The Research Of Several Problems About Pricing Options Based On The Fractional Brownian Motion Environment

The problem of options pricing is an important scope of financial engineering, many scholars have researched this problem in these years. The most famous work is the found of Black-Scholes formula in 1973, which is also became the foundation of research work in the following years. The Black-Scholes formula has an important assumption: the evolution of underlying asset follows the Brownian motion. However, due to these years' study, the security market run following the general fractional Brownian motion instead of the Brownian motion. Therefore, using the fractional Brownian motion to find the options' price is a major research field nowadays.In this paper, it introduced the fractional Brownian motion firstly, and derived the Black-Scholes formula under the fractional Brownian motion. It developed the pricing of perpetual American option with dividend under the fractional Brownian motion, through the study of American option's properties. It gave the European option pricing with Time-varying parameters under the fractional Brownian motion. Through function transformation, it changed this problem to the non Time-varying parameters problem, and it got the European option pricing at the end. Finally, through quoted the empirical analysis result of document[21}, it showed the Chinese stock market doesn't follow the random walk model, besides, it showed the biased random walk and Long-term memory characteristics. This is the starting point for this paper: the underlying asset follows the fractional Brownian motion.