Efficient Importance Sampling for Estimating Lower Tail Probabilities in Financial Models (I, II)
(I) 韓傳祥 教授（清華大學計量財務金融系）(II) 吳慶堂 教授（交通大學應用數學系） 下午 2:00–5:00 鴻經館 107 室Abstract:
(I) Motivated from credit risk modeling, this paper extends the two-dimensional first passage time problem studied by Zhou (2001) to any finite dimension by means of Monte Carlo simulation. We provide an importance sampling method to estimate the joint default probability, and apply the large deviation principle to prove that the proposed importance sampling is asymptotically optimal. Our result is an alternative to the interacting particle systems proposed by Carmona, Fouque, and Vestal (2009).
(II) In credit risk modeling, Gaussian and Student’s t variates arise primarily from the copula method to retain certain correlation structures among defaultable assets. We propose efficient importance sampling algorithms to estimate lower tail probabilities of these two variates in any finite dimension. Variances of importance sampling estimators are shown asymptotically optimal by means of the large deviation theory and a truncation argument. Numerical comparisons with commercial codes, such as mvncdf.m and mvtcdf.m in Matlab, demonstrate robustness and efficiency of our proposed algorithms. Moreover, the flexibility of these algorithms can be seen from an application of probability estimation for the nth-to-default, i.e., the nth order statistic, given a credit portfolio.