution. Thus, in this book, we compute the res-onance coefcients both for Fourier Transforms and for Wavelet Transforms. Forthe measurement of the irregularity of nancial time series, we compute criticalLipschitzHlder exponents, in particular the Hurst Exponent, and the Lvy Sta-bility Alpha, and relate them to Hoskings fractional difference operators, e.g., theFractional Brownian Motion model, which will be our benchmark model.

We use three techniques of nonstationary time series analysis to measuringtime-varying nancial market risk: Range/Scale analysis, windowed Fourier anal-ysis, and wavelet MRA, and we mathematically relate these powerful analytictechniques to classical BoxJenkins time series analysis and Pearsons spectral fre-quency analysis, which both rely on the assumption of stationarity and ergodicity.By empirical examples, we demonstrate the superiority of these advanced tech-niques, which can deal with the occurrence of non-stationarity and non-ergodicity.Themodeling focuswill be again onHoskings fractionally differenced time series,in particular, on Fractional Brownian Motion.

Part III on Term Structure Dynamics is the most adventurous part of the book,delving into the transient phenomena of chaotic risk and of nancial turbulence. Itdenes nancial chaos and demonstrates how such chaos can develop in nancialmarkets. For the rst time, we develop a theory of dynamic cash ow analysis,which allows the modeling of the transient phenomena of nancial chaos and ofturbulence within an adapted nancial framework of term structure analysis andwhich allows the measurement of such phenomena by wavelet MRA.

Financial turbulence is not necessarily a bad phenomenon. We learn that it isactually an efciency enhancing phenomenon that only occurs in the antipersistent,most liquid anchor currency markets. Financial turbulence should, therefore, besharply distinguished from the real bogey of nancial managers: nancial catas-trophe or crisis. A nancial crisis is measured as a discontinuity or singularity ina persistent nancial time series. It is unpredictable and occurs only in persistentnancial markets with low liquidity.

Now, some nancial crises are more dangerous than others. For example, itmay not be dangerous to speed up the trading and price formation activity in anancial market and encounter a crisis, because the nancial market may movethrough a so-called safe nancial crisis. Whereas slowing down trading and priceformation may lead to an unsafe nancial blue sky catastrophe. It may cause

Introduction xxix

a nancial crisis in which the pricing system close to an attractor suddenly headsfor the attractor at innity: the market pricing process breaks down and cantrecover.

Thus, ultimately, this Part III is laying the groundwork for an ongoing, but notyet completed, search for integrity measures for nancial markets, to quantify themargin of safety between a nancial markets attractor and the fractal boundary ofits safety basin. The Lipschitz L and the Hurst H -exponent discussed in Part IIcan be viewed as such an integrity measures. Ive already observed that thesemeasures change very dramatically by basin erosion at a point on the solution pathat which a realistic, inexact nancial pricing system is liable to escape. This isnow known by system safety engineers as the Dover cliff effect. However, acomplete quantication of the margin of safety for nancial markets is not coveredby this book. It will probably have to wait for still more detailed empirical andtheoretical research.

Still, we nd that the statisticians averaging spectral decomposition, which isbased on the ergodic stationarity assumption, has inhibited and slowed down scien-tic progress regarding the investigation of transient structures, such as turbulencevortices in nancial markets. It is also clear that nancial analysis is currentlyshifting from the study of the steady-state solutions of nancial markets to thestudy of their transient behavior. This book is intended to help this transition innance (and economics) along and to speed it up.

For the rst time in nancial-economic analysis, we are looking to measureand engineer the true empirical conditions that ensure the safe and continuousworking of our nancial market pricing systems time locally and not on (time)average. Financial market systems are the complex institutional arrangementsthat guarantee the optimal allocation and most effective and efcient use of ourscarce nancial resources. They are crucial for the proper projection, adoption andintegration of new nancial technology and thereby for the growth in productivitythat raises the living standards of all humankind.

Part IV contains one chapter on Financial Risk Management: the stable kind,the cyclical kind, the turbulent kind and the critical kind. It discusses ExtremeValue Theory and some consequences for the popular Value-at-Risk approach toportfolio and bank management. Insurers try to reduce nancial risk at a cost, bydiversication, using fund management portfolios to reduce the unsystematic riskand by hedging to reduce the systematic risk.

But sometimes nancial market risk cannot be reduced because of its peculiarempirical characteristics of long-term time-dependence and non-stationarity, aphenomenon already studied in the 1960s by Fama and by Samuelson, a winnerof the Nobel Memorial Prize in Economics. Sometimes, we want to have morenancial market risk, because we speculate that more nancial risk may lead tohigher average returns on our investments. Thus, nancial risk management is notonly about reducing risk!

