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Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions

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  • Darae Jeong

    (Korea University)

  • Minhyun Yoo

    (Korea University)

  • Junseok Kim

    (Korea University)

Abstract

We present an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions. The BS equation is a backward parabolic partial differential equation for financial option pricing and hedging. When we solve the BS equation numerically, we typically need an artificial far-field boundary condition such as the Dirichlet, Neumann, linearity, or partial differential equation boundary condition. However, in this paper, we propose an explicit finite difference scheme which does not use a far-field boundary condition to solve the BS equation numerically. The main idea of the proposed method is that we reduce one or two computational grid points and only compute the updated numerical solution on that new grid points at each time step. By using this approach, we do not need a boundary condition. This procedure works because option pricing and computation of the Greeks use the values at a couple of grid points neighboring an interesting spot. To demonstrate the efficiency and accuracy of the new algorithm, we perform the numerical experiments such as pricing and computation of the Greeks of the vanilla call, cash-or-nothing, power, and powered options. The computational results show excellent agreement with analytical solutions.

Suggested Citation

  • Darae Jeong & Minhyun Yoo & Junseok Kim, 2018. "Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions," Computational Economics, Springer;Society for Computational Economics, vol. 51(4), pages 961-972, April.
    • Handle: RePEc:kap:compec:v:51:y:2018:i:4:d:10.1007_s10614-017-9653-0
    DOI: 10.1007/s10614-017-9653-0
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    References listed on IDEAS

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    1. Rahman Farnoosh & Hamidreza Rezazadeh & Amirhossein Sobhani & M. Hossein Beheshti, 2016. "A Numerical Method for Discrete Single Barrier Option Pricing with Time-Dependent Parameters," Computational Economics, Springer;Society for Computational Economics, vol. 48(1), pages 131-145, June.
    2. Peter G Zhang, 1998. "Exotic Options:A Guide to Second Generation Options," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 3800.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. 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.
    5. Mojtaba Hajipour & Alaeddin Malek, 2015. "Efficient High-Order Numerical Methods for Pricing of Options," Computational Economics, Springer;Society for Computational Economics, vol. 45(1), pages 31-47, January.
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    Cited by:

    1. Seda Gulen & Catalin Popescu & Murat Sari, 2019. "A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities," Mathematics, MDPI, vol. 7(8), pages 1-14, August.
    2. Rami Ahmad El-Nabulsi & Waranont Anukool, 2025. "Qualitative financial modelling in fractal dimensions," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 11(1), pages 1-47, December.
    3. Hanbyeol Jang & Sangkwon Kim & Junhee Han & Seongjin Lee & Jungyup Ban & Hyunsoo Han & Chaeyoung Lee & Darae Jeong & Junseok Kim, 2020. "Fast Monte Carlo Simulation for Pricing Equity-Linked Securities," Computational Economics, Springer;Society for Computational Economics, vol. 56(4), pages 865-882, December.
    4. Purba Banerjee & Vasudeva Murthy & Shashi Jain, 2021. "Method of lines for valuation and sensitivities of Bermudan options," Papers 2112.01287, arXiv.org.
    5. Chaeyoung Lee & Jisang Lyu & Eunchae Park & Wonjin Lee & Sangkwon Kim & Darae Jeong & Junseok Kim, 2020. "Super-Fast Computation for the Three-Asset Equity-Linked Securities Using the Finite Difference Method," Mathematics, MDPI, vol. 8(3), pages 1-13, February.
    6. 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.
    7. Sangkwon Kim & Darae Jeong & Chaeyoung Lee & Junseok Kim, 2020. "Finite Difference Method for the Multi-Asset Black–Scholes Equations," Mathematics, MDPI, vol. 8(3), pages 1-17, March.
    8. Mohammad Mehdizadeh Khalsaraei & Ali Shokri & Higinio Ramos & Zahra Mohammadnia & Pari Khakzad, 2022. "A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation," Mathematics, MDPI, vol. 10(11), pages 1-16, May.
    9. Fazlollah Soleymani, 2019. "Efficient Semi-Discretization Techniques for Pricing European and American Basket Options," Computational Economics, Springer;Society for Computational Economics, vol. 53(4), pages 1487-1508, April.

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