Asian Option Pricing Under Fractional Brownian Motion And Jump-diffusion Process

Asian option is a strong path-dependent exotic option whose return on maturity depends on the average of the price of the underlying asset price in the period of validity,thus,to minimize the effects of the price fluctuation and make the Asian option more popular than the conventional options.At present,most of the researches on Asian option pricing are based on the standard Brownian motion,and assuming that the underlying asset price is continuous,and without consideration of transaction cost.While,the underlying asset price presents a kind of distribution of "asymmetric leptokurtic" and “fat tail”,and has self-similarity and long-term dependence.In addition,there are a lot of transaction costs in the real financial market.Therefore,this paper will study the Asian option pricing with proportional transaction costs under the jump-fraction and mixed jump-fraction model.The main contents are as follows:(1)The ?Ito formula of mixed jump-fraction process is deduced by using the fractional ?Ito formula,and the pricing model of the Asian option is obtained by using self-financing dynamic strategy.The value of Asian call option and put option is obtained by solving the pricing model.Finally,the numerical experiment is carried out by using Matlab to discuss the influence of the pricing parameters on the option value,such as Hurst index,jump intensity and stock price.(2)The pricing model of the Asian option with transaction cost is established by using the ?Ito formula of jump-fraction process and self-financing dynamic strategy.Define the form of Leland to simplify the volatility correction factor,so as to simplify the calculation process,then the model is solved by using the method of variable replacement,and the analytic solution of the option value is obtained.The numerical experiments reflect the relationship between option value and Hearst index,jump intensity and transaction rate intuitively.(3)The Asian option pricing model with transaction cost under mixed jump-fraction process is established.The three-dimensional heat conduction equation is reduced into two-dimensional ones by using the method of dimension reduction,and the pricing formula of Asian call option is obtained by solving the classical heat conduction equation,and then the pricing formula of put option is derived.The influences of the Hurst index,transaction rate,risk-free rate and stock price on the option value are discussed in the numerical experiments,and the mixed jump-fractionmodel is closer to the real financial market,and more stable than the jump-fraction model,to a certain extent.