Financial market risk

Financial Market Risk What is nancial market risk? How is it measured and analyzed? Is all nancial market risk dangerous? If not, which risk is hedgeable? These questions, and more, are answered in this comprehensive book written by 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 paradigm of global nancial market risk analysis As such it will be of great interest to students, academics and researchers involved in nancial economics, international nance and business It will also appeal to professionals in international banking institutions 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 of New York and Nomura Research Institute (America), Inc., and Chief Economist of ING Bank, New York He has also been a Professor of Finance at Nanyang Technological University in Singapore and at Adelaide and Deakin Universities in Australia His PhD is from Columbia University in the City of New York Routledge International Studies in Money and Banking Private Banking in Europe Lynn Bicker Bank Deregulation and Monetary Order George Selgin Money in Islam A study in Islamic political economy Masudul Alam Choudhury The Future of European Financial Centres Kirsten Bindemann Payment Systems in Global Perspective Maxwell J Fry, Isaak Kilato, Sandra Roger, Krzysztof Senderowicz, David Sheppard, Francisco Soils and John Trundle What is Money? John Smithin Finance A characteristics approach Edited by David Blake Organisational Change and Retail Finance An ethnographic perspective Richard Harper, Dave Randall and Mark Rounceeld The History of the Bundesbank Lessons for the European Central Bank Jakob de Haan 10 The Euro A challenge and opportunity for nancial markets Published on behalf of Sociộtộ Universitaire Europộenne de Recherches Financiốres (SUERF) Edited by Michael Artis, Axel Weber and Elizabeth Hennessy 11 Central Banking in Eastern Europe Nigel Healey 12 Money, Credit and Price Stability Paul Dalziel 13 Monetary Policy, Capital Flows and Exchange Rates Essays in memory of Maxwell Fry Edited by William Allen and David Dickinson 14 Adapting to Financial Globalisation Published on behalf of Sociộtộ Universitaire Europộenne de Recherches Financiốres (SUERF) Edited by Morten Balling, Eduard H Hochreiter and Elizabeth Hennessy 15 Monetary Macroeconomics A new approach Alvaro Cencini 16 Monetary Stability in Europe Stefan Collignon 17 Technology and Finance Challenges for nancial markets, business strategies and policy makers Published on behalf of Sociộtộ Universitaire Europộenne de Recherches Financiốres (SUERF) Edited by Morten Balling, Frank Lierman and Andrew Mullineux 18 Monetary Unions Theory, History, Public Choice Edited by Forrest H Capie and Geoffrey E Wood 19 HRM and Occupational Health and Safety Carol Boyd 20 Central Banking Systems Compared The ECB, The pre-Euro Bundesbank and the Federal Reserve System Emmanuel Apel 21 A History of Monetary Unions John Chown 22 Dollarization Lessons from Europe for the Americas Edited by Louis-Philippe Rochon and Mario Seccareccia 23 Islamic Economics and Finance: A Glossary, 2nd Edition Muhammad Akram Khan 24 Financial Market Risk Measurement and analysis Cornelis A Los Financial Market Risk Measurement and analysis Cornelis A Los First published 2003 by Routledge 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Routledge 29 West 35th Street, New York, NY 10001 Routledge is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2005 To purchase your own copy of this or any of Taylor & Francis or Routledges collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. â 2003 Cornelis A Los All rights reserved No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Los, Cornelis Albertus, 1951Financial market risk : measurement & analysis / Cornelis A Los p cm (Routledge international studies in money and banking ; 24) Includes bibliographical references and index Hedging (Finance) Risk management I Title II Series HG6024.A3L67 2003 332.015195dc21 ISBN 0-203-98763-2 Master e-book ISBN ISBN 041527866X (Print Edition) 2003040924 To Janie and Klaas Los, Saba and Leopold Haubenstock, and P Kửhne with Gratitude for life, liberty and the pursuit of happiness Contents List of gures List of tables Preface Introduction xiii xix xxi xxvii PART I Financial risk processes Risk asset class, horizon and time 1.1 Introduction 1.2 Uncertainty 1.3 Nonparametric and parametric distributions 17 1.4 Random processes and time series 31 1.5 Software 41 1.6 Exercises 41 Competing nancial market hypotheses 2.1 Introduction 47 2.2 EMH: martingale theory 47 2.3 FMH: fractal theory 53 2.4 Importance of identifying the degree of market efciency 65 2.5 Software 67 2.6 Exercises 67 47 Stable scaling distributions in nance 3.1 Introduction 71 3.2 Afne traces of speculative prices 72 3.3 Invariant properties: stationarity versus scaling 76 3.4 Invariances of (ParetoLộvy) scaling distributions 