Gaussian copula approximations and their applications

We examine the standard Gaussian copula model for correlated defaults (also called the survival copula) and its relationship with the theoretically richer model based on diffusion processes and default thresholds. We show that in a discrete time framework the Gaussian copula can be seen as a simple global approximation to the Brownian copula implied by correlated diffusions. More precisely, the Gaussian copula is larger, in the sense that it induces systematically higher dependency. The result helps clarify some of the peculiar aspects of the standard model. Turning the argument around, the framework allows us to design further applications, capitalizing on the model’s notable tractability. We show that a multiperiod, correlated migrations model based on discrete diffusions can for some purposes be replicated surprisingly well by very simple analytic expressions. As a practical example we show how one can readily compute the distributions of the ”weighted average rating factor” of a pool of credits for any future time period.