On the restriction of the optimal transportation problem to the set of martingale measures with uniform marginals

One of the fundamental problems in mathematical finance is the pricing of derivative assets such as options. In practice, pricing an exotic option, whose value depends on the price evolution of an underlying risky asset, requires a model and then numerical simulations. Having no a priori model for the risky asset, but only the knowledge of its distribution at certain times, we instead look for a lower bound for the option price using the Monge-Kantorovich transportation theory. In this paper, we consider the Monge-Kantorovich problem that is restricted over the set of martingale measure. In order to solve such problem, we first look at sufficient conditions for the existence of an optimal martingale measure. Next, we focus our attention on problems with transports which are two-dimensional real martingale measures with uniform marginals. We then come up with some characterization of the optimizer, using measure-quantization approach.