A local radial basis function method for high-dimensional american option pricing problems
Abstract
In this work, we apply the local Wendland radial basis function (RBF) for solving the time-dependent multi dimensional option pricing nonlinear PDEs. Firstly, cross derivative terms of the PDE are removed with a change of spatial variables based in LDLT factorization of the di_usion matrix. Then, it is discussed that the valuation of a multi-asset option up to 4D can be computed using a modified shape parameter algorithm. In fact, several experiments containing of three and four assets are worked out showing that the results of the presented method are in good agreement with the literature and could be much more accurate once the shape parameter is chosen carefully.
Keyword : radial basis functions, cross derivative elimination, Wendland function, multi-asset problem, American option pricing
How to Cite
Company, R., Egorova, V. N., Jodar, L., & Soleymani, F. (2018). A local radial basis function method for high-dimensional american option pricing problems. Mathematical Modelling and Analysis, 23(1), 117-138. https://doi.org/10.3846/mma.2018.008
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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H.B. Zubair, C.W. Oosterlee and R. Wienands. Multigrid for high-dimensional elliptic partial differential equations on non-equidistant grids. SIAM J. Sci. Comput., 29(4):1613-1636, 2007. https://doi.org/10.1137/060665695
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S. Borovkova, F. Permana and J. V. der Weide. American basket and spread option pricing by a simple binomial tree. The J. Derivatives, 19(4):29-38, 2012. https://doi.org/10.3905/jod.2012.19.4.029
S. Choi and M. Marcozzi. A numerical approach to American currency option valuation. The J. Derivatives, 9(2):19-29, 2001. https://doi.org/10.3905/jod.2001.319172
R. Company, V.N. Egorova, L. Jodar and F. Soleymani. A mixed derivative terms removing method in multi-asset option pricing problems. Appl. Math. Lett., 60:108-114, 2016. https://doi.org/10.1016/j.aml.2016.04.011
G. Fasshauer. Meshfree Approximation Methods with MATLAB. World Scientific Publishing Co., Singapore, 2007. https://doi.org/10.1142/6437
G. Fasshauer and J. Zhang. On choosing optimal shape parameters for RBF approximation. Numer. Algor., 45(1-4):345-368, 2007. https://doi.org/10.1007/s11075-007-9072-8
B. Fornberg and N. Flyer. Solving PDEs with radial basis functions. Acta Numerica, 24:215-258, 2015. https://doi.org/10.1017/S0962492914000130
P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York, NY, USA, 2004.
A. Golbabai, D. Ahamadian and M. Milev. Radial basis functions with application to finance: American put option under jump diffusion. Math. Comput. Modell., 55(3-4):1354-1362, 2012. https://doi.org/10.1016/j.mcm.2011.10.014
H. Gzyl, M. Milev and A. Tagliani. Discontinuous payoff option pricing by Mellin transform: A probabilistic approach. Finance Res. Lett., 20:281-288, 2017. https://doi.org/10.1016/j.frl.2016.10.011
R. Hardy. Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res., 176(8):1905-1915, 1971. https://doi.org/10.1029/JB076i008p01905
Y. Hon and X. Mao. A radial basis function method for solving options pricing models. Finan. Eng., 8:31-49, 1999.
S. U. Islam and I. Ahmad. A comparative analysis of local meshless formulation for multi-asset option models. Eng. Anal. Bound. Elem., 65:159-176, 2016. https://doi.org/10.1016/j.enganabound.2015.12.020
P. Lancaster and M. Tismenetsky. The Theory of Matrices with Applications. Academic Press, USA, 2nd ed. edition, 1985.
E. Larsson, A. Krister and H. Andreas. Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform. J. Comput. Appl. Math., 222(1):175-192, 2008. https://doi.org/10.1016/j.cam.2007.10.039
C. Leentvaar. Pricing multi-asset options with sparse grids. PhD thesis, TU Delft, 2008.
M. Mongillo. Choosing basis functions and shape parameters for radial basis function methods. Technical report, SIAM Undergraduate Research Online (SIURO), 2011.
A. Pascucci. PDE and Martingale Methods in Option Pricing. Springer-Verlag, Italia, 2011. https://doi.org/10.1007/978-88-470-1781-8
U. Pettersson, E. Larsson, G. Marcusson and J. Persson. Improved radial basis function methods for multi-dimensional option pricing. J. Comput. Appl. Math., 222(1):82-93, 2008. https://doi.org/10.1016/j.cam.2007.10.038
C. Reisinger and G. Wittum. Efficient hierarchial approximation of highdimensional option pricing problems. SIAM J. Sci. Comput., 29(1):440-458, 2007. https://doi.org/10.1137/060649616
S. Rippa. An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math., 11(2-3):193-210, 1999. https://doi.org/10.1023/A:1018975909870
V. Shcherbakov and E. Larsson. Radial basis function partition of unity methods for pricing vanilla basket options. Comput. Math. Appl., 71(1):185-200, 2016. https://doi.org/10.1016/j.camwa.2015.11.007
D. Tavella. Quantitative methods in derivatives pricing. John Wiley & Sons, California, 2012.
L. von Sydow, L.J. Hook, E. Larsson, E. Lindstrom, S. Milovanovic, J. Persson, V. Shcherbakov, Y. Shpolyanskiy, S. Siren, J. Toivanen, J. Walden, M. Wiktorsson, J. Levesley, J. Li, C.W. Oosterlee, M.J. Ruijter, A. Toropov and Y. Zhao. BENCHOP - The BENCHmarking project in option pricing. Int. J. Comput. Math., 92(12):2361-2379, 2015. https://doi.org/10.1080/00207160.2015.1072172
H. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math., 4(1):389-396, 1995. https://doi.org/10.1007/BF02123482
M. Yousuf, A.-Q.M. Khaliq and R. Liu. Pricing American options under multistate regime switching with an efficient L-stable method. Int. J. Comput. Math., 92(12):2530-2550, 2015. https://doi.org/10.1080/00207160.2015.1071799
S. Zhu. Compactly supported radial basis functions: how and why? OCCAM report. Technical report, The Mathematical Institute, University of Oxford, 2006.
H.B. Zubair, C.W. Oosterlee and R. Wienands. Multigrid for high-dimensional elliptic partial differential equations on non-equidistant grids. SIAM J. Sci. Comput., 29(4):1613-1636, 2007. https://doi.org/10.1137/060665695
R. Zvan, P. Forsyth and K. Vetzal. Negative coefficients in twofactor option pricing models. J. Comput. Finance, 7(1):37-73, 2003. https://doi.org/10.21314/JCF.2003.096