1 Option Pricing in a Fractional Brownian Motion Environment

The purpose of this paper is to obtain a fractional Black-Scholes formula for the price of an option for every []Tt,0Î, a fractional Black-Scholes equation and a risk-neutral valuation theorem if the underlying is driven by a fractional Brownian motion ( ) 121 , << HtBH. For this purpose we will first prove some results regarding the quasi-conditional expectation, especially the behavior to a Girsanov transform. We will also compare our results with the classical results based on the standard Brownian motion and we conclude that in the case of the fractional Brownian motion the price of the option no longer depends only on tT-