The Study Of Option Pricing Model In Fractional Brownian Motion Environment

Uncertain pricing is one core of financial mathematics study, it involves the theories of modern finance such as asset pricing theory, investment combination theory and the theories of modern mathematics, such as stochastic analyzing and optimizing theory too. Effective management of risk occupies the right evaluation of derivative securities. The critical thing is that the financial derivative securities exist reasonably and develop properly is how to value its fair price.Classical B-S model was established in Brownian motion environment. However, empirical research shows that stock price has long-range dependence as well as self-similar properties, so many researchers began to study stock price with fractional Brownian motion that has these two properties in recent years. Brownian motion is a special case of fractional Brownian motion. Therefore, the study of option pricing in fractional Brownian motion environment is much more wide and practical.This dissertation is intended to study some exotic option pricing problems, so as to establish the mathematic model of option pricing in fractional Brownian motion environment, and the innovation of this dissertation is: First, we get gap option and binary option pricing in fractional Brownian motion environment. Gap option is an exotic option, its earnings doesn't depend on the strike price, but depends on a constant G (called gap). Binary option is also an exotic option, its value depends on whether the price of underlying asset is higher than strike price. Second, we get power option pricing in fractional Brownian motion environment. Power option is an exotic option as well, its payoff function is [ h ( S (T )) ? K ]+, here h ( x )= xα(α> 0 is constant). Then, we extended option pricing of some exotic options.