Handbook of Heavy Tailed Distributions in Finance

Chapter 8

Modelling dependence with copulas and applications to risk management

Paul Embrechts; Filip Lindskog^*; Alexander Mcneil^† http://www.math.ethz.ch/finance    Department of Mathematics, ETHZ, CH-8092 Zürich, Switzerland

Contents

1. Introduction   331

2. Copulas   332

2.1. Mathematical introduction   332

2.2. Sklar's Theorem   334

2.3. The Fréchet-Hoeffding bounds for joint distribution functions   335

2.4. Copulas and random variables   336

3. Dependence concepts   341

3.1. Linear correlation   341

3.2. Perfect dependence   342

3.3. Concordance   343

3.4. Kendall's tau and Spearman's rho   345

3.5. Tail dependence   348

4. Marshall-O lkin copulas   351

4.1. A multivariate extension   352

4.2. Bivariate Marshall-O lkin copulas   354

4.3. A useful modelling ...