Jump - Diffusion Process Option Pricing Model

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 and more attention to the problems on options and investment/consumption. Sincel969, in which the first Economics Nobel prize was awarded, the Nobel prize-winners amounted to 7, for their outstanding contributions in the field of finance and mathematical finance. Effective management of risk occupies the right evalution of derivative securities. The critical thing that the financial 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.Option pricing theory, the important part of modern finance, has promoted the prosperity of financial market. Together with the portfolio selection theory, the capital asset pricing theory, the effectiveness theory of market and acting is-sue, it is regarded as one of the five theory modules in modern finance.This dissertation is intened to study option pricing problems, so as to estab.3.)?lish 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, and to attempt to obtain the results, which are instructive to financial practice and easy to operate. 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 methematical theory.First, the paper summarizes the significance, origin, development, academic trends and research methods of option pricing in Chapter One. And the rest are divided into 4 parts. The first part consists of Chapter Two and Three, which concerns the option pricing research of the stock price variation regularity. Chapter Two establishes the option pricing module that the stock price processes are Possion jump-diffusion processes, duduces the option pricing equation, presents the European option pricing formula, proves that the unobservable parameters are not real transition probability but some other parameters in option pricing formula and also presents the assessment methods of parameters. Chapter Three assumes that jump process is a more common process than Possion process pure birth jump process, sets up pure birth jump-diffusion module of the stock price processes, and deduces the European pricing formula under the assumption of risk-neutral. The second part consists of Chapter Four and Five, which concerns stochastic volatility and stochastic interest rate. ChapterFour deals with the option pricing with stochastic volatility, sets up the stock.4.price dynamics with two states of volatility and deduce the European option pricing formula under the assumption that the stock price processes are continuous and jump risk can抰 be priced. Chapter Five is concerned with the option pricing with stochastic interest rate, sets up the option pricing module with Continuous interest rate and uncontinuous interest rate respectively, and presents the option pricing formula under the assumption that the stock price processes are jump-diffusion processes. The third part is made up of Chapter Six and Seven, which deals with exotic option pricing study. Chapter Six deals with the option pricing of exchange one asset for another, sets up option pricing mo...