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Optimal Skorokhod embeddings with applications to pricing and hedging of double barrier options

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Let B be a Brownian motion and let S and I be its running maximum and minimum processes respectively. Fix a distribution m and positive and negative thresholds U and L. We consider the problem of maximising the probability that S(T) exceeds U and I(T) is less than L, over all stopping times T such that B(T) has distribution m and such that the process B stopped at T is UI. We describe explicitly both the bound and the stopping time which achieves it. We do the same for the minimisation problem. This implies model-free bounds on prices of certain financial derivatives (double barrier one-touch options). Furthermore, similarly to Brown, Hobson and Rogers (2001), in deriving our bounds we construct pathwise inequalities which induce model-free super-hedging (or sub-hedging) strategies for those financial derivatives.

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