Let $(M_{t})_{0leq tleq 1}$ be a continuous martingale with initial law $M_{0}simmu_{0}$ and terminal law $M_{1}simmu_{1}$ and let $S=sup_{0leq tleq 1}M_{t}$. 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. The form of this martingale is motivated by a simple picture. The result is applied to the robust hedging of a forward start digital option.