Model-Independence and 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.

In June 2014 the PI was awarded a START-prize. According to the regulations of the FWF, the project P26736 was therefore formally closed after 11 months, the research is currently continued within the scope of the START-prize. Due to this alteration, this report concerns only the initial stage of the proposed project. Basic theme of this project was a connection between the optimal transport and model- independent finance. 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 explored 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. We were able to make significant advancements in the theory of model- independent finance, we mention two particular results: with our cooperation partners Cox and Huesmann we developed a new systematic method to determine the minimal/maximal model- independent prices of exotic derivatives.Together with Perkowski and Promel we were able to establish a model-independet superreplication theorem. (While results of this type are well-known in classical mathematical finance, it is an important open problem to determine model-independent analogues.) We were able to make significant progress concerning the field of probabilistic inequalities. In cooperation with Nutz we were able to clarify the precise connection between martingale inequalities and their pathwise counterparts. Together with Schachermayer we were able to determine the optimal constant of the BDG-inequality in an important case (a problem that has been open for more than 40 years).