The Robust Pricing–Hedging Duality for American Options in Discrete Time Financial Markets

37 Pages Posted: 29 May 2020

See all articles by Anna Aksamit

Anna Aksamit

affiliation not provided to SSRN

Shuoqing Deng

affiliation not provided to SSRN

Jan Obłój

University of Oxford

Xiaolu Tan

Université Paris Dauphine - CEREMADE

Date Written: July 2019

Abstract

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, 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 nondominated family of probability measures. Our first insight is that, by considering an 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 can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging 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 the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options.

Keywords: American option, dynamic programming principle, Kantorovich duality, martingale optimal transport, measure valued martingale, nondominated model, randomized stopping times, superreplication, weak formulation, Primary: 60G40, 60G05, Secondary: 49M29

Suggested Citation

Aksamit, Anna and Deng, Shuoqing and Obłój, Jan and Tan, Xiaolu, The Robust Pricing–Hedging Duality for American Options in Discrete Time Financial Markets (July 2019). Mathematical Finance, Vol. 29, Issue 3, pp. 861-897, 2019, Available at SSRN: https://ssrn.com/abstract=3613889 or http://dx.doi.org/10.1111/mafi.12199

Anna Aksamit (Contact Author)

affiliation not provided to SSRN

Shuoqing Deng

affiliation not provided to SSRN

Jan Obłój

University of Oxford

Mansfield Road
Oxford, OX1 4AU
United Kingdom

Xiaolu Tan

Université Paris Dauphine - CEREMADE ( email )

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