Indeed, the Chinese pictograph for risk in the following Figure 1 consist of twosymbols: the rst Chinese symbol Wei represents danger, the second symbol Jistands for opportunity.3 Thus, the Chinese dene risk as a combination of danger

xxx Introduction

Figure 1 Risk = Danger (Wei) + Opportunity (Ji).

and opportunity. Greater risk, according to the Chinese, means we have greateropportunity to do well, but also greater danger of doing badly. Interestingly, wewill see that Fourier analysis mathematically teaches us that, in a similar fashion:

risk = volatility = energy = power

Well use Value-at-Risk (VaR) as an initial organizing paradigm for nancialrisk management, contrast it with a few alternative risk paradigms and trace theimplications of L-stable, heavy tail distributions of market pricing for portfoliorisk management. We also show the importance of long-term time dependencefor Value-at-Risk and for modern portfolio management and relate our ndings tothe latest results in Extreme Value Theory. This properly measured approach tonancial risk is of crucial interest to senior nancial risk managers of global banks,insurance and pension funds.

Muchof the illustrativematerial throughout this bookhas beendrawn fromratherrecent research papers in economics, nance, physics and signal processing. It isa feature of advanced nancial market risk measurement and analysis probablymore than in most branches of nance that the details of rather simply speciedtopics like the cash ow dynamics or the frequency of trading in the nancialmarkets are complex and still imperfectly understood. It has been my personalexperience that Ive had difculty in convincing postgraduate students that sometopics I proposed to them have not been fully explained decades ago. It is thus oftenappropriate to use even for introductory purposes (e.g. Chapters 12) topics thatare still the subject of research. This attractive feature of an only partially exploredsubject also makes it easier for this book to serve both as a (challenging) seniorundergraduate text in economics and nance and as a source of relevant technicalinformation for postgraduate nancial researchers and practising professionals inthe nancial services industry.

The descriptions of the gures often contain details that are intended for themore advanced reader who wants to know the particular conditions to which thedata refer. I hope they are detailed enough to convey something of the avorof empirical nancial market risk measurement and analysis. The book is fullyreferenced. Some of the references indicate sources of material of illustrationsor ideas. Other references have been included for the reader who will use the bookas an information source and wishes to follow up a topic in detail. No attempt at

Introduction xxxi

completeness of references has been made, since that would involve far too manyreferences. Ive tried to give appropriate entries into the literature of the varioustopics, more often a recent review or signicant paper, but also sometimes thepioneering research paper, because it was so well written.

Since the primary purpose of this book is to be pedagogical, in the ChapterExercises readers can prepare different cases of nancial market risk and loss,catastrophe and disaster, and trace the implications for their respective manage-ment. All Exercises are preceded by short suggestions of the most appropriatesoftware for themeasurement of nancial risk. All Exercises were tested by senior,master and doctoral students in Tutorials in the School of Economics of AdelaideUniversity and in the Graduate School of Management of Kent State Univer-sity. Together with graduate students Melin Kassabov and Rossitsa Yalamora Iveprepared a solutions manual for all these Chapter Exercises which will be madeavailable via a web site. We have included in Appendix B a simple data set basedon daily prices of the S&P500 stock market index for 1988. Other data sets caneasily be downloaded from the Internet. For example, on his web site, John Hullof the University of Toronto has made available the daily prices of the TSE300,S&P500, FTSE100, CAC40 and Nikkei225 stock market indices for the periodJuly 5, 1994July 10, 1998.

The combined theoretical and practical approach of this book helps the readers(1) to select relevant frameworks for analysis, concepts, tools and techniquesapplied to real nancial market data, and (2) to distinguish between information,knowledge and wisdom in this rapidly adjusting domain of new knowledge.

Notes1 Latin translated: Probability should not be deleted, it should be deduced. The Latin

phrase was formulated by Dr Rudolf E. Kalman, when, on May 3, 1993, he deliveredhis lecture on Stochastic Modeling Without Probability at the Sixth International Sym-posium on Applied Stochastic Models and Data Analysis at the University in Chania,Crete, Greece. There Dr Kalman proved that there is very little, if any, scientic basisfor Haavelmos 1944 presumption of the empirical existence of Kolmogorov probabil-ity. Such an empirical existence has to be deduced from the data to be established asa scientic fact. Kolmogorovs probability theory is still only a theory and has not yeta scientically established support in empirical reality. Science cannot accept Platosdichotomy between true reality and the world we perceive, because if it did, it wouldbecome quickly a religion. True science accepts Aristotles objectivist epistemology.

2 Meyer, Yves (1993) Wavelets: Algorithms & Applications (Translated and revised byRobert D. Ryan), Society for Industrial and Applied Mathematics (SIAM), Philadelphia,PA, p. 119)

3 As I was informed by two of my MBA students at Kent State University: Kang Zhixing(Richard), who was one of my Research Assistants, and Wang Zhengjun.

