Discontinuities in the payoff function (or its derivatives) can cause inaccuracies for numerical schemes when financial contracts are priced. In particular, large errors may occur in the estimation of the hedging parameters. Three methods of dealing with discontinuities are discussed in this article: averaging the initial data, shifting the grid, and a projection method. By themselves, these techniques are not sufficient to restore expected behavior. However, when combined with a special time-stepping method, high accuracy is achieved. Examples are provided for one- and two-factor option pricing problems.