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Copula Methods in Finance (Umberto Cherubini & Elisa Luciano & Walter Vecchiato)

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
1.2.1 Replicating portfolios 3
1.2.2 No-arbitrage and the risk-neutral probability measure 3
1.2.3 No-arbitrage and the objective probability measure 4
1.2.4 Discounting under different probability measures 5
1.2.5 Multiple states of the world 6
1.3 The Black–Scholes model 7
1.3.1 Ito’s lemma 8
1.3.2 Girsanov theorem 9
1.3.3 The martingale property 11
1.3.4 Digital options 12
1.4 Interest rate derivatives 13
1.4.1 Affine factor models 13
1.4.2 Forward martingale measure 15
1.4.3 LIBOR market model 16
1.5 Smile and term structure effects of volatility 18
1.5.1 Stochastic volatility models 18
1.5.2 Local volatility models 19
1.5.3 Implied probability 20
1.6 Incomplete markets 21
1.6.1 Back to utility theory 22
1.6.2 Super-hedging strategies 23
1.7 Credit risk 27
1.7.1 Structural models 28
1.7.2 Reduced form models 31
1.7.3 Implied default probabilities 33

1.7.4 Counterparty risk 36
1.8 Copula methods in finance: a primer 37
1.8.1 Joint probabilities, marginal probabilities and copula functions 38
1.8.2 Copula functions duality 39
1.8.3 Examples of copula functions 39
1.8.4 Copula functions and market comovements 41
1.8.5 Tail dependence 42
1.8.6 Equity-linked products 43
1.8.7 Credit-linked products 44

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
2.3.1 Sklar’s theorem 56
2.3.2 The subcopula in Sklar’s theorem 59
2.3.3 Modeling consequences 60
2.3.4 Sklar’s theorem in financial applications: toward a
non-Black–Scholes world 61
2.4 Copulas as dependence functions: basic facts 70
2.4.1 Independence 70
2.4.2 Comonotonicity 70
2.4.3 Monotone transforms and copula invariance 72
2.4.4 An application: VaR trade-off 73
2.5 Survival copula and joint survival function 75
2.5.1 An application: default probability with exogenous shocks 78
2.6 Density and canonical representation 81
2.7 Bounds for the distribution functions of sum of r.v.s 84
2.7.1 An application: VaR bounds 85

3 Market Comovements and Copula Families 95
3.1 Measures of association 95
3.1.1 Concordance 95
3.1.2 Kendall’s τ 97
3.1.3 Spearman’s ρS 100
3.1.4 Linear correlation 103
3.1.5 Tail dependence 108
3.1.6 Positive quadrant dependency 110
3.2 Parametric families of bivariate copulas 112
3.2.1 The bivariate Gaussian copula 112
3.2.2 The bivariate Student’s t copula 116
3.2.3 The Fr´echet family 118
3.2.4 Archimedean copulas 120
3.2.5 The Marshall–Olkin copula 128

4 Multivariate Copulas 129
4.1 Definition and basic properties 129
4.2 Fr´echet bounds and concordance order: the multidimensional case 133
4.3 Sklar’s theorem and the basic probabilistic interpretation: the multidimensional case 135
4.3.1 Modeling consequences 138
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
4.8.1 The multivariate Gaussian copula 147
4.8.2 The multivariate Student’s t copula 148
4.8.3 The multivariate dispersion copula 149
4.8.4 Archimedean copulas 149

5 Estimation and Calibration from Market Data 153
5.1 Statistical inference for copulas 153
5.2 Exact maximum likelihood method 154
5.2.1 Examples 155
5.3 IFM method 156
5.3.1 Application: estimation of the parametric copula for market data 158
5.4 CML method 160
5.4.1 Application: estimation of the correlation matrix for a Gaussian copula 160
5.5 Non-parametric estimation 161
5.5.1 The empirical copula 161
5.5.2 Kernel copula 162
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
5.9.1 Application to an equity portfolio 178

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
6.3.1 Clayton n-copula 184
6.3.2 Gumbel n-copula 185
6.3.3 Frank n-copula 186
6.4 Marshall and Olkin’s method 188
6.5 Examples of simulations 191

7 Credit Risk Applications 195
7.1 Credit  7.2 Overview of some credit derivatives products 196
7.2.1 Credit default swap 196
7.2.2 Basket default swap 198
7.2.3 Other credit derivatives products 199
7.2.4 Collateralized debt obligation (CDO) 199
7.3 Copula approach 202
7.3.1 Review of single survival time modeling and calibration 202
7.3.2 Multiple survival times: modeling 203
7.3.3 Multiple defaults: calibration 205
7.3.4 Loss distribution and the pricing of CDOs 206
7.3.5 Loss distribution and the pricing of homogeneous basket default swaps 208
7.4 Application: pricing and risk monitoring a CDO 210
7.4.1 Dow Jones EuroStoxx50 CDO 210
7.4.2 Application: basket default swap 210
7.4.3 Empirical application for the EuroStoxx50 CDO 212
7.4.4 EuroStoxx50 pricing and risk monitoring 216
7.4.5 Pricing and risk monitoring of the basket default swaps 221
7.5 Technical appendix 225
7.5.1 Derivation of a multivariate Clayton copula density 225
7.5.2 Derivation of a 4-variate Frank copula density 226
7.5.3 Correlated default times 227
7.5.4 Variance–covariance robust estimation 228
7.5.5 Interest rates and foreign exchange rates in the analysis 229

8 Option Pricing with Copulas 231
8.1 Introduction 231
8.2 Pricing bivariate options in complete markets 232
8.2.1 Copula pricing kernels 232
8.2.2 Alternative pricing techniques 235
8.3 Pricing bivariate options in incomplete markets 239
8.3.1 Fr´echet pricing: super-replication in two dimensions 240
8.3.2 Copula pricing kernel 241
8.4 Pricing vulnerable options 243
8.4.1 Vulnerable digital options 244
8.4.2 Pricing vulnerable call options 246
8.4.3 Pricing vulnerable put options 248
8.4.4 Pricing vulnerable options in practice 250
8.5 Pricing rainbow two-color options 253
8.5.1 Call option on the minimum of two assets 254
8.5.2 Call option on the maximum of two assets 257
8.5.3 Put option on the maximum of two assets 258
8.5.4 Put option on the minimum of two assets 261
8.5.5 Option to exchange 262
8.5.6 Pricing and hedging rainbows with smiles: Everest notes 263
8.6 Pricing barrier options 267
8.6.1 Pricing call barrier options with copulas: the general framework 268
8.6.2 Pricing put barrier option: the general framework 270
8.6.3 Specifying the trigger event 272
8.6.4 Calibrating the dependence structure 276
8.6.5 The reflection copula 276
8.7 Pricing multivariate options: Monte Carlo methods 278
8.7.1 Application: basket option 279

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