Part I

Financial risk processes

1 Risk asset class, horizonand time

1.1 Introduction

1.1.1 Classical market returns assumptions

Most investors, portfolio managers, corporate nancial analysts, investmentbankers, commercial bank loan ofcers, security analysts and bond-rating agenciesare concerned about the uncertainty of the returns on their investment assets, causedby the variability in speculative market prices (market risk) and the instability ofbusiness performance (credit risk) (Alexander, 1999).1

Derivative instruments have made hedging of such risks possible. Hedgingallows the selling of such risks by the hedgers, or suppliers of risk, to thespeculators, or buyers of risk, but only when such risks are systematic, i.e., whenthey show a certain form of inertia or stability. Indeed, the current derivativemarkets are regular markets where stable, i.e., systematic risk is bought and sold.

Unfortunately, all these nancial markets suffer from three major deciencies:(1) Risk is insufciently measured by the conventional second-order moments

(variances and standard deviations). Often one thinks it to be sufcient tomeasure risk by only second-order moments, because of the facile, but erro-neous, assumption of normality (or Gaussianness) of the price distributionsproduced by the market processes of shifting demand and supply curves.

(2) Risk is assumed to be stable and all distribution moments are assumed to beinvariant, i.e., the distributions are assumed to be stationary.

(3) Pricing observations are assumed to exhibit only serial dependencies, whichcan be simply removed by appropriate transformations, like the well-knownRandom Walk, Markov and ARIMA, or (G)ARCH models.

Based on these simplifying assumptions, investment analysis and portfolio theoryhave conventionally described nancial market risk as a function of asset classonly (Greer, 1997; Haugen, 2001, pp. 178184). In a simplifying representation:

portfolio return volatility pp = f (asset class )Figure 1.1 shows the familiar presentation of risk as a function of asset class byIbbotson and Sinqueeld, who have collected annual rates of return as far back

4 Financial risk processes

Geometric mean (%)

Large company stocks

Small company stocks

Long-term corporate bonds

Long-term government bonds

Intermediate-term government bonds

US Treasury bills

Inflation

* The 1933 small company stock total return was 142.9 percent. 90% 90%0%

Arithmeticmean (%)

Standarddeviation (%) Distribution

10.2 12.2 20.3

12.2 17.4 34.6

5.4 5.7 8.4

4.8 5.2 8.8

5.1 5.2 5.7

3.7 3.7 3.3

3.1 3.2 4.6

Figure 1.1 Historical average annual returns and return volatility, 19261995.Source: Stocks, Bonds, Bills and Ination 1996 Yearbook, Ibbotson Associates, Chicago (annuallyupdates work byRoger G. Ibbotson andRexA. Sinqueeld). Usedwith permission. All rights reserved.

as 1926 (Ibbotson and Sinqueeld, 1999). The dispersion of the return distribu-tions, measured by the respective standard deviations, differs by six different assetclasses:

(1) common stocks of large companies;(2) common stocks of small rms;(3) long-term corporate bonds;(4) long-term US government bonds;(5) intermediate-term US government bonds;(6) US Treasury bills.

When an investor wants a higher return combined with more risk, he invests insmall stocks. When he wants less risk and accepts a lower return, he is advised toinvest in cash.

For example, Tobin (1958) made two strong assumptions, which were believedto be true by many followers: rst, that the distributions of portfolio returns are allnormally distributed and, second, that the relationship between the investors port-folio wealth and the utility they derive it from is quadratic of form.2 Under thesetwo conditions, Tobin proves that investors were allowed to choose between port-folios solely on the basis of expected return and variance. Moreover, his liquiditypreference theory, shows that any investment risk level (as dened by the secondmoment of asset returns) can be attained by a linear combination of the market

Risk asset class, horizon and time 5

portfolio and cash, combined with the ability to hold short (borrow) and to holdlong (invest). The market portfolio contains all the non-diversiable systematicrisk, while the cash represents the risk-free asset, of which the return compen-sates for depreciation of value caused by ination. The linear combination of themarket portfolio and cash can create any average return and any risk-premium onewants or needs, under the assumption that the distributions of these investmentreturns are mutually independent over time.

1.1.2 Whats empirically wrong?

Regrettably, there are many things wrong with this oversimplied conceptualiza-tion and modeling of the nancial markets and one has now become alarminglyobvious. Financial disasters are much more common and occur with much higherfrequencies than they should be according to the classical assumptions. Anincomplete but rather convincing listing of nancial disasters can be found inKindleberger (1996). Bernstein (1996) and Bassi et al. (1998) mention manyadditional instances.

The worlds nancial markets exhibit longer term pricing dependencies, whichshow, in aggregated and low frequency trading observations, devastating, butessentially unpredictable aperiodic cyclicity, like the Plagues of the Old Testa-ment or sharp and disastrous discontinuities, like Noahs Flood. On the otherhand, they show, in high frequency trading frequencies, turbulence structures andeddie like condensation and rarefaction patterns. Analysts are now applyinghighly sophisticated mathematical measurement methods from particle physics toidentify such empirical structures. In fact, quite a few nance articles regardingthis topic have recently appeared in physics journals, such as Nature (Potters et al.,1998; Kondor and Kertesz, 1999; Mantegna and Stanley, 2000).

