An improved local radial basis function method for pricing options under the time-fractional Black–Scholes model

The time-fractional Black–Scholes model (T-FBSM) is developed to assess price fluctuations in a correlated fractal transmission system. It is applied to price American and European call and put options on non-dividend-paying stocks. This study focuses on numerically solving the T-FBSM for option pricing using a local compact integrated radial basis function method (LCIRBFM). The temporal discretization is accomplished using the second-order shifted Grünwald scheme, while the spatial derivatives are discretized by using a combination of an integrated RBF interpolation and a compact scheme within a sub-domain (stencil). This approach utilizes second derivatives and nodal function values to construct a link between the RBF weights and the physical domain. The convergence and unconditional stability of the semi-discretized time formulation are proven via the energy method in the $L_2$ space. The proposed method demonstrates efficiency, and the numerical results validate the theoretical formulation.