Analysis of a kernel-based method for some pricing financial options

Document Type : Research Paper

Authors

1 Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran.

2 Department of Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord, Iran.

Abstract

In this paper, we propose a kernel-based method for some pricing financial options. Based on the ideas of the kernel-based approximation and finite-difference discretization, we present an efficient numerical method for solving the generalized Black-Scholes  option pricing models. Utilizing the reproducing property of kernels, we introduce an efficient framework for obtaining cardinal functions. Also, we discuss the solvability of final system to obtain some remarkable results. We provide the error estimate of the proposed kernel-based method and verify its efficiency and accuracy by numerical experiments.

Keywords

References

  • [1] L. Angermann and S. Wang, Convergence of a fitted finite volume method for the penalized Black–Scholes equation governing European and American Option pricing, Numer. Math., 106 (2007), 1–40.
  • [2] F. Baustian, K. Filipov´a, and J. Posp´ıˇsil, Solution of option pricing equations using orthogonal polynomial expan- sion, Appl. Math. (N. Y.), 66 (2021), 553–582.
  • [3] Y. Y. Belov, Inverse problems for partial differential equations, In Inverse Problems for Partial Differential Equations, De Gruyter, 2012.
  • [4] F. Black and M. Scholes, The pricing of options and corporate liabilities, In World Scientific Reference on Con- tingent Claims Analysis in Corporate Finance, 1 (2019), 3–21.
  • [5] Z. Cen and A. Le, A robust finite difference scheme for pricing American put options with singularity-separating method, Numer. Algorithms, 53 (2010), 497–510.
  • [6] Z. Cen and A. Le, A robust and accurate finite difference method for a generalized Black–Scholes equation, J. Comput. Appl. Math., 235 (2011), 3728–3733.
  • [7] Y. Chen, H. Yu, X. Meng, X. Xie, M. Hou, and J. Chevallier, Numerical solving of the generalized Black-Scholes differential equation using Laguerre neural network, Digital Signal Processing, 112 (2021), 103003.
  • [8] R. Company, L. J´odar, and J. R. Pintos, A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets, Math. Comput. Simul., 82 (2012), 1972–1985.
  • [9] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Lect. Notes Pure Appl. Math., (1943), 1–23.
  • [10] J. C. Cox, S. A. Ross, and M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229–263.
  • [11] M. Cui and Y. Lin, Nonlinear numerical analysis in reproducing kernel space, Nova Science Publishers, Inc., 2009.
  • [12] G. E. Fasshauer, Meshfree approximation methods with MATLAB, World Scientific, 2007.
  • [13] G. E. Fasshauer and M. J. McCourt, Kernel-based approximation methods using Matlab, World Scientific Pub- lishing Company, 2015.
  • [14] M. Fardi, A kernel-based pseudo-spectral method for multi-term and distributed order time-fractional diffusion equations, Numerical Methods for Partial Differential Equations, 39 (2023), 2630–2651.
  • [15] M. Fardi, A kernel-based method for solving the time-fractional diffusion equation, Numerical Methods for Partial Differential Equations, 39 (2023), 2719–2733.
  • [16] M. Fardi and M. Ghasemi, Numerical solution of singularly perturbed 2D parabolic initial-boundary-value problems based on reproducing kernel theory: Error and stability analysis, Numerical Methods for Partial Differential Equations, 38 (2022), 876–903.
  • [17] M. Fardi and Y. Khan, Numerical simulation of squeezing Cu-Water nanofluid flow by a kernel-based method, International Journal of Modeling, Simulation, and Scientific Computing, 13 (2021).
  • [18] A. Friedman, Partial differential equations of parabolic type, Courier Dover Publications, 2008.
  • [19] A. Hrennikoff, Solution of problems of elasticity by the framework method, J. Appl. Mech., 8 (1941), A169–A175.
  • [20] J. Hull and A. White, The use of the control variate technique in option pricing, Journal of Financial and Quantitative analysis, 23 (1988), 237–251.
  • [21] M. K. Kadalbajoo, L. P. Tripathi, and P. Arora, A robust nonuniform B-spline collocation method for solving the generalized Black–Scholes equation, IMA J. Numer. Anal., 34 (2014), 252–278.
  • [22] M. K. Kadalbajoo, L. P. Tripathi, and A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black–Scholes equation, Math. Comput. Modelling, 55 (2012), 1483–1505.
  • [23] E. J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computational fluid- dynamics—I surface approximations and partial derivative estimates, Comput. Math. Appl., 19 (1990), 127–145.
  • [24] E. J. Kansa, 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 (1990), 147–161.
  • [25] D. C. Lesmana and S. Wang, An upwind finite difference method for a nonlinear Black–Scholes equation governing European option valuation under transaction costs, Appl. Math. Comput., 219 (2013), 8811–8828.
  • [26] W. K. Liu, S. Li, and H. S. Park, Eighty years of the finite element method: Birth, evolution, and future, Arch. Comput. Methods Eng., (2022), 1–23.
  • [27] R. C. Merton, Theory of rational option pricing, The Bell Journal of economics and management science, 4 (1973), 141–183.
  • [28] D. Prathumwan and K. Trachoo, On the solution of two-dimensional fractional Black–Scholes equation for Euro- pean put option, Adv. Difference Equ., 2020 (2020), 1–9.
  • [29] S. C. S. Rao, Numerical solution of generalized Black–Scholes model, Appl. Math. Comput., 321 (2018), 401–421.
  • [30] P. Roul and V.P. Goura, A new higher order compact finite difference method for generalised Black–Scholes partial differential equation: European call option, J. Comput. Appl. Math., 363 (2020), 464–484.
  • [31] R. Valkov, Fitted finite volume method for a generalized Black–Scholes equation transformed on finite interval, Numer. Algorithms, 65 (2014), 195–220.
  • [32] R. Valkov, Convergence of a finite volume element method for a generalized Black-Scholes equation transformed on finite interval, Numer. Algorithms, 68 (2015), 61–80.
  • [33] S. Wang, A novel fitted finite volume method for the Black–Scholes equation governing option pricing, IMA J. Numer. Anal., 24 (2004), 699–720.
  • [34] Y. Wang, M. Du, F. Tan, Z. Li, and T. Nie, Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions, Appl. Math. Comput., 219 (2013), 5918–5925.
  • [35] D. Wei, Existence, uniqueness, and numerical analysis of solutions of a quasilinear parabolic problem, SIAM J. Numer. Anal., 29 (1992), 484–497.
  • [36] H. Zhang, F. Liu, I. Turner, and Q. Yang, Numerical solution of the time fractional Black–Scholes model governing European options, Comput. Math. Appl., 71 (2016), 1772–1783.
Computational Methods for Differential Equations
Volume 12, Issue 1
January 2024
Pages 16-30

Files

History

  • Receive Date: 17 April 2023
  • Revise Date: 21 June 2023
  • Accept Date: 28 June 2023

Share

How to cite

Statistics