Abstract
We use a sparse grid approach to discretize a multi-dimensional partial integro-differential equation (PIDE) for the deterministic valuation of European put options on Kou’s jump-diffusion processes. We employ a generalized generating system to discretize the respective PIDE by the Galerkin approach and iteratively solve the resulting linear system. Here, we exploit a newly developed recurrence formula, which, together with an implementation of the unidirectional principle for non-local operators, allows us to evaluate the operator application in linear time. Furthermore, we exploit that the condition of the linear system is bounded independently of the number of unknowns. This is due to the use of the Galerkin generating system and the computation of L 2-orthogonal complements. Altogether, we thus obtain a method that is only linear in the number of unknowns of the respective generalized sparse grid discretization. We report on numerical experiments for option pricing with the Kou model in one, two and three dimensions, which demonstrate the optimal complexity of our approach.
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Notes
- 1.
More sophisticated techniques, e.g. space-time sparse grids [17], are available, but this is beyond the scope of this work.
- 2.
This holds for a range of parameters 0 ≤ s < t ≤ r with r being the order of the spline of the space construction. In our case of linear splines r = 2 holds.
- 3.
For s = 0 an additional logarithmic term appears in the error estimate.
- 4.
The discretization in this example includes boundary functions otherwise not used for computation.
- 5.
Note that the numbers \((1),\ldots , (d)\) in the exponents are no powers but indices, indicating that different operations may be carried out in different dimensions.
- 6.
For notational convenience we drop the dimension index (j) in \({\mathbf{A}}_{{l}_{j},{l}_{j} \prime }^{(j)}\) in the following calculations.
- 7.
Here and in the following, we have omitted time related indices and the dependence on other dimensions.
- 8.
In one dimension, full and sparse grids are of course the same.
References
L. Andersen and J. Andreasen. Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Review of Derivatives Research, 4(3):231–262, 2000.
R. Bellman. Adaptive Control Processes: A Guided Tour. Princeton University Press, 1961.
H. Bungartz and M. Griebel. Sparse grids. Acta Numerica, 13:1–123, 2004.
H. Bungartz, A. Heinecke, D. Pflüger, and S. Schraufstetter. Option pricing with a direct adaptive sparse grid approach. Journal of Computational and Applied Mathematics, 2011.
G. Beylkin and M. Mohlenkamp. Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA, 99:10246–10251, 2002.
F. Black and M. Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637–654, 1973.
H. Bungartz. Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation, Fakultät für Informatik, Technische Universität München, November 1992.
R. Balder and C. Zenger. The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci. Comput., 17:631–646, May 1996.
R. Cont and P. Tankov. Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series, 2004.
R. Cont and E. Voltchkova. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal., 43:1596–1626, 2005.
C. Feuersänger. Sparse Grid Methods for Higher Dimensional Approximation. Dissertation, Institut für Numerische Simulation, Universität Bonn, September 2010.
T. Gerstner and M. Griebel. Dimension–adaptive tensor–product quadrature. Computing, 71(1):65–87, 2003.
M. Griebel and H. Harbrecht. On the construction of sparse tensor product spaces. Mathematics of Computations, 2012. to appear. Also available as INS Preprint No. 1104, University of Bonn.
M. Griebel and S. Knapek. Optimized general sparse grid approximation spaces for operator equations. Mathematics of Computations, 78(268):2223–2257, 2009.
M. Griebel and P. Oswald. On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math., 66:449–464, 1994.
M. Griebel and P. Oswald. Tensor product type subspace splitting and multilevel iterative methods for anisotropic problems. Adv. Comput. Math., 4:171–206, 1995.
M. Griebel and D. Oeltz. A sparse grid space-time discretization scheme for parabolic problems. Computing, 81(1):1–34, 2007.
M. Griebel. Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen. Teubner Skripten zur Numerik. Teubner, Stuttgart, 1994.
E. Kaasschieter. Preconditioned conjugate gradients for solving singular systems. Journal of Computational and Applied Mathematics, 24(1–2):265–275, 1988.
S. Kou. A jump-diffusion model for option pricing. Management Science, 48(8):1086–1101, 2002.
S. Kou. Jump-diffusion models for asset pricing in financial engineering. In John R. Birge and Vadim Linetsky, editors, Financial Engineering, volume 15 of Handbooks in Operations Research and Management Science, pages 73–116. Elsevier, 2007.
R. Merton. Theory of rational option pricing. Bell Journal of Economics, 4(1):141–183, 1973.
R. Merton. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3:125–144, 1976.
C. Schwab N. Hilber, S. Kehtari and C. Winter. Wavelet finite element method for option pricing in highdimensional diffusion market models. Research Report 2010-01, Seminar for Applied Mathematics, Swiss Federal Institute of Technology Zurich, 2010.
D. Oeltz. Ein Raum-Zeit Dünngitterverfahren zur Diskretisierung parabolischer Differentialgleichungen. Dissertation, Institut für Numerische Simulation, Universität Bonn, 2006.
C. Reisinger. Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben. Dissertation, Ruprecht-Karls-Universität Heidelberg, 2004.
N. Reich, C. Schwab, and C. Winter. On Kolmogorov equations for anisotropic multivariate Lévy processes. Finance and Stochastics, 14(4):527–567, 2010.
V. Thomée. Galerkin finite element methods for parabolic problems. Springer series in computational mathematics. Springer, 1997.
J. Toivanen. Numerical valuation of European and American options under Kou’s jump-diffusion model. SIAM J. Sci. Comput., 30(4):1949–1970, 2008.
A. Zeiser. Fast matrix-vector multiplication in the sparse-grid Galerkin method. J. Sci. Comput., 47(3):328–346, 2011.
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Griebel, M., Hullmann, A. (2012). An Efficient Sparse Grid Galerkin Approach for the Numerical Valuation of Basket Options Under Kou’s Jump-Diffusion Model. In: Garcke, J., Griebel, M. (eds) Sparse Grids and Applications. Lecture Notes in Computational Science and Engineering, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31703-3_6
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