77 3.5 Zolotarev parametrization of stable distributions 80 3.6 Examples of closed form stable distributions 92 71 446 Financial risk management and based your VaR estimate on the assumed constant standard deviations? Demonstrate your conclusion Notes At this moment there are not yet Hurst or Lipschitz exponent measurements of real estate investment returns available in the nancial literature, but they will soon be published I agree with their conclusion that probability theory relates only to (Las Vegas type roulette, card, one-armed-bandit) game situations But I disagree with their assumption that probability theory has anything to with the empirical world Nobody has ever proved that probability is an empirically observable real world phenomenon, in contrast to randomness or uncertainty (cf Chapter 1) Probability theory does not explain any empirical phenomenon in the real world and is therefore not a scientic theory, but only a philosophical theory Even very recently, McDonald of Northwestern University still assumes in his new text book (McDonald, 2003) that the distribution is normal, so that the Hurst exponent is H = 0.5, despite all the empirical evidence to the contrary Different nancial markets exhibit different degrees of persistence, as I brought to his attention, when I reviewed six chapters of his book a few months before its publication The three gorges are called Qutang, Wu and Xiling Information for this example was gathered from the Internet, in particular from http://21stcenturysciencetech.com/ articles/Three_Gorges.html, and from http://www.chinaonline.co er/ministry_proles/ threegorgesdam.asp EV theory, which was discovered by Stephan Resnick (1987), seems to have been rst applied to VaR by Franỗois Longin in 1996, followed by Jon Danielson, Casper de Vries and their collaborators at the Tinbergen Institute in The Netherlands and at the London School of Economics (LSE) in London, by Paul Embrechts and Alexander McNeil at the ETH Zentrum in Zỹrich, and by Francis Diebold and his associates at the Wharton School in Pennsylvania, USA Because it is often implicitly assumed that the distributions are Gaussian, stationarity is often taken to mean stationarity in the wide sense. (cf Chapter 1) Carl Friedrich Gauss (17771855), German mathematician and scientist is acknowledged to be one of the three leading scientists and mathematicians of all time (the other two are Archimedes and Newton) He was a child prodigy, who taught himself reading and arithmetic by the age of three His outstanding works include the discovery of the method of least squares in 1795, the discovery of non-Euclidean geometry, and important contributions to the theory of numbers During the 1820s, with the collaboration of the physicist Wilhelm Weber, he explored many areas of physics, including magnetism, mechanics, acoustics and optics Similarly, Fama (1965) and Peters (1994, pp 210212) compute an approximate value of Z = 1.66 for the Dow Jones Industrial Index Peters clearly demonstrates the nonconvergence of the volatility of the DJIA, as we have done for the volatility of the S&P500 Index in Chapter This range of Z = 1/L cannot be measured by the Hurst exponent H , but can be measured by the Lipschitz L 10 EMH = Efcient Market Hypothesis (cf Chapter 2) Bibliography Bassi, Franco, Paul Embrechts and Maria Kafetzaki (1998) Risk Management and Quantile Estimation, in Adler, Robert J., Raisa E Feldman and Murad S Taqqu (Eds) (1998), A Practical Guide to Heavy Tails: Statistical Techniques and Applications, Birkhọuser, Boston, 111130 Managing VaR and extreme values 447 Bawa, V S., E J Elton and M J Gruber (1979) Simple Rules for Optimal Portfolio Selection in a Stable Paretian Market, Journal of Finance, 34(4), 10411047 Blum, Peter, and Michel M Dacorogna (2002) Extremal Moves in Daily Foreign Exchange Rates and Risk Limit Setting, Working Paper, August 5, Converium Re, Zỹrich, Switzerland Dacorogna, M M., O V Pictet, U A Mỹller and C G de Vries (2001) Extremal Forex Returns in Extremely Large Data Sets, Working Paper presented at the 2001 Annual Meeting of the European Finance Association at the University of Konstanz Dowd, K (1998) Beyond Value at Risk: The New Science of Risk Management, John Wiley and Sons, New York, NY Dufe, D., and J Pan (1997) An Overview of Value at Risk, Journal of Derivatives, 4(3), 749 Elliott, Robert J., and John van den Hoek (2000) A General Fractional White Noise Theory and Applications in Finance, Quantitative Methods in Finance and Bernoulli Society 2000 Conference (Program, Abstracts and Papers), 58 December, 2000, University of Technology, Sydney, pp 327345 Embrechts, P., C Kluppelberg and T Mikosch (1997) Modeling Extreme Events for Insurance and Finance, Springer Verlag, New York, NY Embrechts, P., S Resnick and G Samorodnitzky (1998) Living on the Edge, RISK, 96100 Fama, Eugene F (1965) Portfolio Analysis in a Stable Paretian Market, Management Science, A, 11(3), 404419 Grau, Wolfdietrich (Ed.) (1999) Guidelines on Market Risk, Volume 5: Stress Testing, ệsterreichische Nationalbank (ệNB), 62 pages Hanley, M (1998) A Catastrophe Too Far, RISK Supplement on Insurance, July Hendricks, D (1996) Evaluation of Value-at-Risk Models Using Historical Data, Economic Policy Review, Federal Reserve Bank of New York, 2, 3969 Hopper, G (1996) Value at Risk: A New Methodology for Measuring Portfolio Risk, Business Review, Federal Reserve Bank of Philadelphia, JulyAugust, 1929 Hua P., and P Wilmott (1997) Crash Courses, RISK, June, 6467 Hull, J C., and A White (1987) The Pricing of Options on Assets with Stochastic Volatilities, Journal of Finance, 42(2), 281300 Hull, J C., and A White (1988) An Analysis of the Bias in Option Pricing Caused by Stochastic Volatility, Advances in Futures and Options Research, 3, 2761 Hull, J C., and A White (1998a) Value at Risk When Daily Changes in Market Variables Are Not Normally Distributed, Journal of Derivatives, 5(3), 919 Hull, J C., and A White (1998b) Incorporating Volatility Updating Into the Historical Simulation Method for Value at Risk, Journal of Risk, 1(1), 519 Jorion, Philippe (1997) Value at Risk: The New Benchmark for Controlling Market Risk, McGraw-Hill, New York, NY Ju, X., and N Pearson (1999) Using Value-at-Risk to Control Risk Taking: How Wrong Can You Be?, Journal of Risk, 1(2), 536 Kindleberger, Charles P (1996) Manias, Panics, and Crashes: A History of Financial Crises, John Wiley and Sons, New York, NY Koedijk, K G., M M Schafgans and G G de Vries (1990) The Tail Index of Exchange Rate Returns, Journal of International Economics, 29(1/2), 93108 Lauridsen, Sarah (2000) Estimation of Value at Risk by Extreme Value Methods, Extremes, 3(2), 107144 448 Financial risk management Litzenberger, R H., D R Beaglehole and C R Reynolds (1996) Assessing Catastrophe Reinsurance Linked Securities as a New Asset Class, Journal of Portfolio Management, 23(2), 7686 Longin, F M (1996) The Asymptotic Distribution of Extreme Value Returns, Journal of Business, 69(3), 383407 Lucas, A., and P Klaasen (1998) Extreme Returns, Downside Risk, and Optimal Asset Allocation, Journal of Portfolio Management, 25(1), 7179 Luciano, Elisa, and Marina Marena (2001) Value at Risk Bounds for Portfolios of Nonnormal Returns, Working Paper, April, University of Turin and International Center for Economic Research (ICER), Turin, Italy Mandelbrot, B B., and J R Wallis (1969) Some Long-Run Properties of Geophysical Records, Water Resources Research, 5(2), 321340 Markowitz, Harry M (1952) Portfolio Selection, Journal of Finance, 7(1), 7791 McCulloch, J Huston (1986) Simple Consistent Estimators of Stable Distribution Parameters, Communications in Statistics Computation and Simulation, 15, 11091136 McCulloch, J Huston (1996) Financial Applications of Stable Distributions, in Maddala, G S., and C R Rao (Eds), Statistical Methods in Finance (Handbook of Statistics), 14, Elsevier, Amsterdam, The Netherlands, pp 393425 McCulloch, J Huston (1997) Measuring Tail Thickness in Order to Estimate the Stable Index : A Critique, Journal of Business and Economic Statistics, 15(1), 7481 McDonald, Robert (2003) Derivatives Markets, Addison-Wesley, New York, NY McNeil, A J (1996), Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory, Mimeo, ETH Zentrum, Zỹrich, pp 56 McNeil, A J (1998), Calculating Quantile Risk Measures for Financial Return Series Using Extreme Value Theory, Mimeo, ETH Zentrum, Zỹrich, Switzerland Peters, Edgar E (1994) Fractal Market Analysis, John Wiley and Sons, New York, NY Rachev, Svetlozar, and Stefan Mittnik (2000) Stable Paretian Models in Finance, John Wiley and Sons, New York, NY Resnick, S (1987) Extreme Values, Regular Variation and Point Processes, Springer Verlag, New York, NY Samuelson, Paul, A (1967) Efcient Portfolio Selection for Pareto-Lộvy Investments, Journal of Financial and Quantitative Analysis, 2(2), 107122 Shafer, Glenn, and Vladimir Vovk (2001) Probability and Finance: Its Only A Game, John Wiley and Sons, New York, NY Sornette, D (1998) Large Deviations and Portfolio Optimization, Physica, A, 256, 251283 Sornette, D (2003) Why Stock Markets Crash: Critical Events in Complex Financial Systems, Princeton University Press, Princeton, NJ Whitcher, B., S D Byers, P Guttorp and D B Percival (2002) Testing for Homogeneity of Variance in Time Series: Long Memory, Wavelets and the Nile River, Water Resources Research, 38(5), 10291039 Wilson, Thomas C (1998) Value at Risk, Chapter in Alexander, Carol (Ed.) (1999), Risk Management and Analysis, Volume 1: Measuring and Modelling Financial Risk, John Wiley and Sons, New York, NY, 61124 Appendix A: original scaling in nancial economics Scale of a, b, c = negative changes of logarithm of price 0.001 0.01 0.1 1.0 1.0 Tail frequency