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.

本研究利用GARCH動態過程的優點捕捉匯率報酬率之異質變異與波動度叢聚性質，並以GARCH動態過程為基礎，考慮跳躍風險服從Lévy過程，再利用特徵函數與快速傅立葉轉換方法推導出GARCH-Lévy動態過程下的歐式匯率選擇權解析解。以日圓兌換美元(JPY/USD)之歐式匯率選擇權為實證資料，比較基準GARCH選擇權評價模型與GARCH-Lévy選擇權評價模型對市場真實價格的配適效果與預測能力。實證結果顯示，考慮跳躍風險為無限活躍之Lévy過程，即GARCH-VG與GARCH-NIG匯率選擇權評價模型，不論是樣本內的評價誤差或是在樣本外的避險誤差皆勝於考慮跳躍風險為有限活躍Lévy過程的GARCH-MJ匯率選擇權評價模型。整體而言，本研究發現進行匯率選擇權之評價時，GARCH-NIG匯率選擇權評價模型有較小的樣本內及樣本外評價誤差。 / In this thesis, we make use of GARCH dynamic to capture volatility clustering and heteroskedasticity in exchange rate. We consider a jump risk which follows Lévy process based on GARCH model. Furthermore, we use characteristic function and fast fourier transform to derive the currency option pricing formula under GARCH-Lévy process. We collect the JPY/USD exchange rate data for our empirical analysis and then compare the goodness of fit and prediction performance between GARCH benchmark and GARCH-Lévy currency option pricing model. The empirical results show that either in-sample pricing error or out-of-sample hedging performance, the infinite-activity Lévy process, GARCH-VG and GARCH-NIG option pricing model is better than finite-activity Lévy process, GARCH-MJ option pricing model. Overall, we find using GARCH-NIG currency option pricing model can achieve the lower in-sample and out-of sample pricing error.

由於違約事件不斷發生以及在財務實證上顯示證券的報酬率有厚尾與高狹峰的現象，本文使用縮減式模型與Lévy過程來評價有信用風險下的可轉換公司債。在Lévy過程中，本研究假設股價服從NIG及VG模型，發現此兩種模型比傳統的GBM模型更符合厚尾現象。此外，在Lévy過程參數估計方面，本文使用最大概似法估計參數，在評價可轉換公司債方面，本研究採用最小平方蒙地卡羅法。本文之實證結果顯示，Lévy模型的績效比傳統GBM模型佳。 / Due to the reason that the default events occurred constantly and still continue taking place, empirical log return distributions exhibit fat tail and excess kurtosis, this paper evaluates convertible bonds under Lévy process with default risk using the reduced-form approach. Under the Lévy process, the underlying stock prices are set to be normal inverse Gaussian (NIG) and variance Gamma (VG) model to capture the jump components. In the empirical analysis, we use the maximum likelihood method to estimate the parameters of Lévy distributions, and apply the least squares Monte Carlo Simulation to price convertible bonds. Five examples are shown in pricing convertible bonds using the traditional model and Lévy model. The empirical results show that the performance of Lévy model is better than the traditional one.

本研究以Lévy過程為模型基礎，考慮Merton Jump及跳躍強度服從Hawkes Process的Merton Jump兩種跳躍風險，利用Particle Filter方法及EM演算法估計出模型參數並計算出對數概似值、AIC及BIC。以S&P500指數為實證資料，比較隨機波動度模型、Variance Gamma模型及兩種不同跳躍風險對市場真實價格的配適效果。實證結果顯示，隨機波動度模型其配適效果勝於Variance Gamma模型，且加入跳躍風險後可使模型配適效果提升，尤其在模型中加入跳躍強度服從Hawkes Process的Merton Jump，其配適效果更勝於Merton Jump。整體而言，本研究發現，以S&P500指數為實證資料時，SVHJ模型有較好的配適效果。 / This paper, based on the Lévy process, considers two kinds of jump risk, Merton Jump and the Merton Jump whose jump intensity follows Hawkes Process. We use Particle Filter method and EM Algorithm to estimate the model parameters and calculate the log-likelihood value, AIC and BIC. We collect the S&P500 index for our empirical analysis and then compare the goodness of fit between the stochastic volatility model, the Variance Gamma model and two different jump risks. The empirical results show that the stochastic volatility model is better than the Variance Gamma model, and it is better to consider the jump risk in the model, especially the Merton Jump whose jump intensity follows Hawkes Process. The goodness of fit is better than Merton Jump. Overall, we find SVHJ model has better goodness of fit when S&P500 index was used as the empirical data.

本研究顯著地發展時間轉換Lévy過程在最適投資組合的運用性。在連續Lévy過程模型設定下，槓桿效果直接地產生跨期波動度不對稱避險需求，而波動度回饋效果則透過槓桿效果間接地發生影響。另外，關於無窮跳躍Lévy過程模型設定部分，槓桿效果仍扮演重要的影響角色，而波動度回饋效果僅在短期投資決策中發生作用。最後，在本研究所提出之一般化隨機波動度不對稱資產報酬動態模型下，得出在無窮跳躍的資產動態模型設定下，擴散項仍為重要的決定項。 / This study significantly extends the applicability of time-changed Lévy processes to the portfolio optimization. The leverage effect directly induces the intertemporal asymmetric volatility hedging demand, while the volatility feedback effect exerts a minor influence via the leverage effect under the pure-continuous time-changed Lévy process. Furthermore, the leverage effect still plays a major role while the volatility feedback effect just works over the short-term investment horizon under the infinite-jump Lévy process. Based on the proposed general stochastic asymmetric volatility asset return model, we conclude that the diffusion term is an essential determinant of financial modeling for index dynamics given infinite-activity jump structure.