Thesis(MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The pricing of plain vanilla options, including early exercise options, such as Bermudan
and American options, forms the basis for the calibration of financial models.
As such, it is important to be able to price these options quickly and accurately.
Empirical studies suggest that asset dynamics have jump components which can be
modelled by exponential Lévy processes. As such models often have characteristic
functions available in closed form, it is possible to use Fourier transform methods,
and particularly, the Fast Fourier Transform, to price such options efficiently. In
this dissertation we investigate and implement four such methods, dubbed the Carr-
Madan method, the convolution method, the COS method and the Fourier spacetime
stepping method. We begin by pricing European options using these Fourier
methods in the Black-Scholes, Variance Gamma and Normal Inverse Gaussian models.
Thereafter, we investigate the pricing of Bermudan and American options in
the Black-Scholes and Variance Gamma models. Throughout, we compare the four
Fourier pricing methods for accuracy and computational efficiency. / AFRIKAANSE OPSOMMING: Die prysbepaling van gewone vanilla opsies, insluitende opsies wat vroeg uitgeoefen
kan word, soos Bermuda-en Amerikaanse opsies, is grondliggend vir die kalibrering
van finansiële modelle. Dit is daarom belangrik dat die pryse van sulke opsies vinnig
en akkuraat bepaal kan word. Empiriese studies toon aan dat batebewegings
sprongkomponente besit, wat gemodelleer kan word met behulp van exponensiëele
Lévyprosesse. Aangesien hierdie modelle dikwels karakteristieke funksies het wat
beskikbaar is in geslote vorm, is dit moontlik om Fourier-transform metodes, en
in besonders die vinnige Fourier-transform, te gebruik om opsiepryse doeltreffend
te bepaal. In hierdie proefskrif ondersoek en implementeer ons vier sulke metodes,
genaamd die Carr-Madan metode, die konvolusiemetode, die COS-metode en die
Fourier ruimte-tydstap metode. Ons begin deur die pryse van Europese opsies
in die Black-Scholes, Gammavariansie (Engels: Variance gamma) en Normaal Invers
Gauss (Engels: Normal Inverse Gaussian)-modelle te bepaal met behulp van
die vier Fourier-metodes. Daarna ondersoek ons die prysbepaling van Bermuda-en
Amerikaanse opsies in die Black-Scholes en Gammavariansiemodelle. Deurlopend
vergelyk ons die vier Fourier-metodes vir akkuraatheid en berekeningsdoeltreffendheid.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/71860 |
| Date | 12 1900 |
| Creators | Fadina, Tolulope Rhoda |
| Contributors | Ouwehand, P. W., Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | 134 p. : ill. |
| Rights | Stellenbosch Univesity |
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