Numerical solution of stochastic models using spectral collocation method

Document Type : Research Paper


Department of Mathematics, Yadegar-e-Imam khomeini (RAH) Share Rey Branch, Islamic Azad University, Tehran, Iran.


In this article, the spectral collocation method based on radial basis functions is used to solve the mentioned models. The advantage of this method is that it converts the equations into a system of algebraic equations. Therefore, we can solve this problem with Newton's method. The purpose of this article is to numerically solve stochastic models such as the Heston model, Vasicek model, Cox-Ingersoll and Ross model, and a model of the Black-Scholes called the Genral Stock model. The method is computationally attractive, and numerical examples confirm the validity and efficiency of the proposed method.


  • [1] A. Alipanah and M. Dehghan, Numerical solution of the nonlinear Fredholm integral equations by positive definite functions, Appl. Math. Comput., 190 (2007), 1754-1761.
  • [2] F. Black and M. Sholes, The Pricing of Options and Corporate Liabilities , Journal of Political Economy, 81 (1973), 637-654.
  • [3] G. Bakshi, C. Cao, and Z. Chen, Empirical performance of alternative option pricing models, The Journal  of Finance, 52(5) (1990), 2003-2049.
  • [4] R. T. Baillie and C. Morana, Modelling long memory and structural breaks in conditional variances: An adaptive FIGARCH approach, Journal of Economic Dynamics and Control, 33(8) (2009), 1577-1992.
  • [5] A. Babaei, H. Jafari, S. Banihashemi, and M. Ahmadi, A Stochastic Mathematical Model for COVID-19 According to Different Age Groups, Applied and computational mathematics, 20(1) (2021), 140-159.
  • [6] T. E. Clark and T. Davig, Decomposing the declining volatility of long-term inflation expectations,Journal of  Economic Dynamics and Control, 35(7) (2011), 981-999.
  • [7] S. R. Chakravarthy and ¨o. erife, Analysis of a Stochastic Model for Crowdsourcing Using Map Arrivals and Phase Type Services, Applied and computational mathematics, 20(3) (2021), 390-407.
  • [8] M. Dehghan and A. Shokri, A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions, Numer. Algor., 52 (2009), 461-477.
  • [9] L. M. DELVES and J. L. MOHAMED, Computational methods for integral  equations, Departement of Statistics  and Computational Mathematics, University of Liverpool,Cambridge University Press (1985), 23-28.
  • [10] M. R. Doostdar, A. R. Vahidi, T. Damercheli, and E. Babolian, A numerical method based on hybrid functions for solving a fractional model of HIV infection of CD4+ T cells, Mathematical Sciences, (2022), 1 -11.
  • [11] M. R. Doostdar, A. R. Vahidi, T. Damercheli, and E. Babolian, A hybrid functions method for solving linear and non-linear systems of ordinary differential equations,Mathematical Communications, 26(2) (2021), 197-213.
  • [12] A. Etheridge, A Course in Financial Calculus, Cambridge University Press, (2002).
  • [13] R. Frank, Scattered data interpolation: tests of some methods, Math. Comput, 38 (1982), 181-199.
  • [14] G. E. Fasshauer, Solving differential equations with radial basis functions:multi-level methods and smoothing, Comp.Math., 11 (1999), 139-159.
  • [15] J. Hull and A. White, The pricing of options on assets with stochastic volatilities, 42(2) (1987), 281-300.
  • [16] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options,Review of Financial Studies, 6(2) (1993), 327-343.
  • [17] F. Hossieni Shekarabi and T. Damercheli, Application of operational matrices in numerical solution of stochastic differential equations in financial mathematics, 4thSeminar of Mathematics and Humanities (Financial Mathe- matics), (2016), 81-85.
  • [18] S. Islam , S. Haq, and A. Ali, A meshfree method for the numerical solution of the RLW equation, Journal of Computational and Applied Mathematics, 223 (2009), 997-1012.
  • [19] A. J. Khattak, S. I. A. Tirmizi, and S.U. Islam, Application of meshfree collocation method to a class of nonlinear partial differential equations, Eng. Anal. Bound. Elem., 33 (2009), 661-667.
  • [20] F. C. Klebaner, Introduction to Stochastic Calculus with Applications, Imperial College Press, London, (2005).
  • [21] M. Kazemi, A. Deep, and A. Yaghoobnia, Application of fixed point theorem on the study of the existence of solutions in some fractional stochastic functional integral equations, Mathematical Sciences, (2022), 1-12.
  • [22] M. Kazemi and A. R. Yaghoobnia, Application of fixed point theorem to solvability of functional stochastic integral equations, Applied Mathematics and Computation, 417 (2022), 126759.
  • [23] A. Melino and S. Turnbull, Pricing foreign currency options with stochastic volatility, Journal of Econometrics, 45 (1990), 239-265.
  • [24] C. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Con- structive Approximation, 2 (1986), 11-22.
  • [25] X. Mao, Stochastic Differential Equation and Application Second Edition, Departement of Statistics and Modelling Science, University of Strathclyde, Glasgow, (2007).
  • [26] K. Maleknejad, M. Khodabin, and M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Model, 55 (2012), 791-800.
  • [27] B. Oksendal, Stochastic Differential Equations An Introduction with Applications, Fifth Edition Corrected Print- ing Springer-Verlag Heidelberg New York, May, 2014.
  • [28] G. M. Phillips and P. J. Taylor, Theory and Application of Numerical Analysis, Academic Press, New York, 1973.
  • [29] L. O. Scott, Option pricing when the variance changes randomly:  Theory,  estimation,  and an application, Journal     of Financial Economics, 22(4) (1987), 419-438.
  • [30] E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: an analytic approach, The Review of Financial Studies, 4(4) (1991), 727-752.
  • [31] R. Schaback, Error Estimates and Condition Numbers for Radial Basis Function Interpolation, Advances in Computational Mathematics, 3 (1995), 251-264.
  • [32] M. Shiralizadeh, A. Alipanah, and M. Mohammadi, A numerical method for KdV equation using rational  radial basis functions, Computational Methods for Differential Equations, 2 (2023), 303-318.
  • [33] A. R. Vahidi, E. Babolian, and Z. Azimzadeh , An Improvement to the Homotopy Perturbation Method for Solving Nonlinear Duffings Equations , Bulletin of the Malaysian Mathematical Sciences Society, 41 (2018), 1105-1117.
  • [34] J. B. Wiggins, Option values under stochastic volatility: Theory and empirical estimates, Journal of Financial Economics, 19(2) (1987), 351-372.
  • [35] H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in Computational Mathematics, 4 (1995), 389-396.
  • [36] M. R. Yaghoti and F. Farshadmoghadam, Choosing the best value of shape parameter in radial basis functions by Leave-P-Out Cross Validation, Computational Methods for Differential Equations, 1 (2023), 108-129.