Duality for American options in non-dominated discrete-time models
The classical pricing-hedging duality for American options with semi-static hedging does not hold in general in the simple formulation inherited from European option set-up. We propose two approaches to recover the duality result. The first approach consists in considering a bigger class of models and rendering an American option a European one. The second way is to relax the static trading and by allowing dynamic trading in the set of vanilla options. As a by-product, it is proved that the dynamic trading of vanilla options does change the optimal pricing-hedging value comparing to the semi-static trading strategy. The connections to classical enlargement of filtration are also discussed. The problem of stopping only at stopping times w.r.t. price process filtration can be related to pseudo-stopping times (as well randomized stopping times) and the immersion property.