This thesis explores option pricing and hedging in a discrete time regime-switching environment. If the regime risk cannot be hedged away, then we cannot ignore this risk and use the Black-Scholes pricing and hedging framework to generate a unique pricing and hedging measure. We develop a risk neutral pricing measure by applying an Esscher Transform to the real world asset price process, with the focus on the issue of incompleteness of the market. The Esscher transform turns out to be a convenient and effective tool for option pricing under the discrete time regime switching models. We apply the pricing measure to both single variate European options and multivariate options. To better understand the effect of the pricing method, we also compared the results with those generated from two other risk neutral methods: the Black-Scholes model, and the natural equivalent martingale method. We further investigate the difference in hedging associated with different pricing measures. This is of interest when the choice of pricing method is uncertain under regime switching models. We compare four hedging strategies: delta hedging for the three risk neutral pricing methods under study, and mean variance hedging. We also develop a more general tool of tail ordering for hedging analysis in a general incomplete market with the uncertainty of the risk neutral measures. As a result of the analysis, we propose that pricing and hedging using the Esscher transform may be an effective strategy for a market where the regime switching process brings uncertainty.

This thesis explores option pricing and hedging in a discrete time regime-switching environment. If the regime risk cannot be hedged away, then we cannot ignore this risk and use the Black-Scholes pricing and hedging framework to generate a unique pricing and hedging measure. We develop a risk neutral pricing measure by applying an Esscher Transform to the real world asset price process, with the focus on the issue of incompleteness of the market. The Esscher transform turns out to be a convenient and effective tool for option pricing under the discrete time regime switching models. We apply the pricing measure to both single variate European options and multivariate options. To better understand the effect of the pricing method, we also compared the results with those generated from two other risk neutral methods: the Black-Scholes model, and the natural equivalent martingale method. We further investigate the difference in hedging associated with different pricing measures. This is of interest when the choice of pricing method is uncertain under regime switching models. We compare four hedging strategies: delta hedging for the three risk neutral pricing methods under study, and mean variance hedging. We also develop a more general tool of tail ordering for hedging analysis in a general incomplete market with the uncertainty of the risk neutral measures. As a result of the analysis, we propose that pricing and hedging using the Esscher transform may be an effective strategy for a market where the regime switching process brings uncertainty.

The Lee-Carter mortality model provides the very first model for modeling the mortality rate with stochastic time and age mortality dynamics. The model is constructed modeling the mortality rate to incorporate both an age effect and a period effect. The Lee-Carter model provides the fundamental set up currently used in most modern mortality modeling. Various extensions of the Lee-Carter model include either adding an extra term for a cohort effect or imposing a stochastic process for mortality dynamics. Although both of these extensions can provide good estimation results for the mortality rate, applying them for the pricing of the mortality/ longevity linked derivatives is not easy. While the current stochastic mortality models are too complicated to be explained and to be implemented, transforming the cohort effect into a stochastic process for the pricing purpose is very difficult. Furthermore, the cohort effect itself sometimes may not be significant. We propose using a new modified Lee-Carter model with a Normal Inverse Gaussian (NIG) Lévy process along with the Esscher transform for the pricing of mortality/ longevity linked derivatives. The modified Lee-Carter model, which applies the Lee-Carter model on the growth rate of mortality rates rather than the level of mortality rates themselves, performs better than the current mortality rate models shown in Mitchell et al (2013). We show that the modified Lee-Carter model also retains a similar stochastic structure to the Lee-Carter model, so it is easy to demonstrate the implication of the model. We proposed the additional NIG Lévy process with Esscher transform assumption that can improve the fit and prediction results by adapting the mortality improvement rate. The resulting mortality rate matches the observed pattern that the mortality rate has been improving due to the advancing development of technology and improvements in the medical care system. The resulting mortality rate is also developed under a martingale measure so it is ready for the direct application of pricing the mortality/longevity linked derivatives, such as q-forward, longevity bond, and mortality catastrophe bond. We also apply our proposed model along with an information theoretic optimization method to construct the pricing procedures for a life settlement. While our proposed model can improve the mortality rate estimation, the application of information theory allows us to incorporate the private health information of a specific policy holder and hence customize the distribution of the death year distribution for the policy holder so as to price the life settlement. The resulting risk premium is close to the practical understanding in the life settlement market.

Dans cette thèse, on s’intéresse à la combinaison des méthodes de réduction de variance et de réduction de la complexité de la méthode Monte Carlo. Dans une première partie de cette thèse, nous considérons un modèle de diffusion continu pour lequel on construit un algorithme adaptatif en appliquant l’importance sampling à la méthode de Romberg Statistique Nous démontrons un théorème central limite de type Lindeberg Feller pour cet algorithme. Dans ce même cadre et dans le même esprit, on applique l’importance sampling à la méthode de Multilevel Monte Carlo et on démontre également un théorème central limite pour l’algorithme adaptatif obtenu. Dans la deuxième partie de cette thèse,on développe le même type d’algorithme pour un modèle non continu à savoir les processus de Lévy. De même, nous démontrons un théorème central limite de type Lindeberg Feller. Des illustrations numériques ont été menées pour les différents algorithmes obtenus dans les deux cadres avec sauts et sans sauts. / In this thesis, we are interested in studying the combination of variance reduction methods and complexity improvement of the Monte Carlo method. In the first part of this thesis,we consider a continuous diffusion model for which we construct an adaptive algorithm by applying importance sampling to Statistical Romberg method. Then, we prove a central limit theorem of Lindeberg-Feller type for this algorithm. In the same setting and in the same spirit, we apply the importance sampling to the Multilevel Monte Carlo method. We also prove a central limit theorem for the obtained adaptive algorithm. In the second part of this thesis, we develop the same type of adaptive algorithm for a discontinuous model namely the Lévy processes and we prove the associated central limit theorem. Numerical simulations are processed for the different obtained algorithms in both settings with and without jumps.

