Robust Pricing and Hedging of American Options in Continuous Time

We consider the robust pricing and hedging of American options in a continuous time setting. We assume asset prices are continuous semimartingales, but we allow for general model uncertainty specification via adapted closed convex constraints on the volatility. We prove the robust pricing-hedging duality. When European options with given prices are available for static trading, we show that duality holds against richer models where these options are traded dynamically. Our proofs rely on probabilistic treatment of randomised stopping times and suitable measure decoupling, and on optimal transport duality. In addition, similarly to the approach of Aksamit et al. (2019) in discrete time, we identify American options with European options on an enlarged space.