| Summary: | 博士 === 國立政治大學 === 統計研究所 === 100 === Under a factor model, computation of the loss density function relies on the estimates of some mixture of the joint default probability and joint survival probability. Monte Carlo simulation is among the most widely used computational tools in such estimation. Nevertheless, general Monte Carlo simulation is an ineffective simulation approach, in particular for rare event aspect and complex dependence between defaults of multiple obligors. So a method to increase efficiency of estimation is necessary.
Importance sampling (IS) seems to be an attractive method to address this problem. Changing the measure of probabilities, IS makes an estimator to be efficient especially for complicated model. Therefore, we consider IS for estimation of tail probability of skew normal copula model. This paper consists of two parts. First, we apply exponential twist, a usual and better IS technique, to conditional probabilities and the factors. However, this procedure does not always guarantee enough variance reduction. Such result indicates the further consideration of choosing IS factor density.
Faced with this problem, a variety of approaches has recently been proposed in the literature ( Capriotti 2008, Glasserman et al 1999, Glasserman and Li 2005). The better choices of IS density can be roughly classified into two kinds of strategies. The first strategy depends on choosing optimal shift. The optimal drift is decided by using different approximation methods. Such strategy is shown in Glasserman et al 1999, or Glasserman and Li 2005.
The second strategy, as shown in Capriotti (2008), considers a family of factor probability densities which depend on a set of real parameters. By formulating in terms of a nonlinear optimization problem, IS density which is not limited the determination of drift is then determinate. The method that searches for the optimal parameters, however, incurs another efficiency problem. To keep the method efficient, particular care for robust parameters estimation needs to be taken in preliminary Monte Carlo simulation. This leads method to be more complicated.
In this paper, we describe an alternative strategy that is straightforward and flexible enough to be applied in Monte Carlo setting. Indeed, our algorithm is not limited to the determination of optimal drift in Gaussian copula model, nor estimation of parameters of factor density. To exploit the similar concept developed for basket default swap valuation in Chiang, Yueh, and Hsie (2007), we provide a reasonable guess of the optimal sampling density and then establish a way different from stochastic approximation to speed up simulation.
Finally, we provide theoretical support for single factor model and take this approach a step further to multifactor case. So we have a rough but fast approximation that execute entirely with Monte Carlo in general situation. We support our approach by some portfolio examples. Numerical results show that such algorithm is more efficient than general Monte Carlo simulation.
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