b+ a+ c+ b a c 0.1 0.01 0.001 0.001 0.01 0.1 1.0 Scale of a+, b+, c+ = positive changes of logarithm of price Figure A.1 Mandelbrots original evidence for scaling in economic pricing processes Source: Reprinted from Mandelbrot 1963b by permission of The University of Chicago Press â 1963 by The University of Chicago Press The original evidence for scaling of the prices in the cotton market was produced by Mandelbrot (1963) and reproduced as Plate 340 in Mandelbrot (1982, p 340), as in Figure A.1 Mandelbrot advanced the hypothesis of an underlying stable distribution on the basis of the observed invariance of the return distribution across different frequencies and the apparent heavy tails of the cotton price distributions Bibliography Cootner, Paul H (Ed.) (1964) The Random Character of Stock Market Prices, The MIT Press, Cambridge, MA Mandelbrot, Benoit B (1963) The Variation of Certain Speculative Prices, The Journal of Business, 36, 394419 and 45, 1972, 542543 Mandelbrot, Benoit B (1982) The Fractal Geometry of Nature, W H Freeman and Co., New York, NY (updated and augmented version, 18th printing, 1999) Appendix B: S&P500 daily closing prices for 1988 The following time series of S&P500 daily closing prices are taken from table 2.7 in Sherry (1992), pp 2932 They form a test set of data to be used in some of the Chapter Exercises The Asian and Latin American FX and stock market data for Chapter are available from the author on CD at cost Daily 255.940 258.630 258.890 261.070 243.400 247.490 245.420 245.810 245.880 252.050 251.880 249.320 242.630 243.140 246.500 252.170 249.570 249.380 252.290 257.070 255.040 255.570 252.210 252.210 250.960 249.100 251.720 256.660 255.950 257.630 259.830 259.210 Obs# 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Daily 257.910 261.610 265.640 265.020 264.430 261.580 262.460 267.820 267.220 267.980 267.880 267.300 267.380 269.430 269.060 263.840 264.940 266.370 266.130 268.650 271.220 271.120 268.740 268.840 268.910 263.350 258.510 258.060 260.070 258.070 258.890 260.140 Obs# 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 Daily 258.510 265.490 266.160 269.430 270.160 271.370 271.570 259.750 259.770 259.210 257.920 256.130 256.420 260.140 262.460 263.930 263.800 262.610 261.330 261.560 263.000 260.320 258.790 257.480 256.540 257.620 253.310 253.850 256.780 258.710 255.390 251.350 Obs# 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 Daily 252.570 253.020 250.830 253.510 253.760 254.630 253.420 262.160 266.690 265.330 266.450 267.050 265.170 271.520 270.200 271.260 271.430 274.300 274.450 269.770 270.680 268.940 271.670 275.660 274.820 273.780 269.060 272.310 270.980 273.500 271.780 275.810 Obs# 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 Daily 272.020 271.780 270.020 270.550 267.850 269.320 270.260 272.050 270.510 268.470 270.000 266.660 263.500 264.680 265.190 262.500 266.020 272.020 272.210 272.060 272.980 271.930 271.150 269.980 266.490 261.900 262.750 262.550 258.690 260.560 260.770 261.030 Obs# 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 Figure B.1 S&P500 daily closing prices taken from table 2.7 in Sherry (1992), pp 2932 Obs# 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Daily 260.240 256.980 257.090 261.130 259.180 259.680 262.330 262.510 261.520 258.350 264.480 265.590 265.870 265.880 266.840 266.470 267.430 269.310 268.130 270.650 268.820 269.730 270.160 269.180 269.760 268.880 268.260 269.080 272.590 271.910 271.380 270.620 Obs# 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 Daily 271.860 272.390 278.070 278.240 277.930 273.980 275.220 275.500 276.410 279.380 276.970 282.880 283.660 282.280 282.380 281.380 277.288 278.530 278.970 279.060 279.060 279.200 276.310 273.930 275.150 273.330 273.690 267.920 267.720 268.340 263.820 264.600 Obs# 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 Daily 266.470 266.220 267.210 269.000 267.230 268.640 270.910 273.700 272.490 271.810 274.930 277.590 278.130 276.590 277.030 276.520 276.310 275.310 274.280 276.290 278.910 277.470 277.380 276.870 277.870 276.830 277.088 279.400 277.720 Index Accardi, Luigi 13 acoustic cash ow motions 368 acoustic spherical waves 411 adaptive wavelet methodology 411 adjustment vortices 262 afne models 294, 3526 algebra, or eld 13; operator 734 Amin, K.I 62 amplitude spectrum 149 aperiodic orbits 31730 Anselmet, F 394 arbitrage 3378, 348 Arbitrage Pricing Theory 106 ARCH processes 11112; statistical properties of 11215 ARFIMA models 107 ARIMA models Arithmetric Brownian Motion 50 ARMA models 103 Arnộodo, A 268, 275 Arnộodo et al Theorem 270 arrhythmias: of nancial markets 278; of the human heart 2535 Asian nancial crisis 36, 231, 263, 3267, 400, 427 asset return distribution 6, 62 Aswan Dam 435 asymptotic covariance matrix 94 atoms collision effect 414 attracting points, detection of 3278 attractor, fractal 3235 autocorrelation functions (ACFs) 