The estimation of copulas : theory and practice

INTRODUCTION Copulas are a way of formalising dependence structures of random vectors. Although they have been known about for a long time (Sklar (1959)), they have been rediscovered relatively recently in applied sciences (biostatistics, reliability, biology, etc). In finance, they have become a standard tool with broad applications: multiasset pricing (especially complex credit derivatives), credit portfolio modelling, risk management, etc. For example, see Li (1999), Patton (2001) and Longin and Solnik (1995). Although the concept of copulas is well understood, it is now recognised that their empirical estimation is a harder and trickier task. Many traps and technical difficulties are present, and these are, most of the time, ignored or underestimated by practitioners. The problem is that the estimation of copulas implies usually that every marginal distribution of the underlying random vectors must be evaluated and plugged into an estimated multivariate distribution. Such a procedure produces unexpected and unusual effects with respect to the usual statistical procedures: non-standard limiting behaviours, noisy estimations, etc (eg, see the discussion in Fermanian and Scaillet, 2005). In this chapter, we focus on the practical issues practitioners are faced with, in particular concerning estimation and visualisation. 1