The main objective of this thesis has been to develop an analysis of the dynamics of exchange rates under two models; one continuous and one allowing for jumps. First we will look at a stochasticdifferential equation with Brownian motion representing the "noise" and later extend this model to incorporate jumps by means of a Gamma process. Some estimation and computation based upon a dataset, consisting of interest rates and exchange rates between Norway and the US, have been done to see how the models would work in practice. Pricing of currency derivatives, in particular currency options and currency forward contracts, will also be investigated.
Exchange rates is essential in many situations. They allow the conversion between domestic and foreign currency and establishes a direct link between a domestic spot price market and a foreignspot price market. It is a process converting foreign market cash flows into domestic currency, and vice versa. An investor operating in the domestic market, who wants to incorporate foreign assets in his portfolio, needs to expand his model to allow for evaluation of foreign currency. Exchange rates also give rise to another important market, the cross-currency derivatives market. Such derivatives serve as important tools in banks and insurance companies to manage or control risk exposure coming from the uncertainty of future exchange rates. Modeling of exchange rates opens up for evaluation of "fair prices" for such derivatives.
The thesis has been divided into 8 chapters. Chapters 1 and 2 are introductory chapters, providing some background on financial derivatives and stochastic analysis in continuous time. Chapter 3introduces our first model, which investigates the dynamics of exchange rates modeled by means of geometric Brownian motion within the Black-Scholes framework. Chapter 4 continues the investigationof this model in a more applicative way through maximum likelihood estimation and computations based on exchange rates between Norway and the US. In Chapter 5 financial derivatives are revisited, the issue now is how their "fair price" should be determined. Chapter 6 providessome stochastic analysis and results based on general Lévy processes to prepare for Chapter 7, where we consider an exponential Lévy process with jumps, represented by a Gamma process, to model exchange rates. Finally, Chapter 8 provides some conclusions and ideas for further extensions of the model, as well as an alternative non-linear model for exchange rates.