Anomalous dynamics of Black–Scholes model time-changed by inverse subordinators

In this paper we consider a generalization of one of the earliest models of an asset price, namely the Black–Scholes model, which captures the subdiffusive nature of an asset price dynamics. We introduce the geometric Brownian motion time-changed by infinitely divisible inverse subordinators, to reflect underlying anomalous diffusion mechanism. In the proposed model the waiting times (periods when the asset price stays motionless) are modeled by general class of infinitely divisible distributions. We find the corresponding Fractional Fokker–Planck equation governing the probability density function of the introduced process. We prove that considered model is arbitrage-free, construct corresponding martingale measure and show that the model is incomplete. We also find formulas for values of European call and put option prices in subdiffusive Black–Scholes model and show how one can approximate them based on Monte Carlo methods. We present some Monte Carlo simulations for the particular case of tempered alpha-stable distribution of waiting times. We compare obtained results with the classical and subdiffusive alpha-stable Black–Scholes prices.