Generalized Jacobi reproducing kernel method in Hilbert spaces for solving the Black-Scholes option pricing problem arising in financial modelling
Abstract
Based on the reproducing kernel Hilbert space method, a new approach is proposed to approximate the solution of the Black-Scholes equation with Dirichlet boundary conditions and introduce the reproducing kernel properties in which the initial conditions of the problem are satisfied. Based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be constructed in the reproducing kernel spaces spanned by the generalized Jacobi basis polynomials. Some new error estimates for application of the method are established. The convergence analysis is established theoretically. The proposed method is successfully used for solving an option pricing problem arising in financial modelling. The ideas and techniques presented in this paper will be useful for solving many other problems.
Keyword : generalized Jacobi polynomials, reproducing kernel Hilbert space method, Black-Scholes equation, Dirichlet boundary conditions, error estimates
How to Cite
Foroutan, M., Ebadian, A., & Fazli, H. R. (2018). Generalized Jacobi reproducing kernel method in Hilbert spaces for solving the Black-Scholes option pricing problem arising in financial modelling. Mathematical Modelling and Analysis, 23(4), 538-553. https://doi.org/10.3846/mma.2018.032
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[20] M. Kaleghi, E. Babolian and S. Abbasbandy. Chebyshev reproducing kernel method: application to two-point boundary value problems. Advances in differential equation, 2017(1):26. https://doi.org/10.1186/s13662-017-1089-2.
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[28] Q. Xue, J. Niu, D. Yu and C. Ran. An improved reproducing kernel method for fredholm integro-differential type two-point boundary value problems. International Journal of Computer Mathematics,0(0):1–9, 2017. https://doi.org/10.1080/00207160.2017.1322201.
[29] C.P. Zhang, J. Niu and Y.Z. Lin. Numerical solutions for the three-point boundary value problem of nonlinear fractional differential equations. Abstract and Applied Analysis, 2012:1–16, 2012. https://doi.org/10.1155/2012/360631.
[2] P. Amster, C.G. Aberbuj, M.C. Mariani and D. Rial. A Black-Scholes option pricing model with transaction costs. Journal of Mathematical Analysis and Applications, 303 (2):685–695, 2005. https://doi.org/10.1016/j.jmaa.2004.08.067
[3] O. Abu Arqub. The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations. Mathematical Methods in the Applied Sciences, 39(15):4549–4562, 201 https://doi.org/10.1002/mma.3884
[4] O. Abu Arqub. Adaptation of reproducing kernel algorithm for solving fuzzy FredholmVolterra integrodifferential equations. Neural Computing and Applications, 28(7):1591–1610, 2017 https://doi.org/10.1007/s00521-015-2110-x
[5] O. Abu Arqub. Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions. Computers and Mathematics with Applications, 73(6):1243–1261, 2017. https://doi.org/10.1016/j.camwa.2016.11.032
[6] O. Abu Arqub and M. Al-Smadi. Numerical algorithm for solving twopoint, second-order periodic boundary value problems for mixed integro- differential equations. Applied Mathematics and Computation, 243:911–922, 2014. https://doi.org/10.1016/j.amc.2014.06.063.
[7] O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat. Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Computing, 20(8):3283–3302, 2016. https://doi.org/10.1007/s00500-015-1707-4.
[8] O. Abu Arqub and H. Rashaideh. The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs. Neural Computing and Applications, pp. 1–12, 2017. https://doi.org/10.1007/s00521-017-2845-7.
[9] F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637–654, 1973. https://doi.org/10.1086/260062.
[10] M.G. Cui and F.Z. Geng. A computational method for solving one-dimensional variable-coefficient Burgers equation. Applied Mathematics and Computation, 188(2):1389–1401, 2007. https://doi.org/10.1016/j.amc.2006.11.005.
[11] M.G. Cui and F.Z. Geng. Solving singular two-point boundary problem in reproducing kernel space. Journal of Computational and Applied Mathematics, 205(1):6–15, 2007. https://doi.org/10.1016/j.cam.2006.04.037.
[12] R. Figueroa and M.R. Grossinho. On some nonlinear boundary value problems related to a Black-Scholes model with transaction costs. Boundary Value Problems, 2015:145, 2017. https://doi.org/10.1186/s13661-015-0410-9.