First, well quickly learn that the uncertainty of the investment returns is a muchwider concept than just the volatility of the prices as measured by second-ordermoments. Higher order moments, like skewness and kurtosis, play an under-estimated, but a very important role. For example, the distributions of investmentreturns exhibit positive biases, because of the termination of nonperforming busi-nesses and the continuing life of performing ones. There is a nancial need tosucceed and not to fail. Thus, the return distributions are positively skewed. Inaddition, the tails of the rate of return distribution returns are fatter, i.e., the outlyingreturns are more prevalent, than normally expected.

Second, we will observe that the stationarity of the investment returns can-not be so easily assumed, since we empirically observed that the distributions ofinvestment returns change over time. Overwhelming empirical evidence has nowaccumulated that volatility, i.e., the standard deviation of price or rate of returnchanges, which in Modern Portfolio Theory (MPT) measures the risk of assetsand portfolios, is not time-invariant. Even worse, it also does not exhibit trendsor any form of stability! As Peters (1994, pp. 143158) shows: both realizedand implied volatilities are antipersistent. An antipersistent time series reversesitselfmore often than a normal or neutral time series.3 This phenomenonmay occur


becausemarkets develop their institutional frameworks andmature, thereby chang-ing the constraints of their nancial pricing processes. These cash ow constraintsdetermine the behavioral regimes of the dynamic pricing processes of which thevolatilities become turbulent.

Third, wend that intertemporal dependencies cannot easily beltered out of theobserved pricing series by simple serial correlation (ARIMA)models. The randompricing processes cannot be so easily reduced to independent white noise series,since nancial pricing series exhibit global dependencies, due to an intricate pat-tern of widely differing investment horizons of nancial institution. For example,do long-term or short-term bonds have the largest variance of return? The answerto this question depends on the time horizon of investors. Commercial banks haveshort-term liabilities in the form of deposits. These institutions minimize risk bymatching these liabilities with short-term investments. On the other hand, pensionfunds and life insurance companies have long-term liabilities. If they are concernedat all about their survival, they will immunize their portfolios and view invest-ments in long-term bonds as less risky than short-term investments (Haugen, 2001,pp. 358359). Such scaling patterns of differing investment horizons introducelong-term dependencies among the rates of return of the various asset classes.

Consequently, Mann and Walds (1943) conventional econometric assumptionof serial dependence for time series can be shown to be empirically false. Global,long-term dependence plays a pervasive and important role. Thus, it is morecomprehensive and justiable to present nancial market risk, in a simplifyingrepresentation, as follows.

asset return distribution P = f (asset class , horizon , time t)

Not only is the rate of return distribution produced by speculative markets depen-dent on the asset classes and on the time horizons of the investors, but thisdistribution function may be time-varying, as indicated by the time t-argument.This empirical reality, which only now starts to become properly modeled(Bouchaud and Potters, 2000), has serious consequences for portfolio manage-ment and investment analysis. Tobins (1958) liquidity preference theory is clearlytoo simple to adequately reect all these dimensions of risk. The simple, static,2-dimensional return-risk tradeoff, on which classical MPT is based, will have tobe replaced by multidimensional and dynamic return-risk tradeoffs, as was earliersuggested in Los (1998, 2000b).

Example 1 A ne example of the time-dependence of price distributions is thewell-documented strong time-dependence of the standard deviation or volatility ofstock price changes (Schwert, 1989).

This rst chapter contains many concepts and denitions to acquire a proper ana-lytic and technical lingo for the remainder of this book, and to review basicstatistical analysis. It forms the prolegomena of the main body of our discus-sion. In particular, well review Kolmogorovs axiomatic (set-theoretic) denition


of probability and of random processes, the real world denition of frequencydistributions and of observed time series, and the summarizing characterization ofthese time series by their moments and cumulants.

1.2 UncertaintyThere is no doubt in the mind of physicists that uncertainty, like relativity, is ofan absolutely fundamental nature, that admits no exceptions. The world could noteven physically exist without uncertainty:

One of the fundamental consequences of uncertainty is the very size of atoms,which, without it, would collapse to an innitesimal point.

(Schroeder, 1991, p. 113)

In mathematics, the theory of Hilbert bases and (linear) operator algebra ledto the formulation of the Uncertainty Principle (Meyer, 1985), which we judi-ciously and fruitfully exploited in our preceding book on Computational Finance:A Scientic Perspective (Los, 2001). But for the development of nancial risktheory we may need a somewhat broader denition.

According toWebstersNew Universal Unabridged Dictionary (Deluxe SecondEdition, Dorset and Baber, 1983, p. 1990):

uncer tainty=the quality or state of being uncertain; lack of certainty; doubt

and

uncer tain(1) not certainly known; questionable; problematical;(2) vague; not denite or determined;(3) doubtful; not having certain knowledge; not sure;(4) ambiguous;(5) not steady or constant; varying;(6) liable to change or vary; not dependable or reliable.

