  Recent work of Dupire and Carr and Lee has highlighted the importance of
understanding the Skorokhod embedding originally proposed by Root for the
model-independent hedging of variance options. Root's work shows that there
exists a barrier from which one may define a stopping time which solves the
Skorokhod embedding problem. This construction has the remarkable property,
proved by Rost, that it minimizes the variance of the stopping time among all
solutions. In this work, we prove a characterization of Root's barrier in terms
of the solution to a variational inequality, and we give an alternative proof
of the optimality property which has an important consequence for the
construction of subhedging strategies in the financial context.
