Copula Methods in Finance

Umberto CherubiniElisa Luciano

and

Walter Vecchiato

Innodata0470863455.jpg

Copula Methods in Finance

Wiley Finance Series

Investment Risk ManagementYen Yee Chong

Understanding International Bank RiskAndrew Fight

Global Credit Management: An Executive SummaryRon Wells

Currency OverlayNeil Record

Fixed Income Strategy: A Practitioners Guide to Riding the CurveTamara Mast Henderson

Active Investment ManagementCharles Jackson

Option TheoryPeter James

The Simple Rules of Risk: Revisiting the Art of Risk ManagementErik Banks

Capital Asset Investment: Strategy, Tactics and ToolsAnthony F. Herbst

Brand AssetsTony Tollington

Swaps and other DerivativesRichard Flavell

Currency Strategy: A Practitioners Guide to Currency Trading, Hedging and ForecastingCallum Henderson

The Investors Guide to Economic FundamentalsJohn Calverley

Measuring Market RiskKevin Dowd

An Introduction to Market Risk ManagementKevin Dowd

Behavioural FinanceJames Montier

Asset Management: Equities DemystifiedShanta Acharya

An Introduction to Capital Markets: Products, Strategies, ParticipantsAndrew M. Chisholm

Hedge Funds: Myths and LimitsFrancois-Serge Lhabitant

The Managers Concise Guide to RiskJihad S. Nader

Securities Operations: A Guide to Trade and Position ManagementMichael Simmons

Modeling, Measuring and Hedging Operational RiskMarcelo Cruz

Monte Carlo Methods in FinancePeter Jackel

Building and Using Dynamic Interest Rate ModelsKen Kortanek and Vladimir Medvedev

Structured Equity Derivatives: The Definitive Guide to Exotic Options and Structured NotesHarry Kat

Advanced Modelling in Finance Using Excel and VBAMary Jackson and Mike Staunton

Operational Risk: Measurement and ModellingJack King

Advanced Credit Risk Analysis: Financial Approaches and Mathematical Models to Assess, Price and Manage Credit RiskDidier Cossin and Hugues Pirotte

Risk Management and Analysis vol. 1: Measuring and Modelling Financial RiskCarol Alexander (ed.)

Risk Management and Analysis vol. 2: New Markets and ProductsCarol Alexander (ed.)

Copula Methods in Finance

Umberto CherubiniElisa Luciano

and

Walter Vecchiato

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Library of Congress Cataloging-in-Publication Data

Cherubini, Umberto.Copula methods in finance / Umberto Cherubini, Elisa Luciano, and Walter Vecchiato.

p. cm.ISBN 0-470-86344-7 (alk. paper)1. FinanceMathematical models. I. Luciano, Elisa. II. Vecchiato, Walter. III. Title.

HG106.C49 2004332.01519535 dc22

2004002624

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0-470-86344-7

Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, IndiaPrinted and bound in Great Britain by TJ International, Padstow, Cornwall, UKThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.

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Contents

Preface xi

List of Common Symbols and Notations xv

1 Derivatives Pricing, Hedging and Risk Management: The State of the Art 1

1.1 Introduction 11.2 Derivative pricing basics: the binomial model 2

1.2.1 Replicating portfolios 31.2.2 No-arbitrage and the risk-neutral probability measure 31.2.3 No-arbitrage and the objective probability measure 41.2.4 Discounting under different probability measures 51.2.5 Multiple states of the world 6

1.3 The BlackScholes model 71.3.1 Itos lemma 81.3.2 Girsanov theorem 91.3.3 The martingale property 111.3.4 Digital options 12

1.4 Interest rate derivatives 131.4.1 Affine factor models 131.4.2 Forward martingale measure 151.4.3 LIBOR market model 16

1.5 Smile and term structure effects of volatility 181.5.1 Stochastic volatility models 181.5.2 Local volatility models 191.5.3 Implied probability 20

