This study is devoted to the development of a characteristic method for the solution of the Black and Scholes equation for the valuation of European Option contracts. The scheme uses continuous test and trial functions which are piecewise cubic in space within the framework of the Eulerian Lagrangian localized adjoint method (ELLAM). The method symmetrized the governing equations and generates accu rate option values as well as option Greeks even if large time steps are used. We present numerical experiments which illustrate the performance of this method and show that it compares favorably to many well received and widely known methods including the standard ELLAM method which uses piecewise linear test and trial functions.