Similarly, in modern risk theory, we distinguish three different, but closelyrelated concepts: randomness, chaos and probability.4 Lets explain what each ofthese concepts mean and discuss their limitations.

1.2.1 Randomness = irregularityEssentially, from Websters Dictionary, we have the following informal denitionfor randomness:5

randomness = the state of being haphazard, not unique, irregular

Thus, these denitions are based on the regular use of the word randomness inthe English language (cf. Bernstein, 1996).


1.2.1.1 Degree of irregularityWhat is regular is dened and xed and clearly determined. But how irregularis the absence of determinedness? Recently, Pincus and Singer (1996) asked thequestion: what is the degree of irregularity and how dowemeasure it? One extremeis the certainty of being xed, of being unique, a constant, and having thus no spec-trum at all. The other extreme, the ultimate state of irregularity, is when somethingis indistinguishable from background noise, that has no spectral features, i.e., noisethat covers the whole spectrum.

For example, white noise has a at spectrum. Thus, it exhibits a specic, dis-tinguished spectral feature and, therefore, cannot be called irregular or randombackground noise. In between these two extremes we nd degrees of irregularitythat can be described by variously shaped spectra. Each irregular series has its ownspectrum, be it a Fourier spectrum for a stationary series; or a changing spectrumfor a nonstationary series, to be analyzed by either windowed Fourier Transformsor by Wavelet Transforms, depending on how fast the changes occur. One nowcomputes even a singularity spectrum for observational series, which show manydiscontinuities or jumps, as we will discuss in Chapter 8.

1.2.1.2 Measures of sequential irregularityIn nancial risk theory we are not interested in irregularity per se, but in dynamicirregularity, i.e., in irregularity as it manifests itself over time. For example,how irregular are the prices produced by a market pricing mechanism over time?Figure 1.2 provides some examples of nancial market price series and their ratesof return (Mittnik et al., 1998, p. 84):(1) the daily AMEX Composite index from September 1, 1988 to July 28, 1994,

with T = 1,810 observations;(2) the daily AMEX OIL Composite index from September 1, 1988 to July 28,

1994, with T = 1,810 observations; and(3) the daily DEM/USD Exchange Rate from January 2, 1973 to July 28, 1994

with T = 5,401 observations.For the past hundred years, since Bacheliers PhD thesis of 1900, in which he

described speculative price formation as a Random Walk, people have attemptedto describe the degree of irregularity of market pricing and of related investmentreturns (Bachelier, 1900). The currently best known rational measures of suchirregularity are the Lipschitz exponents (such as the HlderHurst exponents),which will be discussed in Chapter 4.

1.2.2 Pseudorandomness versus genuine randomness

People working with computers often sloppily talk about their systems randomnumber generator and the randomnumbers it produces. But numbers calculatedby a computer through a deterministic process, cannot, by denition, be random.Given knowledge of the algorithm used to create the numbers and its original

AMEX returns%

1989

3

2

1

0

1

2

3

4

1990 1991 1992 1993 1994 1995 1996

AMEX composite

1989

300

180

1.5

2.0

2.5

3.0

3.5

200

220

240

260

280

300

350

400

450

500

550

1990 1991 1992 1993 1994 1995 1996

AMEX oil composite

1989 1990 1991 1992 1993 1994 1995 1996

4

2

0

2

4

6

1989 1990 1991 1992 1993 1994 1995 1996

AMEX oil returns%

1975 1980 1985 1990 1995

DEMUSD exchange rate

1975 1980 1985 1990 1995

DEMUSD returns%8

6

4

2

0

2

4

6

8

Figure 1.2 Levels and returns of empirical nancial time series: AMEXstock andoil indicesand DEMUSD exchange rate.


state or seed, you can predict all the numbers returned by subsequent calls to thealgorithm, whereas with genuinely random numbers, knowledge of one numberor of an arbitrarily long sequence of numbers is of no use whatsoever in predictingthe next number to be generated.

Computer-generated random numbers are more properly referred to aspseudorandom numbers, and pseudorandom sequences of such numbers(Goldreich, 1999). A variety of clever algorithms have been developed that gener-ate sequences of numbers which pass every statistical test used to distinguish ran-domsequences from those containing somepattern or internal order. Ahigh-qualitypseudorandom sequence generator generates data that are indistinguishable from asequence of bytes chosen at random. Indistinguishable, but not genuinely random!

We no longer have to use pseudorandom number generators. There are nowsystems to collect genuine random numbers, generated by a process fundamentallygoverned by the inherent uncertainty in the quantum mechanical laws of nature,directly to your computer in a variety of forms.