1.6 Incomplete markets 211.6.1 Back to utility theory 221.6.2 Super-hedging strategies 23

1.7 Credit risk 271.7.1 Structural models 281.7.2 Reduced form models 311.7.3 Implied default probabilities 33

vi Contents

1.7.4 Counterparty risk 361.8 Copula methods in finance: a primer 37

1.8.1 Joint probabilities, marginal probabilities and copula functions 381.8.2 Copula functions duality 391.8.3 Examples of copula functions 391.8.4 Copula functions and market comovements 411.8.5 Tail dependence 421.8.6 Equity-linked products 431.8.7 Credit-linked products 44

2 Bivariate Copula Functions 49

2.1 Definition and properties 492.2 Frechet bounds and concordance order 522.3 Sklars theorem and the probabilistic interpretation of copulas 56

2.3.1 Sklars theorem 562.3.2 The subcopula in Sklars theorem 592.3.3 Modeling consequences 602.3.4 Sklars theorem in financial applications: toward a

non-BlackScholes world 612.4 Copulas as dependence functions: basic facts 70

2.4.1 Independence 702.4.2 Comonotonicity 702.4.3 Monotone transforms and copula invariance 722.4.4 An application: VaR trade-off 73

2.5 Survival copula and joint survival function 752.5.1 An application: default probability with exogenous shocks 78

2.6 Density and canonical representation 812.7 Bounds for the distribution functions of sum of r.v.s 84

2.7.1 An application: VaR bounds 852.8 Appendix 87

3 Market Comovements and Copula Families 95

3.1 Measures of association 953.1.1 Concordance 953.1.2 Kendalls 973.1.3 Spearmans S 1003.1.4 Linear correlation 1033.1.5 Tail dependence 1083.1.6 Positive quadrant dependency 110

3.2 Parametric families of bivariate copulas 1123.2.1 The bivariate Gaussian copula 1123.2.2 The bivariate Students t copula 1163.2.3 The Frechet family 1183.2.4 Archimedean copulas 1203.2.5 The MarshallOlkin copula 128

Contents vii

4 Multivariate Copulas 129

4.1 Definition and basic properties 1294.2 Frechet bounds and concordance order: the multidimensional case 1334.3 Sklars theorem and the basic probabilistic interpretation: the multidimen-

sional case 1354.3.1 Modeling consequences 138

4.4 Survival copula and joint survival function 1404.5 Density and canonical representation of a multidimensional copula 1444.6 Bounds for distribution functions of sums of n random variables 1454.7 Multivariate dependence 1464.8 Parametric families of n-dimensional copulas 147

4.8.1 The multivariate Gaussian copula 1474.8.2 The multivariate Students t copula 1484.8.3 The multivariate dispersion copula 1494.8.4 Archimedean copulas 149

5 Estimation and Calibration from Market Data 153

5.1 Statistical inference for copulas 1535.2 Exact maximum likelihood method 154

5.2.1 Examples 1555.3 IFM method 156

5.3.1 Application: estimation of the parametric copula for market data 1585.4 CML method 160

5.4.1 Application: estimation of the correlation matrix for a Gaussiancopula 160

5.5 Non-parametric estimation 1615.5.1 The empirical copula 1615.5.2 Kernel copula 162

5.6 Calibration method by using sample dependence measures 1725.7 Application 1745.8 Evaluation criteria for copulas 1765.9 Conditional copula 177

5.9.1 Application to an equity portfolio 178

6 Simulation of Market Scenarios 181

6.1 Monte Carlo application with copulas 1816.2 Simulation methods for elliptical copulas 1816.3 Conditional sampling 182

6.3.1 Clayton n-copula 1846.3.2 Gumbel n-copula 1856.3.3 Frank n-copula 186

6.4 Marshall and Olkins method 1886.5 Examples of simulations 191

7 Credit Risk Applications 195

7.1 Credit derivatives 195

viii Contents