Robust pricing--hedging duality for American options in discrete time financial markets

TL;DR

This paper establishes a robust duality framework for pricing and hedging American options in discrete-time markets, using space enlargements to recover duality even under model uncertainty and dynamic trading constraints.

Contribution

It introduces a universal space enlargement approach to recover duality for American options, applicable to both classical and robust market models with dynamic and static assets.

Findings

Duality can be recovered via space enlargement methods.

Applicable to classical and robust market models.

Duality holds in Bouchard and Nutz's and Beiglb"ock et al.'s frameworks.

Abstract

We investigate pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, e.g. a family of European options, only statically. In the first part of the paper we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a non-dominated family of probability measures. Our first insight is that by considering a (universal) enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of the original problem. Our second insight is that lack of duality is caused by the lack of dynamic consistency and hence a different enlargement with dynamic consistency is sufficient to recover duality: it is enough to consider (fictitious) extensions of the market in which all the assets are traded dynamically. In the second part of the paper we study two important examples of robust framework: the setup of Bouchard and Nutz (2015) and the martingale optimal transport setup of Beiglb\"ock et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.