Martingale Optimal Transport and Utility Maximization

This PhD dissertation presents two independent research topics dealing with contemporary issues from financial mathematics, the second one being composed of two distinct problems. In the first part we study the question of martingale optimal transport, which comes from the questions of no-arbitrage optimal bounds of liabilities. We first consider the question in discret time of the existence of a martingale law with given marginals. This result was first proved by Strassen (1965) and is the starting point of martingale optimal transport. We provide a new proof of this theorem based on utility maximization technics, adapted from a proof of the fundamental theorem of asset pricing by Rogers. We then consider the question of martingale optimal transport in continuous time, introduced in the framework of lookback options by Galichon, Henry-Labordere et Touzi. We first establish a partial duality result concerning the robust superhedging of any contingent claim. For that purpose, we adapt recent technics developed by Neufeld and Nutz in the context of martingale optimal transport. In a second time we study a robust utility maximization of a contingent claim with exponential utility in the context of martingale optimal transport, and we deduce its robust utility indifference price, given that the underlying's dynamic has a constant and well-known sharpe ratio. We prove that this robust utility indifference price is equal to the robust superhedging price. The second part of this disseration considers first the problem of optimal liquidation of an indivisible asset. We study the advantage that an agent can take from having a dynamic trading strategy in an orthogonal asset. The question of its influence on the optimal liquidation rule is asked. We then provide examples illustrating our results. The last chapter of this thesis concerns the utility indifference price of a European option in the context of small transaction costs. We use technics developed by Soner and Touzi to obtain an asymptotic expansion of the Merton value functions with and without the option. These expansions are obtained by using homogenization technics. We formally obtain a system of equations verified by the values involved in the expansion and show rigorously that they are solutions. We then deduce an asymptotic expansion of the utility indifference price.