P1: TIX/OSW P2: c - leseprobe.buch.de Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. Fabozzi ... Collateralized Debt Obligations: Structures and Analysis,

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Financial Models withLevy Processes andVolatility Clustering

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The Frank J. Fabozzi SeriesFixed Income Securities, Second Edition by Frank J. FabozziFocus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. AbateHandbook of Global Fixed Income Calculations by Dragomir KrginManaging a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. FabozziReal Options and Option-Embedded Securities by William T. MooreCapital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. FabozziThe Exchange-Traded Funds Manual by Gary L. GastineauProfessional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. FabozziInvesting in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia PilarinuHandbook of Alternative Assets by Mark J. P. AnsonThe Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad ChoudhryThe Handbook of Financial Instruments edited by Frank J. FabozziInterest Rate, Term Structure, and Valuation Modeling edited by Frank J. FabozziInvestment Performance Measurement by Bruce J. FeibelThe Handbook of Equity Style Management edited by T. Daniel Coggin and Frank J. FabozziThe Theory and Practice of Investment Management edited by Frank J. Fabozzi and Harry M. MarkowitzFoundations of Economic Value Added, Second Edition by James L. GrantFinancial Management and Analysis, Second Edition by Frank J. Fabozzi and Pamela P. PetersonMeasuring and Controlling Interest Rate and Credit Risk, Second Edition by Frank J. Fabozzi, Steven V. Mann, and

Moorad ChoudhryProfessional Perspectives on Fixed Income Portfolio Management, Volume 4 edited by Frank J. FabozziThe Handbook of European Fixed Income Securities edited by Frank J. Fabozzi and Moorad ChoudhryThe Handbook of European Structured Financial Products edited by Frank J. Fabozzi and Moorad ChoudhryThe Mathematics of Financial Modeling and Investment Management by Sergio M. Focardi and Frank J. FabozziShort Selling: Strategies, Risks, and Rewards edited by Frank J. FabozziThe Real Estate Investment Handbook by G. Timothy Haight and Daniel SingerMarket Neutral Strategies edited by Bruce I. Jacobs and Kenneth N. LevySecurities Finance: Securities Lending and Repurchase Agreements edited by Frank J. Fabozzi and Steven V. MannFat-Tailed and Skewed Asset Return Distributions by Svetlozar T. Rachev, Christian Menn, and Frank J. FabozziFinancial Modeling of the Equity Market: From CAPM to Cointegration by Frank J. Fabozzi, Sergio M. Focardi, and

Petter N. KolmAdvanced Bond Portfolio Management: Best Practices in Modeling and Strategies edited by Frank J. Fabozzi, Lionel

Martellini, and Philippe PriauletAnalysis of Financial Statements, Second Edition by Pamela P. Peterson and Frank J. FabozziCollateralized Debt Obligations: Structures and Analysis, Second Edition by Douglas J. Lucas, Laurie S. Goodman, and

Frank J. FabozziHandbook of Alternative Assets, Second Edition by Mark J. P. AnsonIntroduction to Structured Finance by Frank J. Fabozzi, Henry A. Davis, and Moorad ChoudhryFinancial Econometrics by Svetlozar T. Rachev, Stefan Mittnik, Frank J. Fabozzi, Sergio M. Focardi, and Teo JasicDevelopments in Collateralized Debt Obligations: New Products and Insights by Douglas J. Lucas, Laurie S. Goodman,

Frank J. Fabozzi, and Rebecca J. ManningRobust Portfolio Optimization and Management by Frank J. Fabozzi, Petter N. Kolm, Dessislava A. Pachamanova, and

Sergio M. FocardiAdvanced Stochastic Models, Risk Assessment, and Portfolio Optimizations by Svetlozar T. Rachev, Stogan V. Stoyanov,

and Frank J. FabozziHow to Select Investment Managers and Evaluate Performance by G. Timothy Haight, Stephen O. Morrell, and Glenn

