Efficient implicit scheme with positivity preserving and smoothing properties

Using classical finite difference schemes often generates numerical drawbacks such as spurious oscillations in the solution of the famous Black-Scholes partial differential equation. We analyze the fully implicit scheme, frequently used numerical method in Finance, that in the presence of discontinuous payoff and low volatility arises spurious oscillations. We propose a modification of this scheme so that we guarantee a smooth numerical solution, free of spurious oscillations and satisfies the positivity requirement, as is demanded for the financial solution of the Black-Scholes equation. The method is used, within the strategy suggested by Rannacher, only in few initial time steps in the presence of discontinuous initial conditions. As a consequence, although the method is low order accurate, it returns a spurious oscillations free solution. Next, starting from the smooth initial condition obtained, any other family of arbitrary higher order schemes may be used.