Numerically pricing American and European options using a time fractional Black–Scholes model in financial decision-making

The time fractional Black–Scholes equation (TFBSE) is designed to evaluate price fluctuations within a correlated fractal transmission system. This model prices American or European put and call options on non-dividend-paying stocks. Reliable and efficient numerical techniques are essential for solving fractional differential models due to the global characteristics of fractional calculus. This paper focuses on the numerical solution for the TFBSE for American and European option pricing models by means of the local meshless radial basis function (RBF) interpolation. This problem is temporally approximated using a finite difference scheme with $2-\beta$ order accuracy for $0<\beta <1$, and spatially discretized using the localizing RBF partition of unity method (LRBFPUM). The theoretical discussion confirms the convergence analysis and unconditional stability of the semi time-discretized formulation in the perspective of the $H^1$-norm. A main disadvantage of global RBF-based (GRBF) methods is high computational burden required to solve large linear systems. The LRBFPUM overcomes the ill-conditioning that arises in the GRBF methods. It allows for significant sparsification of the algebraic system, leading to a lower condition number and reduced computational effort, while keeping high accuracy. Numerical examples and applications highlight the accuracy of the LRBFPUM technique and confirm the theoretical prediction.