E. RossBayesian Methods in Finance by Svetlozar T. Rachev, John S. J. Hsu, Biliana S. Bagasheva, and Frank J. FabozziThe Handbook of Municipal Bonds edited by Sylvan G. Feldstein and Frank J. FabozziSubprime Mortgage Credit Derivatives by Laurie S. Goodman, Shumin Li, Douglas J. Lucas, Thomas A. Zimmerman,

and Frank J. FabozziIntroduction to Securitization by Frank J. Fabozzi and Vinod KothariStructured Products and Related Credit Derivatives edited by Brian P. Lancaster, Glenn M. Schultz, and Frank J. FabozziHandbook of Finance: Volume I: Financial Markets and Instruments edited by Frank J. FabozziHandbook of Finance: Volume II: Financial Management and Asset Management edited by Frank J. FabozziHandbook of Finance: Volume III: Valuation, Financial Modeling, and Quantitative Tools edited by Frank J. FabozziFinance: Capital Markets, Financial Management, and Investment Management by Frank J. Fabozzi and Pamela Peterson-

DrakeActive Private Equity Real Estate Strategy edited by David J. LynnFoundations and Applications of the Time Value of Money by Pamela Peterson-Drake and Frank J. FabozziLeveraged Finance: Concepts, Methods, and Trading of High-Yield Bonds, Loans, and Derivatives by Stephen Antczak,

Douglas Lucas, and Frank J. FabozziModern Financial Systems: Theory and Applications by Edwin NeaveInstitutional Investment Management: Equity and Bond Portfolio Strategies and Applications by Frank J. FabozziQuantitative Equity Investing: Techniques and Strategies by Frank J. Fabozzi, Sergio M. Focardi, and Petter N. KolmSimulation and Optimization in Finance: Modeling with MATLAB, @Risk, or VBA by Dessislava A. Pachamanova and

Frank J. Fabozzi

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Financial Models withLevy Processes andVolatility Clustering

SVETLOZAR T. RACHEVYOUNG SHIN KIM

MICHELE LEONARDO BIANCHIFRANK J. FABOZZI

John Wiley & Sons, Inc.

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Copyright C© 2011 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Limit of Liability/Disclaimer of Warranty: While the publisher and author have used theirbest efforts in preparing this book, they make no representations or warranties with respect tothe accuracy or completeness of the contents of this book and specifically disclaim any impliedwarranties of merchantability or fitness for a particular purpose. No warranty may be createdor extended by sales representatives or written sales materials. The advice and strategiescontained herein may not be suitable for your situation. You should consult with aprofessional where appropriate. Neither the publisher nor author shall be liable for any loss ofprofit or any other commercial damages, including but not limited to special, incidental,consequential, or other damages.

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Library of Congress Cataloging-in-Publication Data:

Financial models with Levy processes and volatility clustering / Svetlozar T. Rachev . . . [et al.].p. cm.—(The Frank J. Fabozzi series)

Includes index.ISBN 978-0-470-48235-3 (cloth); 978-0-470-93716-7 (ebk);978-0-470-93726-6 (ebk); 978-1-118-00670-2 (ebk)

1. Capital assets pricing model. 2. Levy processes. 3. Finance—Mathematical models.4. Probabilities. I. Rachev, S. T. (Svetlozar Todorov)

HG4637.F56 2011332′.0415015192—dc22 2010033299

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

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STRTo my grandchildren Iliana, Zoya, and Svetlozar

YSKTo my wife Myung-Ja and my son Minseob

MLBTo my wife Giorgia

FJFTo my wife Donna and my children Francesco,

Patricia, and Karly

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Contents

Preface xv

About the Authors xix

CHAPTER 1Introduction 11.1 The Need for Better Financial Modeling of Asset Prices 11.2 The Family of Stable Distribution and Its Properties 5

1.2.1 Parameterization of the Stable Distribution 51.2.2 Desirable Properties of the Stable Distributions 71.2.3 Considerations in the Use of the Stable

Distribution 81.3 Option Pricing with Volatility Clustering 9

1.3.1 Non-Gaussian GARCH Models 111.4 Model Dependencies 121.5 Monte Carlo 131.6 Organization of the Book 14

References 15

CHAPTER 2Probability Distributions 192.1 Basic Concepts 192.2 Discrete Probability Distributions 20

2.2.1 Bernoulli Distribution 212.2.2 Binomial Distribution 212.2.3 Poisson Distribution 22

2.3 Continuous Probability Distributions 222.3.1 Probability Distribution Function,

Probability Density Function, andCumulative Distribution Function 23

2.3.2 Normal Distribution 262.3.3 Exponential Distribution 282.3.4 Gamma Distribution 28

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viii CONTENTS

2.3.5 Variance Gamma Distribution 292.3.6 Inverse Gaussian Distribution 30

2.4 Statistic Moments and Quantiles 302.4.1 Location 312.4.2 Dispersion 312.4.3 Asymmetry 312.4.4 Concentration in Tails 322.4.5 Statistical Moments 322.4.6 Quantiles 342.4.7 Sample Moments 35

