Robust Quantitative Finance and Martingale Optimal Transport

Abstract

The robust approach has been a prominent area of research within modern mathematical finance since the early 2000s. To this day, it remains a rich area of study, with applications ranging from the study of risk measures, to economic decision theory, and in particular to pricing path-dependent derivatives. In this thesis, the latter of these areas will be explored in detail. In particular, the link between pricing and hedging exotic options and the celebrated Skorokhod embedding theorem will be made apparent. We will see that certain solutions to the Skorokhod embedding problem enjoying specific optimality conditions can be used to derive maximal price bounds for a class of exotic options. In the final chapters of the thesis, a treatment of the martingale optimal transport problem will be given. We will explore how certain classical notions in the standard optimal transport problem are manifest in the martingale case; this includes a sufficiency and uniqueness result for the problem to be well-posed, and a proof showing that, given certain conditions on the chosen cost function, there is no duality gap between the corresponding primal and dual problems. The thesis will conclude with a discussion of recent developments towards numerically solving the martingale optimal transport problem, which is of particular relevance for practitioners in the financial industry. This includes exploring a convergence result for a suitably relaxed version of the problem, which is then able to be solved via classical linear programming techniques. We will also study an algorithm for constructing an optimal martingale transport plan, in the case where the given marginal measures are atomic.