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  • 1. Springer Finance Editorial Board M. Avellaneda G. Barone-Adesi M. Broadie M.H.A. Davis E. Derman C. Klppelberg E. Kopp W. Schachermayer
  • 2. Springer Finance Springer Finance is a programme of books aimed at students, academics and practitioners working on increasingly technical approaches to the analysis of nancial markets. It aims to cover a variety of topics, not only mathematical nance but foreign exchanges, term structure, risk management, portfolio theory, equity derivatives, and nancial economics. M. Ammann, Credit Risk Valuation: Methods, Models, and Application (2001) K. Back, A Course in Derivative Securities: Introduction to Theory and Computation (2005) E. Barucci, Financial Markets Theory. Equilibrium, Efciency and Information (2003) T.R. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and Hedging (2002) N.H. Bingham and R. Kiesel, Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives (1998, 2nd ed. 2004) D. Brigo and F. Mercurio, Interest Rate Models: Theory and Practice (2001) R. Buff, Uncertain Volatility Models-Theory and Application (2002) R.A. Dana and M. Jeanblanc, Financial Markets in Continuous Time (2002) G. Deboeck and T. Kohonen (Editors), Visual Explorations in Finance with Self-Organizing Maps (1998) R.J. Elliott and P.E. Kopp, Mathematics of Financial Markets (1999, 2nd ed. 2005) H. Geman, D. Madan, S.R. Pliska and T. Vorst (Editors), Mathematical Finance- Bachelier Congress 2000 (2001) M. Gundlach, F. Lehrbass (Editors), CreditRisk+ in the Banking Industry (2004) B.P. Kellerhals, Asset Pricing (2004) Y.-K. Kwok, Mathematical Models of Financial Derivatives (1998) M. Klpmann, Irrational Exuberance Reconsidered (2004) P. Malliavin and A. Thalmaier, Stochastic Calculus of Variations in Mathematical Finance (2005) A. Meucci, Risk and Asset Allocation (2005) A. Pelsser, Efcient Methods for Valuing Interest Rate Derivatives (2000) J.-L. Prigent, Weak Convergence of Financial Markets (2003) B. Schmid, Credit Risk Pricing Models (2004) S.E. Shreve, Stochastic Calculus for Finance I (2004) S.E. Shreve, Stochastic Calculus for Finance II (2004) M. Yor, Exponential Functionals of Brownian Motion and Related Processes (2001) R. Zagst, Interest-Rate Management (2002) Y.-L. Zhu, X. Wu, I.-L. Chern, Derivative Securities and Difference Methods (2004) A. Ziegler, Incomplete Information and Heterogeneous Beliefs in Continuous-time Finance (2003) A. Ziegler, A Game Theory Analysis of Options (2004)
  • 3. Attilio Meucci Risk and Asset Allocation With 141 Figures 123
  • 4. Attilio Meucci Lehman Brothers, Inc. 745 Seventh Avenue New York, NY 10019 USA e-mail: attilio meucci@symmys.com Mathematics Subject Classication (2000): 15-xx, 46-xx, 62-xx, 65-xx, 90-xx JEL Classication: C1, C3, C4, C5, C6, C8, G0, G1 Library of Congress Control Number: 2005922398 ISBN-10 3-540-22213-8 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-22213-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands Scientic WorkPlace is a trademark of MacKichan Software, Inc. and is used with permission. MATLAB is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This books use or discussion of MATLAB software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB software. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production, Heidelberg Cover illustration: courtesy of Linda Gaylord Typesetting by the author Printed on acid-free paper 41/sz - 5 4 3 2 1 0
  • 5. to my true love, should she come
  • 6. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV Audience and style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII Structure of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII A guided tour by means of a simplistic example . . . . . . . . . . . . . XIX Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XXVI Part I The statistics of asset allocation 1 Univariate statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Building blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Summary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.3 Higher-order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.4 Graphical representations . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Taxonomy of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.2 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.3 Cauchy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.4 Student t distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.5 Lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.6 Gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.7 Empirical distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.T Technical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . www 2 Multivariate statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 Building blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2 Factorization of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.1 Marginal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.2 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .www1.E Exercises . . .. . . . . . . .
  • 7. VIII Contents 2.3 Dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4 Shape summary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4.1 Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.4.3 Location-dispersion ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . 54 2.4.4 Higher-order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.5 Dependence summary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5.1 Measures of dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5.2 Measures of concordance . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.5.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.6 Taxonomy of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.6.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.6.2 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.6.3 Student t distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.6.4 Cauchy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.6.5 Log-distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.6.6 Wishart distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.6.7 Empirical distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.6.8 Order statistics . . . . . . . . .

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