Martingale option pricing

We show that our earlier generalization of the Black–Scholes partial differential equation (pde) for variable diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, the equivalence of Black–Scholes to a Martingale was proven for the case of the Gaussian returns model by Harrison and Kreps, but we prove it for a much larger class of returns models where the returns diffusion coefficient depends irreducibly on both returns x and time t. That option prices blow up if fat tails in logarithmic returns x are included in market return is also proven.