Numerical solution of ψ-Hilfer fractional Black–Scholes equations via space–time spectral collocation method

Abstract

Trivially, the time-fractional Black–Scholes (FBS) equation is utilized to describe the behavior of the option pricing in financial markets. This work is intended as an attempt to introduce the ψ-Hilfer fractional Black–Scholes (ψ-HFBS) equation. First, we concentrate on demonstrating the existence of the solution to the ψ-HFBS equations. Second, a numerical scheme is presented for solving the equation given the appropriate initial and boundary conditions. The approximate solutions are considered as linear combinations of the Lagrange functions with unidentified coefficients. By collocating the considered equation together with the boundary and initial conditions at Chebyshev-Gauss-Lobato (CGL) points, it will be converted to a system of linear algebraic equations. Next, we have proved the convergence of the approach. Finally, some test problems are given in order to indicate the suggested method.

Keywords

• 26A33
• 34A08
• 34A30