On uniform tail expansions of bivariate copulas
The theory of copulas provides a useful tool for modelling dependence in risk management. The goal of this paper is to describe the tail behaviour of bivariate copulas and its role in modelling extreme events. We say that a bivariate copula has a uniform lower tail expansion if near the origin it can be approximated by a homogeneous function $L(u,v)$ of degree 1; and it is said to have a uniform upper tail expansion if the associated survival copula has a lower tail expansion. In this paper we (1) introduce the notion of the uniform tail expansion of a bivariate copula; (2) describe the main properties of the leading part $L(u,v)$ like two-monotonicity or concavity; (3) determine the set of all possible leading parts $L(u,v)$; (4) compute the leading parts of the uniform tail expansions for the most popular copulas like gaussian, archimedean or BEV; (5) apply uniform tail expansions in estimating the extreme risk of a portfolio consisting of long positions in risky assets.