Copula Methods in Finance

Copula Methods in Finance is the first book to address the mathematics of copula functions illustrated with finance applications. It explains copulas by means of applications to major topics in derivative pricing and credit risk analysis. Examples include pricing of the main exotic derivatives (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.

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.

1 Derivatives Pricing, Hedging and Risk Management.

1.1 Introduction.

1.2 Derivative pricing basics: the binomial model.

1.2.1 Replicating portfolios.

1.2.2 No-arbitrage and the risk-neutral probability measures.

1.2.3 No-arbitrage and the objective probability measure.

1.2.4 Discounting under different probability measures.

1.2.5 Multiple states of the world.

1.3 The Black and Scholes model.

1.3.1 Ito's lemma.

1.3.2 Girsanov theorem.

1.3.3 The martingale property.

1.3.4 Digital options.

1.4 Interest rate derivatives.

1.4.1 Affine factor models.

1.4.2 Forward martingale measure.

1.4.3 Libor market model.

1.5 Smile and term structure effects of volatility.

1.5.1 Stochastic volatility models.

1.5.2 Local volatility models.

1.5.3 Implied probability.

1.6 Incomplete markets.

1.6.1 Back to utility theory.

1.6.2 Super-hedging strategies.

1.7 Credit risk.

1.7.1 Structural models.

1.7.2 Reduced form models.

1.7.3 Implied default probabilities.

1.7.4 Counterpart risk.

1.8 Copula methods in finance: a primer.

1.8.1 Joint probabilities, marginal probabilities and copula functions.

1.8.2 Copula functions duality.

1.8.3 Examples of copula functions.

1.8.4 Copula functions and market co-movements.

1.8.5 Tail dependence.

1.8.6 Equity-linked products.

1.8.7 Credit-linked products.

2 Bivariate Copula Functions.

2.1 Definition and properties.

2.2 Frechet bounds and concordance order.

2.3 Sklar's theorem and the probabilistic interpretation of copulas.

2.3.1 Sklar's theorem.

2.3.2 The subcopula in Sklar's theorem.

2.3.3 Modelling consequences.

2.3.4 Sklar's theorem in financial applications: towards a non-Black-Scholes world.

2.4 Copulas as dependence functions: basic facts.

2.4.1 Independency.

2.4.2 Comonotonicity.

2.4.3 Monotone transforms and copula invariancy.

2.4.4 An application: VaR trade-off.

2.5 Survival copula and joint survival function.

2.5.1 An application: default probability with exogenous shocks.

2.6 Density and canonical representation.

2.7 Bounds for the distribution functions of sum of rvs.

2.7.1 An application: VaR bounds.

3 Market Comovements and Copula Families.

3.1 Measures of association.

3.1.1 Concordance.

3.1.2 Kendall's.

3.1.3 Spearman's .

3.1.4 Linear correlation.

3.1.5 Tail dependence.

3.1.6 Positive quadrant dependency.

3.2 Parametric families of bivariate copulas.

3.2.1 The Bivariate Gaussian copula.

3.2.2 The Bivariate Student's t copula.

3.2.3 The Frechet family.

3.2.4 Archimedean copulas.

3.2.5 The Marshall-Olkin copula.

4 Multivariate Copulas.

4.1 Definition and basic properties.

4.2 Frechet bounds and concordance order: the multidimensional case.

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

4.3.1 Modelling consequences.

4.4 Survival copula and joint survival function.

4.5 Density and canonical representation of a multidimensional copula.

4.6 Bounds for distribution functions of sums of n rvs.

4.7 Multivariate Dependence.

4.8 Parametric families of n-dimensional copulas.

4.8.1 The Multivariate Gaussian copula.

4.8.2 The Multivariate Student's t copula.

4.8.3 The Multivariate dispersion copula.

4.8.4 Archimedean Copulas.

7

5 Estimation and Calibration from Market Data.

5.1 Statistical Inference for Copulas.

5.2 Exact Maximum Likelihood Method.

5.2.1 Examples.

5.3 IFM method.

5.3.1 Application: estimation of the parametric copula for market data.

5.4 CML method.

5.4.1 Application: Estimation of the correlation matrix for a gaussian copula.

5.5 Non Parametric Estimation.

5.5.1 The empirical copula.

5.5.2 Kernel copula.

5.6 Calibration method by using sample dependence measures.

5.7 Application.

5.8 Evaluation criteria for copulas.

5.9 Conditional Copula.

5.9.1 Application to an equity portfolio.

6 Simulation of Market Scenarios.

6.1 Monte Carlo Application with Copulas.

6.2 Simulation methods for Elliptical copulas.

6.3 Conditional Sampling.

6.3.1 Clayton n-Copula.

6.3.2 Gumbel n-Copula.

6.3.3 Frank n-Copula.

6.4 Marshall and Olkin's method.

6.5 Examples of simulations.

7 Credit Risk Applications.

7.1 Credit Derivatives.

7.2 Overview of some credit derivatives products.

7.3 Copula approach.

7.3.1 Review of single survival time modelling and calibration.

7.3.2 Multiple survival times: modelling.

7.3.3 Multiple defaults: calibration.

7.3.4 Loss distribution and the pricing of CDOs.

7.3.5 Loss distribution and the pricing of homogeneous Basket Default Swaps.

7.4 Application: Pricing and Risk Monitoring a CDO.

7.4.1 Dow Jones EuroStoxx50 CDO.

7.4.2 Application: Basket Default Swap.

7.4.3 Empirical Application for the Eurostoxx50 CDO.

7.4.4 Eurostoxx50 pricing and risk monitoring.

7.4.5 Pricing and Risk Monitoring of the Basket Default Swaps.

7.5 Technical Appendix.

7.5.1 Derivation of a multivariate Clayton copula density.

7.5.2 Derivation of a 4-variate Frank copula density.

7.5.3 Correlated Default Times.

7.5.4 Variance - Covariance Robust Estimation.

7.5.5 Interest rates and foreign exchange rates in the analysis.

8 Option Pricing with Copulas.

8.1 Introduction.

8.2 Pricing bivariate options in complete markets.

8.2.1 Copula pricing kernels.

8.2.2 Alternative pricing techniques.

8.3 Pricing bivariate options in incomplete markets.

8.3.1 Frechet pricing: super-replication in two dimensions.

8.3.2 Copula pricing kernel.

8.4 Pricing vulnerable options.

8.4.1 Vulnerable digital options.

8.4.2 Pricing vulnerable call options.

8.4.3 Pricing vulnerable put options.

8.4.4 Pricing vulnerable options in practice.

8.5 Pricing rainbow two colours options.

8.5.1 Call option on the minimum of two assets.

8.5.2 Call options on the maximum of two assets.

8.5.3 Put options on maximum of two assets.

8.5.4 Put options on the minimum between two assets.

8.5.5 Option to exchange.

8.5.6 Pricing and hedging rainbows with smiles.

8.6 Pricing barrier options.

8.6.1 Pricing call barrier options with copulas: the general framework.

8.6.2 Pricing put barrier option: the general framework.

8.6.3 Specifying the trigger event.

8.6.4 Calibrating the dependence structure.

8.6.5 The reflection copula.

8.7 Pricing multivariate options: Monte Carlo methods.

8.7.1 Application: Basket Option.

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Copula Methods in Finance

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