Lookback Option Pricing Based On Mixed Fractional Brownian Motion

The look-back option depends on path strongly.Its benefit on the maturity date depends on the maximum or minimum value of entire risk assets over the life,which makes the option income of the look-back option at the closing date of the is higher than other options.So the price of the look-back option is very expensive.Therefore,it is very meaningful for us to further study the price of the look-back option.The current academic studies on the look-back option are based on the assumption that the price of the look-back option is under the geometric Brownian motion.However,in the actual financial transaction,assert returns is distributed the feature that has aiguille large remaining part,and the price of the underlying asset in the financial market is not continuous but which will occur intermittently "jump" case,which makes the studies on the options under the geometric Brownian motion inconsistent with the actual situation.Therefore,in this paper,we will study the price of the European look-back put option under three models: the Mixed Brownian motion? the Mixed Brownian motion with transaction costs and mixed jump � diffusion and the fractional Brownian motion.The main results are as follows:(1)The pricing of lookback option is studied when the price dynamics of underlying asset are under the Mixed Brownian motion model.It is assumed that the stock price is driven by a geometric Brown motion process and fractional Brownian motion process.Applying the Ito lemma and the risk-hedging principle,the option pricing mathematical differential equations and corresponded boundary conditions of the European look-back put option under the Mixed Brownian motion is derived.Then by using the differential method,the article has transformed the differential equations into the differential equations.Finally,the effectiveness is tested through a numerical example.(2)We study the more realistic problem,on the basis of the above study,by making use of the principle of risk-neutral,In order to characterize the actual option price changes in financial markets more precisely,we join the Poisson jump-diffusion process in this article to establish of the pricing model of the European look-back put option under mixed jump � diffusion and the fractional Brownian motion.First of all,applying the Ito lemma and the risk-hedging principle,the integral equation and corresponded boundary conditions of the European look-back put option under the mixed jump-diffusion and the fractional Brownian motionis derived.Adopting the Taylor expansion,the inte gral equation is transformed into partial differential equation.Then the difference differential and implicit difference are obtained for the partial differential equations respectively,and the corresponding iterative equations are obtained.By variable substitution,the differential term for the transformed is approximated by means of constructing explicit discrete format and the stability and consistency of the scheme is analyzed.Finally,we employ the Matlab software to discuss the impact of market parameter values on the lookback put option by a numerical example.(3)The pricing of lookback option with transaction costs is studied when the price dynamics of underlying asset are governed by the Mixed Brownian motion process.Adopting the method of risk-hedging principle,difference equations for the lookback put option prices is obtained.It is very difficult for us to get the analytic solutions of the differential equations,and the strike price at the expiration date for the lookback option is uncertain,so we need to reduce the dimension of the resulting model through variable substitution,then construct Crank-N icolson scheme to get the numerical solution of the equations.Finally,we discuss the convergence of the numerical scheme and analyze the influence of values of transaction costs,Hurst and other parameters on values of lookback put option.