10711, 1223, 136, 381 autocovariance functions 1635, 237 Available Cash Flow Risk 3467 Babbs, S.H 295 Bachelier, L 8, 52 backwardation, normal 656 Bacry, E 269, 275 bagpipe frequencies 1502 Baillie, R.T 115 Balduzzi, P.S.R 352 Bank of America 428 Bankers Trust 428 bank risk model approach 428 basins of attraction 326 basis, sinusoidal 174, 190; dyadic 208; orthonormal 205; Riesz 205; wavelet 208 Basle Accords 4289 Batten, Jonathan 78, 224 Bayes theorem 38 Beaglehole, D.R 295, 355 BelousovZhabotinsky reaction 327 Bendjoya, Ph 403 Benoit 1.3 Fractal System Analysis 128, 186, 224, 281, 3301 Bernoulli equation 3445 bifurcation 11, 32, 2936, 302, 3079, 329 billiard balls, motion of 31819 binomial cascade 276 binomial option pricing model 35 Birkhoffs theorem 40 Black, Fischer 34 black noise 126, 128, 400, 427, 443 BlackScholes pricing model 35, 59, 2925, 347, 407 blue noise 168, 254, 427 Bollerslev, T 11217 passim Box, G.E.P 136 brown noise 110, 1225, 168, 427 Burgers equation 404, 4068 Burrus, Sidney 223 business cycles 102 454 Index Capital Asset Pricing Model (CAPM) 80, 106, 341, 443 cascade models 389, 401 cash ow 3379; continuous 342; laminar and turbulent 339, 368, 370; perfect liquidity of 366; rate 341; steady 342; stress and strain in 3637; theory of 3407; see also discounted cash ows Cash Flow Risk (CFR) 3467 cash ow viscosity see illiquidity; coefcient of 365 cash ow vortices 359 cash risk equation see Bernoulli equation catastrophe, nancial 337, 340, 4257 Cauchy distribution 923, 440 CauchySchwarz inequality 243 Central Limit Theorem, Generalized (GCLT) 912, 440 certainty, almost 15 Chang, E.C 66 chaos 1112, 32, 272, 28999, 319, 3248; complete 31217, 329 Chapman, David A 338 characteristic functions 823, 86, 95, 173 Chemical and Chase Bank 428 Chichilnisky, G 340 chirps 177, 1812, 2012, 209, 242 Chi-squared distribution 2512 Citibank 428 Claerbout, Jon 281, 373 clarinet frequencies 1502 closure 206 cobweb processes 301 Coifman, Ronald 192 color categorization of randomness 1223 compact support property 222 complement, orthogonal 218 complexity, science of 28992, 295 conditional probability 389 cone of inuence 2446, 25960 condence levels 4289 conjugate mirror lter 215 Constantinides, G 355 contagion 262, 380, 400 Continuous Wavelet Transform (CWT) 192, 1956, 2445 contour dynamics 401 convolution theorems 1612 Cooley, J.W 160 correlation dimensions 272 cotango 66 covariance functions 13740, 163, 352 covariance as time convolution 141 Cowles, Alfred III 65 Cox, J 35, 352, 354 crash-o-phobia 28 crises, nancial 340; see also Asian nancial crisis cross-covariance functions 138 cumulants 1820, 378 cyclicity: aperiodic 1056, 155, 231, 234, 253, 292, 308, 314, 321, 3258; of nancial markets 5; of time series from economic models 102 Dacorogna, M.M 438 Daubechies, Ingrid 1801, 192, 209, 21423 passim, 242 day traders 342, 392, 400 delta-hedging 4323 dependence 41, 10511, 164; see also linear dependence; series dependence derivatives 3, 1213, 106, 356 Desnyanski, V.N 401 Devils staircase 2734 diffusion models 3526; wavelet solutions of 402414 dilation 193 dilation equations 214 dimension, Euclidean 266; Hausdorff 2667; information 272; correlation 272; skewness 278; kurtosis 273 discontinuous data 2314 discounted cash ows 348 discount factors 350 Discrete Wavelet Transforms (DWTs) 20711, 250 distribution, asset return 6; Cauchy 92; functions 17; Gaussian 92; heavy tailed 88; kurtosis of 21; Lộvy 93; location of 21; parametrized stable 82, 857; Pareto 88; scale of 26; scaling 76; skewness of 21; stable 812; symmetrically stable 81 diversication of portfolios 4424 domain of attraction 92 Doppler effect 390 double-entry bookkeeping 348 Dufe, D 355 Dusak, K 66 dyadic nesting 214 dyadic scaling 21314 dynamic inertia 32 Efcient Market Hypothesis 47, 512 Eidgenửssische Technische Hochschule (ETH) 2301 eigenfunctions 1734 Index 455 eigenvalues 173, 367 Elliott, Robert J 118 Ellis, Craig 224 Ellsberg paradox 1617 El Niủo 326 end effects 78 Engle, Robert F 111 environmental uncertainty 32930 equivalent time duration 172 ergodicity 3940, 51, 163, 371, 3812 Euclidian geometry 57 Eulers equation 412 Everson, R 395 exceedences, nancial 438 EXCEL 41, 330, 372, 414, 445 exogenous variables 36 expansion 205; wavelet 207 expansion sets 2056 exponential distribution 202 exponential functions 174 extreme values: scarcity of 4323; theory of 4379 extreme risk 4337 Fama, Eugene F 47, 49, 51, 94, 106 FamaSamuelson proposition 4414 familiar attractors 325 Farge, Marie 372, 383, 3956, 4001 Fast Fourier Transform 83, 160 Fast Wavelet Transform 209 Feigenbaum, M.J 3078 Feigenbaum diagram 295 Fickian scaling 76 g-tree plots 295 lter banks 208; high-pass 208; low-pass 208 nancial pressure, concept of 3446, 400 nancial risk spectrum 167 Flandrin, Patrick 237, 2503, 272 foreign exchange rates, volatility of 4267 Fourier, J.B.J 145 Fourier resonance