Copula Methods in Finance by Umberto Cherubini, Elisa Luciano, Walter Vecchiato

Copula Methods in Finance by Umberto Cherubini, Elisa Luciano, Walter Vecchiato
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Copula Methods in Finance

by Umberto Cherubini, Elisa Luciano, Walter Vecchiato


310 pages July 2004


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.


Preface. List of Common Symbols and Notations.

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

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 measure.

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每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 Counterparty 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 comovements.

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 Fréchet 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 Modeling consequences.

2.3.4 Sklar*s theorem in financial applications: toward a non-Black每Scholes world.

2.4 Copulas as dependence functions: basic facts.

2.4.1 Independence.

2.4.2 Comonotonicity.

2.4.3 Monotone transforms and copula invariance.

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 r.v.s.

2.7.1 An application: VaR bounds.

2.8 Appendix.

3 Market Comovements and Copula Families.

3.1 Measures of association.

3.1.1 Concordance.

3.1.2 Kendall*s 而.

3.1.3 Spearman*s 老S.

3.1.4 Linear correlation.

3.1.5 Tail dependence.

3.1.6 Positive quadrant dependency.

3.2 Parametric families of bivariate copula.

3.2.1 The bivariate Gaussian copula.

3.2.2 The bivariate Student*s t copula.

3.2.3 The Fr∩echet family.

3.2.4 Archimedean copulas.

3.2.5 The Marshall每Olkin copula.

4 Multivariate Copulas.

4.1 Definition and basic properties.

4.2 Fréchet bounds and concordance order: the multidimensional case.

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

4.3.1 Modeling 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 random variables.

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.

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.

7.2.1 Credit default swap.

7.2.2 Basket default swap.

7.2.3 Other credit derivatives products.

7.2.4 Collateralized debt obligation (CDO).

7.3 Copula approach.

7.3.1 Review of single survival time modeling and calibration.

7.3.2 Multiple survival times: modeling.

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每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 Frćicing: 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-color options.

8.5.1 Call option on the minimum of two assets.

8.5.2 Call option on the maximum of two assets.

8.5.3 Put option on the maximum of two assets.

8.5.4 Put option on the minimum of two assets.

8.5.5 Option to exchange.

8.5.6 Pricing and hedging rainbows with smiles: Everest notes.

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.




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 谷cole Nationale Sup谷rieure de Cachan, Paris, and at the 谷cole Sup谷rieure en Sciences Informatiques, Universit谷 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.