Option Pricing Under Fractional Brownian Motion Environment

In this paper we consider two problems on option pricing under fractional Brownian motion environment: exchange option pricing under fractional Brownian motion environment, and European option pricing with transaction costs under fractional Brownian motion environment.First of all, we set up the model of exchange option using the theory of European option pricing under fractional Brownian motion environment, then obtain the exchange Call and Put option pricing formulas using insurance actuarial method, and point out that the traditional exchange option pricing formula which was obtained by Margrabe(1978) is a special case of our pricing formula. Besides, we compare the formula to the general fractional B-S formula, and point out that the general fractional B-S formula is also a special case of our result. Then we construct an investment portfolio to get the Call-Put parity of exchange option under fractional Brownian motion environment. At the end of the second part, we obtain the exchange option pricing formula with dividends under fractional Brownian motion environment.Secondly, in the third part, we obtain the European option pricing formula with transaction costs under fractional Brownian motion environment, through amending the volatility of the stock in the frictionless market. As an application of this formula, we also consider the problem of Deductible option pricing, and then give the option pricing formula.