A novel local meshless scheme based on the radial basis function for pricing multi-asset options

Document Type : Research Paper


Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, Iran.


A novel local meshless scheme based on the radial basis function (RBF) is introduced in this article for price multi-asset options of even European and American types based on the Black-Scholes model. The proposed approach is obtained by using operator splitting and repeating the schemes of Richardson extrapolation in the time direction and coupling the RBF technology with a finite-difference (FD) method that leads to extremely sparse matrices in the spatial direction. Therefore, it is free of the ill-conditioned difficulties that are typical of the standard RBF approximation. We have used a strong iterative idea named the stabilized Bi-conjugate gradient process (BiCGSTAB) to solve highly sparse systems raised by the new approach. Moreover, based on a review performed in the current study, the presented scheme is unconditionally stable in the case of independent assets when spatial discretization nodes are equispaced. As seen in numerical experiments, it has a low computational cost and generates higher accuracy. Finally, the proposed local RBF scheme is very versatile so that it can be used easily for solving numerous models and obstacles not just in the finance sector, as well as in other fields of engineering and science.


  • [1]         J. Amani Rad, P. Kourosh, and S. Abbasbandy, Local weak form meshless techniques based on the radial point in- terpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options, Comm Nonlinear Sci Numer Simulat., 22 (2015), 1178–1200.
  • [2]         J. Amani Rad, P. Kourosh, and L. V. Ballestra, Pricing European and American options by radial basis point interpolation, Appl Math Comput., 251 (2015), 363–377.
  • [3]         L. V. Ballestra and G. Pacelli, Pricing European and American options with two stochastic factors: a highly efficient radial basis function approach, J Econ Dynam Contr., 37 (2013), 1142–1167.
  • [4]         S. Banei and K. Shanazari, Solving the forward-backward heat equation with a non-overlapping domain decompo- sition method based on multiquadric RBF meshfree method, Computational Methods for Differential Equations, em 9(4) (2021), 1083–1099.
  • [5]         V. Bayona, M. Moscoso, and M. Carretero, Manuel KindelanRBF-FD formulas and convergence properties, J Comput Phys., 229 (2010), 8281–8295.
  • [6]         F. Black and M. Scholes, The pricing of options and corporate liabilities, J Polit Econ., 81 (1973), 637–659.
  • [7]         M. D. Buhmann, Radial Basis Functions: Theory and Implementation. University of Gissen, Cambridge Univer- sity Press, 2004.
  • [8]         S. Chantasiriwan, Investigation of the use of radial basis functions in local collocation method for solving diffusion problems, Int Commun Heat Mass Transfer., 31 (2004), 1095–1104.
  • [9]         W. Cheney, An Introduction to Approximation Theory (2rd ed.), New York: AMS Cheslea Publishing: American Mathematical Society, 2000.
  • [10]       T. A. Driscoll and B. Fornberg, Interpolation in the limit of increasingly flat radial basis functions, Comput Math Appl., 43 (2002), 413–422.
  • [11]       B. Fornberg, Calculation of weights in finite difference formulas, SIAM Rev., 40 (1998), 685–691.
  • [12]       B. Fornberg, G. B. Wright, and E. Larsson, Some observations regarding interpolants in the limit of flat radial basis functions, Comput Math Appl., 47 (2004), 37–55.
  • [13]       A. Golbabai, D. Ahmadian, and M. Milev, Radial basis functions with application to finance: American put option under jump diffusion, Math Comput Model., 55 (2012), 1354–1362.
  • [14]       Y. C. Hon, A quasi-radial basis functions method for American options pricing, Comput Math Appl., 43 (2002), 513–524.
  • [15]       J. C. Hull, Options, futures, Other derivatives (7rd ed.), University of Toronto .Prentice Hall, 2002.
