Research On Applications Of Martingale Method In Contingent Claim Pricing

Contingent claims pricing is one of the kernel problems in financial calculus, whose development promotes many branches of mathematical field. The theory of contingent claims pricing has a direct influence on the innovating of financial tools and the effective performance of financial market. It also has wide applications to the investment strategies of the companies, the evaluation of the research object and the risky management in the financial institution. Meanwhile, martingale analysis is one of the most powerful tools in congtingent claims pricing. Thus using martingale method to study option-pricing not only has important theoretic significance but also the practical value.This dissertation mainly studies some option-pricing problems under exponential Ornstein-Uhlenbeck process model by means of mathematical tools such as martingale theory and stochastic analysis, and also provides the pricing formulas of European options with change exercise price and geometric average Asian options and lookback options. At the same time, it also discusses power payoffs options and power-function innovative options. The following are this paper's main results and innovations:(1) This dissertation proposes a new hypothesis of stock-price model, namely, choosing an exponential O-U process which can reflect fluctuation of appreciation rate of the stock price to describe the underlying stock-price, and then establishes option pricing model based on exponential O-U process. The problem of option pricing whose exercise price is a random variable is taken into account. Besides, European option pricing formula with change exercise price and the put-call parity relation are obtained.(2) The innovative types of financial derivatives--exotic options are taken into consideration, of which the most representative geometric average Asian options and lookback options driven by O-U process are studied.(3) Under the framework of equivalent martingale measures, European power function innovative options are discussed. This dissertation assumes that the risk-free rate of interest, the expected rate of return, the volatility for the stock price are deterministic functions of the time. Using the martingale method proposed by Harrison and Kreps(1979), closed options pricing formulas are gained, and then improved and promoted.