Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Diffusive Regime

In statistical physics, anomalous diffusion plays an important role, whose applications have been found in many areas. In this paper, we introduce a composite-diffusive fractional Brownian motion X _α,H (t)=X _H (S _α (t)), 0<α,H<1, driven by anomalous diffusions as a model of asset prices and discuss the corresponding fractional Fokker-Planck equation and Black-Scholes formula. We obtain the fractional Fokker-Planck equation governing the dynamics of the probability density function of the composite-diffusive fractional Brownian motion and find the Black-Scholes differential equation driven by the stock asset X _α,H (t) and the corresponding Black-Scholes formula for the fair prices of European option.