coefcients 1439, 3078, 404 Fourier series 1423 Fourier spectra 1601 Fourier Transforms (FTs) 18, 82, 111, 155, 238, 241, 381; algebraic properties of 1589; for aperiodic variables 15669; see also Windowed Fourier Transforms fractal attractors see strange attractors fractal dimensions 26671 Fractal Market Hypothesis 47, 5664 fractal objects 578 fractal pricing processes 4404 Fractal System Analysis 128 Fractional Brownian Motion (FBM) 107, 11011, 11720, 124, 126, 135, 1589, 1678, 190, 2368, 264, 2689, 290; average risk spectrum of 1669; and Value-at-Risk 4389 fractional difference operators 745 fractionally differenced processes 107 frames of reference 147 frame theory 2045; tight 205; redundant 205 frequency convolution 1612 frequency spectra 135, 14952 Frisch, Uriel 236, 2689, 383, 393, 394 fundamental frequency and fundamental angular frequency 143 function, characteristic 18; joint characteristic 18; orthogonal 146; orthonormal 147; of rapid decay 157; regularly varying 110; of slow growth 157 Gỏbor, D 155, 175, 177, 179 Gỏbor atom 1757 Gỏbor Transform 174, 180; see also Windowed Fourier Transform Gỏbor wavelets 209 Galerkin nite elements methodology 338, 340, 348, 380, 403, 40814; wavelet solutions of 40314 gamma function 88, 1089, 139 (G)ARCH processes 11117 Gaussian distributions 2031 passim, 72, 440, 443 Geometric Brownian Motion (GBM) 47, 74, 135, 159, 236, 294, 338; timewarped 273, 2767 Gibbs partition function 269, 272, 394 Gibbs phenomenon 1434, 4038 Gilmore, R 292, 328 global dependence and global independence 10511, 164 Goldstein, R 355 Gonỗalvốs, Paulo 250, 269 Granger, C.W.J 102 Grassberger, P 308 Gray, R.V 66 Gregg, Mike 359 Grossman, Alex 192 Haar, Alfrộd 208, 21516, 220, 223 harmonic amplitudes 143 456 Index Hausdorff dimension 2667, 308, 325, 390, 3934 Heal, G.M 340 heartbeat 2535 Heath, D 352 Heaviside function 4046 heavy-tailed distributions 889, 93, 96, 316, 432 hedge funds 3378 hedging 3, 34, 65, 4323; dynamic 174 Heisenberg, Werner 40, 171, 180 Heisenberg box 177, 179, 198, 200, 202, 250 H-exponent 1208, 16873, 203, 224, 2389, 2778, 292, 2969, 314, 317, 433; homogeneous 268; of monofractal time series 25065 Hicks, John 656 high-pass lters 211 Hilbert space 204, 208, 212, 218, 243 holographs 175 Holton, Glyn A 53, 56 Hong Yan 338 horizon analysis 56; see also investment horizons Hosking, J.R.M 107 hotbits 10 Houthakker, H.S 65 Hull, J.C 27, 295, 352 Hurst, Harold Edwin 106, 1201, 433, 435; see also H-exponent Hurst noise 121 Hwang, W.L 2467 hydrology 340, 4335 Ibbotson, Roger G 34 idealization theorem 83 IGARCH models 112, 117 illiquidity 3378, 3627; coefcient of 365 Ilmanen, A 34 impulse functions 15960 incompressible nancial markets 343 inertial zone 381 infrared catastrophe 126 Ingersửll, J.E 354 institutional investors 342, 392, 400 intermittency 106, 195, 203, 2912, 30913, 326, 32930, 340, 346, 387; of nancial turbulence 3967; see also wavelet intermittency investment horizons 6, 2932, 538 passim, 106, 392, 4278 irregularity 222; degrees of 8, 238; measurement of 235; sequential irregularity exponents 1236 irrotational nancial markets 343 Itụs lemma 294, 353 Jaffards theorem 2437 James, Jessica 295, 355, 358 Jenkins, G.M 136 Jorion, Philippe 428 Joseph effect 106 JP Morgan (company) 428, 430 Julia set 578 Kalman lter 105 Kaplan, Lance M 252, 272 Karnosky, D.S 339 Karuppiah, Jeyanthi 263, 290 K-distribution 202 Kennedy, D.P 355 kernels 175, 412 Keynes, John Maynard 656 Koch snowake 267 Kolmogorov, A.N 1315, 323, 236, 279, 340, 3812, 38892, 3989 Kolmogorov distance 94 Kửrner, T.W 158 Kuo, C.-C Jay 252, 272 kurtosis 23, 29, 84, 2645, 398 kurtosis dimension 273 Kyaw, Nyonyo 2567 laminar cash ow 339, 368 laughter 1825 Legendre Transform 2701, 275, 390, 394, 399 lepto-kurtosis 23, 433 Lộvy, Paul 31, 80, 440, 441 Lộvy distribution 93 Li, T.-Y 312 Likelihood Function 94 linear combinations of assets and liabilities 348 linear dependence and linear independence 50, 138, 164 linear systems 73 line spectrum 149 Lipschitz exponent 8, 23543, 250, 268, 359 liquidity 35861; perfect 366 liquidity management 444 liquidity preference theory 46 Lo, Andrew W 121, 253, 290 logarithmic transformation of data 95 Index 457 logistic parabola regimes 291319, 32430 lognormal distribution 268, 31 Longstaff, F.A 352, 355 Long-Term Capital Management (LTCM) 291, 3378, 400 long-term dependent random processes 110 Los, Cornelis A 67, 13, 74, 263, 290, 33940, 349 low-pass lters 21011 Luenberger, David G 355 Lui, David T.W 60 McCulloch, Huston 94, 97, 445 MacKinlay, A Craig 121, 253, 290 Mallat, Stộphane 192, 20810, 21315, 2378, 243, 2467, 269, 275 Mallats Theorem 243 Mallat and Meyer MRA design 215 Mandelbrot, Benoit 30, 47, 52, 64, 78, 80, 89, 1067, 1201, 155, 236, 265, 278, 290, 321, 337, 340, 391, 393, 3989, 444 Mann, H.B Mantegna, Rosario N 116, 1701, 292, 392 market efciency, degree of 656 market-makers prots 