Before revision:  By using the stochastic control approach initiated by Galichon, Henry-Labordere and Touzi, we recover the optimal hedging strategies and the corresponding prices given by Vallois ' embeddings to the SEP . Furthermore, we extend the analysis to the two-marginal case .
After revision:  By using the stochastic control approach initiated by Galichon, Henry-Labordere and Touzi, we recover the optimal hedging strategies and the corresponding prices in the one-marginal case. As a by-product, we recover the property that the Vallois solutions to the Skorokhod embedding problem (SEP) maximize and minimize the expectation of any convex function of the local time . Furthermore, we extend the analysis to the two-marginal case .