Root’s and Rost’s Embeddings: Construction, Optimality and Applications to Variance Options

Abstract

Root’s solution (Root [1969]) to the Skorokhod embedding problem can be describedas the first hitting time of a space-time process (Xt, t) on a so-called barrier, charac-terised by certain properties, such that the stopped underlying process X has a givendistribution. Recent work of Dupire [2005] and Carr and Lee [2010] has highlighted theimportance of understanding the Root’s solution for the model-independent hedging ofvariance options.We consider the problem of finding Root’s solutions when the underlying process is atime-homogeneous diffusion with a given initial distribution in one dimension. We areinterested in constructing Root’s solution by partial differential equations. We beginby showing that, under some mild conditions, constructing Root’s solution is equiv-alent to solving a specialized parabolic free boundary problem in the case where theunderlying process is a Brownian motion starting at 0. This result is then extended totime-homogeneous diffusions. Replacing some conditions needed in the free boundaryconstruction, we then also consider the construction of Root’s solutions by variationalinequalities. Finally we consider the optimality and applications of Root’s solutions.Unlike the existing proof of optimality (Rost [1976]), which relies on potential theory,an alternative proof is given by finding a path-wise inequality which has an impor-tant application for the construction of subhedging strategies in the financial context.In addition, we also consider these questions, construction and optimality, for Rost’ssolution, which is also known as the reverse of the Root’s solution.

Date of Award	31 Dec 2011
Original language	English
Awarding Institution	University of Bath
Supervisor	Alex Cox (Supervisor)