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The Heat-Kernel Most-Likely-Path Approximation

In: Finance at Fields

Author

Listed:
  • JIM GATHERAL

    (Department of Mathematics, Baruch College, CUNY, One Bernard Baruch Way, New York, NY 10010, USA)

  • TAI-HO WANG

    (Department of Mathematics, Baruch College, CUNY, One Bernard Baruch Way, New York, NY 10010, USA)

Abstract

In this article, we derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. We confirm the improved performance of our new approximation relative to existing approximations in an explicit computation using a realistic S&P500 local volatility function.

Suggested Citation

  • Jim Gatheral & Tai-Ho Wang, 2012. "The Heat-Kernel Most-Likely-Path Approximation," World Scientific Book Chapters, in: Matheus R Grasselli & Lane P Hughston (ed.), Finance at Fields, chapter 17, pages 389-406, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789814407892_0017
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    Citations

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    Cited by:

    1. Leif Döring & Blanka Horvath & Josef Teichmann, 2017. "Functional Analytic (Ir-)Regularity Properties Of Sabr-Type Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-48, May.
    2. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    3. Dan Pirjol & Jing Wang & Lingjiong Zhu, 2017. "Short Maturity Forward Start Asian Options in Local Volatility Models," Papers 1710.03160, arXiv.org.
    4. Hossein Jafari & Ghazaleh Rahimi, 2019. "Small-Time Asymptotics In Geometric Asian Options For A Stochastic Volatility Jump-Diffusion Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(02), pages 1-19, March.
    5. Lingjiong Zhu, 2015. "Options with Extreme Strikes," Risks, MDPI, vol. 3(3), pages 1-16, July.
    6. Lingjiong Zhu, 2015. "Short maturity options for Azéma–Yor martingales," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 1-32, December.
    7. Thomas Mazzoni, 2018. "Asymptotic Expansion of Risk-Neutral Pricing Density," IJFS, MDPI, vol. 6(1), pages 1-26, March.
    8. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2018. "Most-Likely-Path In Asian Option Pricing Under Local Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(05), pages 1-32, August.
    9. Dan Pirjol & Lingjiong Zhu, 2017. "Short Maturity Asian Options for the CEV Model," Papers 1702.03382, arXiv.org.
    10. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2017. "Most-likely-path in Asian option pricing under local volatility models," Papers 1706.02408, arXiv.org, revised Aug 2018.
    11. Olesya Grishchenko & Xiao Han & Victor Nistor, 2018. "A volatility-of-volatility expansion of the option prices in the SABR stochastic volatility model," Papers 1812.09904, arXiv.org.
    12. Siyan Zhang & Anna L. Mazzucato & Victor Nistor, 2016. "Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility Model," Papers 1605.03097, arXiv.org.
    13. Stefano De Marco & Peter Friz, 2013. "Varadhan's formula, conditioned diffusions, and local volatilities," Papers 1311.1545, arXiv.org, revised Jun 2016.

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