Saddlepoint approximations for credit portfolio distributions with applications in equity risk management

Abstract

We study saddlepoint approximations to the tail-distribution for credit portfolio losses in continuous time intensity based models under conditional independent homogeneous settings. In such models, conditional on the filtration generated by the individual default intensity up to time t, the conditional number of defaults distribution (in the portfolio) will be a binomial distribution that is a function of a factor Z_t which typically is the integrated default intensity up to time t. This will lead to an explicit closed-form solution of the saddlepoint equation for each point used in the number of defaults distribution when conditioning on the factor Z_t, and we hence do not have to solve the saddlepoint equation numerically. The ordo-complexity of our algorithm computing the whole distribution for the number of defaults will be linear in the portfolio size, which is a dramatic improvement compared to e.g. recursive methods which have a quadratic ordo-complexity in the portfolio size. The individual default intensities can be arbitrary as long as they are conditionally independent given the factor Z_t in a homogeneous portfolio. We also outline how our method for computing the number of defaults distribution can be extend to heterogeneous portfolios. Furthermore, we show that all our results can be extended to hold for any factor copula model. We give several numerical applications and in particular, in a setting where the individual default intensities follow a CIR process we study both the tail distribution and the number of defaults distribution. We then repeat similar numerical studies in a one-factor Gaussian copula model. We also numerically benchmark our saddlepoint method to other computational methods. Finally, we apply of our saddlepoint method to efficiently investigate Value-at-Risk for equity portfolios where the individual stock prices have simultaneous downward jumps at the defaults of an exogenous group of defaultable entities driven by a one-factor Gaussian copula model were we focus on Value-at-Risk as function of the default correlation parameter in the one-factor Gaussian copula model.