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.

住宅抵押貸款保險(Mortgage Insurance)為管理違約風險的重要工具，在2008年次級房貸風暴後更加受到金融機構的關注。為了能更準確且更有效率的預測房價及合理評價住宅抵押貸款保險，本文延續Christoffersen, Heston and Jacobs (2006)對股票報酬率的研究，提出新的GARCH模型，利用Inverse Gaussian分配取代常態分配來捕捉房價序列中存在的自我相關以及典型現象(stylized facts)，並且同時考慮房價市場中所隱含的價格跳躍現象。本文將新模型命名為IG-GARJI模型，以便和傳統GARCH模型作區分。由於傳統的GARCH模型在計算保險價格時，通常不存在封閉解，必須藉由模擬的方法來計算價格，會增加預測的誤差，本文提供IG-GARJI模型半封閉解以增進預測效率與準確度，並利用Bühlmann et al. (1996)提出的Esscher transform方法找出其風險中立機率測度，而後運用Heston and Nandi (2000)提出之遞迴方法，找出適合的住宅抵押貸款保險評價模型。實證結果顯示，在新建房屋市場中，使用Inverse Gaussian分配會比常態分配的表現要好；對於非新建房屋，不同模型間沒有顯著的差異。另外，本文亦引用Bardhan, Karapandža, and Urošević (2006)的觀點，利用不同評價模型來比較若房屋所有權無法及時轉換時，對住宅抵押貸款保險價格帶來的影響，為住宅抵押貸款保險提供更準確的評價方法。 / Mortgage insurance products represent an attractive alternative for managing default risk. After the subprime crisis in 2008, more and more financial institutions have paid highly attention on the credit risk and default risk in mortgage market. For the purpose of giving a more accurate and more efficient model in forecasting the house price and evaluate mortgage insurance contracts properly, we follow Christoffersen, Heston and Jacobs (2006) approach to propose a new GARCH model with Inverse Gaussian innovation instead of normal distribution which is capable of capturing the auto-correlated characteristic as well as the stylized facts revealed in house price series. In addition, we consider the jump risk within the model, which is widely discussed in the house market. In order to separate our new model from traditional GARCH model, we named our model IG-GARJI model. Generally, traditional GARCH model do not exist an analytical solution, it may increase the prediction error with respect to the simulation procedure for evaluating mortgage insurance. We propose a semi-analytical solution of our model to enhance the efficiency and accuracy. Furthermore, our approach is implemented the Esscher transform introduced by Bühlmann et al. (1996) to identify a martingale measure. Then use the recursive procedure proposed by Heston and Nandi (2000) to evaluate the mortgage insurance contract. The empirical results indicate that the model with Inverse Gaussian distribution gives better performance than the model with normal distribution in newly-built house market and we could not find any significant difference between each model in previously occupied house market. Moreover, we follow Bardhan, Karapandža, and Urošević (2006) approach to investigate the impact on the mortgage insurance premium due to the legal efficiency. Our model gives another alternative to value the mortgage contracts.

本文實證探討黃金期貨報酬率的特性並在標的黃金期貨價格遵循狀態轉換跳躍擴散過程時實現美式選擇權之評價。在這樣的動態過程下，跳躍事件被一個複合普瓦松過程與對數常態跳躍振幅所描述，以及狀態轉換到達強度是由一個其狀態代表經濟狀態的隱藏馬可夫鏈所捕捉。考量不同的跳躍風險假設，我們使用Merton測度與Esscher轉換推導出在一個不完全市場設定下的風險中立黃金期貨價格動態過程。為了達到所需的精確度，最小平方蒙地卡羅法被用來近似美式黃金期貨選擇權的價值。基於實際市場資料，我們提供實證與數值結果來說明這個動態模型的優點。 / This dissertation empirically investigates the characteristics of gold futures returns and achieves the valuation of American-style options when the underlying gold futures price follows a regime-switching jump-diffusion process. Under such dynamics, the jump events are described as a compound Poisson process with a log-normal jump amplitude, and the regime-switching arrival intensity is captured by a hidden Markov chain whose states represent the economic states. Considering the different jump risk assumptions, we use the Merton measure and Esscher transform to derive risk-neutral gold futures price dynamics under an incomplete market setting. To achieve a desired accuracy level, the least-squares Monte Carlo method is used to approximate the values of American gold futures options. Our empirical and numerical results based on actual market data are provided to illustrate the advantages of this dynamic model.