Fractional Brownian Motion In Asset Pricing

Compared with the standard Brownian motion, the fractional Brownian motion has a long-term dependent nature.It makes fractional Brownian motion can be used to characterize or describe long-term relevance of stock price, financial volatility and "fat tail "distribution of asset yields. So the fractional Brownian motion is widely used in the financial market pricing and many other areas.The main topic of this paper is the application in asset pricing of fractional Brownian motion.Based on the previous work,this paper will attempt to deduce pricing formula of the continuous geometric average Asian option an European power options,and this paper will obtain dynamic CAPM from dynamic linear evaluation rule in fractional Brownian environment.The main contents of this paper are as follows:For continuous geometric average Asian option pricing formula, in order to characterize the long-term correlation of the stock price better, many researchers expanded the underlying asset price process from standard Brownian motion to fractional Brownian motion, but interest rates and other parameters is constant. Some researchers extended interest rates model to the stochastic interest rates model, but they just deduced standard European option pricing formula.In this part of paper, we will deduce the pricing formula of continuous geometric average Asian option with path-dependent.Under the assumption of the stock price obeying the stochastic differential equation driven by fractional Brownian motion and interest rate obeying fractional Vasicek model, we obtain the general pricing formula of continuous geometric average Asian option.It is one of the focus of this paper, and it is also an important application of fractional Brownian motion in this paper.Power option is a typical new exotic option, whose value is a linear function of the power function of the underlying asset price.Under the assumption of the stock price obeying Fractional O-U process and the interest rate obeying fractional Vasicek model, using Risk-neutral pricing formula and the theory of stochastic analysis.We obtain the general pricing formula of European power options.About Capital Asset Pricing Model, many scholars are working to extend this model.Based on a number of previous studies, and combined with fractional Girsanov theorem and Clark-Haussmann-Ocone theorem, this paper will obtain dynamic CAPM from dynamic linear evaluation rule in fractional Brownian environment.