We aim to generalize the duality results of Bouchard and Nutz (2015) to the case of American options. By introducing an enlarged canonical space, we reformulate the superhedging problem for American options as a problem for European options . Then in a discrete time market with finitely many liquid options, we show that the minimum superhedging cost of an American option equals to the supremum of the expectation of the payoff at all (weak) stopping times and under a suitable family of martingale measures. Moreover, by taking the limit on the number of liquid options, we obtain a new class of martingale optimal transport problems as well as a Kantorovich duality result .
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 .
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aim to generalize the duality results of Bouchard and Nutz (2015) to the case of American options. By introducing an enlarged canonical
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
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reformulate the superhedging problem for
can see
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a problem for European options . Then in a discrete time market with finitely many liquid options, we show that the minimum superhedging cost of an American option equals to the supremum of
European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of
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expectation of the payoff at all (weak) stopping times and under a suitable family of martingale measures. Moreover, by taking the limit on the number of liquid options, we obtain a new class of
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
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problems as well as a Kantorovich duality result
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