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.

Bassamboo, A., Juneja, S., and Zeevi, A. (2008), Portfolio credit risk
with extremal dependence: Asymptotic analysis and efficient simulation,
Operations Research, 56(3), 593--606.

Bruyere, R., Cont, R., and Smart, G. (2006), Credit Derivatives and
Structured Credit: A Guide for Investors, Chichester, UK: Wiley.

Chaplin, G. (2005), Credit Derivatives: Risk Management, Trading &
Investing, Chichester, UK: Wiley.

Chen, Z. and Glasserman, P. (2008), Fast pricing of basket default
swaps, Operations Research, 56(2), 286--303.

Chiang, M.H., Yueh, M.L., and Hsieh, M.H. (2007), An efficient
algorithm for basket default swap valuation, Journal of
Derivatives, 15(2), 8--19.

Glasserman, P. (2004), Monte Carlo Methods in Financial Engineering,
volume~53 of Stochastic Modelling and Applied Probability, New York:
Springer Verlag.

Glasserman, P. and Li, J. (2005), Importance sampling for portfolio
credit risk, Management Science, 51(11), 1643--1656.

Gupton, G.M., Finger, C.C., and Bhatia, M. (1997), Credit Metrics
Technical Document, New York: J.P. Morgan & Co.

Hull, J. and White, A. (2004), Valuation of a cdo and an nth to
default cds without monte carlo simulation, Journal of Derivatives,
12(2), 8--23.

Joshi, M.S. and Kainth, D. (2004), Rapid and accurate development of
prices and greeks for nth to default credit swaps in the {Li model,
Quantitative Finance, 4, 266--275.

Kalemanova, A., Schmid, B., and Werner, R. (2007), The normal inverse
gaussian distribution for synthetic cdo pricing, The Journal of
Derivatives, 14(3), 80--94.

Laurent, J.P. and Gregory, J. (2005), Basket default swaps, {CDOs and
factor copulas, Journal of Risk, 7(4), 103--122.

Li, D.X. (2000), On default correlation: A copula function approach,
Journal of Fixed Income, 9, 43--54.

Lindskog, F. and RiskLab, ETH (2000), Modelling dependence with
copulas and applications to risk management, Swiss Federal Institute
of Technology Zurich.

Mashal, R. and Zeevi, A. (2002), Beyond correlation: Extreme
co-movements between financial assets, Working paper, Columbia
University.

Schonbucher, P.J. (2003), Credit Derivatives Pricing Models:
Models, Pricing and Implementation, New York: Wiley.

Zheng, H. (2006), Efficient hybrid methods for portfolio credit
derivatives, Quantitative Finance, 6(4), 349--357.