Stratified importance sampling for a Bernoulli mixture model of portfolio credit risk

Bernoulli mixture model is a general framework by which most existing models of portfolio credit risk can be represented. In the model, the default probability of an obligor is determined by a set of latent factors. The model allows various types of joint default probability of obligors. For the model, we propose an importance sampling scheme to estimate the tail loss probability. We consider the case that there are several types of default events of obligors leading to large losses. In such a case, the optimal importance distribution leading to frequent outcomes of a typical default event of large loss is different from those of other typical default events. We stratify the sample space of defaults of obligors according to the defaults of some obligors with large exposures, and propose to sample from an importance distribution chosen optimally for each stratum. We show that the stratified importance sampling is more efficient than the importance sampling without stratification in terms of variance reduction under a condition. For the optimal choice of importance distribution for each stratum, we apply the cross entropy minimization method and the exponential twisting. For the case that the importance distribution of latent factors is confined to the family of multivariate normal mixtures, it is hard to find the optimal parameter which is the solution of a cross entropy minimization problem. We implement an EM-algorithm to solve the problem. Numerical results are given to compare the performance of the proposed scheme with the crude Monte Carlo simulation and the importance sampling without stratification.