Optimal Stopping Methods for Multidimensional Problems, Pricing and Hedging of American Options

In Chapter 1, we give an introduction to all subsequent chapters in the thesis. In Chapter 2, we first introduce the underlying stochastic process, notations and some formulae used in the thesis. We then collect some classical results of optimal stopping and free boundary problems, including the solution of the American option pricing problem under the classical Black and Scholes model. In Chapter 3, we consider the seller of a perpetual American put option who can hedge her portfolio once, until the underlying stock price leaves a certain range of values (a,b). We determine optimal trading boundaries as functions of the initial stock holding, and an optimal hedging strategy for a bond/stock portfolio. Optimality here refers to minimal variance of the hedging error at the (random) time when the stock leaves the interval (a,b). Our study leads to analytical expressions for both the optimal boundaries and the optimal stock holding, which can be evaluated numerically with no effort. In Chapter 4, we study pricing of American put options on the Black and Scholes market with a stochastic interest rate and finite-time maturity. We prove that the option value is a C¹ function of the initial time, interest rate and stock price. By means of Itô calculus we rigorously derive the option's early exercise premium formula and the associated hedging portfolio. We prove the existence of an optimal exercise boundary splitting the state space into continuation and stopping region. The boundary has a parametrisation as a jointly continuous function of time and stock price, and it is the unique solution to an integral equation which we compute numerically. Our results hold for a large class of interest rate models including CIR and Vasicek models. We show a numerical study of the option price and the optimal exercise boundary for Vasicek model. In Chapter 5, we derive a change of variable formula for C¹ functions U: R+ x Rᵐ → R whose second order spatial derivatives may explode and not be integrable in the neighbourhood of a surface R+ x Rᵐˉ¹ → R that splits the state space into two sets C and D. The formula is tailored for applications in problems of optimal stopping where it is generally very hard to control the second order derivatives of the value function near the optimal stopping boundary. Differently to other existing results on similar topics we only require that the surface b be monotonic in each variable and we formally obtain the same expression as the classical Itô's formula. In Chapter 6, we provide sufficient conditions under which a two dimensional (time-space) optimal stopping surface, arising from a general three dimensional optimal stopping problem, is continuous. We require mild local regularity assumptions on the coefficients of the dynamics of the underlying process, the gain function and the value function. Further, we assume monotonicity of the optimal stopping surface in each variable.