European Option Pricing With Both Fixed And Proportional Transaction Costs Under The Fractional Black-scholes Model

In classic Black-Scholes option pricing model, the movement of asset price is assumedby following the geometric Brownian motion, so the movement of underlying assets pricehave the same properties as geometric Brownian motion have (such as Markov property andmartingale etc.), that is the future asset price at a certain moment only related to the currentprice, and has nothing to do with the price of the past for a period of time. But a large numberof empirical studies have shown that the underlying assets price do not follow random walk,but present the scaling laws and long-range dependence. So people put forward usingfractional Brownian motion process with characteristics of these two views instead of thegeometric Brownian motion of classical Black-Scholes model, better illustrated in process ofthe movement of asset price.In a real financial market with transaction costs, if the investors trade continuously thenthey will face huge sums and cannot be ignored of the transaction costs. Thus, we assume thatthe trading occurs in a discrete time.In a discrete time, we assume that the investors are bounded rational investors. Theyadopt anchoring-adjustment strategy in making decisions by Delta-hedging argument underthe mean-self-financing. Using a replicating portfolio with underlying asset and the risklessbond, we obtain the pricing formulas of European option pricing with both fixed andproportional transaction costs under the fractional Black-Scholes model.With this model, we have the minimal rehedging-time-interval of an option under bothfixed and proportional transaction costs, and also have the corresponding European optionprice.At last, through the numerical analyzing we get that both fixed and proportionaltransaction costs have a significant impact on potion pricing.