Uporaba Lévyjevih procesov v finančni matematiki : doktorska disertacija

slovenian

doktorska dela

2019

American options   ESG   financial derivatives   GMWB   Lévy processes   stochastic differential equations   stochastic integrals   variable annuities

Univerza v Ljubljani, Fakulteta za matematiko in fiziko
(Obvezni izvod spletne publikacije)

When it comes to modelling prices on the capital markets the Lévy processes and other processes with jumps became very popular in the last decade. They are used in risk management and also in pricing options and other financial derivatives. The theory of such processes is known for a while, but in financial mathematics these processes are used since the late 80's of the 20th century. Since then there are numerous research papers deriving properties for ?$\alpha$?-stable, gamma, normal inverse Gaussian and generalized hyperbolic processes. All of these are now used for pricing derivatives because Gaussian distributions and Brownian motion do not mimic the motion of the financial markets well enough as has been shown by empirical studies. When processes incorporate jumps in addition to the continuous parts they become much more difficult to work with, even on the theoretical level. Methods of stochastic analysis become even more complicated when the jump term exists and because of that there are fewer methods to solve stochastic differential equations and even less analytic ones. On the other hand many researchers increasingly recognize the weaknesses of the Monte Carlo methods, which have been most used in the past. Therefore researchers started developing fast and direct approach methods by which one can obtain arbitrarily accurate results without randomness. In our case we will use such methods for solving partial differential equations. We will use the method based on discretization and finite elements which was developed in the past years for American option pricing. For the finite elements basis we will use wavelets which allows us fast computation since the obtained matrix is sparse. In addition one can easily increase accuracy due to the multiresolution technique. In the thesis we first introduce Lévy processes and some important results in the field. Next we present a part of the stochastic analysis for Lévy processes which is important for the pricing of financial derivatives and give special emphasis to the Itô formula for Lévy processes. We continue with the description of some financial derivatives and insurance products. In the continuation we give a method for solving the differential equations and methods of numerical approximation and integration. Finally, we apply these methods and theoretical results to the pricing of financial derivatives, economic scenario generators and pricing variable annuities, especially the product GMWB. Our introduced approach for the latter seems to be new. In the appendix we describe Bessel functions which are used in some special Lévy processes. We also present another algorithm for the pricing of the GMWB, namely dynamic programming algorithm with which we compare our numerical results.

12.05.2023