European Option Pricing With Proportional Transaction Costs

With the continuous development of financial market,financial derivatives has been widely promotion and application.As option is an important part in the derivatives,it is particularly important to reasonably price these options.The assumptions of Black-Scholes option pricing model based on continuous trading and without transaction costs are too idealistic,making there exist a certain deviation of option pricing.Later,scholars began to study option pricing in the discrete time trade case,one of the most prominent is the Leland model,the model is based on Black-Scholes model and considered paying the proportional transaction costs,and considered that the Delta hedging strategy is occurred during the interval [t,t+δt].Because trading is in discrete time occasions,the portfolio with stock and risk-free asset replicating option will generate hedging error and the hedging error is large enough that does not allow to ignore.We study image of the hedging risk(hedging error) and the investors’risk preference for the option pricing with the proportional transaction costs in this paper.Assuming that the stock price St satisfy St=S0eμt+σBt,t∈[0,T]where So is a constant;μ-the expected rate of return, σ-the volatility,{Bt}t∈[0,T]is the standard Brownian motion.In the presence of proportional transaction costs,this paper uses the generalized Delta hedge strategy:Getting the corresponding partial differential equation of the European call option C In particular, when8t is enough small, we approximately have the following formula: C(t,S)=C0(S,t)+O(δt)=SN(d1)-Xe-r(T-t)N(d2)+O(δt), N(·) is the standard normal distribution function.For the above option pricing formula C,we have done the numerical comparison with Leland option pricing formula. The numerical results showed that new option pricing formula to the hedging rate for portfolio is superior to Leland option pricing formula in certain circumstances,this suggests that hedging error and investors’preferences have important effects on option pricing;In the end of the paper,we use behavioral finance point to explain phenomenon of implied volatility smile.