Structural properties of multi-period martingale optimal transport problems and applications

This paper develops new tools to study the structural properties of solutions to multi-period martingale optimal transport (MOT) problems. More precisely, conditions are obtained on how and when two-period martingale couplings may be glued together to obtain multi-period martingales and which among these gluings are optimal for particular MOT problems. Together with a novel linearization of the optimal cost as certain terms vanish, these gluing are used to obtain a complete characterization of limiting solutions in a three-period problem as the interaction between two of the variables vanishes. For the full three-period problem, several structural and uniqueness results under a variety of different assumptions on the marginals and cost function are also obtained. For high-dimensional input data, these approximation methods, if compared with classic direct numerical approximations, are cost-efficient. To illustrate the practicality of these results approximate model independent upper and lower bounds are computed for options prices depending on Amazon stock prices at three different times.