  We consider model-free pricing of digital options, which pay out if the
underlying asset has crossed both upper and lower barriers. We make only weak
assumptions about the underlying process (typically continuity), but assume
that the initial prices of call options with the same maturity and all strikes
are known. Under such circumstances, we are able to give upper and lower bounds
on the arbitrage-free prices of the relevant options, and further, using
techniques from the theory of Skorokhod embeddings, to show that these bounds
are tight. Additionally, martingale inequalities are derived, which provide the
trading strategies with which we are able to realise any potential arbitrages.
We show that, depending of the risk aversion of the investor, the resulting
hedging strategies can outperform significantly the standard delta/vega-hedging
in presence of market frictions and/or model misspecification.
