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We discuss model-free pricing of digital options, which pay out depending on whether the underlying asset has crossed upper and lower levels. 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. Treating this market data as input, 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.
Joint work with Alexander Cox (University of Bath)