The Fast Fourier Transform (FFT) has been used in many scientific and engineering applications. The use of FFT for financial derivatives has been gaining momentum in the recent past. In this thesis, i) we have improved a recently proposed model of FFT for pricing financial derivatives to help design an efficient parallel algorithm. The improved mathematical model put forth in our research bridges a gap between quantitative approaches for the option pricing problem and practical implementation of such approaches on modern computer architectures. The thesis goes further by proving that the improved model of fast Fourier transform for option pricing produces accurate option values. ii) We have developed a parallel algorithm for the FFT using the classical Cooley-Tukey algorithm and improved this algorithm by introducing a data swapping technique that brings data closer to the respective processors and hence reduces the communication overhead to a large extent leading to better performance of the parallel algorithm. We have tested the new algorithm on a 20 node SunFire 6800 high performance computing system and compared the new algorithm with the traditional Cooley-Tukey algorithm. Option values are calculated for various strike prices with a proper selection of strike-price spacing to ensure fine-grid integration for FFT computation as well as to maximize the number of strikes lying in the desired region of the stock price. Compared to the traditional Cooley-Tukey algorithm, the current algorithm with data swapping performs better by more than 15% for large data sizes. In the rapidly changing market place, these improvements could mean a lot for an investor or financial institution because obtaining faster results offers a competitive advantages.

This thesis examines the empirical performance of four Affine Jump Diffusion models in pricing and hedging S&P 500 Index options: the Black Scholes (BS) model, Heston???s Stochastic Volatility (SV) model, a Stochastic Volatility Price Jump (SVJ) model and a Stochastic Volatility Price-Volatility Jump (SVJJ) model. The SVJJ model structure allows for simultaneous jumps in price and volatility processes, with correlated jump size distributions. To the best of our knowledge this is the first empirical study to test the hedging performance of the SVJJ model. As part of our research we derive the SVJJ model minimum variance hedge ratio. We find the SVJ model displays the best price prediction. The SV model lacks the structural complexity to eliminate Black Scholes pricing biases, whereas our results indicate the SVJJ model suffers from overfitting. Despite significant evidence from in and out-of-sample pricing that the SV and SVJ models were better specified than the BS model, this did not result in an improvement in dynamic hedging performance. Overall the BS delta hedge and SV minimum variance hedge produced the lowest errors, although their performance across moneyness-maturity categories differed greatly. The SVJ model???s results were surprisingly poor given its superior performance in out-of-sample pricing. We attribute the inadequate performance of the jump models to the lower hedging ratios these models provided, which may be a result of the negative expected jump sizes.

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.

[pt] O apreçamento de opções é um assunto muito importante nos dias de hoje. Métodos probabilisticos são necessários para fazer o apreçamento neutro ao risco. Usaremos o método de Siu et al. para duas classes de GARCHs, o FC-GARCH e a mistura de GARCHs Em ambos os modelos nós encontramos a versão neutra ao risco do modelo que é necessária para a precificação de contratos, em dois diferentes casos, quando o ruído é normal e quando é shifted gamma. Fizemos também simulações para ilustrar e comparamos os resultados com o valor de Black Scholes, verificamos a existência de smile e fizemos uma análise de sensibilidade nos parâmetros. / [en] Option pricing is a very important issue nowadays. The use of probabilistic methods is required for risk neutral pricing. Here we apply the method of Siu et al. for two classes of GARCHs, viz., the FC-GARCH and the Mixture of GARCHs. In both models we derive the risk neutral version of the model which is essential for pricing contracts, in two different cases, when the noise is normal as well as when it is shifted gamma. We also performed simulations with both models and compared to the benchmark Black Scholes model, checked for the smile effect and made some sensibility analysis in the parameters.

