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Copula Methods in Finance


The evaluation and risk measurement of portfolios of complex non-linear positions and non-normal risk factors has become a major nightmare for people working in the structured finance business. Dealing with "fat tails" and "smile effects", as well as the typical asymmetric shape of default risk has rapidly made obsolete the traditional linear correlation tools. In this new environment, the copula functions methodology has become the most significant new technique to handle the co-movement between markets and risk factors in a flexible way. This is the first book addressing copula functions from the viewpoint of mathematical finance applications. The method is to explain copulas by means of applications to major topics in derivative pricing and credit risk analysis, with the target to make the reader able to device her own application, following the strategies illustrated throughout the book. Examples include pricing of the main exotic derivatives typically included in commonly traded structured finance products (barrier, basket, rainbow options), as well as risk management issues. Particular focus is given to the pricing of asset-backed securities and basket credit derivative products and the evaluation of counterparty risk in derivative transactions.

Copula Methods in Finance provides:

  • Rigorous treatment of the mathematics of copula functions, illustrated with financial applications
  • Complete analysis of estimation and simulation issues applied to market data
  • Credit-linked structured products applications: CDO and basket credit derivatives
  • Equity-linked structured product applications: barrier, rainbow and basket derivatives
  • Counterparty risk in derivative transactions: vulnerable option pricing

UMBERTO CHERUBINI is Associate Professor of Mathematical Finance at the University of Bologna, and partner in Polyhedron Computational Finance, Florence, Italy. He is fellow of FERC, Cass Business School, London and Ente Einaudi, Bank of Italy, Rome. He has also taught graduate finance courses at Catholic University in Milan, Hitotsubashi University in Tokyo, and is supervisor of the Market Risk Area at the risk management education program of the Italian Banking Association (ABI). He is a member of the independent screening committee of TLX, the new Italian structured products market. Before joining the academia, he was with the Economic Research Department of Banca Commerciale Italiana, where he was Head of the Risk Management Unit.

ELISA LUCIANO, Ph.D., is Full Professor of Mathematical Finance at the University of Turin (Italy), Fellow of ICER, Turin, and Associate Fellow of FERC, Cass Business School, London. She also teaches at the École Nationale Supérieure de Cachan, Paris, and at the École Supérieure en Sciences Informatiques, Université de Nice-Sophia Antipolis, France. Her main research interest is Quantitative Finance, with special emphasis on portfolio selection and risk measurement. She has published extensively in Academic journals, including the Journal of Finance and Applied Mathematical Finance.

WALTER VECCHIATO is Head of Risk Management and Research at Veneto Banca in Montebelluna Treviso, Italy. Previously he was Head of Credit Derivatives Analysis at Banca Intesa in Milan, Italy. He was also Professor of Applied Statistics in University of Pavia, Italy and he was Visiting Researcher in Financial Econometrics at University of California at San Diego, La Jolla. He enhanced his research with the presence of Nobel Economic Sciences 2003 award winner Professor Robert F. Engle. He has written and published on quantitative finance and risk management techniques. He is a referee for many academic and practitioner journals and a frequent speaker for many symposiums on Finance worldwide.

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 1

1.2 Derivative pricing basics: the binomial model 2

1.3 The Black-Scholes model 7

1.4 Interest rate derivatives 13

1.5 Smile and term structure effects of volatility 18

1.6 Incomplete markets 21

1.7 Credit risk 27

1.8 Copula methods in finance: a primer 37

2 Bivariate Copula Functions 49

2.1 Definition and properties 49

2.2 Fr´echet bounds and concordance order 52

2.3 Sklar’s theorem and the probabilistic interpretation of copulas 56

2.4 Copulas as dependence functions: basic facts 70

2.5 Survival copula and joint survival function 75

2.6 Density and canonical representation 81

2.7 Bounds for the distribution functions of sum of r.v.s 84

2.8 Appendix 87

3 Market Comovements and Copula Families 95

3.1 Measures of association 95

3.2 Parametric families of bivariate copulas 112

4 Multivariate Copulas 129

4.1 Definition and basic properties 129

4.2 Frechet bounds and concordance order: the multidimensional case 133

4.3 Sklar's theorem and the basic probabilistic interpretation: the multidimensional case 135

4.4 Survival copula and joint survival function 140

4.5 Density and canonical representation of a multidimensional copula 144

4.6 Bounds for distribution functions of sums of n random variables 145

4.7 Multivariate dependence 146

4.8 Parametric families of n-dimensional copulas 147

5 Estimation and Calibration from Market Data 153

5.1 Statistical inference for copulas 153

5.2 Exact maximum likelihood method 154

5.3 IFM method 156

5.4 CML method 160

5.5 Non-parametric estimation 161

5.6 Calibration method by using sample dependence measures 172

5.7 Application 174

5.8 Evaluation criteria for copulas 176

5.9 Conditional copula 177

6 Simulation of Market Scenarios 181

6.1 Monte Carlo application with copulas 181

6.2 Simulation methods for elliptical copulas 181

6.3 Conditional sampling 182

6.4 Marshall and Olkin’s method 188

6.5 Examples of simulations 191

7 Credit Risk Applications 195

7.1 Credit derivatives 195

7.2 Overview of some credit derivatives products 196

7.3 Copula approach 202

7.4 Application: pricing and risk monitoring a CDO 210

7.5 Technical appendix 225

8 Option Pricing with Copulas 231

8.1 Introduction 231

8.2 Pricing bivariate options in complete markets 232

8.3 Pricing bivariate options in incomplete markets 239

8.4 Pricing vulnerable options 243

8.5 Pricing rainbow two-color options 253

8.6 Pricing barrier options 267

8.7 Pricing multivariate options: Monte Carlo methods 278

Bibliography 281

Index 289