In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and apply it to price various types of arithmetic Asian options, as well as plain vanilla options (both european and american style) on stocks paying known cash dividends. We find prices which are accurate to $\sim 0.1%$ in about 10ms on a Pentium 233MHz computer and to $\sim 0.001%$ in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.

13 pages, 5 tables, LaTeX 2e, v2: added convergence proof