Copula function in

multivariate dependency analysis

Igor Hlivka

MUFJ Securities International, LONDON

Copula is a constructor function for multivariate distribution from univariate marginals. It is a method to link univariate samples, not necessarily from identical distributions, into joint multivariate distributions.  In this way, copulas are more generic and flexible functions to study dependency arising from multivariate distributions.

Conceptually, copulas are based on transformation of the underlying marginal into new derived variable with uniform distribution. Consequently, any multivariate distribution can be expressed in the form of copula function. If each marginal is continuous then copula is unique. Sklar in 1959 was the first to point this out.

Copulas represent a broad set of functions and they generally differ by (i) number of dependency factors and (ii) construction complexity. The choose of copula depends on the nature of the multivariate study and fitting objectives to an underlying data.

Copulas have been originally applied in reliability and actuarial analysis. For the past 10 years dependency feature in copulas attracted much attention in the financial risk modeling and they were heavily used in valuation of now defunct collateralised debt obligations (CDO). Heavy dependence on one particular copula class - Gaussian - is now frequently cited as a reason for the credit-led financial crisis in the past several years. Nevertheless, copulas have remained a useful tool for dependency analysis and have been recently re-introduced into finance industry for other asset classes and derivative products.