×

A reduced-order model based on integrated radial basis functions with partition of unity method for option pricing under jump-diffusion models. (English) Zbl 1537.91356


MSC:

91G60 Numerical methods (including Monte Carlo methods)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65D12 Numerical radial basis function approximation
91G20 Derivative securities (option pricing, hedging, etc.)

Software:

Matlab
Full Text: DOI

References:

[1] Costabile, M.; Leccadito, A.; Massabó, I.; Russo, E., Option pricing under regime-switching jump-diffusion models, J Comput Appl Math, 256, 152-167 (2014) · Zbl 1314.91206
[2] Seydel, R., Tools for computational finance (2009), Springer: Springer Berlin · Zbl 1160.91017
[3] Carr, P.; Madan, D. B., Option valuation using the fast Fourier transform, J Comput Finance, 2, 61-73 (1999)
[4] Merton, R. C., Option pricing when underlying stock returns are discontinuous, J Financ Econ, 3, 1-2, 125-144 (1976) · Zbl 1131.91344
[5] Kou, S. G., A jump-diffusion model for option pricing, Manage Sci, 48, 8, 1086-1101 (2002) · Zbl 1216.91039
[6] Andersen, L.; Andreasen, J., Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev Deriv Res, 4, 3, 231-262 (2000) · Zbl 1274.91398
[7] Almendral, A.; Oosterlee, C. W., Numerical valuation of options with jumps in the underlying, Appl Numer Math, 53, 1, 1-18 (2005) · Zbl 1117.91028
[8] Zhang, K.; Wang, S., Pricing options under jump diffusion processes with fitted finite volume method, Appl Math Comput, 201, 1-2, 398-413 (2008) · Zbl 1142.91576
[9] Ngounda, E.; Patidar, K. C.; Pindza, E., Contour integral method for European options with jumps, Commun Nonlinear Sci Numer Simul, 18, 3, 478-492 (2013) · Zbl 1278.91182
[10] d’Halluin, Y.; Forsyth, P. A.; Vetzal, K. R., Robust numerical methods for contingent claims under jump diffusion processes, IMA J Numer Anal, 25, 87-112 (2005) · Zbl 1134.91405
[11] Briani, M.; Natalini, R.; Russo, G., Implicit-explicit numerical schemes for jump-diffusion processes, Calcolo, 44, 1, 33-57 (2007) · Zbl 1150.65033
[12] Kwon, Y.; Lee, Y., A second-order tridiagonal method for American options under jump-diffusion models, SIAM J Sci Comput, 33, 4, 1860-1872 (2011) · Zbl 1227.91034
[13] Tangman, D.; Gopaul, A.; Bhuruth, M., Exponential time integration and Chebychev discretisation schemes for fast pricing of options, Appl Numer Math, 58, 9, 1309-1319 (2008) · Zbl 1151.91546
[14] Kadalbajoo, M. K.; Kumar, A.; Tripathi, L. P., A radial basis function based implicit-explicit method for option pricing under jump-diffusion models, Appl Numer Math, 110, 159-173 (2016) · Zbl 1348.91285
[15] Shirzadi, M.; Dehghan, M.; Bastani, A. F., Optimal uniform error estimates for moving least-squares collocation with application to option pricing under jump-diffusion processes, Numer Methods Partial Differential Equations, 37, 1, 98-117 (2021) · Zbl 1535.65238
[16] Chen, Y.; Xiao, A.; Wang, W., An IMEX-BDF2 compact scheme for pricing options under regime-switching jump-diffusion models, Math Methods Appl Sci, 42, 8, 2646-2663 (2019) · Zbl 1417.65150
[17] Shirzadi, M.; Dehghan, M.; Bastani, A. F., On the pricing of multi-asset options under jump-diffusion processes using meshfree moving least-squares approximation, Common Non Sci Numer Simul, 84, Article 105160 pp. (2020) · Zbl 1463.91203
[18] Dehghan, M.; Pourghanbar, S., Solution of the black-scholes equation for pricing of barrier option, Z Natforsch A, 66, 5, 289-296 (2011)
[19] Kazemi, S. M.M.; Dehghan, M.; Bastani, A. F., Asymptotic expansion of solutions to the Black-Scholes equation arising from American option pricing near the expiry, J Comput Appl Math, 311, 11-37 (2017) · Zbl 1352.91034
[20] Kazemi, S. M.M.; Dehghan, M.; Bastani, A. F., On a new family of radial basis functions: mathematical analysis and applications to option pricing, J Comput Appl Math, 328, 75-100 (2018) · Zbl 1372.65283
[21] Dehghan, M., Solution of a partial integro-differential equation arising from viscoelasticity, Int J Comput Math, 83, 1, 123-129 (2006) · Zbl 1087.65119
[22] Haghi, M.; Mollapourasl, R.; Vanmaele, M., An RBF-FD method for pricing American options under jump-diffusion models, Comput Math Appl, 76, 10, 2434-2459 (2018) · Zbl 1442.91100
[23] Company, R.; Egorova, V. N.; Jódar, L., An ETD method for multi-asset American option pricing under jump-diffusion model, Math Methods Appl Sci, 46, 9, 10332-10347 (2023) · Zbl 1531.91273