  • [16]       M. K. Kadalbajoo, A. Kumar, and L. P. Tripathi, Application of the local radial basis function-based finite difference method for pricing American options, Int J of Comput Math., 92 (2015), 1608–1624.
  • [17]       E. J. Kansa and Y. C. Hon, Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Comput Math Appl., 39 (2000), 123–137.
  • [18]       A. Khaliq, G. Fasshauer, and D. Voss, Using meshfree approximation for multi-asset American option problems, J Chin Inst Eng., 27 (2004), 563–571.
  • [19]       H. Mesgarani, S. Ahanj, and Y. Esmaeelzade Aghdam, Numerical investigation of the time-fractional Black- Scholes equation with barrier choice of regulating European option, Journal of Mathematical Modeling., (2021) 1-10.
  • [20]       H. Mesgarani, A. Beiranvand and Y. Esmaeelzade Aghdam, The impact of the Chebyshev collocation method on solutions of the time-fractional Black-Scholes, Mathematical Sciences., 15 (2) (2021), 1–13.
  • [21]       V. Mohammadi, M. Dehghan, and S. De Marchi, Numerical simulation of a prostate tumor growth model by the RBF-FD scheme and a semi-implicit time discretization, Journal of Computational and Applied Mathematics., 388 (2021), 113314.
  • [22]       V. Mohammadi, D. Mirzaei, and M. Dehghan, Numerical simulation and error estimation of the time-dependent Allen-Cahn equation on surfaces with radial basis functions, Journal of Scientific Computing., 79 (2019), 493–516.
  • [23]       U. Petterssona, E. Larssona, G. Marcussonb, and J. Perssonc, Improved radial basis function methods for multi- dimensional option pricing, J Computl Appl Math., 222 (2008), 82–93.
  • [24]       M. Safarpoor and A. Shirzadi, A localized RBF-MLPG method for numerical study of heat and mass transfer equations in elliptic fins, Engineering Analysis with Boundary Elements., 98 (2019), 35–45.
  • [25]       M. Safarpoor and A. Shirzadi, Numerical investigation based on radial basis function–finite-difference (RBF–FD) method for solving the Stokes-Darcy equations, Engineering with Computers., 37 (2021), 909–920.
  • [26]       M. Safarpoor, F. Takhtabnoos, and A. Shirzadi, A localized RBF-MLPG method and its application to elliptic PDEs, Engineering with Computers., 36 (2020), 171–183.
  • [27]       A. A. Saib, D. Y. Tangman, and M. A. Bhuruth, New radial basis functions method for pricing American options under Merton’s jump-diffusion model, Intl J Comput Math., 89 (2012), 1164–1185.
  • [28]       S. A. Sarra and E. J. Kansa, Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations, Tech Science Press, 2009.
  • [29]       S. A. Sarra and D. Sturgill, A random variable shape parameter strategy for radial basis function approximation methods, Eng Anal Bound Elem., 33 (2009), 1239–1245.
  • [30]       E. Shivanian and A. Jafarabadi, Numerical investigation based on a local meshless radial point interpolation for solving coupled nonlinear reaction-diffusion system, Computational Methods for Differential Equations., 9 (2) (2021), 358–374.
  • [31]       F. Takhtabnoos and A. Shirzadi, A Local Strong form Meshless Method for Solving 2D time-Dependent Schr¨odinger Equations, Mathematical researches., 4 (2) (2019), 1–12.
  • [32]       D. Tavella and C. Randall, Pricing Financial Instruments: The Finite Difference Approach, John Wiley & Sons. New York, 2000.
  • [33]       H. V. Vorst, BCGSTAB: a fast and smoothly converging variant of BCG for the solution of nonsymmetric linear systems, SIAM J Sci Stat Comput., 18 (1992), 631–644.
  • [34]       H. Wendland, Scattered Data Approximation, Cambridge University Press, 2005.
  • [35]       P. Wilmott, S. Howison, and J. Dewynne, Option Pricing: Mathematical Models and Computations, Oxford Financial Press, Oxford, 1995.
  • [36]       P. Wilmott, Introduces Quantitative Finance, John Wiley & Sons, 2007.
  • [37]       P. Wilmott, The Theory and Practice of Financial Engineering, John Wiley & Sons, 1998.