Pricing European Double Barrier Option with Moving Barriers Under a Fractional Black–Scholes Model

Analysis of financial time series shows the existence of the long memory in financial markets. Fractional stochastic models can be a suitable tool for capturing phenomena that have memory and inheritance properties. In this paper, we consider a fractional Black–Scholes equation for pricing double barrier options with moving barriers that underlying asset price follows a fractional-order stochastic differential equation. In contrast to the classical Black–Scholes model, where the parameters are considered constant, in this study, we consider the parameters to be time-dependent and also assume that dividend yield is paid over the life of the option. Because of the complexity of such problems, they have no closed-form solution. The main objective of this study is to obtain a numerical solution using the implicit difference scheme to determine the value of double barrier options with moving barriers and parameters as functions. Also, we approximate the hedging quantities Delta and Gamma of double barrier options using the implicit difference scheme. Finally, the numerical results illustrate that the introduced scheme is accurate and efficient.