Measures of Extremal Dependence

In this chapter, we outline how multivariate extreme value theory and in particular extremal dependence can help to quantify extreme quantile regions concerning multivariate data. We briefly discuss some properties of multivariate extreme value distributions and show that a transformation allows us to decouple modelling of the marginal distributions from that of the dependence structure. We consider different definitions of extreme quantile regions that extend concepts of univariate quantiles to the multivariate case and discuss approaches to estimate those sets in the bivariate case. Finally, we apply the presented results by investigating the dependence structure of large losses in different financial asset classes.