Subdiffusive fractional Brownian motion regime for pricing currency options under transaction costs

Abstract

A new framework for pricing the European currency option is developed in the case where the spot exchange rate fellows a time-changed fractional Brownian motion. An analytic formula for pricing European foreign currency option is proposed by a mean self-financing delta-hedging argument in a discrete time setting. The minimal price of a currency option under transaction costs is obtained as time-step $\Delta t=\left(\frac{t^{\beta-1}}{\Gamma(\beta)}\right)^{-1}\left(\frac{2}{\pi}\right)^{\frac{1}{2H}}\left(\frac{\alpha}{\sigma}\right)^{\frac{1}{H}}$ , which can be used as the actual price of an option. In addition, we also show that time-step and long-range dependence have a significant impact on option pricing.