Root's barrier: Construction, optimality and applications to variance options

TL;DR

This paper characterizes Root's barrier using variational inequalities and demonstrates its optimality in minimizing stopping time variance, with significant implications for model-independent hedging of variance options.

Contribution

It provides a new variational inequality characterization of Root's barrier and offers an alternative proof of its optimality property.

Findings

Root's barrier characterized by a variational inequality

Alternative proof of the barrier's optimality

Implications for subhedging strategies in finance

Abstract

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.