[13] M.R. Foroutan, A. Ebadian and R. Asadi. Reproducing kernel method in Hilbert spaces for solving the linear and nonlinear four-point boundary value problems. International Journal of Computer Mathematics, pp. 1–15, 2017. https://doi.org/10.1080/00207160.2017.1366464.
[14] M.R. Foroutan, A. Ebadian and S. Najafzadeh. Analysis of unsteady stagnation point flow over a shrinking sheet and solving the equation with rational Chebyshev functions. Mathematical Methods in the Applied Sciences, 40(7):2610–2622, 2017. https://doi.org/10.1002/mma.4185.
[15] F.Z. Geng. Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method. Applied Mathematics and Computation, 215(6):2095–2102, 2009. https://doi.org/10.1016/j.amc.2009.08.002.
[16] F.Z. Geng and M.G. Cui. Solving singular nonlinear secondorder periodic boundary value problems in the reproducing kernel space. Applied Mathematics and Computation, 192(2):389–398, 2007. https://doi.org/10.1016/j.amc.2007.03.016.
[17] F.Z. Geng, S.P. Qian and S. Li. A numerical method for sinqularly perturbed turning point problems with an interior layer. Journal of Computational and Applied Mathematics, 255:97–105, 2014. https://doi.org/10.1016/j.cam.2013.04.040.
[18] M.R. Grossinho and E. Morais. A note on a stationary problem for a Black-Scholes equation with transaction costs. International Journal of Pure and Applied Mathematics, 51:579–587, 2009.
[19] M. R. Grossinho and E. Morais. A fully nonlinear problem arising in financial modelling. Boundary Value Problems, 2013:146, 2013. https://doi.org/10.1186/1687-2770-2013-146.
[20] M. Kaleghi, E. Babolian and S. Abbasbandy. Chebyshev reproducing kernel method: application to two-point boundary value problems. Advances in differential equation, 2017(1):26. https://doi.org/10.1186/s13662-017-1089-2.
[21] L. Lara. A numerical method for solving a system of nonautonomous linear ordinary differential equations. Applied Mathematics and Computation, 170(1):86–94, 2005. https://doi.org/10.1016/j.amc.2004.10.097.
[22] Y.Z. Lin, J. Niu and M.G. Cui. A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space. Applied Mathematics and Computation, 218 (14):7362–7368, 2012. https://doi.org/10.1016/j.amc.2011.11.009.
[23] J. Niu, Y.Z. Lin and C.P. Zhang. Approximate solution of nonlinear multi-point boundary value problem on the half-line. Mathemathical Modelling and Analysis, 17(2):190–202, 2012. https://doi.org/10.3846/13926292.2012.660889.
[24] J. Niu, Y.Z. Lin and C.P. Zhang. Numerical solution of nonlinear three-point boundary value problem on the positive half-line. Mathematical Methods in the Applied Sciences, 35(13):1601–1610, 2012. https://doi.org/10.1002/mma.2549.
[25] S. Saitoh and Y. Sawano. Theory of reproducing kernels and applications. Springer, 2016. https://doi.org/10.1007/978-981-10-0530-5.
[26] J. Shen and T. Tang. Spectral and High-Order Methods with Applications. Science Press, 2006. (Beijing)
[27] M.Q. Xu, Y.Z. Lin and Y.H.Wang. A new algorithm for nonlinear fourth order multi-point boundary value problems. Applied Mathematics and Computation, 274:163–168, 2016. https://doi.org/10.1016/j.amc.2015.10.041.
[28] Q. Xue, J. Niu, D. Yu and C. Ran. An improved reproducing kernel method for fredholm integro-differential type two-point boundary value problems. International Journal of Computer Mathematics,0(0):1–9, 2017. https://doi.org/10.1080/00207160.2017.1322201.
[29] C.P. Zhang, J. Niu and Y.Z. Lin. Numerical solutions for the three-point boundary value problem of nonlinear fractional differential equations. Abstract and Applied Analysis, 2012:1–16, 2012. https://doi.org/10.1155/2012/360631.