Galerkin approach by certain shifted Jacobi polynomials for solving the time-fractional Black-Scholes equation

This work presents a spectral Galerkin approach for solving the time-fractional Black-Scholes equation (TFBSE) used in option pricing models, considering memory effects. We use certain shifted Jacobi polynomials as the basis functions. To implement the proposed solution, some properties of the shifted Jacobi polynomials transform the problem with its underlying conditions into an efficiently treated system of equations. A thorough error analysis is performed in weighted Sobolev spaces to determine the convergence rates of the numerical scheme. We present several examples with known exact solutions to demonstrate the method’s effectiveness. The numerical results show that the proposed method is applicable and efficient, making it a powerful tool for solving fractional-order financial models.