a combination of non-standard method and laplace transform for the black-scholes equation

Numerical methods based on the standard finite difference approach is consistent with the original differential equation and guarantee convergence of the discretesolution to the exact one, but they impose severe restrictions on the time step and inthe presence of discontinuous payoff and low volatility, essential qualitative propertiesof the solution are not transferred to the numerical solution. spurious oscillations andnegative values might occur in the solution. in this paper, we propose a combination ofthe laplace transform method and the nonstandard finite-difference method to solvingthe black-scholes equation. the new scheme is positivity preserving, conditionally stable, consistent, and convergent. compared with other standard numerical methods suchas the mixed method, our results indicate that a properly implemented version of ourscheme is useful for the numerical integration of the considered blach-scholes equation.the results obtained indicate that non-standard difference schemes may be promisingfor solving problems that may have an impact