Options Pricing In A Market Structure Combined Model With Stochastic Jump Risks And Random Intensity

In the early 1970s, Black and Scholes have successfully deduced the Partial Differential Equation (P.D.E.) for a vanilla derivatives based on the underlying stock with non-divident paying using the non-arbitrage principle and the portfolio approach in the celebrated paper entitled " The Pricing of Options and Corporate Liabilities ", and obtained the closed-form solutions of prices for the European call option and put option by view of the P.D.E above, that is, the celebrated Black-Scholes option pricing formula in which the underlying stock is assumed to be followed the geometrical Brownian Motion. Hereafter, the underlying stock's price following the Jump-diffusion model is firstly introduced by Merton who considered the stock's price rose suddenly falls suddenly and also obtained the closed -form solution for the European option. With the develop of the option pricing research works, some scholars considered the stochastic volatility model in which the underlying stock's instantaneous volatility is assumed to be satisfied another stochastic process related to the underlying stock. However, there are many uncertainty factors and unexpected events which cause the jump risks occurring and influencing both the underlying stock's prices and the market structure changing such as interest rates, volatility, etc. Therefore, considering the options pricing under the market structure with stochastic risks jump and random intensity is very meaningful works and whose research result remains lack, this paper considers the European options and American option pricing under this combined model, the main works are as follows:Chapter 1 introduces briefly the research significances on the financial devivatives, option pricing literatures, and the structure of essay.Chapter 2 considers the European option pricing under the market structure confined model with stochastic risks jump and random intensity. Using the Fourier inverse transform and the Feynman-Kac formula to the partial differential equation for the European stock option price, the closed-form solutions for the European options are obtained. Finally, some numerical results are proved to analyze the effects of both the jumping parameters or not on the option price.Chapter 3 based on the financial market model in chapter 2, we consider the pricing American options. First, applying the compound option pricing ideas, the pricing formulas of the Bermuda with one to three predetermined executable dates are obtained. Second, the approximated pricing formula of the American put option is obtained by view of the Richardson interpolation method. Finally, Some examples are provided to explain the impacts of model parameters on the option price.Chapter 4 sums up our main results in the article, and provides the future direction of further research works.