Martingale Optimal Transport for Americans

A central goal in mathematical finance is to build a clear, rational framework for determining fair prices of financial derivatives contracts whose value depends on other assets, like stocks or bonds. This mostly involves creating mathematical models that describe how asset prices move in ways that match what we actually observe in the market, especially the prices of similar derivatives already being traded. These models then serve as the basis for pricing new contracts. A key principle in this process is the idea of no-arbitrage: in a well-functioning market, there should not be any risk-free opportunities to make a profit. Typically, there are many possible models that fit the data while respecting the no-arbitrage principle, and they can suggest very different prices. A major challenge is understanding the impact of this model uncertainty, especially quantifying the risk involved in choosing one model over another. This is the focus of robust finance. Our approach tackles these challenges using optimal transport theory, which provides a powerful and structured way to measure and manage such risks. Over the past decade, both principal investigators have played key roles in developing this theory within the context of mathematical finance, helping to establish it as its own area of research. So far, this analysis has mostly focused on relatively simple financial derivatives. However, real-world financial products are becoming increasingly complex, and a deeper understanding is needed. Specifically, most derivatives on individual stocks have an American exercise structure, meaning they can be exercised at any time before maturity, as opposed to only at maturity as in the European case. This feature significantly complicates pricing and hedging, especially in a robust setting. This project aims to break through these current limitations by developing a systematic approach for selecting models that align with richer and more detailed market data and for evaluating the risks that come with those choices. We address the problem from both a theoretical and a numerical perspective, aiming not only to establish rigorous results but also to develop algorithms and methods that are practically implementable. A central aspect of this work is properly accounting for the role that information plays in pricing and risk assessment by considering causal extensions of classical transport theory.