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Multiple Shooting Method for Solving Black–Scholes Equation

Author

Listed:
  • Somayeh Abdi-Mazraeh

    (Azarbaijan Shahid Madani University
    University of Tabriz)

  • Ali Khani

    (Azarbaijan Shahid Madani University)

  • Safar Irandoust-Pakchin

    (University of Tabriz)

Abstract

In this paper, the Black–Scholes (B–S) model for the pricing of the European and the barrier call options are considered, which yields a partial differential problem. First, A numerical technique based on Crank–Nicolson (C–N) method is used to discretisize the time domain. Consequently, the partial differential equation will be converted to a system of an ordinary differential equation (ODE). Then, the multiple shooting method combined with Lagrange polynomials is utilized to solve the ODEs. Regarding the convergence order of the approximate solution which normally decreases due to the non-smooth properties of the option’s payoff (at the strike price), in this study, the equipped C–N scheme with variable step size strategy is applied for the time discretization. As a result, the variable step size strategy prevents the error propagation by controlling the error at each time step and increases the computational speed by raising the step size in the smooth points of the domain. In order to implement the variable step size, an algorithm is presented. In addition, the stability of the presented method is analyzed. The extracted numerical results represent the accuracy and efficiency of the proposed method.

Suggested Citation

  • Somayeh Abdi-Mazraeh & Ali Khani & Safar Irandoust-Pakchin, 2020. "Multiple Shooting Method for Solving Black–Scholes Equation," Computational Economics, Springer;Society for Computational Economics, vol. 56(4), pages 723-746, December.
    • Handle: RePEc:kap:compec:v:56:y:2020:i:4:d:10.1007_s10614-019-09940-9
    DOI: 10.1007/s10614-019-09940-9
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    References listed on IDEAS

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    1. Golbabai, A. & Ballestra, L.V. & Ahmadian, D., 2013. "Superconvergence of the finite element solutions of the Black–Scholes equation," Finance Research Letters, Elsevier, vol. 10(1), pages 17-26.
    2. Javidi, M. & Golbabai, A., 2009. "A new domain decomposition algorithm for generalized Burger’s–Huxley equation based on Chebyshev polynomials and preconditioning," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 849-857.
    3. Higham,Desmond J., 2004. "An Introduction to Financial Option Valuation," Cambridge Books, Cambridge University Press, number 9780521547574.
    4. Rad, Jamal Amani & Parand, Kourosh & Ballestra, Luca Vincenzo, 2015. "Pricing European and American options by radial basis point interpolation," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 363-377.
    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    6. Persson, Jonas & von Sydow, Lina, 2010. "Pricing American options using a space-time adaptive finite difference method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(9), pages 1922-1935.
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    Cited by:

    1. Chaeyoung Lee & Soobin Kwak & Youngjin Hwang & Junseok Kim, 2023. "Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions," Computational Economics, Springer;Society for Computational Economics, vol. 61(3), pages 1207-1224, March.

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