Pricing Barrier Options With Transaction Costs Under The Fractional Black-Scholes Model

The financial markets are regarded as complex and nonlinear dynamic systems in the last few years, a series of studies have also found that many financial market time series display scaling laws and long-range dependence. Therefore, many researchers began to study the movement of stock price and accordingly option pricing with fractional Brownian motion which has these two properties in recent years. Classical Black-Scholes model was established in Brownian motion environment, but Brownian motion is a special case of fractional Brownian motion. Therefore, the study of option pricing in fractional Brownian motion environment is much more wide and practical. From economics, the fractional Brownian motion can be well illustrated in the momentum effect. Furthermore, Classical Black-Scholes model has solved European option's pricing in efficient market successfully. Nevertheless, if the investors trade continuously in the occasion needing to pay transaction costs, they have to face a lot of transaction costs.In a discrete-time setting, this paper deals with the problem of barrier options pricing by the fractional Black-Scholes model with transaction costs. First, the pricing problems of eight kinds of European barrier options by the fractional Black-Scholes model with transaction costs are discussed, their pricing formulae and call- put parity are derived. In particular, the minimal price of the European down-and-in call option under transaction costs is obtained, which can be used as the actual price of an option. Then, we discussed the relationship between the time scale and the implied volatility smile phenomenon, our results showed that time scalingδt plays an important role in determining the shape of implied volatility function and the different options having the different hedging frequencies is another reason for the implied volatility smile. About this question's research, we not yet see any similarly reports at present. Furthermore, with European down-and-out and down-and-in call options as an example, we discussed the impact and characteristics of the scale and long-time dependence on the barrier options. In contrast to the classical Black-Scholes model, we know that time scalingδt and Hurst exponent H play an important role in option pricing with transaction costs and that barrier option pricing is scaling-dependent.