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Reconstructing the Local Volatility Surface from Market Option Prices

Author

Listed:
  • Soobin Kwak

    (Department of Mathematics, Korea University, Seoul 02841, Korea)

  • Youngjin Hwang

    (Department of Mathematics, Korea University, Seoul 02841, Korea)

  • Yongho Choi

    (Department of Computer & Information Engineering (Information Security), Daegu University, Gyeongsan-si 38453, Korea)

  • Jian Wang

    (School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China)

  • Sangkwon Kim

    (Department of Mathematics, Korea University, Seoul 02841, Korea)

  • Junseok Kim

    (Department of Mathematics, Korea University, Seoul 02841, Korea)

Abstract

We present an efficient and accurate computational algorithm for reconstructing a local volatility surface from given market option prices. The local volatility surface is dependent on the values of both the time and underlying asset. We use the generalized Black–Scholes (BS) equation and finite difference method (FDM) to numerically solve the generalized BS equation. We reconstruct the local volatility function, which provides the best fit between the theoretical and market option prices by minimizing a cost function that is a quadratic representation of the difference between the two option prices. This is an inverse problem in which we want to calculate a local volatility function consistent with the observed market prices. To achieve robust computation, we place the sample points of the unknown volatility function in the middle of the expiration dates. We perform various numerical experiments to confirm the simplicity, robustness, and accuracy of the proposed method in reconstructing the local volatility function.

Suggested Citation

  • Soobin Kwak & Youngjin Hwang & Yongho Choi & Jian Wang & Sangkwon Kim & Junseok Kim, 2022. "Reconstructing the Local Volatility Surface from Market Option Prices," Mathematics, MDPI, vol. 10(14), pages 1-12, July.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2537-:d:868205
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    References listed on IDEAS

    as
    1. Kathrin Hellmuth & Christian Klingenberg, 2022. "Computing Black Scholes with Uncertain Volatility-A Machine Learning Approach," Papers 2202.07378, arXiv.org.
    2. 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).
    3. Kathrin Hellmuth & Christian Klingenberg, 2022. "Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach," Mathematics, MDPI, vol. 10(3), pages 1-20, February.
    4. Kim, Seong-Tae & Kim, Hyun-Gyoon & Kim, Jeong-Hoon, 2021. "ELS pricing and hedging in a fractional Brownian motion environment," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    5. Yuzi Jin & Jian Wang & Sangkwon Kim & Youngjin Heo & Changwoo Yoo & Youngrock Kim & Junseok Kim & Darae Jeong, 2018. "Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model," Discrete Dynamics in Nature and Society, Hindawi, vol. 2018, pages 1-9, May.
    6. Ahmet Murat Ozbayoglu & Mehmet Ugur Gudelek & Omer Berat Sezer, 2020. "Deep Learning for Financial Applications : A Survey," Papers 2002.05786, arXiv.org.
    7. Jian Geng & I. Michael Navon & Xiao Chen, 2014. "Non-parametric calibration of the local volatility surface for European options using a second-order Tikhonov regularization," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 73-85, January.
    8. Vinicius Albani & Adriano De Cezaro & Jorge P. Zubelli, 2017. "Convex Regularization Of Local Volatility Estimation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-37, February.
    9. 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.
    10. Kim, Sangkwon & Kim, Junseok, 2021. "Robust and accurate construction of the local volatility surface using the Black–Scholes equation," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
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