محمد مهدی زاده خالسرایی

Analytical solution of generalized Black-Scholes models are not available, therefore, numerical simulations are of fundamental importance in gaining some useful insights into the solutions. Numerical methods based on standard finite difference approach, most of the time the essential qualitative properties of the solution are not transferred to the numerical solution. Therefore, this might result in a calamitous erroneous outcome. One way to overcome this disadvantage is to use nonstandard finite difference methods. In this paper, we propose a non-standard finite difference method for solving the generalized Black-Scholes equation. In constructing the new scheme, we use a nonlocal approximation in the reaction term of the generalized Black-Scholes equation combined with an implicit time step technique. The new scheme is positivity preserving, conditionally stable, consistent, and the order of the scheme with respect to the space variable is two. Numerical investigations were conducted to validate these results. Furthermore, the obtained results are more accurate than other existing results in the literature.