Application & Studies On The Option Pricing Under The Stock Price Processes Are Jump And Diffusion Process

The theory and practice of derivative securities have been developed very fast in the recent twenty years all over the world. Many mathematical scientists and financial economists pay more attention to the problems on options and investment consumption .Since 1969, in which the first Economics Nobel prize was awarded, there are great many Nobel prize-winners have been awarded for their outstanding contributions in the field of finance and mathematical finance. Effective management of risk occupies the right evaluation of derivative securities. The critical thing that the finance derivative securities exist reasonably and develop properly is how to value its fair price. Among all the pricing systems, the investigation of option pricing is most extensive. The reasons for this are: 1) Compared to other derivative securities, option is easy to price. 2) Many derivative securities appear in the form of option. 3) The pricing principles are same to all sorts of derivative securities, so it is possible to find pricing theory of common derivative securities through the option pricing methods.This dissertation is intend to study option pricing problems, so as to establish the mathematic module of option pricing with jump diffusion process by means of mathematical tools such as martingale theory and stochastic analysis, to deduce the option pricing equation and reasonable value; At the same time, some helpful mathematical conclusions will be reached through the research, trying to display the dialectical relationship between mathematics and finance from an aspect, that is to say: mathematics is the powerful tool for financial research, and vice versa, financial practice promotes the development of mathematical theory.This dissertation is divided into four chapters: Chapter one is preface, which summarizes the significance, origin, development.In the chapter two, I introduce the existing models of stock price, and mostly aim at the Black-Scholes option pricing present its foreground, expand, problems and remediation.In the chapter three, the variation in price of stock not only has a continuous of time, but also has brusque variation (release of corporation recombine, an unexpected economic figure, major political changes or natural disaster). We call the discontinuous function of time"jump", under the hypothesis of stock price submitting to jump-diffusion process model, we gets its option pricing formula is: Besides, in the base we get the European call option pricing formula with jump-diffusion process: At last introduces the"Thomas C. Schelling's spot"theory under the game theory frame to prove the inevitable trend of real estate mortgage development. According to the property of the insurance money the real estate mortgage insurance is regarded as the put options, Then it has established the model and carries on the solution, and discover the conclusion is tally with practical instance.The chapter four summarized this dissertation and prospect.