Example 2 Hotbits are random numbers generated by timing successive pairsof radioactive decay electrons or beta particles. These particles are producedby the spontaneous transformation of neutrons (with charge 0) in the nucleus ofKrypton-85 into pairs of protons (with charge +1) and free electrons (= betaparticles with charge 1). The free electrons, or beta rays, are then detectedby a GeigerMller tube in a simple radiation monitor (Figure 1.3) interfaced toa computer. The unstable nucleus of the radioactive Krypton-85 (the 85 meansthere are a total of 85 protons and neutrons in the atom) spontaneously turns intothe stable nucleus of the non-radioactive Rubidium-85, which still has a sum of85 protons and neutrons, and a beta particle is emitted with an energy of 687kiloelectron volts (keV), resulting in no net difference in charge:

85Kr 85 Rb + + (1.1)

In this case, a gamma ray is also emitted with an energy of 514 keV, carrying awaysome of the energy. Gamma rays are nothing other than photons particles oflight, just carrying a lot more energy than visible light. Krypton-85 has a half-lifeof 10.73 years. This is called its half-life, since every 10.73 years half of a verylarge number of Krypton-85 nuclei present at the start of the period have decayedinto Rubidium-85. But there is no way, even in principle, to predict when a givenatom of Krypton-85 will decay into Rubidium-85. It has a 50/50 chance of doingso in the next 10.73 years, but thats all we can say. The inherent uncertainty ofsuch decay time is genuinely random. Since the time of any given decay is random,the interval between two consecutive decays is also genuinely random (not unlikebetween two nancial transactions). Using the Geiger teller, we can now measurethe lengths of the uncertain intervals after the fact and thus collect genuinelyrandom numbers. We measure a pair of these intervals, and emit a zero or onebit based on the relative length of the two intervals. To create each random bit,we wait until the rst count occurs, then measure the time, T1, until the next. We


Monitor 5 radiation monitor Detect: Alpha, Beta, Gamma, X-raysFeatures: Easy to read analog meters, red light count, anti-saturation circuitry and audible beeperSpecifications:Ranges: 1, 10, 100 and BATT (battery check)Power: One 9 volt alkaline battery provides up to 2,000 hours of operation at normal background levelsTemperature: 20C to +50CDetector: Uncompensated halogen-quenched with 1.52.0 mg/cm2 mica end windowEnergy sensitivity: Detects Alpha down to 2.5 MeV through the end window; typical efficiency at 3.6 MeV is greater than 80%. Detects 50 KeV Beta at 35% typical efficiency; 150 KeV is typically 75%. Detects Gamma and X-rays down to 10 KeV through the end window, 40 KeV minimum through the case.Meter reading: 050 mR/hr and 050,000 CPM, or 0500 uSv/hr and 050 mR/hr

Monitor 5 Meter C31,475 $299.00

Figure 1.3 Simple and relatively inexpensive radiation monitor.

then wait for a third pulse and measure T2, yielding a pair of durations. If theyrethe same, we throw away the measurement and try again. Otherwise if T1 is lessthan T2 we emit a zero bit; if T1 is greater than T2, a one bit. In practice, to avoidany residual bias resulting from nonrandom systematic errors in the apparatus ormeasuring process consistently favoring one state, the sense of the comparisonbetween T1 and T2 is reversed for consecutive bits.

1.2.3 Chaos = deterministic dynamic nonuniquenessChaos is a special form of irregularity. It means that at a certain time something,that was certain and unique, suddenly can become nonunique, although it remainsvery well determined. The dynamic system can have more than one equilibriumstate to be in, because of equilibrium state bifurcations (Lorenz, 2001). How manytimes the system orbits or jumps through a set of equilibrium states depends onthe nonlinear constraints imposed on the dynamic system. However, these separateequilibrium states of the same system can be perfectly well identied, determinedand described, likeH2Omolecules orbiting through the two coexisting equilibriumstates of ice oating in liquid water. The molecule is either in the ice state, orin the water state, and it orbits through these two states as time progresses(Prigogine, 1997).

chaos = deterministic dynamic nonuniqueness


Even more precisely, complete chaos is the coexistence of an innite numberof unstable deterministic equilibrium orbits, through which the dynamic systemcycles. Chaos is a form of inexactness, since there is clearly nonuniquenessor aperiodic cyclicity.6 But it is not pseudorandomness, since there coexist morethan one equilibrium state of being at the same time, while with pseudorandomnessthere is only one equilibrium state of being at the same time. However, chaos isdeterministic, since there is no doubtwhat these distinct, but coexisting equilibriumorbit states are: they are perfectly well determined. The set of such distinct butcoexisting aperiodic equilibrium states is called the strange attractor. Alreadymore than a decade ago, chaos is asserted to have been observed in speculativemarket prices on the trading oor (Savit, 1988, 1994). For a way of visualizing thedistinction between a chaotic nancial time series and a truly random or irregularone, cf. Chapter 9.

1.2.4 Probability = complete set of relative frequenciesProbability is a very well dened and constraint form of randomness. In math-ematics, probability is the ratio of the chances favoring a certain state to all thechances for and against it. Thus, probability is a rational measure: it measuresrelative frequencies. It counts the number of times of being in state A relative tothe total number of states, i.e., the sum of the number of times of being in state Arelative to the number of times of being in the A and non-A states.

