Option Pricing In Fractional Brownian Motion With Some Jumps

Option pricing is a very important and complex problem in Financial engineering,and many scholars have studied on it.As a landmark in option pricing work,since the advent of the classical Black-Scholes formula[3],it has been widely recognized and lay the foundation for the following research.But in recent years some scholars have found that the behavior of underlying assets is not totally fit for the standard Brown Motion,but more in line with the Fractional Brownian Motion.So this discovery leads more people to the research of Fractional Brownian Motion.Our paper has mainly constructed two share(underlying asset)price models,and given the option pricing formula separately.The first share price model is (?)(0.3)Here the jump process Nt represents the unexpected risk.We give the pricing formula for a new class of options-power options.However,due to the existence of Nt,the whole market is no longer complete.So we use the measure changing skills and quasi-martingale pricing method to give a risk-neutral pricing formula.The second share price model is(?)(0.4)Here we introduce the Levy process Zt.Since Brown motion and Poisson jump process can be Levy process,and Levy process can be departed into a Brown motion and a pure jump process,model(0.4)is actually the generalized version of the model(0.3).But there is one problem,the Levy market is not complete either.To solve this,we introduce the third assets for the market-the power jump assets,just as[19]do.(?),Here (?)And we proof that the market is complete on the condition of (?).Finally,we give option price formula of the standard European call based on the model(0.4)and several simulations price performance.