Conditional Tail Dependence for Bivariate Copulas with Applications in Finance

Associations between variables can be measured in many ways such as correlation, concordance, dependence, and so on. Researchers and practitioners recently are interested in the associations between variables at extreme values in which large amount of profits or losses in financial industry are considered. Sibuya (1960) proposed a conditional dependence structure, called tail dependence coefficient, to measure the asymptotic dependency between variables. This coefficient has become a standard measurement of associations between variables at extreme values.

A conditional tail dependence is a structure that involves the joint and marginal distributions of random variables at their extreme values. Different conditions of identifying extreme events are used to capture various behaviors of random variables approaching to their limiting situations. Existing tail dependence coefficients consider the co-movements of two variables at the same rate towards extremes. Such measures may well describe the dependence of two similar variables at the extremes. However, it may be more reasonable to consider the asymptotic dependence behavior for two not so similar variables at different rates.

In this research, we propose a functional tail dependence structure between two variables at their extreme values in the sense that the rates approaching to extremes are functionally different. Furthermore, we propose a generalized tail dependence structure in which each tail may approach extreme values by itself. We obtain definite solutions of such proposed tail dependence coefficients for six commonly used copulas under mild assumptions. In addition, empirical studies are carried out for financial data.