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

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.

Robust bounds for the American put

“…The study of American-style claims in a robust framework was initiated by Neuberger [25]; see also Hobson and Neuberger [20], Bayraktar and Zhou [3] and Aksamit et al [1]. (There is also a paper by Cox and Hoeggerl [10] which asks about the possible shapes of the price of an American put, considered as a function of strike, given the prices of co-maturing European puts.)…”

The Robust Superreplication Problem: A Dynamic Approach

“…Here, we consider an abstract general setup and allow any d-tuple of traded assets, for a finite d. We may expect that the level of uncertainty regarding different assets may differ and this would be reflected in P. However it is crucial that all the assets are traded dynamically. From a theoretical standpoint, this is both necessary to obtain a dynamic programming principle for the superhedging prices and without loss of generality in the sense that any Bouchard and Nutz [2015] setup where some assets are only available for trading at time 0 can be lifted to a setup with dynamic trading in all assets in a way which does not introduce arbitrage and does not affect time-0 superhedging prices, see Aksamit et al [2018]. From a practical standpoint, this is not a significant assumption as we may only consider liquidly traded assets.…”

Transport Plans with Domain Constraints

“…Remark 4.1. The result still holds and the proof still goes through with minor adjustments, if we weaken/replace the assumption by:(1) there exists T ⊂ [0, 1] that is dense in [0, 1], such that (A s ∩ C [s,t] )• (X s , X t ) −1 is convex and closed for any s, t ∈ T with s < t; (2) A s ∩ C [s,t] is weakly compact for any s, t ∈ T with s < t; (3) the consistency assumption (4.1).Example 4.1 (Martingale measures with volatility uncertainty).Let Ω = C[0, 1] with X = R d . Assume α t has a finite first moment for any t ∈ [0, 1].…”

“…In this section, we consider two cases Ω = C[0,1] or D[0, 1]. For t ∈ [0, 1], let Ω t = C[0, t], D[0, t]when Ω = C[0, 1], D[0, 1] respectively, Ω x t ⊂ Ω t be the set of paths starting from position x ∈ X.…”