Super-replication with nonlinear transaction costs and volatility uncertainty

We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a dual characterization of super-replication prices as a supremum of penalized expectations for the contingent claim's payoff. We also describe the scaling limit of this dual representation when the number of trading periods increases to infinity. Hence, this paper complements the results in [Finance Stoch. 17 (2013) 447-475] and [Ann. Appl. Probab. 5 (1995) 198-221] for the case of model uncertainty.