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 Black􀀀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 Black􀀀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.

In this thesis we present the Laplace transform method of option pricing and it's realization, also compare it with another methods. We consider vanilla and exotic options, but more attention we pay to the two-asset correlation options. We chose the one of the modifications of Black-Scholes model, the Kou double exponential jump-diffusion model with the double exponential distribution of jumps, as model of the underlying stock prices development. The computations was done by the Laplace transform and it's inversion by the Euler method. We will present in details proof of finding Laplace transforms of put and call two-asset correlation options, the calculations of the moment generation function of the jump-diffusion by Levy-Khintchine formulae in cases without jumps and with independent jumps, and direct calculation of the risk-neutral expectation by solving double integral. Our work also contains the programme code for two-asset correlation call and put options. We will show the realization of our programme in the real data. As a result we see how our model complies on the NASDAQ OMX Stock-holm Market, considering the two-asset correlation options on three cases by stock prices of Handelsbanken, Ericsson and index OMXS30.