Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: In this thesis, we discuss and apply the Monte Carlo and integral transform methods in
pricing options. These methods have proved to be very e ective in the valuation of options
especially when acceleration techniques are introduced. By rst pricing European call
options we have motivated the use of these methods in pricing arithmetic Asian options
which have proved to be di cult to price and hedge under the Blackô€€€Scholes framework.
The arithmetic average of the prices in this framework, is a sum of correlated lognormal
distributions whose distribution does not admit a simple analytic expression. However,
many approaches have been reported in the academic literature for pricing these options.
We provide a hedging strategy by manipulating the results by Geman and Yor [42] for
continuous xed strike arithmetic Asian call options. We then derive a double Laplace
transform formula for pricing continuous Asian call options following the approach by Fu
et al. [39]. By applying the multi-Laguerre and iterated Talbot inversion techniques for
Laplace transforms to the resulting pricing formula we obtain the option prices. Finally,
we discuss the shortcomings of using the Laplace transform in pricing options. / AFRIKAANSE OPSOMMING: In hierdie tesis bespreek ons Monte Carlo- en integraaltransform metodes om die pryse van
nansi ele opsies te bepaal. Hierdie metodes is baie e ektief, veral wanneer versnellingsmetodes
ingevoer word. Ons bepaal eers die pryse van Europese opsies as motivering, voordat
ons die bostaande metodes gebruik vir prysbepaling van Asiatiese opsies met rekenkundige
gemiddeldes, wat baie moeiliker is om te hanteer in die Blackô€€€Scholes raamwerk. Die
rekenkundige gemiddelde van batepryse in hierdie raamwerk is 'n som van gekorreleerde
lognormale distribusies wie se distribusie nie oor 'n eenvoudige analitiese vorm beskik nie.
Daar is egter talle benaderings vir die prysbepaling van hierdie opsies in die akademiese
literatuur. Ons bied 'n verskansingsstrategie vir Asiatiese opsies in kontinue tyd met 'n
vaste trefprys aan deur die resultate van Geman en Yor [42] te manipuleer. Daarna volg
ons Fu et al. [39] om 'n dubbele Laplace transform formule vir die pryse af te lei. Deur
toepassing van multi-Laguerre en herhaalde Talbotinversie tegnieke vir Laplace transforms
op hierdie formule, bepaal ons dan die opsiepryse. Ons sluit af met 'n bespreking van die
tekortkominge van die gebruik van die Laplace transform vir prysbepaling.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/4292 |
| Date | 03 1900 |
| Creators | Chibawara, Trust |
| Contributors | Ouwehand, P. W., University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 90 p. |
| Rights | University of Stellenbosch |
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