Maximun Of Brownian Motion And Exotic Option

The maximum of Brownian motion with drift is often encountered in the pricing of exotic option, we apply the theory of the maximum of Brownian motion to barrier option and lookback option, and obtain their pricing formulas.In this paper two problems on Exotic option pricing was considered emphatically: pricing up-out call European option with varied barrier on underlying assets driven by the exponential Ornstein-Uklenbeck stochastic process ,giving a new method of option pricing on lookback option with fixed executive price.Firstly, The distribution of the maximum of Brownian motion with drift and the joint distribution of it with the terminal value of its underlying Brownian motion were given by the reflection principle and Girsanov theorem. Then the above results were applied to barrier option, thus a simple and effective piecing expressions was derived. On the basis of it, we refered to the exponential Ornstein-Uklenbeck stochastic process model. Under the model, parameterαenable the stock price to drop after the stock price rises to a certain height. Thus the model conformed to the actual character of stock price much more. The price formula of up-out call options with varied barriers under Ornstein-Uklenbeck process was obtained by Girsanov theorem, martingale theory and variable substitution.Secondly,In the chapter four, The pricing of lookback option with fixed executive price was discussed. Firstly the density function of maximum of Brownian motion was obtained through its distribution, then we carried on the integral to the density function, thus produced its pricing formula. It is a simple and direct-viewing new method .