Formulation and analysis of an implicit non-standard finite difference scheme for the Black-Scholes option pricing model

This paper introduces a novel numerical scheme for estimating option prices under the Black-Scholes (B-S) model. The proposed method utilizes a non-standard finite difference (NSFD) approach that incorporates the powerful techniques of methods of sub-equation and exact finite difference (EFD). The proposed technique exhibits several positive characteristics: It preserves positivity by design, works with large step sizes, ensures dynamic consistency, and enhances stability. Notably, its implicit scheme and construction ensures that the fundamental properties of the solution are accurately captured. Finally, some numerical simulations are provided to demonstrate the effectiveness of the proposed implicit NSFD scheme.