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

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 delta t = (2/pi)(1/2H) (k/sigma)(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 sigma root 2 (2/pi)(1/2 -1/4H) (k/sigma)(1-1/2H) to replace the volatility sigma, where k/sigma < (pi/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. (C) 2009 Elsevier B.V. Ail rights reserved.