an efficient nonstandard numerical method with positivity preserving property

classical explicit finite difference schemes are unsuitable for the solution of the famous black-scholes partial differential equation, since they impose severe restrictions on the time step. furthermore, they may produce spurious oscillations in the solution. we propose a new scheme that is free of spurious oscillations and guarantees the positivity of the solution for arbitrary stepsizes. the proposed method is constructed based on a nonstandard discretization of the spatial derivatives and is applicable to black-scholes equation in the presence of discontinues initial conditions.