An Efficient Numerical Scheme to Approach the Time Fractional Black-Scholes Model Using Orthogonal Gegenbauer Polynomials

This paper proposes an efficient procedure to estimate the fractional Black-Scholes model in time-dependent on the market prices of European options using the composition of the orthogonal Gegenbauer polynomials (GB polynomials) and the approximation of the fractional derivative dependent on the Caputo derivative. First, the payoff function's singularity leads to a slow convergence in time. So, We construct an accurate and fast numerical method based on the improved method to restore the convergence rate. The derivative operational matrices for orthogonal GB polynomials are obtained by the fractional Caputo-type derivative. The implementation of this algorithm has high accuracy. The advantage of the numerical method is the orthogonality of GB polynomials and operational matrices, which reduces computation time and increases speed. Finally, three numerical examples are provided to illustrate the validity, and the numerical experiments are presented to illustrate the accuracy and efficiency of the proposed method.