Joint Distributions of Time to Default with Application to the Pricing of Credit Derivatives

Abstract

Modeling portfolio credit risk involves the default dependencies between the individual securities in a portfolio. The copula is a common approach to construct it. It parameterizes the joint distribution of individual defaults independently of their marginal distributions. The current market standard model is the Gaussian copula. It assumes the event of default depends on both the common systematic risk factor and the individually idiosyncratic risk factor. The dependence of the defaults is induced by modeling a vector of latent variables that have a multivariate normal distribution. First we study the loss distribution of the large portfolio by using the Gaussian copula. We derive several asymptotic approximations to the loss distribution of the Gaussian copula. Every approximation is compared with the exact loss distribution by using the Hellinger distance between their probability density functions.

However, the tail dependence of the extreme events such as credit defaults can not be captured by the Gaussian copula. We develop the Student's t copula process to define an appropriate dependence structure of the defaults. The dependence of the default events is induced by modeling a vector of latent variables that have a multivariate Student's t distribution. We study the loss distribution of the large portfolio by using the Student's t copula. We also derive several asymptotic approximations to the loss distribution of the Student's t copula. We compare every approximation to the exact loss distribution by using the Hellinger distance between their probability density functions.

The market data of iTraxx Europe Series 4 (5-year) is investigated by using both the Gaussian copula and the Student's t copula. We say the Student's t copula works better than the Gaussian copula to describe the dependence of the extreme events with an extra parameter, the degrees of freedom of the Student's t copula. The parameters are estimated by using the weighted least squares of the mark to market. The base correlation curve implied from the Gaussian copula is skewed. The base correlation curve implied from the Student's t copula is also skewed unless we allow varying degrees of freedom. Since the implied base correlations of the individual tranches are not consistent, it is impossible to interpolate the base correlation for a non-standard tranche. A less skewed implied base correlation curve will be one of our interests in the future study.