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

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Rapid Numerical Solution of Jump Diffusion Models

Masterarbeit Wirtschaftsmathematik

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

Supervision

• Prof. Dr. Matthias Ehrhardt

Description

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.

Keywords

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

References:

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2. A. Anderson and J. Andresen, Jump diffusion process: Volatility smile fitting and numerical methods for option pricing, Rev. Derivatives Res. 4 (2000), 231-262.
3. D. Bates, Jumps and stochastic volatility: the exchange rate processes implicit in Deutschemark options, Rev. Fin. Studies 9 (1996), 69-107.
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.
5. C. Chiarella, B. Kang, G.H. Meyer and A. Ziogas, The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines, Int. J. of Theor. and Appl. Finance 12 (2009), 393-425.
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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.