Abstracts

In credit risk modeling, Gaussian and Student t variates arise primarily from the copula method for retaining certain correlation structures among defaultable assets. We propose efficient importance sampling algorithms to estimate lower tail probabilities of these two variates in any finite dimension. Variances of importance sampling estimators are shown to be zero asymptotically by means of the large deviation theory and a truncation argument. Moreover these algorithms are suitable for parallel computing so that their standard errors can be dramatically reduced by small standard deviation and large sample size. Numerical comparisons with commercial codes, such as mvncdf.m and mvtcdf.m in Matlab, demonstrate robustness and efficiency of our proposed algorithms. Furthermore, generalizations of these algorithms can be seen from the following two applications: (1) probability estimation for the nth-to-default, i.e., the nth order statistic, given a credit portfolio of homogeneous assets, and (2) the default probability estimation of an inhomogeneous portfolio. The latter is related to a relative entropy minimization problem.