Cholesky Decomposition is usually used to deal with the correlation problem among a financial product's underlying assets. However, Cholesky Decomposition inherently suffers from the requirement that all eigenvalues must be positive. Therefore, Cholesky Decomposition can't work very well when the number of the underlying assets is high. The report takes a diffrent approach called spectral Decomposition in attempt to solve the problem. But it turns out that although Spectral Decomposition can meet the requirement of all-positive eigenvalue, the decomposision error will be larger as the number of underlying asset getting larger. Thus, although Spectral Decomposition does offer some help, it works better when the number of underlying assets is not very large.

一籃子信用違約交換評價上並不存在公式解，一般是用蒙地卡羅模擬來推估商品價格，然而，因為蒙地卡羅執行速度較慢，往往會需要能夠大規模運行的計算資源以及高成本的硬體，為了減少成本和提高蒙地卡羅的效率就必須從其演算法改進，於是本文利用Chiang et al.(2007)所提出的一籃子信用違約交換演算法來提升一籃子信用違約交換的評價效率，但是該方法採用多元常態分佈假設下的Factor gaussian copula模型進行評價，並不符合市場實際金融市場資料具有不對稱的偏態現象，尤其對未來的環境危機發生的頻率不斷增加，極端事件可能出現的機會也越來越高，基於此問題，本文將Factor t copula、Factor clayton copula、Factor NIG copula以及Modify factor NIG copula與重要性抽樣演算法結合來提昇商品評價的準確度，並且分析各模型與該演算法結合的效果。

相較於單一信用違約交換只能對單一信用標的進行信用保護，一籃子信用違約交換則能對一籃子的信用標的進行信用保護。此種產品的評價決定於一籃子信用標的實體的聯合機率分配，因此多個標的資產間違約相關性的衡量，對於一籃子信用違約交換的評價和風險管理是相當重要的課題。 　　在一個資產池中，有時可以將其切割成兩個以上的群體，各群體間彼此相互獨立，而在各群內彼此相依。我們將其視為在多因子模型下的特例，此模型提供我們更具彈性的方式去建立資產之間彼此的相關性。 　　在這篇文章中，我們主要以 Chiang, Yueh, and Hsieh (2007) 在單因子模型下所提出來的方法為基礎，將其延伸至多因子的模型下的特例。藉由選擇一個合適的(IS)分配，在每一次的模擬中必定會有k個違約事件發生；因此我們獲得一個有效率的方法對一籃子違約交換進行評價，此演算法不僅簡單並且其變異數較蒙地卡羅小。 / In contrast to a single name credit default swaps which provides credit protection for a single underlying, a basket credit default swap extends the credit protection to portfolio of obligors with the restriction that the default of only one underlying is compensated. The price of the products depends on the joint default probability of the underlying in the credit portfolio. Thus, the modeling of default correlation, default risk and expected loss is a key issue for the valuation and risk management of basket default swaps. Sometimes a pool of underlying obligors can have two or more separate groups, between those they are unrelated, but in each part they are related. The special cases provide more flexible way to construct the correlation between two or more underlying obligors. In this paper, our approach is based on the construction of importance sampling (IS) method proposed by Chiang, Yueh and Hsieh (2007) under one-factor model, and then we extend the model to a special case under the multi-factor model. By the appropriate choice of the importance sampling distribution, we establish a way of ensuring that for every path generated, k default events always take place. Then we can obtain an efficiency algorithm for basket default swap valuation. The algorithm is simple to implement and it also guarantees variance reduction.