Bergische Universität Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Angewandte Mathematik - Numerische Analysis


Rapid Numerical Solution of Jump Diffusion Models

Masterarbeit Wirtschaftsmathematik

Master's Thesis in Financial Mathematics, Halmstad University, Sweden



In this thesis we discuss the proper numerical treatment of partial integro differential equations (PIDE) that appear as pricing equations for options on assets that follow jump-diffusion processes. Usually these PIDEs need a high computational effort due to the involved integrals.
Although Fast Fourier Transform (FFT) methods (pioneered by Andersen and Andreasen [1]) can be applied to evaluate these jump-integrals, the usage of the FFT induces additional errors such that the jump integral must be discretized on a much finer grid than the one for the differential operators (cf. Y. d'Halluin et al.). Moreover, usually implicit time stepping methods are considered and hence the FFT is inside some iterative solution procedure (i.e. a micro-iteration) which makes the benefits using the FFT relatively small.
Consequently, in this thesis we will present alternative approaches without using a FFT.


Option Pricing, Stochastic Volatility, Lévy Processes, Bates Model, operator splitting technique, Jump-diffusion process, finite elements, nonlinear partial integro differential equation (PIDE)


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  4. G. Cheang, C. Chiarella and A. Ziogas, The representation of American options prices under stochastic volatility and jump-diffusion dynamics, Research Paper Series 256, Quantitative Finance Research Centre, University of Technology, Sydney, 2009.
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  14. S. McKee, D.P. Wall and S.K. Wilson, An Alternating Direction Implicit Scheme for Parabolic Equations with Mixed Derivative and Convective Terms, J. Comput. Phys. 126 (1996), 64-76.
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  16. V. Surkov, Option Pricing using Fourier Space time-stepping Framework, Ph.D. thesis, University of Toronto, 2009.
  17. J. Toivanen, A Componentwise Splitting Method for Pricing American Options under the Bates Model, Applied and Numerical Partial Differential Equations: Scientific Computing, Simulation, Optimization and Control in a Multidisciplinary Context, W. Fitzgibbon, Yu. Kuznetsov, P. Neittaanmäki, J. Periaux, O. Pironneau (eds.), Computational and Methods in Applied Sciences 15 (2010), Springer, pp. 213-227.

University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Applied Mathematics & Numerical Analysis Group

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