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Double no-touch options, contracts which pay out a fixed amount provided anunderlying asset remains within a given interval, are commonly traded,particularly in FX markets. In this work, we establish model-free bounds on theprice of these options based on the prices of more liquidly traded options(call and digital call options). Key steps are the construction of super- andsub-hedging strategies to establish the bounds, and the use of Skorokhodembedding techniques to show the bounds are the best possible.  In addition to establishing rigorous bounds, we consider carefully what ismeant by arbitrage in settings where there is no {\it a priori} knownprobability measure. We discuss two natural extensions of the notion ofarbitrage, weak arbitrage and weak free lunch with vanishing risk, which areneeded to establish equivalence between the lack of arbitrage and the existenceof a market model.

Robust pricing and hedging of double no-touch options