Saddlepoint approximations for credit portfolios with stochastic recoveries

We study saddlepoint approximations to the tail-distribution for different credit portfolio losses in continuous time intensity based models which stochastic recoveries, 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 study the credit portfolio loss distribution with random recoveries. In particular, under the assumption that the stochastic recoveries are conditional binomial distributions correlated with the default times conditional on the factor Z_t, we derive very convenient semi closed-form expression for the credit portfolio loss distribution. Our algorithm for computing the tail-distribution at a point x for the credit portfolio loss with these random recoveries will have a ordo-complexity which is linear in x. Furthermore, we show that all our results, both for the number of defaults distribution and portfolio loss distribution with random recoveries, can be extended to hold for any factor copula model. In the case when the stochastic recoveries are independent of the default times, we give an example of how our method with random recoveries can be adapted to intensity based contagion models (which falls outside the family of conditional independent credit portfolio models). Finally, we give several numerical applications and in particular, in a setting where the individual default intensities follow a CIR process we study the time evolution of Value-at-Risk (i.e. VaR as function of time) both with constant and stochastic recoveries correlated with the default times. We then repeat similar numerical studies in a one-factor Gaussian copula model. We also numerically benchmark our method to other computational methods.