Lookback Option Pricing with the COS Method

Abstract

In this thesis we describe the use of the COS method, a numerical option pricing method based on characteristic functions, developed by Fang and Oosterlee. The novelty in my thesis is the adaptation of this method to price lookback options, an option style highly dependent on the maximum of a chosen stochastic process. The main focus of this thesis is therefore the description of the characteristic function of the maximum of a Geometric Brownian Motion process. To showcase the effectiveness of the COS method we have devised a Python implementation, comparing COS-estimates with both known option values and Monte Carlo simulations. We will find that the COS method is a highly efficient pricing method, far outperforming for example Monte Carlo estimates. As a final chapter this thesis also includes a draft for an alternative method of describing the maximum of a process. This method, applicable to a broad range of stochastic processes known as exponential L´ evy processes, recursively defines the characteristic function of a process’ maximum. While this chapter is only a proof of concept, this method does give a very promising outlook to a much broader application of the COS method for lookback option pricing.