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A practical finite difference method for the three-dimensional Black–Scholes equation

Author

Listed:
  • Kim, Junseok
  • Kim, Taekkeun
  • Jo, Jaehyun
  • Choi, Yongho
  • Lee, Seunggyu
  • Hwang, Hyeongseok
  • Yoo, Minhyun
  • Jeong, Darae

Abstract

In this paper, we develop a fast and accurate numerical method for pricing of the three-asset equity-linked securities options. The option pricing model is based on the Black–Scholes partial differential equation. The model is discretized by using a non-uniform finite difference method and the resulting discrete equations are solved by using an operator splitting method. For fast and accurate calculation, we put more grid points near the singularity of the nonsmooth payoff function. To demonstrate the accuracy and efficiency of the proposed numerical method, we compare the results of the method with those from Monte Carlo simulation in terms of computational cost and accuracy. The numerical results show that the cost of the proposed method is comparable to that of the Monte Carlo simulation and it provides more stable hedging parameters such as the Greeks.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:252:y:2016:i:1:p:183-190
    DOI: 10.1016/j.ejor.2015.12.012
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    References listed on IDEAS

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    1. Bandi, Chaithanya & Bertsimas, Dimitris, 2014. "Robust option pricing," European Journal of Operational Research, Elsevier, vol. 239(3), pages 842-853.
    2. 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.
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    7. Marroquı´n-Martı´nez, Naroa & Moreno, Manuel, 2013. "Optimizing bounds on security prices in incomplete markets. Does stochastic volatility specification matter?," European Journal of Operational Research, Elsevier, vol. 225(3), pages 429-442.
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    Cited by:

    1. Panumart Sawangtong & Kamonchat Trachoo & Wannika Sawangtong & Benchawan Wiwattanapataphee, 2018. "The Analytical Solution for the Black-Scholes Equation with Two Assets in the Liouville-Caputo Fractional Derivative Sense," Mathematics, MDPI, vol. 6(8), pages 1-14, July.
    2. Jang Hanbyeol & Wang Jian & Kim Junseok, 2019. "Equity-linked security pricing and Greeks at arbitrary intermediate times using Brownian bridge," Monte Carlo Methods and Applications, De Gruyter, vol. 25(4), pages 291-305, December.
    3. 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.
    4. 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).
    5. Černá, Dana & Fiňková, Kateřina, 2024. "Option pricing under multifactor Black–Scholes model using orthogonal spline wavelets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 309-340.
    6. 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).
    7. 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.
    8. Wang, Jian & Yan, Yan & Chen, Wenbing & Shao, Wei & Wang, Jian & Tang, Weiwei, 2021. "Equity-linked securities option pricing by fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).

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