重要抽樣法在估計高維度聯合違約機率的應用

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。

關鍵字

聯合違約；稀有事件模擬；高效能重要抽樣法

並列摘要

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.

並列關鍵字

joint default event ； rare event simulation ； efficient importance sampling

參考文獻

[1] V. Acharya, R. Engle, and M. Richardson. Capital shortfall: A new approach to ranking and regulating systemic risks. The American Economic Review, 102(3):59– 64, 2012.

[3] R. Carmona, J.-P. Fouque, and D. Vestal. Interacting particle systems for the com- putation of rare credit portfolio losses. Finance and Stochastics, 13(4):613–633, 2009.

[4] A. Dembo and O. Zeitouni. Large deviations techniques and applications, vol- ume 38. Springer Science & Business Media, 2009.

[6] M. Fischer et al. On the form of the large deviation rate function for the empirical measures of weakly interacting systems. Bernoulli, 20(4):1765–1801, 2014.

[7] A. Friedman. Variational principles and free-boundary problems. Courier Corpora- tion, 2010.

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重要抽樣法在估計高維度聯合違約機率的應用

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