Thus, the basic problem of using a probability measure to describe a degreeof uncertainty is that one has to know the complete universe of states that mayoccur to compute the relative frequency or probability of a particular event. Butthe denition of the concept of uncertainty already indicates that were doubtfulor ignorant of what may happen. In other words, the very fact that we cannotcompletely know the extent of the universe fromwhich the event is drawnprecludesthe use of probability measures in most of real life. Probability only plays a rolein games that have completely predened rules. Most real life situations are notlike well-dened games. Often we dont know which gaming rules in a particularsituation apply or what possibly such rules may be. Science deals with attemptingto discover such gaming rules and the relative frequencies by which particularevents occur. In other words, the probability distribution of a particular event isitself a phenomenon that science tries to discover and identify. It is pseudoscienceto presume and predene such probability distributions before the nite empiricaldata sets have been analyzed.

Remark 3 Earlier nancial analysts claimed that probability is a necessaryconcept for the pricing of derivatives and that the broader concept of uncertaintywould not do. Nowadays it is acknowledged that probability is not needed for thepricing of options, because the prices of derivatives can be replicated by the linearcombinations of the prices of portfolios of fundamental assets. The prices of suchfundamental assets are uncertain and not probabilistic, as this book demonstrates.Even bond prices are uncertain, when discontinuous credit or default risk is takeninto account. However, one can always give an ex post probabilistic interpretation


to the prices of derivatives, once they have been priced on the basis of replicationportfolios consisting of fundamental assets.

1.2.4.1 Kolmogorovs axiomatic probability

In 1933, the Russian mathematician Kolmogorov provided one particularaxiomatic denition of probability, using set theory (Kolmogorov, 1933; cf. alsoPapoulis, 1984, for a complete treatment). Since Kolmogorov, particular non-Kolmogorovian denitions of probability have been discovered, inter alia, by theItalian mathematician Luigi Accardi, a student of Kolmogorov and now Professorof Mathematics at the University of Rome. Thus, currently, at least two denitionsof probability coexist. The denition of probability is no longer unique! However,we will still discuss Kolmogorovs set-theoretic denition of probability, since itis the most familiar. But it is also a somewhat misleading representation of realworld randomness, since it is based on simple integer counting of events withina preselected or presumed universe. This discussion assists with making theconnection to the modern approach to randomness, i.e., to our broader concept ofrandomness as uncountable irregularity.

To provide the axiomatic denition of probability in the proper context of settheory, we assume some familiaritywith the Boolean notions of sets, complements,intersections and unions, as well as with the notion of function. The following pre-sentation by a series of denitions sequentially building on each other, beginningwith the fundamental set denitions, follows an earlier tutorial exposition pre-pared for the Federal Reserve Bank of New York by Los (1982) (cf. also Chowand Teicher, 1978).Denition 4 A space is a nonempty set which serves as the ultimate referenceset, or the universe.

Denition 5 A nonempty family of subsets, say A, is said to be an algebra, oreld, if for A1, A2 A, Ai A, i = 1, 2 and A1 A2 A, i.e., A is closedunder the Boolean set operations of complements and unions.

Here Ai means non-Ai , i.e., the complement of Ai .

Denition 6 Let A be an algebra dened on , then if for Ai Ai=1

Ai A (1.2)

A is said to be a -algebra (or Borel-eld).Thus, A is closed under Boolean set operations and sequential limits.

Denition 7 Let be a space and A a -algebra relative to . Then, theset (,A) is said to be a measurable space and the set of A is said to bea measurable set.


Denition 8 Let be a space and A a nonempty class of subsets of . A setfunction on A is a real valued function on A, i.e., if A1 A,then (A1) is areal number.

Denition 9 Let be a set function onA. LetA A and {Ai : i = 1, 2, . . .}is a collection of disjoint sets such that

A =i=1

Ai (1.3)

and

(A) =i=1

(Ai) (1.4)

then is said to be -additive, or countably additive.

Denition 10 A nonnegative, -additive set function , dened on a class A,which contains the empty set , such that

() = 0 (1.5)is said to be a measure. If is a measure on a -algebra of subsets of, the triplet(,A, ) is said to be a measure space. If, in addition,

() = 1 (1.6)then the triplet (,A, ) is said to be a probability space, and is denoted(,A, P ), where P is the probability measure, which mapsA to the real numbersbetween 0 and 1.

Denition 11 Let (,A, P ) be a probability space. An event is simply a setA A. The real number P(A) is said to be the probability (-measure) attachedto the event A, or, for short, the probability of A. Events A of probability zero,i.e., such that

P(A) = 0 (1.7)are called null events, or null sets.

Remark 12 The monotone property of a measure ensures that0 = P {} P {A} P {} = 1 (1.8)

Denition 13 The probability space (,A, P ) is said to be complete, if anysubset of a set in G with P(A) = 0 also belongs to A.


