Numerical methods based on the standard finite difference approach is consistent with the original differential equation and guarantee convergence of the discrete solution to the exact one, but they impose severe restrictions on the time step and in the presence of discontinuous payoff and low volatility, essential qualitative properties of the solution are not transferred to the numerical solution. Spurious oscillations and negative values might occur in the solution. In this paper, we propose a combination of the Laplace transform method and the nonstandard finite-difference method to solving the Black-Scholes equation. The new scheme is positivity preserving, conditionally stable, consistent, and convergent. Compared with other standard numerical methods such as the Mixed method, our results indicate that a properly implemented version of our scheme is useful for the numerical integration of the considered Blach-Scholes equation. The results obtained indicate that non-standard difference schemes may be promising for solving problems that may have an impact.
