Simulating Exchangeable Multivariate Archimedean Copulas and its Applications

Multivariate exchangeable Archimedean copulas are one of the most popular classes of copulas that are used in actuarial science and finance for modelling risk dependencies and for using them to quantify the magnitude of tail dependence. Owing to the increase in popularity of copulas to measure dependent risks, generating multivariate copulas has become a very crucial exercise. Current methods for generating multivariate Archimedean copulas could become a very difficult task as the number of dimension increases. The resulting analytical procedures suggested in the existing literature do not offer much guidance for practical implementation. This paper presents an algorithm for generating multivariate exchangeable Archimedean copulas based on a multivariate extension of a bivariate result. A procedure for generating bivariate Archimedean copulas has been originally proposed in Genest and Rivest (1993) and again later described in Nelsen (1999) and Embrechts, et al. (2002a, 2002b). Using a proof that is simply based on fundamental Jacobian techniques for deriving distributions of transformed random variables, we are able to extend the bivariate result into the multivariate case allowing us to develop an interesting algorithm to generate exchangeable Archimedean copulas. As auxiliary results, we are able to derive the distribution function of an n-dimensional Archimedean copula, a result already known in Genest and Rivest (2001) but our approach of proving this result is based on a different perspective. This paper focuses on this class of copulas that has one generating function and one parameter that characterizes the dependence structure of the joint distribution function. ∗The authors thank the Australian Research Council through the Discovery Grant DP0345036 and the UNSW Actuarial Foundation of the Institute of Actuaries of Australia for financial support. Corresponding author. †Please email at [email protected] for inquiries regarding this paper.