Option Pricing In A Random Environment

Option theory is one of the greatest findings in the area of the world's economics in the 20th century. The research on option pricing theory is focused on the following two aspects: one is how to design new option to satisfy the changing investment demand, the other is how to price the more and more complicated options. In 1973, based upon the stock's lognormal distribution and hedging continously, Black and Scholes launched the famous B-S pricing model in efficient market successfully. Later, this model is improved and generalized by some home and abroad scientists. This paper discuss the option pricing by synthesising several factors, including the influence of random environment.In this thesis, under the assumptions that the volatility, dividend and the risk-free rate are all known functions of time and there exist transaction costs and exchange rate, we first study how to price options if the price variation of underlying stock is not a continuous stochastic process, but a jump diffusion stochastic process. By hedging strategy, the risk caused by stock normal volatility will be hedged, then a European option pricing equation and formulae whose underlying stock pricing process is mixed process are gotten. Then, considering the risk bringed by Poisson jumps, the option pricing equation is gotten by minimal-variance hedging strategy. After this, we study option pricing when stock prices are influenced by random environment. For this model a European option pricing formulae is gained by methods of equivalent martingale measure.Morever, we give an improved option pricing formulae in the random environment, including the change of volatility, risk-free rate and appreciation rate, and give a European call option pricing formulae in the special random environment .