Duality Methods for Barrier-type Solutions to the Skorokhod Embedding Problem

Abstract

The Skorokhod embedding problem is to find a stopping time of a Brownian motion, W, for which the stopped process has a given distribution. The Root, Rost, and cave embedding solutions to the problem can be seen as the first hitting time for (Wt,t) of regions known as barriers, inverse barriers, and cave barriers, respectively. In this thesis we present three ways of approaching the embedding problem, and apply the methods to these barrier-type solutions. Specifically, we consider infinite dimensional linear optimisation problems in both discrete and continuous time, and we also reformulate into an optimisation constrained by backwards stochastic differential equations and then solve using techniques from stochastic optimal control.For certain financial derivatives it is well known that there is an optimal Skorokhod embedding problem which corresponds to finding a model-independent upper bound on the price of the contingent claim. With this application in mind, the embedding problem has the dual problem of finding the minimal cost of a superhedging portfolio for the option. The methods developed in this thesis enable us to explore the rela- tion between the primal and dual problems, and, in the applications above, find dual optimisers. We also introduce a new barrier-type embedding, known as a K-cave em- bedding, which has the property of maximising the price of a European call option on a leveraged exchange traded fund. For the cave and K-cave embeddings the attainment of an optimal superhedging strategy is needed to find the optimal barriers. Unlike in the cases of Root and Rost, there are not unique cave, or K-cave barriers which embed a given distribution and in this way these are the first examples of embeddings which are not uniquely determined by their geometric structure.

Date of Award	3 Oct 2018
Original language	English
Awarding Institution	University of Bath
Supervisor	Alex Cox (Supervisor)