[24] Cerna, D., Wavelet method for sensitivity analysis of European options under merton jump-diffusion model, (AIP conference proceedings, vol. 2425, no. 1 (2022), AIP Publishing LLC), Article 110004 pp.
[25] Fakharany, M.; Egorova, V. N.; Company, R., Numerical valuation of two-asset options under jump diffusion models using Gauss-Hermite quadrature, J Comput Appl Math, 330, 822-834 (2018) · Zbl 1415.91316
[26] Patel, K. S.; Mehra, M., Fourth-order compact finite difference scheme for American option pricing under regime-switching jump-diffusion models, Int J Appl Comput Math, 3, Suppl 1, 547-567 (2017)
[27] Ikonen, S.; Toivanen, J., Operator splitting methods for American option pricing, Appl Math Lett, 17, 7, 809-814 (2004) · Zbl 1063.65081
[28] Ikonen, S.; Toivanen, J., Efficient numerical methods for pricing American options under stochastic volatility, Numer Methods Partial Differential Equations, 24, 1, 104-126 (2008) · Zbl 1152.91516
[29] Hardy, R., Multiquadric equations of topography and other irregular surfaces, J Geophys Res, 76, 8, 1905-1915 (1971)
[30] Hardy, R., Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968-1988, Comput Math Appl, 19, 8-9, 163-208 (1990) · Zbl 0692.65003
[31] Buhmann, M., Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics, Constr Approx, 6, 1, 21-34 (1990) · Zbl 0682.41007
[32] Madych, W.; Nelson, S., Multivariate interpolation and conditionally positive definite functions. II, Math Comp, 54, 189, 211-230 (1990) · Zbl 0859.41004
[33] Kansa, E., Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-I. Surface approximations and partial derivative estimates, Comput Math Appl, 19, 8-9, 127-145 (1990) · Zbl 0692.76003
[34] Kansa, E., Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput Math Appl, 19, 8-9, 147-161 (1990) · Zbl 0850.76048
[35] Mai-Duy, N.; Tanner, R. I., A collocation method based on one-dimensional RBF interpolation scheme for solving PDEs, Internat J Numer Methods H, 17, 2, 165-186 (2007) · Zbl 1231.76188
[36] Mai-Duy, N.; Tran-Cong, T., A multidomain integrated radial basis function collocation method for elliptic problems, Numer Methods Partial Differential Equations, 24, 5, 1301-1320 (2008) · Zbl 1147.65099
[37] Mai-Duy, N.; Tran-Cong, T., An efficient indirect RBFN-based method for numerical solution of PDEs, Numer Methods Partial Differential Equations, 21, 4, 770-790 (2005) · Zbl 1077.65125
[38] Mai-Duy, N.; Tran-Cong, T., A compact five-point stencil based on integrated RBFs for 2D second-order differential problems, J Comput Phys, 235, 302-321 (2013)
[39] Sarra, S. A., Integrated multiquadric radial basis function approximation methods, Comput Math Appl, 51, 1283-1296 (2006) · Zbl 1146.65327
[40] Shu, C.; Wu, Y. L., Integrated radial basis functions-based differential quadrature method and its performance, Internat J Numer Methods Fluids, 53, 6, 969-984 (2007) · Zbl 1109.65025
[41] Thai-Quang, N.; Mai-Duy, N.; Tran, C.-D.; Tran-Cong, T., High-order alternating direction implicit method based on compact integrated-RBF approximations for unsteady/steady convection-diffusion equations, CMES Comput Model Eng Sci, 89, 3, 189-220 (2012) · Zbl 1357.65138
[42] Chan, R. T.L.; Hubbert, S., Options pricing under the one-dimensional jump-diffusion model using the radial basis function interpolation scheme, Rev Deriv Res, 17, 161-189 (2014) · Zbl 1303.91189
[43] Brummelhuis, R.; Chan, R. T., A radial basis function scheme for option pricing in exponential Lévy models, Appl Math Finance, 21, 3, 238-269 (2014) · Zbl 1395.91433
[44] Baxter, B.; Brummelhuis, R., Convergence estimates for stationary radial basis function interpolation and for semi-discrete collocation-schemes, J Fourier Anal Appl, 28, 3 (2022), [https://doi.org/10.1007/s00041-022-09945-3] · Zbl 1511.41001
[45] Toivanen, J., A high-order front-tracking finite difference method for pricing American options under jump-diffusion models, J Comput Finance, 13, 61-79 (2010) · Zbl 1284.91575
[46] Franke, R.; Nielson, G., Smooth interpolation of large sets of scattered data, Internat J Numer Methods Engrg, 15, 11, 1691-1704 (1980) · Zbl 0444.65011
[47] McLain, D. H., Two dimensional interpolation from random data, Comput J, 19, 2, 178-181 (1976) · Zbl 0321.65009