2.5 Characteristic Function 352.6 Joint Probability Distributions 39

2.6.1 Conditional Probability 392.6.2 Joint Probability Distribution Defined 402.6.3 Marginal Distribution 412.6.4 Dependence of Random Variables 412.6.5 Covariance and Correlation 422.6.6 Multivariate Normal Distribution 432.6.7 Elliptical Distributions 462.6.8 Copula Functions 47

2.7 Summary 54References 54

CHAPTER 3Stable and Tempered Stable Distributions 573.1 α-Stable Distribution 58

3.1.1 Definition of an α-Stable Random Variable 583.1.2 Useful Properties of an α-Stable Random Variable 613.1.3 Smoothly Truncated Stable Distribution 63

3.2 Tempered Stable Distributions 653.2.1 Classical Tempered Stable Distribution 653.2.2 Generalized Classical Tempered Stable

Distribution 683.2.3 Modified Tempered Stable Distribution 693.2.4 Normal Tempered Stable Distribution 703.2.5 Kim-Rachev Tempered Stable Distribution 733.2.6 Rapidly Decreasing Tempered Stable Distribution 75

3.3 Infinitely Divisible Distributions 763.3.1 Exponential Moments 80

3.4 Summary 82


Contents ix

3.5 Appendix 823.5.1 The Hypergeometric Function 833.5.2 The Confluent Hypergeometric Function 83References 84

CHAPTER 4Stochastic Processes in Continuous Time 874.1 Some Preliminaries 884.2 Poisson Process 88

4.2.1 Compounded Poisson Process 894.3 Pure Jump Process 89

4.3.1 Gamma Process 924.3.2 Inverse Gaussian Process 924.3.3 Variance Gamma Process 924.3.4 α-Stable Process 934.3.5 Tempered Stable Process 94

4.4 Brownian Motion 954.4.1 Arithmetic Brownian Motion 994.4.2 Geometric Brownian Motion 99

4.5 Time-Changed Brownian Motion 1004.5.1 Variance Gamma Process 1014.5.2 Normal Inverse Gaussian Process 1024.5.3 Normal Tempered Stable Process 103

4.6 Levy Process 1044.7 Summary 105

References 106

CHAPTER 5Conditional Expectation and Change of Measure 1075.1 Events, σ -Fields, and Filtration 1075.2 Conditional Expectation 1095.3 Change of Measures 111

5.3.1 Equivalent Probability Measure 1115.3.2 Change of Measure for Continuous-Time

Processes 1135.3.3 Change of Measure in Tempered Stable Processes 117


CHAPTER 6Exponential Levy Models 1236.1 Exponential Levy Models 123


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6.2 Fitting α-Stable and Tempered Stable Distributions 1266.2.1 Fitting the Characteristic Function 1266.2.2 Maximum Likelihood Estimation with

Numerical Approximation of the Density Function 1276.2.3 Assessing the Goodness of Fit 127

6.3 Illustration: Parameter Estimation for TemperedStable Distributions 131

6.4 Summary 1356.5 Appendix: Numerical Approximation of Probability

Density and Cumulative Distribution Functions 1356.5.1 Numerical Method for the Fourier Transform 139References 140

CHAPTER 7Option Pricing in Exponential Levy Models 1417.1 Option Contract 1417.2 Boundary Conditions for the Price of an Option 1427.3 No-Arbitrage Pricing and Equivalent Martingale Measure 1457.4 Option Pricing under the Black-Scholes Model 1487.5 European Option Pricing under Exponential

Tempered Stable Models 1497.5.1 Illustration: Implied Volatility 1527.5.2 Illustration: Calibrating Risk-Neutral Parameters 1537.5.3 Illustration: Calibrating Market Parameters

and Risk-Neutral Parameters Together 1617.6 Subordinated Stock Price Model 164

7.6.1 Stochastic Volatility Levy Process Model 1667.7 Summary 167

References 167

CHAPTER 8Simulation 1698.1 Random Number Generators 170

8.1.1 Uniform Distributions 1708.1.2 Discrete Distributions 1728.1.3 Continuous Nonuniform Distributions 1728.1.4 Simulation of Particular Distributions 177

