From Doob's inequality to robust hedging

The limitations of specific models of financial markets, such as the Black Scholes model, are increasingly noted. On the other hand, the model-free approach has recently gained in interest. Given the prices of plain vanilla options, this approach allows to deduce the possible prices of certain exotic options, using only the principle of no arbitrage. For example, for exotic options pertaining to the maximal price of an underlying asset during a given period, we find an interesting connection to the classical Doob inequalities pertaining to the maximal function of a martingale. We present a pathwise proof of these inequalities which allows for a financial interpretation as a model-free super-hedge. We alsopresent a recent general theorem, due to B. Bouchard and M. Nutz, relating martingale inequalities with pathwise superhedges.