過去實證研究發現，資產的動態過程存在不連續的跳躍與大波動伴隨大波動的波動度叢聚現象而造成資產報酬分配呈現出厚尾與高狹峰的情況，然而，此現象並不能完全被傳統所使用幾何布朗運動模型與跳躍擴散模型給解釋。因此，本文設定資產模型服從Lévy 過程中Generalized Hyperbolic (GH)的normal inverse Gaussian(NIG) 和 variance gamma (VG)兩個模型，然而，Lévy 過程是一個跳躍過程，是屬於一個不完備的市場，這將使得平賭測度並非唯一，因此，本文將採用Gerber 和 Shiu (1994)所提的Esscher 轉換來求得平賭測度。關於美式選擇權將採用LongStaff and Schwartz (2001)所提的最小平方蒙地卡羅模擬法來評價美式選擇權。實證結果發現VG有較好的評價績效，此外，進一步探討流動性與價內外的情況對於評價誤差的影響，亦發現部分流動性高的樣本就較小的評價誤差；此外，價外的選擇權其評價誤差最大。另一方面從交易的觀點來看，次貸風暴後交易對手信用風險愈來愈受到重視，此外，近年來由於巨災事件的頻傳，使得傳統保險公司風險移轉的方式，漸漸透過資本市場發行衍生性商品來進行籌資，以彌補其在巨災發生時所承擔的損失。因此，透過發行衍生性商品來進行籌資，必須考量交易對手的信用風險，否則交易對手違約，就無法獲得額外的資金挹注，因此，本文評價巨災權益賣權，並考量交易對手信用風險對於其價格的影響。 / In the traditional models such as geometric Brownian motion model or the Merton jump diffusion model can’t fully depict the distributions of return for financial securities and the those return always have heavy tail and leptokurtic phenomena due to the price jump or volatilities of return changing over time. Hence, the first article uses two time-changed Lévy models: (1) normal inverse Gaussian model and (2) variance gamma model to capture the dynamics of asset for pricing American option. In order to deal with the early-exercised problem of the American option, we use the LSM to determine the optimal striking point until maturity. In the empirical analyses, we can find the VG model have better performance than the other three models in some cases. In addition, with the comparison the pricing performance under different liquidity and moneyness conditions, we also find in some samples increasing the liquidity really can reduce the pricing errors, at the same time, the maximum pricing errors appears in the OTM samples in all cases. The global subprime crisis during 2008 and 2009 arouses much more attention of the counterparty risk and the number of default varies with economic condition. Hence, we investigate the counterparty risk impact on the price of the catastrophe equity put with a Markov-modulated default intensity model in the second study. At the same time, we also extend the stochastic interest rate setting in Jaimungal and Wang (2006) and relax some restrictive assumption of Black-Scholes model by taking the regime-switching effects of the economic status, then use the Markov-modulated processes to model the dynamics of the underlying asset and interest rate. In the numerical analyses, we illustrate the impact of the recovery rate, time to maturity, jump intensity of the equity and default intensity of counterparty on the CatEPut price.