Before revision:  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 .
After revision:  Moreover, by taking the limit on the number of liquid options, we obtain a new class of 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 .