Thesis (MSc)--Stellenbosch University, 2015. / ENGLISH ABSTRACT : The standard-Heath–Jarrow–Morton (HJM) framework is well-known for its application to
pricing and hedging interest rate derivatives. This study implemented the extended HJM
framework introduced by Eberlein and Raible (1999), in which a Brownian motion (BM) is
replaced by a wide class of processes with jumps. In particular, the HJM driven by the generalised
hyperbolic processes was studied. This approach was motivated by empirical evidence
proving that models driven by a Brownian motion have several shortcomings, such as inability
to incorporate jumps and leptokurticity into the price dynamics. Non-homogeneous Lévy
processes and the change of measure techniques necessary for simplification and derivation of
pricing formulae were also investigated. For robustness in numerical valuation, several transform
methods were investigated and compared in terms of speed and accuracy. The models
were calibrated to liquid South African data (ATM) interest rate caps using two methods of
optimisation, namely the simulated annealing and secant-Levenberg–Marquardt methods.
Two numerical valuation approaches had been implemented in this study, the COS method
and the fractional fast Fourier transform (FrFT), and were compared to the existing methods
in the context. Our numerical results showed that these two methods are quite efficient and
very competitive. We have chose the COS method for calibration due to its rapidly speed and
we have suggested a suitable approach for truncating the integration range to address the
problems it has with short-maturity options. Our calibration results provided a nearly perfect
fit, such that it was difficult to decide which model has a better fit to the current market state.
Finally, all the implementations were done in MATLAB and the codes included in appendices. / AFRIKAANSE OPSOMMING : Die standaard-Heath–Jarrow–Morton-raamwerk (kortom die HJM-raamwerk) is daarvoor bekend
dat dit op die prysbepaling en verskansing van afgeleide finansiële instrumente vir rentekoerse
toegepas kan word. Hierdie studie het die uitgebreide HJM-raamwerk geïmplementeer
wat deur Eberlein en Raible (1999) bekendgestel is en waarin ’n Brown-beweging deur ’n
breë klas prosesse met spronge vervang word. In die besonder is die HJM wat deur veralgemeende
hiperboliese prosesse gedryf word ondersoek. Hierdie benadering is gemotiveer
deur empiriese bewyse dat modelle wat deur ’n Brown-beweging gedryf word verskeie tekortkominge
het, soos die onvermoë om spronge en leptokurtose in prysdinamika te inkorporeer.
Nie-homogene Lévy-prosesse en die maatveranderingstegnieke wat vir die vereenvoudiging en
afleiding van prysbepalingsformules nodig is, is ook ondersoek. Vir robuustheid in numeriese
waardasie is verskeie transformmetodes ondersoek en ten opsigte van spoed en akkuraatheid
vergelyk. Die modelle is vir likiede Suid-Afrikaanse data vir boperke van rentekoerse sonder
intrinsieke waarde gekalibreer deur twee optimiseringsmetodes te gebruik, naamlik die gesimuleerde
uitgloeimetode en die sekans-Levenberg–Marquardt-metode.
Twee benaderings tot numeriese waardasie is in hierdie studie gebruik, naamlik die kosinusmetode
en die fraksionele vinnige Fourier-transform, en met bestaande metodes in die konteks
vergelyk. Die numeriese resultate het getoon dat hierdie twee metodes redelik doeltreffend
en uiters mededingend is. Ons het op grond van die motiveringspoed van die kosinus-metode
daardie metode vir kalibrering gekies en ’n geskikte benadering tot die trunkering van die
integrasiereeks voorgestel ten einde die probleem ten opsigte van opsies met kort uitkeringstermyne
op te los. Die kalibreringsresultate het ’n byna perfekte passing gelewer, sodat dit
moeilik was om te besluit watter model die huidige marksituasie die beste pas. Ten slotte is
alle implementerings in MATLAB gedoen en die kodes in bylaes ingesluit.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/96783 |
| Date | 03 1900 |
| Creators | Alfeus, Mesias |
| Contributors | Ouwehand, P. W., Stellenbosch University. Faculty of Science. Department of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | x, 123 pages : illustrations |
| Rights | Stellenbosch University |
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