Copula Methods in Finance

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