The solution of the Black-Scholes partial differential equation determines the option price, respectively according to the used initial conditions. In the computation of the fair price of an option, it is a natural demand that the resulting numerical approximations, should be nonnegative. Numerical methods based on standard finite difference approach such as fully implicit, Crank-Nicolson, and semi-implicit schemes are powerful tools for pricing. They are usually consistent with the original differential equation and guarantee convergency of the discrete solution to the exact one, but 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 The application of a nonstandard finite difference method and investigation of its positivity preserving and smoothing properties for pricing European call options with a discrete double barrier is the subject of this paper. The new scheme is positivity preserving, conditionally stable, consistent, and convergent. Some numerical experiments have been performed to illustrate the efficiency of the new scheme
