Robust Bounds for Forward Start Options

We consider the problem of finding a model-free upper bound on the price of a forward-start straddle with payoff |PT2 −PT1 |. The bound depends on the prices of vanilla call and put options with maturities T1 and T2, but does not rely on any modelling assumptions concerning the dynamics of the underlying. The bound can be enforced by a super-replicating strategy involving puts, calls and a forward transaction. We find an upper bound, and a model which is consistent with T1 and T2 vanilla option prices, for which the model-based price of the straddle is equal to the upper bound. This proves that the bound is best possible. For lognormal marginals we show that the upper bound is at most 30% higher than the Black-Scholes price. The problem can be recast as finding the solution to a Skorokhod embedding problem with non-trivial initial law so as to maximise E|Bτ −B0|.