重要抽樣法在估計高維度聯合違約機率的應用

Abstract

我們探討估計稀有事件機率的模擬方法，特別針對「聯合違約」事 件於不同模型中的狀況。我們所提供的方法主要基於重要抽樣法，是 一種藉由增加抽到稀有事件的比率，來減低抽樣誤差的技術。本篇論 文主要分為兩部分。在第一部分，我們提出在四個不同模型下，決定 重要抽樣法機率測度的方式。其中，這些方法引入大離差理論的結果， 可以被證明是「漸進式最佳」的。此外，大部分模型在高維度狀況並 不擁有解析解，而我們的結果皆適用在多維度的情形。第二部分，我 們將先前的結果拿來應用在更為複雜的模型上，可以看到在幾個例子， 皆有出色的數值成果，同時計算上仍保持一定效率。

We discuss the simulation method for estimating the probabilities of rare events, especially the “joint default” probabilities under different models. The methods we provide are based on importance sampling, which is a variance re- duction technique used to increase the number of samples reaching the “rare” region. There are two parts in this thesis. For the first part, we provide several importance sampling schemes, where the “asymptotically optimal” (or efficient) property is proved by applying the large deviation theory. Also, the results could be applied to the multi-dimensional scene, where most of the probabilities interested do not have an explicit expression. For the second part, we apply our idea to the model that is more complicated. It is shown that we have some satisfying results while keeping the computation efficient.