A standard problem in the field of computational finance is that of pricing derivative securities. This is often accomplished by estimating an expected value of a functional of a stochastic process, defined by a stochastic differential equation (SDE). In such a setting the random sampling algorithm Monte Carlo (MC) is useful, where paths of the process are sampled. However, MC in its standard form (SMC) is inherently slow. Additionally, if the analytical solution to the underlying SDE is not available, a numerical approximation of the process is necessary, adding another layer of computational complexity to the SMC algorithm. Thus, the computational cost of achieving a certain level of accuracy of the estimation using SMC may be relatively high. In this thesis we introduce and review the theory of the SMC method, with and without the need of numerical approximation for path simulation. Two numerical methods for path approximation are introduced: the Euler–Maruyama method and Milstein's method. Moreover, we also introduce and review the theory of a relatively new (2008) MC method – the multilevel Monte Carlo (MLMC) method – which is only applicable when paths are approximated. This method boldly claims that it can – under certain conditions – eradicate the additional complexity stemming from the approximation of paths. With this in mind, we wish to see whether this claim holds when pricing a European call option, where the underlying stock process is modelled by geometric Brownian motion. We also want to compare the performance of MLMC in this scenario to that of SMC, with and without path approximation. Two numerical experiments are performed. The first to determine the optimal implementation of MLMC, a static or adaptive approach. The second to illustrate the difference in performance of adaptive MLMC and SMC – depending on the used numerical method and whether the analytical solution is available. The results show that SMC is inferior to adaptive MLMC if numerical approximation of paths is needed, and that adaptive MLMC seems to meet the complexity of SMC with an analytical solution. However, while the complexity of adaptive MLMC is impressive, it cannot quite compensate for the additional cost of approximating paths, ending up roughly ten times slower than SMC with an analytical solution.

This study has a twofold purpose: to demonstrate the use of the Marquardt compromise method in estimating the unknown parameters contained in the probability call-option pricing models and to test empirically the following models: the Boness, the Black-Scholes, the Merton proportional dividend, the Ingersoll differential tax, and the Ingersoll proportional dividend and differential tax.

Efficient numerical methods for option pricing is an active field of research. This project has the goal to examine possible ways to improve an established method of numerical pricing. The method is based on an adaptive finite difference method in price and uses the backwards differentiation formula of order 2, BDF2, in time. The project will focus on improvements to the time integration through implementation of discontinuous Galerkin methods, dG. Empirical convergence and accuracy results are obtained for equidistant dG-methods up to order 3 and performance is compared to BDF2. The dG-methods do not succeed in outperforming the BDF2-method when comparing accuracy to time for computation, but they do match the performance. Possible ways for improvements are suggested.

This thesis contains results on convergence studies for different stencils of radial basis function generated finite difference (RBF-FD) method applied to solving Black-Scholes equation for pricing European call options. The results experimentally confirm the theoretical convergence rates for smooth payoff functions with stencils of size 3, 5 and 7 in one- dimensional problems, and 9, 13 and 25 in two- dimensional problems. Moreover, it is shown how different terms in the equation can be approximated individually using the proposed method and then combined into a discrete approximation of the entire spatial differential operator. This new version of the RBF-FD method, where each term has been approximated individually, has been compared to the classical method and the outcome did not show any significant performance advantages. Nevertheless, the results also showed that the second order derivative was the hardest one to approximate accurately and this poses an important finding for the future development of the method.

This paper examines different combinations of close-field and far-field boundary conditions for solving the 2D Black-Scholes model using finite difference methods in space. The combinations were also tested for different parameter settings. The research showed that in the area close to the strike price, the error was not particularly affected by the boundary conditions but rather by the characteristics of the problem itself. The main differences in error for the combinations of conditions are located close to the boundaries. However, if the computational domain for some reason has to be reduced, e.g. to save computational time, the boundary conditions will play an important role on the error in the area close to the strike price. Based on the findings presented in this report, Dirichlet boundary condition on the far- field boundary together with no boundary condition on the close-field is the best combination. If any of those are not applicable, the linearity condition should be used on that boundary instead.

This thesis is devoted to the study of different stochastic processes which have a common feature: they are Markov-modulated, which means that their evolution rules depend on the state occupied by an underlying Markov process. In the first part of this thesis, we analyse the stationary distribution and various first passage problems for Markov-modulated Brownian motions (MMBMs) as well as for two extensions: MMBMs with jumps and MMBMs modified by a temporary change of regime upon visits to level zero. The second part of this thesis is devoted to the use of Markov-modulated processes in mathematical finance, more precisely for the calculation of different option prices. We use a Fourier transform approach to price different European options (vanilla, exchange and quanto options) in the case where the value of the considered risky assets evolves like the exponential of a Markov-modulated Lévy process. The third part of this thesis is devoted to the study of some stochastic epidemic processes, namely the SIR processes. In our models, a Markov process is used to modulate the behaviour of the individuals who bring the disease. We use different martingale approaches as well as matrix analytic methods to obtain various information about the state of the population when the epidemic is over. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished