Abstract. Suppose we are given a double continuum (in time and strike) of discounted option prices, or equivalently a set of measures which is increasing in convex order. Given sufficient regularity, Dupire showed how to construct a time-inhomogeneous martingale diffusion which is consistent with those prices. But are there other martingales with the same 1-marginals? (In the case of Gaussian marginals this is the fake Brownian motion problem.) In this talk we show that the answer to the question above is yes. Amongst the class of martingales with a given set of marginals we construct the process with smallest possible expected total variation.