Robust replication under model uncertainty

We consider the robust hedging problem in which an investor wants to super-hedge an option in the framework of uncertainty in a model of a stock price process. More specifically, the investor knows that the stock price process is H-self-similar with H∈(1/2,1), and that the log-returns are Gaussian. This leads to two natural but mutually exclusive hypotheses both being self-contained to fix the probabilistic model for the stock price. Namely, the investor may assume that either the market is efficient, i.e. the stock price process is a semimartingale, or that the centred log-returns are stationary. We show that to be able to super-hedge a convex European vanilla-type option robustly the investor must assume that the markets are efficient. If it turns out that if the other hypothesis of stationarity of the log-returns is true, then the investor can actually super-hedge the option as well as receive a net hedging profit.