Pricing of Bi-direction European Option Under the mixed Brownian-fractional Brownian model

Assuming that the stock price obeys the stochastic differential equation driven by the mixed Brownian-fractional Brownian motion, we establish the mathematical model for the financial market in the mixed Brownian-fractional Brownian motion setting with Hurst parameter greater than 0.5. Under the fractional risk neutral measure, we get the unique equivalent measure by using fractional Girsanov theorem. With quasi-martingale method, we obtain the general pricing formula for the Bi-direction European option, which makes the fractional Brownian motion as an especial case.