Polynomial optimization : Applications in finance

Abstract

This thesis discusses the theory of modern polynomial optimization and its applications in the field of finance. From a theoretical point of view, special attention is directed towards examining and proving the finite convergence of the so-called Lasserre's hierarchy which serves as the backbone of the polynomial optimization procedure detailed in this thesis.

The first two sections of the thesis mainly deal with the relevant background theory. (Positive) polynomials, moment problems, and polynomial optimization are introduced. The culmination of the second section is the proof of finite convergence of Lasserre's hierarchy.

The third section provides an overview of algorithmic implementation of the polynomial optimization methodology. Relevant algorithms are described in detail and various issues pertaining to the implementation are discussed.

The fourth section consists of numerical examples from the field of finance and beyond. Each example starts with a problem statement and an overview of how the question tackled can be stated as a polynomial optimization problem. Explicit numerical examples illustrate the proposed methods. Most importantly, it is acknowledged that via the polynomial optimization procedure one is able to obtain numerically verified global optimal solutions to problems which are often solved in earlier literature using various heuristic (quasi-global) methods. The fifth section concludes the thesis.