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This paper addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, which is also known in the literature as the first kind of non-arbitrage. For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and random time for which the resulting stopped model fulfills NUPBR condition. The second main result characterises the random time models that preserve the NUPBR property after stopping for any market model. These results are elaborated in a very general market model, and we also pay attention to some particular and practical models. The analysis that drives these results is based on new stochastic developments in semimartingale theory with progressive enlargement. Furthermore, we construct explicit martingale densities (deflators) for some classes of local martingales when stopped at random time.Remark 2.5. Proposition 2.3 implies that for any process X and any finite stopping time σ, the two concepts of NUPBR(H) (the current concept and the one of the literature) coincide for X σ .Below, we prove that, in the case of infinite horizon, the current NUPBR condition is stable under localization, while this is not the case for the NUPBR condition defined in the literature.Proposition 2.6. Let X be an H adapted process. Then, the following assertions are equivalent. (a) There exists a sequence (T n ) n≥1 of H-stopping times that increases to +∞, such that for each n ≥ 1, there exists a probability Q n on (Ω, H Tn ) such that Q n ∼ P and X Tn satisfies NUPBR(H) under Q n . (b) X satisfies NUPBR(H). (c) There exists an H-predictable process φ, such that 0 < φ ≤ 1 and (φ X) satisfies NUPBR(H).Proof. The proof for (a)⇐⇒(b) follows from the stability of NUPBR condition for a finite horizon under localization which is due to [41] (see also [8] for further discussion about this issue), and the fact that the NUPBR condition is stable under any equivalent probability change. The proof of (b)⇒(c) is trivial and is omitted. To prove the reverse, we assume that (c) holds. Then Proposition 2.3 implies the existence of an H-predictable process ψ such that 0 < ψ ≤ 1 and a positive H-local martingale Z = E(N ) such that Z(ψφ X) is a local martingale. Since ψφ is predictable and 0 < ψφ ≤ 1, we deduce that S satisfies NUPBR(H). This ends the proof of the proposition.We end this section with a simple, but useful result for predictable process with finite variation.Lemma 2.7. Let X be an H-predictable process with finite variation. Then X satisfies NUPBR(H) if and only if X ≡ X 0 (i.e. the process X is constant).Proof. It is obvious that if X ≡ X 0 , then X satisfies NUPBR(H). Suppose that X satisfies NUPBR(H). Consider a positive H-local martingale Y , and an H-predictable process θ such that 0 < θ ≤ 1 and Y (θ X) is a local martingale. Let (T n ) n≥1 be a sequence of H-stopping times that increases to +∞ such that Y Tn and Y Tn (θ...

This paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping at the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature -up to our knowledge-proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore, we construct explicitly "the-after-τ " local martingale deflators for a large class of initial models (i.e. models in the small filtration) that are already risk-neutralized.

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.

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