We investigate the simple option pricing problem, either for Vanilla or Digital european options. As compared to the work of Saint-Pierre et al., we have a more specific theory - we are unable to extend it to exotic options - with more analytical results. In particular, we obtain a representation theorem for the Value function, both in continuous and discrete trading, yielding a fast algorithm for the discrete trading case, together with a convergence theorem of this discrete trading value toward the continuous one as the step size vanishes.