Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model

Abstract

This paper deals with the problem of discrete time option pricing by the fractional Black–Scholes model with transaction costs. By a mean self-financing delta-hedging argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price C min ( t , S t ) of an option under transaction costs is obtained as timestep δ t = ( 2 π ) 1 2 H ( k σ ) 1 H , which can be used as the actual price of an option. In fact, C min ( t , S t ) is an adjustment to the volatility in the Black–Scholes formula by using the modified volatility σ 2 ( 2 π ) 1 2 − 1 4 H ( k σ ) 1 − 1 2 H to replace the volatility σ , where k σ < ( π 2 ) 1 2 , H > 1 2 is the Hurst exponent, and k is a proportional transaction cost parameter. In addition, we also show that timestep and long-range dependence have a significant impact on option pricing.