Pricing of Option with Power Payoff Driven by Mixed Fractional Brownian Motion

Assuming that the stock price obeys the stochastic differential equation driven by mixed fractional Brownian motion, we establish the mathematical model for the financial market in mixed 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 European call option with power payoff, which makes the fractional Brownian motion as an especial case. At same time, we get the explicit expression for the European put option with power payoff and the call-put parity.