The theory of optimal transport has already proved to be very successful to solve many important economic issues. Recent progresses were motivated by applications to robust pricing and hedging problems in mathematical finance. This course is divided in two parts.

Part I - Filippo Santambrogio
Abstract Optimal Transport, Main Concepts and Techniques

This first part is dedicated to the presentation of the Monge and Monge-Kantorovich problems in deterministic settings. The important mathematical tools, key results, and main applications in economy will be discussed.

Part II - Nizar Touzi & Pierre Henry-Labordère
Applications to robust pricing and hedging problems in mathematical finance

This second part concerns extensions of the results obtained in deterministic settings to cases where the transport is done through a semi-martingale. We will in particular discuss an important application in mathematical finance: the derivation of robust no-arbitrage bounds for exotic options, given the implied volatility curve of some maturity.

(slides) (Lecture notes)

The course gives an overview of the Skorokhod embedding problem and its applications to robust pricing and hedging in mathematical finance. The general embedding problem is: given a process find a stopping time such that the stopped process has a given distribution. It has been a fascinating and active field of research in probability for over 50 years now. We will introduce the general embedding problem and discuss results on existence of solutions. Subsequently, we will present the principal methods used to obtain solutions leading to several explicit constructions. We will then show how these different constructions may be characterised by their additional optimal properties. These are then translated into robust pricing and hedging of exotic options given prices of co-maturing European call options. Furthermore, we will show how the markets implied by the optimal solutions allow us to guess the robust super-/sub- hedging strategies. The link with the optimal transport approach will be discussed.