Any probability space (,A, P ) can always be completed!Denition 14 A property is said to hold almost certainly (a.c.) [also: almostsurely (a.s.), almost everywhere (a.e.), or with probability one (w.p.1)], if it holdseverywhere, except possibly on a null set A, i.e., a set A such that P(A) = 0.

After so many preparatory denitions, we have nally arrived at the classical,Kolmogorovs, axiomatic denition of a random variable (r.v.).Denition 15 Let (,A, P ) be a complete probability space. A measurablefunction X: R (where R is the real line) is said to be a r.v. if

P(A) = 0, A = {t : |X(t)| = } (1.9)Surprisingly, perhaps, Kolmogorovs apparently innocuous denition of r.v.

excludes a lot of irregular events, e.g., all singularities and discontinuities. Thus,nowadays, Kolmogorovs denition is considered decient by sophisticated realworld mathematicians, such as Mandelbrot (1982), Pincus, Kalman and Singer(Kalman, 1996; Pincus and Singer, 1996; Pincus andKalman, 1997). For example,we can observe and measure (= count the frequency of) special irregular events,called singularities. The occurrence of singularities is a frequent occurrence inthe real world and not all singularities are alike. Thus, in Chapter 8 we will discussthe concept of a singularity spectrum and how we can measure it. Therefore, weneed a new non-Kolmogorov concept of a random variable, which effectivelyasserts

P(A) 0, A = {t : |X(t)| = } (1.10)

1.2.4.2 Empirical real world: relative frequencyIn the empirical world, we determine the probability of an irregular event bymeasuring how relatively often it occurs. We measure its relative frequency ofoccurrence. We do NOT require that the absolute measure of that event is nite.Thus, for empirical simplicity, as long as we set the total number of measuredevents (= the measured universe equal to unity), we can always dene:

probability = relative frequency of eventsBased on the preceding discussion of why probability measures a very specializedform of randomness, we categorically state that it is more in agreement with theempirical world to dene:

randomness = irregularityand not to bound the describing and measuring functions of such irregularity, incontrast to what Kolmogorovs probability axioms require. In fact, we can nowmeasure the degree of randomness of a price or rate of return process, bymeasuringthe degree of irregularity of such a process (Pincus and Singer, 1996).


1.2.5 The Ellsberg Paradox

Peters (1999) uses the Ellsberg Paradox to illustrate the essential differencebetween uncertainty and probabilistic risk.7 In the Ellsberg Paradox, you are shownan urn that contains 90 balls. Of these, 30 balls are red, and the remaining 60 ballsare an unknown mixture of white and blue balls. One ball is to be drawn from theurn, and you are paid an amount of money if a particular color ball is chosen. Youare given two payoff options to choose from as in Table 1.1.

Look over Options 1 and 2 in Table 1.1 and decide which you would chooseand keep your choice in mind. Most people choose Option 1 for the set of payoffsin Table 1.1. We will soon see why.

Next, turn to the two other options that are offered in Table 1.2. The drawingwill be of the same urn, with the same mixture of red, white and blue balls, asbefore.

Which of these two new options would you now choose? Be honest! Mostpeople choose Option 4 for the second set in Table 1.2. But why?

In Option 1, you know for certain that you have a 13 probability of winning. Butyou have no idea of the probability of winning in Option 2. It could be anywherebetween zero and 23 , i.e., it could be zero or higher than the

13 of red. This demon-

strates that most people prefer to go with the odds they know, instead of to choosefor uncertainty.

Option 4 is chosen for the same reason. You know that Option 4 has a 23 prob-ability, because 60 of the 90 balls are either white or blue, but you do not knowthe odds of nding a red or a blue ball, which can be anywhere between 13 and 1.Again, most people prefer to go with the odds they know, instead of to confront ordeal with uncertainty.

Now, by itself, each choice appears rational. Remember, though, that you chosebothOptions 1 and 4. Here is where the Paradox comes into play. Choosing Option1 over Option 2 means that you believe that a red ball is more likely to be drawnthan a white ball. However, choosing Option 4 over Option 3 implies that you

Table 1.1 Ellsberg Paradox payoffs: Options 1 and 2

Red ($) White ($) Blue ($)Option 1 100 0 0Option 2 0 100 0

Table 1.2 Ellsberg Paradox payoffs: Options 3 and 4

Red ($) White ($) Blue ($)Option 3 100 0 100Option 4 0 100 100


believe that white or blue is more likely than red or blue and thus you believethat white ismore likely than redwhen choosingOption 4. Consequently, choosingbothOption 4 andOption 1 under the same conditions is inconsistent, since the twobeliefs supporting the respective choices are in conict with each other, accordingto the tenets of subjective probability, so loved by rational Bayesian statisticians.

Why do the majority of people choose Options 1 and 4? Because, when facedwith true uncertainty, we are more comfortable with what we know, than whatwe dont know. Thus, uncertainty is very different from probabilistic risk, whichis known. Probabilistic risk depends on the concept of known odds. The oddsare known and calculable for prob

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