This paper shows how to handle ill-posed option valuation problems with finite differences. Approximating the (multidimensional) Black-Scholes PDE by a difference scheme, the solution often shows numerical oscillations. It is known that dispersion is a reason for these wiggles. Moreover the paper demonstrates how to suppress numerical dispersion and therewith how to use finite difference approximations in general settings. Pointing out the properties, especially the disadvantages, of common variable transformations in the Black-Scholes context. The results will be extended for options on baskets of multiple assets. Finally a new transformation is derived to reduce the computational effort calculating basket options with two and three assets.