Robust Hedging and Optimal Transport

Optimal transport as a mathematical field goes back to Monge (1783) and Kantorovich (1942) who coined its modern formulation. Recently it experienced an impressive development prompted by Brenier`s theorem and McCann`s milestone PhD-thesis. The field is now famous for its striking applications in various areas ranging from mathematical physics and the theory of PDEs to geometric and functional inequalities. The area of model independent / robust finance is much younger, but has been rapidly evolving into an independent field in the last years. The over-confidence in mathematical models and the failure to account for model-risk which often prevails among practitioners have been frequently blamed for their infamous role in financial crises. The aim of model-independent finance is to understand and quantify the effects of model-ambiguity. We will explore the connections between these previously unrelated fields which were very recently discovered by Galichon, Henry-Labordere, Penkner, Touzi, and the principal investigator. Roughly speaking, the starting point is the following principle: interpret the evolution of a martingale S as a way to transport distributions from earlier time instances to later ones. This martingale transport problem occurs naturally in model-independent finance, where S represents the price of a financial asset. Market-data essentially yields information about the distributions of S at particular time instances. What remains unknown is how S moves from one point in time to the next. The problem to determine the range of prices consistent with market-data is thus closely related to the optimal transport problem. Optimal transport provides a duality theory for model independent finance. Results obtained recently by Henry-Labordere, Penkner and the PI are just a first step; it is a main goal to develop the dual part of the martingale transport problem in the required generality. Applications of this approach go beyond mathematical finance. Using the dual viewpoint, new elementary proofs for the classical inequalities of Doob and Burkholder-Davis-Gundy were found by the PI and his collaborators. This line of attack is ideal for an entire array of applications to martingale- and analytic inequalities. Transport theory has well-developed optimality criteria. Following work of the PI and his collaborators it is possible to translate them into a Variational Principle for martingale transport, with applications from Brenier`s Theorem to the Skorokhod embedding problem. This principle and extensions will provide a systematic approach to the model-independent pricing problem, yielding a numerically tractable method to obtain sharp robust bounds, as well as characterizations of the cases of equality in martingale-inequalities.

The roots of optimal transport as a mathematical field can be traced back to Monge and Kantorovich who coined its modern formulation. More recently it experienced an impressive development prompted by Brenier's theorem and McCann's milestone PhD-thesis. The field is now famous for its striking applications in various areas ranging from mathematical physics and the theory of PDEs to geometric and functional inequalities. The area of model independent pricing is much younger, but has been rapidly evolving into an independent field within mathematical finance since Hobson's seminal paper on the look-back option (1998). While classical mathematical finance accurately determines a specific price for a financial derivative by making assumptions about the underlying model, the area of model independent finance rather seeks to derive conservative but robust bounds. In the START Project we developed connections between these previously unrelated fields. Roughly speaking, the starting point is the following principle: interpret the evolution of a martingale S as a way to transport distributions from earlier time instances to later ones. This martingale transport problem occurs naturally in model-independent finance, where S represents the price of a financial asset. Market-data essentially yields information about the distributions of S at particular time instances. What remains unknown is how S moves from one point in time to the next. The problems to determine minimal/maximal prices and optimal trading/hedging strategies is closely related to the main tasks of optimal transport. Transport theory has well-developed optimality criteria. In the course of the project we have developed analogous results, so called monotonicity principles which play a similar role in model-independent finance. They allow us to characterize minimal and maximal, respectively financial models for a variety of combinations of market environments and financial derivatives. We have thus obtained a systematic method to calculate robust pricing bounds together with corresponding hedging strategies in general circumstances. Notably, the techniques developed in the START project have implications beyond mathematical finance, e.g. concerning the Skorokhod embedding problem where the task is to embed a specific distribution into a Brownian motion. Many different authors have constructed various specific solutions since the problem was first posed more than 60 years ago. The transport based approach developed in the START project yields for the first time a universal approach that allows to derive all previously known optimal solutions to the Skorokhod problem as well as an abundance of new solutions.