Pricing Quanto Options, Quanto Reset And Quanto Extremum Options In A Jump-difussion Model

Options are financial derivatives, which have been widely used in hedging , risk management and price speculation in 70's. Since Black and Scholes made a major breakthrough in the pricing options in 1973, the option market has been developed quickly. Pricing option theory has become the key of modernal mathematical finance, and many research results have been gained. However, these results are almost based on the Black-Scholes model where the underlying asset price is assumed to follow geometry Brown motion. With the deepness of study in financial practice, especially, serious concerning on the recent rare financial events and many questions of financial reform, etc. the Black-Scholes model is found to be unable to fit the changes in the financial market. Recently, many empirical studies show that there exist systemstic difference between the Black-Scholes model and prictice. In 1976, Merton firstly established a compound Poisson jump-diffusion model where the jump risks are unsystematic and jump magnitude of the log of the asset price is assumed to be a normal distribution, and considered option pricing of European option and gained even more practicable results. Because a jump-difussion Model can picture (asymmetric leptokurtic) features of the asset returns, it becomes a hot model in research. However, studies are not enough.The quanto options are contracts which invest to foreign assets and their payoffs depend on not only the price of foreign asset, but also the affect of exchange rate and domestic and foreign interest rates. They have been widely used in international finance, trade or other investment field. Quanto reset options or quanto extremum options add reset or extremum features to the vanilla quanto options contracts and have double advantages of quanto and reset or extremum, thereby further strengthen investor's ability to guard against financial risks. In this thesis, we discuss option pricing of quanto European options, quanto reset options and quanto extremum options on two conditions: 1) foreign stock price (exchange rate) follows a jump-diffusion Model; 2) interest rates are constant or stochastic respectively. First, we derive the closed form solutions for the above options respectively by using martingale method. Second, we compare and analyze these results in this model by the help of numerical examples. Our main contributions are as follow:Chapter 1 first provides an introduction to the necessity and significance of research on option pricing and elaborates the academic literature. Second, we introduce some correlative preparatory knowledge. Furthermore, we provide our motivations and main study topics in this dissertation.In Chapter 2, we first derive the analytic price formulas for four types of quanto European call options on the condition that foreign stock price follows a jump-difussion process and the domestic and foreign interest rates are constant. Second, we compare these results with those in Black-Scholes model. Furthermore, we assume that exchange rate also follows another jump-difussion process. Similarly, we derive the analytic price formulas for four types of quanto European call options and compare these results with the former.In Chaper 3, we extend the results in Chapter 2 to more general occasions where the domestic and foreign interest rates follow the Hull-White stochastic model. Similarly, we derive the closed form solutions for four types of quanto European call options on the condition that only stock price follows a jump-difussion process or further exchange rate also follows another jump-difussion process. Finally, we give the computational example.In Chaper 4, we apply the market model in Chapter 2 and Chapter 3 to the pricing of quanto reset options. Analogously, we derive the analytic price formulas for four types of the quanto reset European call options and test the effects of reset features to options price.In Chaper 5, we apply the market model in Chapter 2 and Chapter 3 to the pricing of quanto extremum options. In the same way, we derive the analytic price formulas for four types of the quanto maximum European call options and compare these results in Merton model with those in Black-Scholes model.