Large deviations and fast simulation of multifactor portfolio credit risk

Monte Carlo simulation is widely used to estimate portfolio credit risk. We develop large deviations results and asymptotically optimal importance sampling techniques for fast simulation in the multifactor Gaussian copula model.; The measurement of portfolio credit risk focuses on rare but significant large-loss events. We investigate rare event asymptotics for loss distributions in the widely used Gaussian copula model of portfolio credit risk. We establish logarithmic limits for the tail of the loss distribution in two limiting regimes: the first limiting regime is based on letting the individual default probabilities decrease toward zero while the second limit examines the tail of the loss distribution at increasingly high loss thresholds. Both limits are also based on letting the size of the portfolio increase to infinity. Our analysis reveals a qualitative distinction between the two cases: in the small default probability regime, the tail of the loss distribution decreases exponentially, but in the large loss threshold regime the decay is consistent with a power law. This indicates that the dependence between defaults imposed by the Gaussian copula is qualitatively different for portfolios of high-quality and lower-quality credits.; We also investigate fast simulation methods for the estimation of portfolio credit risk in the Gaussian copula model. We develop an importance sampling technique to address the rare-event simulation problem of estimating probabilities of large losses.; We focus on difficulties arising in multifactor models, in which different combinations of factor movements and obligor defaults can produce large losses. To address these difficulties, our method combines multiple importance sampling distributions, each associated with a shift in the mean of a set of underlying factors. We characterize "optimal" mean shifts. Finding these points is both a combinatorial problem and a convex optimization problem, so we address computational aspects of this step as well. We establish asymptotic optimality results for our method, ultimately showing that---unlike standard simulation---it remains efficient as the event of interest becomes rarer.; Using the results obtained for the Gaussian copula model, we suggest an importance sampling procedure for the t-copula model. We use a version of the multivariate t-distribution that can be expressed as a ratio of a multivariate normal random vector and a chi-square random variable. From the large deviations result for the Gaussian model, we devise an importance sampling change of measure for the chi-square distribution. Then, conditional on the chi-square random variable, we apply the importance sampling procedure developed for the Gaussian model. We overcome the difficulty of varying factor loading coefficients by applying stratification to the sampling from the chi-square distribution. We support our importance sampling procedure for t-copula by numerical examples.