8.2 Simulation Techniques for Levy Processes 1828.2.1 Taking Care of Small Jumps 1838.2.2 Series Representation: A General Framework 1868.2.3 Rosinsky Rejection Method 1918.2.4 α-Stable Processes 192


Contents xi

8.3 Tempered Stable Processes 1938.3.1 Kim-Rachev Tempered Stable Case 1968.3.2 Classical Tempered Stable Case 198

8.4 Tempered Infinitely Divisible Processes 1998.4.1 Rapidly Decreasing Tempered Stable Case 2018.4.2 Modified Tempered Stable Case 202

8.5 Time-Changed Brownian Motion 2038.5.1 Classical Tempered Stable Processes 2058.5.2 Variance Gamma and Skewed Variance

Gamma Processes 2068.5.3 Normal Tempered Stable Processes 2078.5.4 Normal Inverse Gaussian Processes 208

8.6 Monte Carlo Methods 2098.6.1 Variance Reduction Techniques 2108.6.2 A Nonparametric Monte Carlo Method 2148.6.3 A Monte Carlo Example 216Appendix 217References 220

CHAPTER 9Multi-Tail t-Distribution 2259.1 Introduction 2259.2 Principal Component Analysis 227

9.2.1 Principal Component Tail Functions 2289.2.2 Density of a Multi-Tail t Random Variable 231

9.3 Estimating Parameters 2329.3.1 Estimation of the Dispersion Matrix 2339.3.2 Estimation of the Parameter Set � 233

9.4 Empirical Results 2379.4.1 Comparison to Other Models 2379.4.2 Two-Dimensional Analysis 2389.4.3 Multi-Tail t Model Check for the DAX 242


CHAPTER 10Non-Gaussian Portfolio Allocation 24710.1 Introduction 24710.2 Multifactor Linear Model 24810.3 Modeling Dependencies 25110.4 Average Value-at-Risk 25310.5 Optimal Portfolios 255


xii CONTENTS

10.6 The Algorithm 25710.7 An Empirical Test 25910.8 Summary 268

References 269

CHAPTER 11Normal GARCH models 27111.1 Introduction 27111.2 GARCH Dynamics with Normal Innovation 27211.3 Market Estimation 27511.4 Risk-Neutral Estimation 278

11.4.1 Out-of-Sample Performance 28211.5 Summary 285

References 285

CHAPTER 12Smoothly Truncated Stable GARCH Models 28712.1 Introduction 28712.2 A Generalized NGARCH Option Pricing Model 28812.3 Empirical Analysis 291

12.3.1 Results under the Objective Probability Measure 29212.3.2 Explaining S&P 500 Option Prices 296


CHAPTER 13Infinitely Divisible GARCH Models 30913.1 Stock Price Dynamic 31113.2 Risk-Neutral Dynamic 31213.3 Non-Normal Infinitely Divisible GARCH 315

13.3.1 Classical Tempered Stable Model 31513.3.2 Generalized Tempered Stable Model 31713.3.3 Kim-Rachev Model 31913.3.4 Rapidly Decreasing Tempered Stable Model 32213.3.5 Inverse Gaussian Model 32413.3.6 Skewed Variance Gamma Model 32613.3.7 Normal Inverse Gaussian Model 329

13.4 Simulate Infinitely Divisible GARCH 331Appendix 332References 334


Contents xiii

CHAPTER 14Option Pricing with Monte Carlo Methods 33714.1 Introduction 33714.2 Data Set 338

14.2.1 Market Estimation 33914.3 Performance of Option Pricing Models 346

14.3.1 In-Sample 34614.3.2 Out-of-Sample 352


CHAPTER 15American Option Pricing with Monte Carlo Methods 35715.1 American Option Pricing in Discrete Time 35815.2 The Least Squares Monte Carlo Method 35915.3 LSM Method in GARCH Option Pricing Model 36415.4 Empirical Illustration 36515.5 Summary 372

References 372

Index 373

Documents

P1: TIX/OSW P2: c - leseprobe.buch.de Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. Fabozzi ... Collateralized Debt Obligations: Structures and Analysis,