4323 Markov processes 39, 1035, 242 Markowitz, Harry M 72, 4404 martingale theory 41, 4752, 106 MATLABđ Higher-Order Spectral Analysis (HOSA) Toolbox 152, 182, 186 MATLABđ Signal Processing Toolbox 152, 186 MATLABđ Statistics Toolbox 41 MATLABđ Wavelet Toolbox 224, 281, 373 maxima lines 2467, 269 meso-kurtosis 23 meteorological processes 299, 326, 340 Meyer, Yves 1912, 215, 394 Mindlin, G 292, 328 Mittnik, Stefan 92 mixing processes 1023 Modern Portfolio Theory 56, 440 modulus maximum 2467, 26970 moments: existence of 85; fractional absolute 89; generation of 1820; higher-order 5; of parametric distributions 2031; vanishing 194, 221, 26970 monofractal time series 26 Monte Carlo methods 252 Morgan Stanley 437 Morgenstern, O 102 Morlet, J 1912 Morlet wavelet 258 Morton, A.J 62 mother wavelets 219 MRA design 215 MRA equations 21317; for wavelets 219 MRA, Mallats 20910, 214 Mỹller, Ulrich A 72, 290 multifractality 64, 259, 26573, 27781 Multiresolution Analysis (MRA) 1912, 20720, 2301, 2368, 2902, 317, 328, 340, 3889, 3946, 407; design properties of systems 2213; of multifractal price series 26580; of turbulence 398402 musical frequencies 1502 Muzy, J.F 275 NavierStokes equations 236, 291, 295, 300, 31719, 330, 338, 340, 347, 381, 383, 3889, 397, 4012, 407, 41114 Nelson, C.R 3578 net present value 65 Newtonian cash ows 367 Newtons equation of state 3834 Newtons second law of ows 363 Nicolis, Gregoire 292 Noah effect 106 noise, color categorization of 89, 1228, 1678, 278, 299, 4267 Nolan, John 80, 947, 445 normal distribution norm of time series 2045 Novikov, E.A 401 October 19th, 1987 (stock market crash) 289 odds 17 oil reserves, geographical distribution of 27880 Olsen and Associates 230 operator 734; afne 75; rst difference 74; fractional difference 74; linear 73; time-invariant 173 OrnsteinUhlenbeck process 353 orthogonality 1469, 1935, 2057, 214, 21823 Osborne, A 272 oscillation sequences 3016 458 Index PanAgora Asset Management 290 parabolic deterministic processes 291 parametric and nonparametric distributions 1731, 4317 ParetoLộvy scaling distributions 7680 Parisi, G 236, 2689, 393, 394 Parsevals theorem 1489, 162, 197, 218, 2234, 381; formula 179; tiling 223 partition functions 26972 Pawelzik, K 290 Pearson, Neil D 338 perfectly efcient markets 343 period-doubling 3017 periodicity 142 periodic orbits 3078 Perrier, V 403 persistence and antipersistence 10911, 1212, 126, 262, 268, 289, 297, 337, 392, 397; in Asian markets 2635; in Latin American markets 25663; persistence regimes 295300, 4267 Peters, Edgar E 5, 16, 77, 85, 89, 121, 290, 337, 392, 3967, 400, 444 phase angles 143 phase spectrum 149 Philippine pesos 361 Pincus, Steve pink noise 89, 125, 262, 427 platy-kurtosis 23, 316 portfolio theory 72 power laws 768 power spectral density (PSD) 142, 162, 1645, 237 predictability of nancial series 32, 2901, 324, 330 Prigogine, Ilya 292 principal component analysis 352 probabilistic risk 17 probability: chaotic 3267; concept of 78, 1213; conditional 389; denition of 1315; fractionally differenced 107; (G)ARCH 111; long-term dependent 110 process, rst- and second-order Markov 103; stochastic 32; strong-mixing 103 Provenzale, A 272 pseudorandom numbers 1012 put-call parity 61 quadratic map see logistic parabola regimes Rachev, Svetlozar 92 radiation monitors 10 Ramsey, James B 193 random eld models 355 randomness: color categorization of 1223; concept of 78, 15; degrees of 238; genuine and pseudo 812 random variable, axiomatic denition 15; independent 49; uncorrelated 50 Random Walk process 506, 74, 76, 255, 399, 400 Range/Scale analysis 1202 rate of return, total 53 Rayleigh distribution 202 recurrence plots 327 red noise 124, 168 regular time series 238 regularity conditions 241 Reproducible Research 224, 281, 373 Reynolds number 346, 381, 3889, 399; nancial 369; wavelet 371 Richardson, L.F 339, 389 Riesz basis 205, 214 risk: average 164; denition of 59; dependence on asset class 34, 55; extreme 71; local measures of 174, 179, 200, 23750, 2659, 317, 344, 371; long-term 1356; market price of 3523; spectrum 164, 269; systematic risk contents 149, 1625, 174, 2045 risk-free assets risk spectra 1659, 203, 269; examples of 2738 Roll, R 94 Ross, S.A 354 R/S measure 106 -additive 14 -algebra(s) 43; current of 32 Sadourney, R 395 Samuelson, Paul A 51; see also FamaSamuelson proposition Santa-Clara, P 355 scalar product 137 scaling functions 21023 passim; coefcients of 21417, 270; in relation to wavelets 21720 scalegram 202, 250; of heart arrhythmias 2535; of stock prices 2556 scalograms 136, 181, 186, 192, 196204, 252, 25866 passim, 31617, 328, 3712 Schroeder, Manfred 7, 126, 293, 396 Schwartz, E.S 352 Index 459 Schwert, G.W self-organization 3017 self-similarity 2378, 2656, 3078, 325, 389, 397 semi-logarithmic plots 170 series dependence 1025 set function 14 Sharkovskii, A.N 313 Sharkovskii Theorem 313 shearing 3634 