Copula Methods in Finance

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.

"...This book is of great use for researchers as well as practitioners..." (Statistical Papers, July 2005)

• Preface xiList of Common Symbols and Notations xv1 Derivatives Pricing, Hedging and Risk Management: The State of the Art 11.1 Introduction 11.2 Derivative pricing basics: the binomial model 21.3 The Black–Scholes model 71.4 Interest rate derivatives 131.5 Smile and term structure effects of volatility 181.6 Incomplete markets 211.7 Credit risk 271.8 Copula methods in finance: a primer 372 Bivariate Copula Functions 492.1 Definition and properties 492.2 Fr´echet bounds and concordance order 522.3 Sklar’s theorem and the probabilistic interpretation of copulas 562.4 Copulas as dependence functions: basic facts 702.5 Survival copula and joint survival function 752.6 Density and canonical representation 812.7 Bounds for the distribution functions of sum of r.v.s 842.8 Appendix 873 Market Comovements and Copula Families 953.1 Measures of association 953.2 Parametric families of bivariate copulas 1124 Multivariate Copulas 1294.1 Definition and basic properties 1294.2 Frechet bounds and concordance order: the multidimensional case 1334.3 Sklar's theorem and the basic probabilistic interpretation: the multidimensional case 1354.4 Survival copula and joint survival function 1404.5 Density and canonical representation of a multidimensional copula 1444.6 Bounds for distribution functions of sums of n random variables 1454.7 Multivariate dependence 1464.8 Parametric families of n-dimensional copulas 1475 Estimation and Calibration from Market Data 1535.1 Statistical inference for copulas 1535.2 Exact maximum likelihood method 1545.3 IFM method 1565.4 CML method 1605.5 Non-parametric estimation 1615.6 Calibration method by using sample dependence measures 1725.7 Application 1745.8 Evaluation criteria for copulas 1765.9 Conditional copula 1776 Simulation of Market Scenarios 1816.1 Monte Carlo application with copulas 1816.2 Simulation methods for elliptical copulas 1816.3 Conditional sampling 1826.4 Marshall and Olkin’s method 1886.5 Examples of simulations 1917 Credit Risk Applications 1957.1 Credit derivatives 1957.2 Overview of some credit derivatives products 1967.3 Copula approach 2027.4 Application: pricing and risk monitoring a CDO 2107.5 Technical appendix 2258 Option Pricing with Copulas 2318.1 Introduction 2318.2 Pricing bivariate options in complete markets 2328.3 Pricing bivariate options in incomplete markets 2398.4 Pricing vulnerable options 2438.5 Pricing rainbow two-color options 2538.6 Pricing barrier options 2678.7 Pricing multivariate options: Monte Carlo methods 278Bibliography 281Index 289