[48] Shepard D. A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM national conference. 1968, p. 517-24.
[49] Wendland, H., Fast evaluation of radial basis functions: methods based on partition of unity, (Approximation theory X (2002), Vanderbilt University Press: Vanderbilt University Press Nashville, TN), 473-483 · Zbl 1031.65022
[50] Fasshauer, G. E., Meshfree approximation methods with MATLAB. interdisciplinary mathematical sciences, 6 (2007), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1123.65001
[51] Babuska, I.; Melenk, J. M., The partition of unity method, Internat J Numer Methods Engrg, 40, 4, 727-758 (1997) · Zbl 0949.65117
[52] Larsson, E.; Heryudono, A., A partition of unity radial basis function collocation method for partial differential equations, Comput Math Appl, 75, 11, 4066-4090 (2018) · Zbl 1419.65078
[53] Chaturantabut, S., Dimension reduction for unsteady nonlinear partial differential equations via empirical interpolation methods, ProQuest (2009)
[54] Chaturantabut, S.; Sorensen, D. C., A state space error estimate for POD-DEIM nonlinear model reduction, SIAM J Numer Anal, 50, 1, 46-63 (2012) · Zbl 1237.93035
[55] Ravindran, S., Reduced-order adaptive controllers for fluid flows using POD, J Sci Comput, 15, 4, 457-478 (2000) · Zbl 1048.76016
[56] Fang, F.; Pain, C.; Navon, I.; Gorman, G.; Piggott, M.; Allison, P., A pod reduced order unstructured mesh ocean modelling method for moderate Reynolds ’s number flows, Ocean Model, 28, 1, 127-136 (2009)
[57] Fox-Kemper, B., Eddies and friction: Removal of vorticity from the wind-driven gyre (2003), Massachusetts Institute of Technology: Massachusetts Institute of Technology MIT
[58] Ravindran, S. S., A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Internat J Numer Methods Fluids, 34, 5, 425-448 (2000) · Zbl 1005.76020
[59] Stefanescu, R.; Navon, I. M., POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model, J Comput Phys, 237, 95-114 (2013) · Zbl 1286.76106
[60] Stefanescu, R.; Sandu, A.; Navon, I. M., Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations, Internat J Numer Methods Fluids, 76, 8, 497-521 (2014)
[61] Xiao, D.; Fang, F.; Pain, C.; Navon, I., Domain decomposition non-intrusive reduced order modelling of non-linear flow dynamics, Comput Methods Appl, 182, 15-27 (2019) · Zbl 1411.76072
[62] Xiao, D.; Lin, Z.; Fang, F.; Pain, C.; Navon, I. M.; Salinas, P., Non-intrusive reduced-order modeling for multiphase porous media flows using Smolyak sparse grids, Internat J Numer Methods Fluids, 83, 2, 205-219 (2017)
[63] Dehghan, M.; Abbaszadeh, M.; Khodadadian, A.; Heitzinger, C., Galerkin proper orthogonal decomposition-reduced order method (POD-rom) for solving generalized Swift-Hohenberg equation, Internat J Numer Methods H, 29, 8, 2642-2665 (2019)
[64] Abbaszadeh, M.; Dehghan, M., A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation, Appl Numer Math, 158, 271-291 (2020) · Zbl 1452.65145
[65] Kwon, Y.; Lee, Y., A second-order finite difference method for option pricing under jump-diffusion models, SIAM J Numer Anal, 49, 6, 2598-2617 (2011) · Zbl 1232.91712
[66] Bialecki, R. A.; Kassab, A. J.; Fic, A., Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis, Internat J Numer Methods Engrg, 62, 774-797 (2005) · Zbl 1092.80010
[67] Sun, P.; Luo, Z.; Zhou, Y., Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations, Appl Numer Math, 60, 154-164 (2010) · Zbl 1193.65159
[68] Chaturantabut, S.; Sorensen, D. C., Nonlinear model reduction via discrete empirical interpolation, SIAM J Sci Comput, 32, 5, 2737-2764 (2010) · Zbl 1217.65169
[69] Zhang, X.; Xiang, H., A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems, Int J Heat Mass Transfer, 84, 729-739 (2015)
[70] Buchan, A. G.; Calloo, A.; Goffin, M. G.; Dargaville, S.; Fang, F.; Pain, C. C., A POD reduced order model for resolving angular direction in neutron/photon transport problems, J Comput Phys, 296, 138-157 (2015) · Zbl 1352.82030
[71] Toivanen, J., Numerical valuation of European and American options under Kou’s jump-diffusion model, SIAM J Sci Comput, 30, 1949-1970 (2008) · Zbl 1178.35225
[72] Salmi, S.; Toivanen, J., An iterative method for pricing American options under jump-diffusion models, Appl Numer Math, 61, 7, 821-831 (2011) · Zbl 1213.91164
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.