An improved approximate method for solving two-dimensional time-fractional-order Black-Scholes model: a finite difference approach

In this paper, we considered the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense. The Black-Scholes model was an important tool in the financial market, used for determining option prices in the European-style market. However, finding a closed-form analytical solution for the fractional-order partial differential equation was challenging. To address this, we introduced an improved finite difference method for approximating the solution of the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense, based on the Crank-Nicolson finite difference method. This method combined the concepts of the finite difference method for solving the multidimensional Black-Scholes model and the finite difference method for solving the fractional-order heat equation. We analyzed the conditional stability and the order of convergence. Furthermore, numerical examples were provided to illustrate the determination of option prices.