The robust pricing–hedging duality for American options in discrete time financial markets

Anna Aksamit,Shuoqing Deng,Xiaolu Tan,Jan Obłój

doi:10.1111/mafi.12199

Anna Aksamit, Shuoqing Deng + Show 2 more

Open Access

https://doi.org/10.1111/mafi.12199

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Abstract

AbstractWe 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.

Highlights

In a complete market, where every contingent claim can be perfectly replicated by a self-financing trading strategy, the option price is given by its replication cost, under the no-arbitrage assumption
In the classical dominated model, the no-arbitrage condition is proved to be equivalent to the existence of the equivalent martingale measures, by the so-called first fundamental theorem of asset pricing, see e.g. Delbaen & Schachermayer [13], Follmer & Schied [18], etc
The super-replication problem is related to a model-free pricing problem, i.e. the supremum of the expectations of the payoff under a suitable family of “martingale measures” models

Summary

Introduction

In a complete market, where every contingent claim can be perfectly replicated by a self-financing trading strategy, the option price is given by its replication cost, under the no-arbitrage assumption. For the continuous time model under “volatility Another branch of literature studied the superhedging problem using dynamic strategy on the underlying risky asset as well as the static strategy on a given set (finite or infinite) of liquid options. Neuberger [27] considered a discrete time, discrete space market with presence of liquid European vanilla options, and obtained a duality result by using a “weak” dual formulation. This approach has recently been exploited and presented with more fruitful results in Hobson & Neuberger [24].

Setting

Main results

Further discussions

Proofs

A Appendix