This paper links two applications of Extreme Value Theory (EVT) to analyze Taiwanese financial markets: 1. computation of Value at Risk (VaR) and Expected Shortfall (ES) 2. estimates of cross-market dependence under extreme events. Daily data from the Taiwan Stock Exchange Capitalization Weight Stock Index (TAIEX) and the foreign exchange rate, USD/NTD, are employed to analyze the behavior of each return and the dependence structure between the foreign exchange market and the equity market. In the univariate case, when computing risk measures, EVT provides us a more accurate way to estimate VaR. In bivariate case, when measuring extremal dependence, the results of whole period data show the extremal dependence between two markets is asymptotically independent, and the analyses of subperiods illustrate that the relation is slightly dependent in specific periods. Therefore, there is no significant evidence that extreme events appeared in one market (the equity market or the foreign exchange market) will affect another in Taiwan.

在針對投資組合之信用風險模擬時，如何選取適當的模型來解釋資產間的相依程度是非常重要的。最常用來解釋投資組合的模型為常態關聯結構模型，但近年來發現t關聯結構模型更適合用在解釋投資組合間的相依程度。蒙地卡羅法在針對信用風險模擬上是一個很實用的工具，但是其缺點是模擬時間久且對於發生極端情況時，將不易得到結果，導致其效率過低。而此時，重要性取樣法則是一個很適合用來針對信用風險模擬所使用的工具，其優點在於模擬時間短，且針對極端值也能夠模擬出結果。 本篇文章將蒙地卡羅法作為比較的基準，以Glasserman, and Li (Management Science, 51(11), 1643-1656, 2005) 所提出的二階段重要性取樣法，我們稱為GIS，以及將Chiang et al. (Journal of Derivatives, 15(2), 8-19, 2007) 所提出的重要性取樣法加以改良，我們稱為MIS，針對bassamboo et al. (Operations Research, 56(3), 593-606, 2008) 所提出的極值相依模型，也就是t關聯結構模型進行模擬研究，並根據模擬出來的數值結果判斷重要性取樣法的估計效益，此外，我們也會對常態關聯結構模型進行模擬。依據模擬結果我們發現到，整體而言，在模擬時間上，MIS法所花費的時間較GIS法來得少，在準確率方面，MIS法一樣是比GIS法來的準確，也較為穩定，且MIS法所達到的變異數縮減效果更佳。

蒙地卡羅模擬是在組合信用風險的管理上相當實用的計算工具。衡量組合信用風險時，必須以適當的模型描述資產間的相依性。常態關聯結構是目前最廣為使用的模型，但實證研究認為 t 關聯結構更適合用於配適金融市場的資料。在本文中，我們採用 Bassamboo et al. (2008) 提出的極值相依模型建立 t 關聯結構用以捕捉資產之間的相關性。同時，為增進蒙地卡羅法之收斂速度，我們以 Chiang et al. (2007) 的重要性取樣法為基礎，將其拓展到極值相依模型下，並提出兩階段的重要性取樣技巧確保使用此方法估計一籃子信用違約時，所有模擬路徑均會發生信用事件。數值結果顯示，所提出的演算法皆達變異數縮減。而在模型自由度較低或是資產池較大的情況下，兩階段的重要性取樣法將會有更佳的估計效率。我們也以同樣的思路，提出用以估計投資組合損失機率的演算法。雖然所提出的演算法經過重要性取樣的技巧後仍無法使得欲估計的事件在所有模擬路徑下都會發生，但數值結果仍顯示所提出的方法估計效率遠遠優於傳統蒙地卡羅法。 / Monte Carlo simulation is a useful tool on portfolio credit risk management. When measuring portfolio credit risk, one should choose an appropriate model to characterize the dependence among all assets. Normal copula is the most widely used mechanism to capture this dependence structure, however, some emperical studies suggest that $t$-copula provides a better fit to market data than normal copula does. In this article, we use extremal depence model proposed by Bassamboo et al. (2008) to construct $t$-copula. We also extend the importance sampling (IS) procedure proposed by Chiang et al. (2007) to evaluate basket credit default swaps (BDS) with extremal dependence and introduce a two-step IS algorithm which ensures credit events always take place for every simulation path. Numerical results show that the proposed methods achieve variance reduction. If the model has lower degree of freedom, or the portfolio size is larger, the two-step IS method is more efficient. Following the same idea, we also propose algorithms to estimate the probability of portfolio losses. Althought the desired events may not occur for some simulations, even if the IS technique is applied, numerical results still show that the proposed method is much better than crude Monte Carlo.