Short-Term Fourier Transforms 199 Shuster, H 290 Siegel, A.F 3578 Singer, B 8, 339 single-index models 35 singularities 15, 52, 185, 1967, 2338, 247, 265, 26972, 326, 398; measurement of 24650 singularity spectra 127 Sinqueeld, Rex A 34 skewness 23, 31, 273 Slộzak, E 403 Sứrensen, C 352 Sornette, D 355 span of a basis set 206, 212 space, complete 14; measurable 13; probability 14 spectral analysis 102, 105, 3078, 367 spectral bandwidth, equivalent 172 spectral density and spectral exponent 111 spectrograms 136, 142, 155, 17986, 192, 2012 spectrum, amplitude and phase of 149, 160; multifractal 268; risk 164; support of 268 speculation 3, 65, 427 speech 1823 splines 357 spot rate curve see term structure of interest rates stability spectra 1268 stable distributions 31, 712, 803, 96; closed form 923; examples of 86; general properties of 835; Value-at-Risk for 42831 STABLE.EXE software 445 Stanley, H Eugene 116, 1701, 292, 392 stationarity, concept of 33, 76; in the strict sense 33; in the wide sense 34 stationary processes 338 steady states 298306, 311, 31416, 320, 325 steady-state equilibria 32830 Stevens, Peter S 385 stochastic string models 355 strange attractors 12, 279, 2923, 3236 stress testing 432 summary statistics 18 symmetric scaling functions and wavelets 2223 system, denition of 73; dynamic 32; invertible 73; linear 73; MRA 221; nonlinear 316323; time-invariant 73 tail properties of distributions 88 Tang, Gordon Y.N 60 tangent bifurcation 306, 309 tapers 157 Taylor expansion 23840 Telser, L.G 66 Tenney, M.S 295, 355 tensor algebra 339, 349 term structure gradient (TSG) 364 term structure of interest rates 34951; parametrized models of 3518 Three Gorges Dam 4337 Tice, J 294, 355 time convolution 1401, 161 time dependence 323, 56 time duration, equivalent 172 time-invariant systems 73 time warping 230, 273, 2768 Tobin, James 4, 6, 444 Toon scripts 224, 281, 373 total rate of return on investment 53 trading time 77 transient chaos 203 translation 193 Tukey, J.W 160 turbulence 195, 236, 28992, 326, 33740, 346, 36870, 3803; advanced research of 4002; denition of 3589; nancial, informal theory of 3878; homogeneous or monofractal 38993; heterogeneous or multifractal 3936; measurement and simulation of 38895; multifractals in modeling of 27881; Multiresolution Analysis of 398402; physical, informal theory of 3837; simulation of 3968; wavelet representation of 402 ultra-unstable distributions 128 uncertainty 7, 17 Uncertainty Principle 7, 1717, 180, 192, 198 universal order of period lengths 31314 460 Index Urbach, Richard 290, 292 Value-at-Risk (VaR) of investment 425; denition of 429; and fractal pricing processes 4404; and Fractional Brownian Motion 4389; for parametric distributions 4317; for stable distributions 42831 van den Hoek, John 118 Van Ness, John W 107 Vasicek, O.A 2945, 3523 velocity elds 389 Venturi cash ow channel 345 Verhulst, Pierre-Franỗois 293 violin frequencies 1502 viscosity of cash ow 3625 visualization plots 196 volatility matrices 60; denition of 53; Normalized Random Walk 55; portfolio return 3; smile 247; stochastic 441; swaps 64; time dependence of vortices 298, 317, 359, 380, 38593, 400 Wald, A Wallis, J.R 121 Wavelab 224, 281, 373 wavelet, Daubechies 217, 220; Gỏbor 209; Gaussian 219; Haar 208; orthogonal 195; orthonormal 195 triangle 209, 220 wavelet analysis 1902, 325, 338; atom 193; regularity conditions 2412; in relation to scaling functions 21720; resonance coefcients 212; of transient pricing 192209; usefulness of 223 wavelet decomposition tree 210 wavelet expansions 2067 wavelet lters 210 wavelet nancial Reynolds number 371 wavelet generation coefcients 21920 wavelet intermittency 3712; nancial 371 wavelet Multiresolution Analysis see Multiresolution Analysis wavelet resonance coefcients 21921, 243, 2508 wavelet spectrum, nancial 371 Wavelet Transforms 157, 1745, 238, 2427, 2659, 383 Webber, Nick 2945, 355, 358 weighted mixtures 7880 White, A 27, 295, 352 white noise 1668, 186 Wickerhauser, Victor 192 WienerKhinchin Theorem 142, 1623, 167 Windowed Fourier Transforms (WFTs) 1556, 17386, 199, 328; see also Gỏbor Transform Yangtze River 4336 Yorke, J.A 312 Zabusky, Norman 395 zero coupon bonds 3505 passim Zhang, Zhifeng 193 Zimin, V 401 Zolotarev parameterization 802, 857, 91, 123, 437, 440 .. .Financial Market Risk What is financial market risk? How is it measured and analyzed? Is all financial market risk dangerous? If not, which risk is hedgeable? These questions,... distributions of financial market risk we actually measure? (3) Is all financial market risk dangerous or can we distinguish between “safe” financial market risk and “dangerous” financial market risk? For example,... basic understanding of financial market risk, we must ask at least four fundamental questions: (1) What is financial market risk? (2) How we measure financial market risk? For example, which frequency