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UM E-Theses Collection

 Title : applications of Fourier analysis to European option pricing     English Abstract Recently some efficient numerical methods based on Fourier Analysis wereproposed. In this thesis, I summarize the Carr & Madan method and Fang& Oosterlee method respectively. Furthermore, I proposed new methods andnumerical schemes to improve and develop their methods.Carr & Madan method is the first method to use Fast Fourier Transformalgorithm on option pricing, and the overall complexity is O(N log N). Althoughthis method is fast, it has a quite large error when the strike pricesare small. My main work in this thesis is to develop a new method whichcombines the quadrature technique and their method. The method is accurateand stable even the strike prices are small. The theoretical analysisshows that the overall complexity of this new method is still O(N log N).The main idea of Fang & Oosterlee method is to price the option basedon Fourier cosine expansion, the convergence rate of this method is exponentialand the computational complexity is linear. The disadvantage of theirmethod is being inefficient when the option prices are calculated with manydifferent strike prices. My main work is to develop a new efficient methodwhich is based on their method. Moreover, I apply the Fourier sine expansionand Fourier series expansion to the option pricing, then I compare thosemethods in this thesis.In Chapter 1, Section 1.1, I introduce some basic knowledge of Europeanoptions. Section 1.2, I give the definition of L´evy model, and show someExponential L´evy models for the financial markets, I also give the characteristicfunction of these models. Section 1.3, I change the pricing problemsto quadrature problem, and the definition of characteristic function are alsoobtained.In Chapter 2, Section 2.1, I show the basic idea of Fourier series, cosineand sine expansions, the definition of algebraic convergence rate, geometryconvergence rate and some convergence theorems. Section 2.2, I will givethe definitions of Fourier Transform and Inverse Fourier Transform. Section2.3, I introduce the idea of Fast Fourier Transform (FFT), and some usefultheorems of FFT.In Chapter 3, Section 3.1, I summarize the Carr & Madan method andpoint out the problem of this method. Section 3.2, I give the error analysisof Carr & Madan method which comes from their paper i.e. [4]. Section 3.3,I propose our method which combines the quadrature technique and Carr &Madan method, and our method is more accurate and stable. Section 3.4,the Numerical Experiments show the comparison of our method and Carr &Madan method in some models which are given in Section 2.2.In Chapter 4, Section 4.1, I summarize the Fang & Oosterlee method.Section 4.2, I will give the error analysis of Fang & Oosterlee method whichcomes from their paper i.e. [9]. Section 4.3, I propose our method whichincreases the efficiency of their method for the option prices calculated withmany different strike prices. I also applied other expansions which are givenin Section 2.1 to European option pricing. Section 4.4 is the Numerical Experiments. 2009 U, Sio Chong Faculty of Science and Technology Department of Mathematics M.Sc. Options (Finance) -- Prices -- Mathematical models Options (Finance) -- Prices -- Europe -- Mathematical models Fourier analysis Mathematics -- Department of Mathematics Ding, Deng b2148263 1/F Zone C