Estimating Joint Default Probability by Efficient Importance Sampling with Applications from Bottom Up

This paper provides a unified approach to estimate the probability of joint default under classical models in the bottom up approach. Starting from a toy model defined on the Gaussian random variable in one dimension, we develop an importance sampling scheme and consider it variance approximation problem with a small scale. By means of the large deviation principle, the importance sampling is proved to be efficient, justified by numerical experiments in credit risk applications such as VaR and C-VaR estimation. The same approach is applicable to construct importance sampling schemes for high dimensional problems including some factor copula models in reduced form and structural-form models. In particular, the large deviation principle is applied to prove that all these importance sampling schemes are efficient for rare event simulation. Extensive numerical examples demonstrate the efficiency and stability of importance sampling. When stochastic correlation or stochastic volatility arising from structural-form models, the importance sampling proposed cannot be applied to estimate the joint default probability. We overcome this difficulty by a combination of the singular perturbation approximation with the importance sampling scheme. A numerical example for computing the loss density function of a credit portfolio confirms the efficiency and stability of this new importance sampling method in high dimensions. ∗Department of Quantitative Finance, National Tsing Hua University, Hsinchu, Taiwan, 30013, ROC, [email protected]. Work supported by NSC 97-2115-M-007-002MY2, Taiwan. Ackowledgements: NCTS, National Tsing-Hua University and TIMS, National Taiwan University.