The Power Option Pricing Under The Fractional Brownian

The B-S option pricing formula which is proposed by Black and Scholes in1973has been widely applied to the pricing analysis of derivative securities in financial market. Scholars at home and abroad have already done a lot of research work and obtained many meaningful results. But in recent years, a large number of empirical research results show that stock price in market does not obey geometric Brownian motion. However, it presents a distribution of "spike, fat tail". The volatility of stock price is not driven by a random walk. What’s more, it has longtime-related and self-similar characteristics at different time. So geometric Brownian motion can not well characterize these features of stock price. While, fractional Brownian motion exactly has longtime-related and self-similar characteristics. Therefore, using fractional Brownian motion to study the option pricing problem has more practical significance.Power option is an important new option. This paper develops reasonable pricing for power option with Hurst parameter H,1/2<H<1. First, based on the environment that stock price follows fractional Brownian motion, we study pricing formula of power option with dividends. In addition, dividens and interest rate rely on the time which is non-random. Second, based on Ho-Lee model, we derive the power option pricing formula with stochastic interest rate in the environment that stock price follows fractional Brownian motion. Then, in the environment that stock price follows fractional Brownian motion, we derive the pricing formula of power option with jump-diffusion. Finally, using Theorem1pricing formula we give the numerical results of power option.