IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v182y2021icp690-704.html
   My bibliography  Save this article

Optimal non-uniform finite difference grids for the Black–Scholes equations

Author

Listed:
  • Lyu, Jisang
  • Park, Eunchae
  • Kim, Sangkwon
  • Lee, Wonjin
  • Lee, Chaeyoung
  • Yoon, Sungha
  • Park, Jintae
  • Kim, Junseok

Abstract

In this article, we present optimal non-uniform finite difference grids for the Black–Scholes (BS) equation. The finite difference method is mainly used using a uniform mesh, and it takes considerable time to price several options under the BS equation. The higher the dimension is, the worse the problem becomes. In our proposed method, we obtain an optimal non-uniform grid from a uniform grid by repeatedly removing a grid point having a minimum error based on the numerical solution on the grid including that point. We perform several numerical tests with one-, two- and three-dimensional BS equations. Computational tests are conducted for both cash-or-nothing and equity-linked security (ELS) options. The optimal non-uniform grid is especially useful in the three-dimensional case because the option prices can be efficiently computed with a small number of grid points.

Suggested Citation

  • Lyu, Jisang & Park, Eunchae & Kim, Sangkwon & Lee, Wonjin & Lee, Chaeyoung & Yoon, Sungha & Park, Jintae & Kim, Junseok, 2021. "Optimal non-uniform finite difference grids for the Black–Scholes equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 690-704.
    • Handle: RePEc:eee:matcom:v:182:y:2021:i:c:p:690-704
    DOI: 10.1016/j.matcom.2020.12.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475420304596
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2020.12.002?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kim, Junseok & Kim, Taekkeun & Jo, Jaehyun & Choi, Yongho & Lee, Seunggyu & Hwang, Hyeongseok & Yoo, Minhyun & Jeong, Darae, 2016. "A practical finite difference method for the three-dimensional Black–Scholes equation," European Journal of Operational Research, Elsevier, vol. 252(1), pages 183-190.
    2. Gongqiu Zhang & Lingfei Li, 2019. "Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior," Operations Research, INFORMS, vol. 67(2), pages 407-427, March.
    3. Bustamante, M. & Contreras, M., 2016. "Multi-asset Black–Scholes model as a variable second class constrained dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 540-572.
    4. Darae Jeong & Seungsuk Seo & Hyeongseok Hwang & Dongsun Lee & Yongho Choi & Junseok Kim, 2015. "Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-10, March.
    5. Milovanović, Slobodan & von Sydow, Lina, 2020. "A high order method for pricing of financial derivatives using Radial Basis Function generated Finite Differences," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 174(C), pages 205-217.
    6. Slobodan Milovanovi'c & Lina von Sydow, 2018. "A High Order Method for Pricing of Financial Derivatives using Radial Basis Function generated Finite Differences," Papers 1808.05890, arXiv.org, revised Aug 2018.
    7. Chen, Wen & Wang, Song, 2020. "A 2nd-order ADI finite difference method for a 2D fractional Black–Scholes equation governing European two asset option pricing," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 171(C), pages 279-293.
    8. Min Huang & Guo Luo, 2019. "A simple and efficient numerical method for pricing discretely monitored early-exercise options," Papers 1905.13407, arXiv.org, revised Jun 2019.
    9. Al–Zhour, Zeyad & Barfeie, Mahdiar & Soleymani, Fazlollah & Tohidi, Emran, 2019. "A computational method to price with transaction costs under the nonlinear Black–Scholes model," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 291-301.
    10. Yingzi Chen & Wansheng Wang & Aiguo Xiao, 2019. "An Efficient Algorithm for Options Under Merton’s Jump-Diffusion Model on Nonuniform Grids," Computational Economics, Springer;Society for Computational Economics, vol. 53(4), pages 1565-1591, April.
    11. Darae Jeong & Minhyun Yoo & Changwoo Yoo & Junseok Kim, 2019. "A Hybrid Monte Carlo and Finite Difference Method for Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 111-124, January.
    12. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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.
    2. 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.
    3. Cho, Junhyun & Kim, Yejin & Lee, Sungchul, 2022. "An accurate and stable numerical method for option hedge parameters," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    4. 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.
    5. Elisa Alòs & Maria Elvira Mancino & Tai-Ho Wang, 2019. "Volatility and volatility-linked derivatives: estimation, modeling, and pricing," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 321-349, December.
    6. Wang, Jian & Wen, Shuai & Yang, Mengdie & Shao, Wei, 2022. "Practical finite difference method for solving multi-dimensional black-Scholes model in fractal market," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    7. Elena-Corina Cipu, 2019. "Duality Results in Quasiinvex Variational Control Problems with Curvilinear Integral Functionals," Mathematics, MDPI, vol. 7(9), pages 1-9, September.
    8. Hanno Gottschalk & Marco Reese, 2021. "An Analytical Study in Multi-physics and Multi-criteria Shape Optimization," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 486-512, May.
    9. Karel Van Bockstal, 2020. "Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)," Mathematics, MDPI, vol. 8(8), pages 1-16, August.
    10. Savin Treanţă, 2019. "On Locally and Globally Optimal Solutions in Scalar Variational Control Problems," Mathematics, MDPI, vol. 7(9), pages 1-8, September.
    11. Darvishi, M.T. & Najafi, Mohammad & Wazwaz, Abdul-Majid, 2021. "Conformable space-time fractional nonlinear (1+1)-dimensional Schrödinger-type models and their traveling wave solutions," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    12. Bahareh Afhami & Mohsen Rezapour & Mohsen Madadi & Vahed Maroufy, 2021. "Dynamic investment portfolio optimization using a Multivariate Merton Model with Correlated Jump Risk," Papers 2104.11594, arXiv.org.
    13. Ivan Francisco Yupanqui Tello & Alain Vande Wouwer & Daniel Coutinho, 2021. "A Concise Review of State Estimation Techniques for Partial Differential Equation Systems," Mathematics, MDPI, vol. 9(24), pages 1-15, December.
    14. Christian Klein & Julien Riton & Nikola Stoilov, 2021. "Multi-domain spectral approach for the Hilbert transform on the real line," Partial Differential Equations and Applications, Springer, vol. 2(3), pages 1-19, June.
    15. Marco Cirant & Roberto Gianni & Paola Mannucci, 2020. "Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games," Dynamic Games and Applications, Springer, vol. 10(1), pages 100-119, March.
    16. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1046-1062.
    17. Christian Kuehn & Cinzia Soresina, 2020. "Numerical continuation for a fast-reaction system and its cross-diffusion limit," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-26, April.
    18. Zaiping Zhu & Andres Iglesias & Liqi Zhou & Lihua You & Jianjun Zhang, 2022. "PDE-Based 3D Surface Reconstruction from Multi-View 2D Images," Mathematics, MDPI, vol. 10(4), pages 1-17, February.
    19. Wensheng Yang & Jingtang Ma & Zhenyu Cui, 2021. "Analysis of Markov chain approximation for Asian options and occupation-time derivatives: Greeks and convergence rates," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(2), pages 359-412, April.
    20. Wei Zhou & Xingxing Hao & Kaidi Wang & Zhenyang Zhang & Yongxiang Yu & Haonan Su & Kang Li & Xin Cao & Arjan Kuijper, 2020. "Improved estimation of motion blur parameters for restoration from a single image," PLOS ONE, Public Library of Science, vol. 15(9), pages 1-21, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:182:y:2021:i:c:p:690-704. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.