The Minimum Maximum of a Continuous Martingale with Given Initial and Terminal Laws

Let (Mt)0≤t≤1 be a continuous martingale with initial lawM0 ∼ μ0, and terminal lawM1 ∼ μ1, and let S = sup 0≤t≤1 Mt. In this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures, on the law of S. We give an explicit construction of this bound. Furthermore a martingale is constructed which attains this minimum by solving a Skorokhod embedding problem. A simple picture motivates the form of this martingale. The result is applied to the robust hedging of a forward start digital option.