在因子關聯結構模型下用高效率權重取樣估計聯合違約機率

Abstract

權重取樣是常見的蒙地卡羅方法之一，可用於有效的估計罕見事件。 此方法加重罕見事件發生之機率，所以適用於估計一些罕見之違約相關事件。 此方法也同時容易應用於處理不同的違約事件上，但主要的問題在於選擇有效的權重取樣，不但加重罕見事件機率，也同時縮小估計值變異數。 在多變量的架構下，縮小變異數的問題相當複雜，也多數無直接解答。此論文提出有效的權重取樣演算法，同時的增加罕見事件機率並縮小估計量變易數，在 Large Deviation Theory 下縮小變異數，然後將此演算法用於常見的信用商品上，並提出數據說明此演算法有效的增進速度及估計準確度。

Importance sampling is a commonly used technique to improve Monte Carlo methods, especially in working with rare events. It is designed to increase the probability of sampling from rare events and is therefore well-suited for estimating default related items in various products given the rarity of default events. It is also simple to implement and versatile in that in can be easily extended to estimate different items. But the main challenge is selecting an importance sampling scheme that not only increases the probability of rare events but also effectively reduces the variance of the estimate. Under the multivariate framework when multiple entities are involved, variance reduction becomes even more challenging as there is no closed form solution for such optimization problem. In this study, we propose an effective importance sampling algorithm that both increases the probability of rare events and reduce variance of estimates. We consider the problem of variance reduction under the framework of Large Deviation Theory, and establish an efficient importance sampling estimator that can be applied to evaluating default events. Then we extend this importance sampling scheme to another popular type of default event and incorporate it into a conditional importance sampling scheme. Our numerical results confirm that the proposed algorithms for direct importance sampling and conditional importance sampling are more efficient in terms of variance reduction. Our algorithms are overall more robust under different specified initial conditions.