Optimal Transport Martingale and Utility Maximization Problems.

Summary This thesis presents two main independent research topics, the last one grouping two distinct problems. In the first part we focus on the martingale optimal transport problem, whose primary goal is to find no-arbitrage bounds for any option. We are first interested in the discrete time question of the existence of a probability law under which the canonical process is martingale, having two fixed marginal laws. This result due to Strassen (1965) is the starting point for the primal martingale optimal transport problem. We give a proof based on financial techniques of utility maximization, adapting a method developed by Rogers to prove the fundamental asset pricing theorem. These techniques correspond to a discretized time version of the martingale optimal transport. We then consider the continuous-time martingale optimal transport problem introduced in the lookback options framework by Galichon, Henry-Labordère and Touzi. We start by establishing a partial duality result concerning the robust over-coverage of any option. We adapt recent work by Neufeld and Nutz to the optimal martingale transport. We then study the robust utility maximization problem of any option with an exponential utility function in the framework of martingale optimal transport, and derive the robust utility indifference price, under a dynamic where the sharpe ratio is constant and known. In particular, we prove that this robust utility indifference price is equal to the robust over-coverage price. The second part of this thesis deals first with an optimal liquidation problem of an indivisible asset. We study the profitability of adding a strategy of buying and selling an asset orthogonal to the first one on the optimal liquidation strategy of the indivisible asset. We then provide some illustrative examples. The last chapter of this thesis concerns the indifference utility pricing problem of a European option in the presence of small transaction costs. We draw on recent work by Soner and Touzi to obtain asymptotic developments of the value functions of the Merton problems with and without the option. These developments are obtained using homogenization techniques. We formally obtain a system of equations verified by the components of the problem and we check that they are indeed solutions. Finally, we deduce an asymptotic development of the desired indifference utility price.