IDEAS home Printed from https://ideas.repec.org/a/taf/quantf/v25y2025i3p471-489.html

Pricing and calibration in the 4-factor path-dependent volatility model

Author

Listed:
  • Guido Gazzani
  • Julien Guyon

Abstract

We consider the path-dependent volatility (PDV) model of Guyon and Lekeufack (2023), where the instantaneous volatility is a linear combination of a weighted sum of past returns and the square root of a weighted sum of past squared returns. We discuss the influence of an additional parameter that unlocks enough volatility on the upside to reproduce the implied volatility smiles of S&P 500 and VIX options. This PDV model, motivated by empirical studies, comes with computational challenges, especially in relation to VIX options pricing and calibration. We propose an accurate pathwise neural network approximation of the VIX which leverages on the Markovianity of the 4-factor version of the model. The VIX is learned pathwise as a function of the Markovian factors and the model parameters. We use this approximation to tackle the joint calibration of S&P 500 and VIX options, quickly sample VIX paths, and price derivatives that jointly depend on S&P 500 and VIX. As an interesting aside, we also show that this time-homogeneous, low-parametric, Markovian PDV model is able to fit the whole surface of S&P 500 implied volatilities remarkably well.

Suggested Citation

  • Guido Gazzani & Julien Guyon, 2025. "Pricing and calibration in the 4-factor path-dependent volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 25(3), pages 471-489, March.
    • Handle: RePEc:taf:quantf:v:25:y:2025:i:3:p:471-489
    DOI: 10.1080/14697688.2025.2472892
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/14697688.2025.2472892
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/14697688.2025.2472892?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Thomas Kokholm & Martin Stisen, 2015. "Joint pricing of VIX and SPX options with stochastic volatility and jump models," Journal of Risk Finance, Emerald Group Publishing, vol. 16(1), pages 27-48, January.
    2. Thomas Kokholm & Martin Stisen, 2015. "Joint pricing of VIX and SPX options with stochastic volatility and jump models," Journal of Risk Finance, Emerald Group Publishing Limited, vol. 16(1), pages 27-48, January.
    3. Julien Guyon & Jordan Lekeufack, 2023. "Volatility is (mostly) path-dependent," Post-Print hal-04373380, HAL.
    4. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2024. "The rough Hawkes Heston stochastic volatility model," Post-Print hal-03827332, HAL.
    5. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    6. Julien Guyon & Jordan Lekeufack, 2023. "Volatility is (mostly) path-dependent," Quantitative Finance, Taylor & Francis Journals, vol. 23(9), pages 1221-1258, September.
    7. Jim Gatheral & Paul Jusselin & Mathieu Rosenbaum, 2020. "The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem," Papers 2001.01789, arXiv.org.
    8. Mathieu Rosenbaum & Jianfei Zhang, 2021. "Deep calibration of the quadratic rough Heston model," Papers 2107.01611, arXiv.org, revised May 2022.
    9. Gilles Zumbach, 2009. "Time reversal invariance in finance," Quantitative Finance, Taylor & Francis Journals, vol. 9(5), pages 505-515.
    10. Christa Cuchiero & Josef Teichmann, 2019. "Markovian lifts of positive semidefinite affine Volterra-type processes," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 407-448, December.
    11. Christa Cuchiero & Josef Teichmann, 2019. "Markovian lifts of positive semidefinite affine Volterra type processes," Papers 1907.01917, arXiv.org, revised Sep 2019.
    12. Andrea Pascucci & Paolo Foschi, 2005. "Calibration of the Hobson&Rogers model: empirical tests," Finance 0509020, University Library of Munich, Germany.
    13. Enrique Sentana, 1995. "Quadratic ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 62(4), pages 639-661.
    14. Julien Guyon, 2020. "Inversion of convex ordering in the VIX market," Quantitative Finance, Taylor & Francis Journals, vol. 20(10), pages 1597-1623, October.
    15. Sana Ben Hamida & Rama Cont, 2005. "Recovering Volatility from Option Prices by Evolutionary Optimization," Post-Print hal-02490586, HAL.
    16. Pacati, Claudio & Pompa, Gabriele & Renò, Roberto, 2018. "Smiling twice: The Heston++ model," Journal of Banking & Finance, Elsevier, vol. 96(C), pages 185-206.
    17. Léo Parent, 2023. "The EWMA Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 23(1), pages 71-93, January.
    18. Gilles Zumbach, 2010. "Volatility conditional on price trends," Quantitative Finance, Taylor & Francis Journals, vol. 10(4), pages 431-442.
    19. Julien Guyon & Florian Bourgey, 2024. "Fast Exact Joint S&P 500/VIX Smile Calibration in Discrete and Continuous Time," Post-Print hal-03932808, HAL.
    20. Bondi, Alessandro & Livieri, Giulia & Pulido, Sergio, 2024. "Affine Volterra processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 168(C).
    21. Julien Guyon & Scander Mustapha, 2023. "Neural Joint S&P 500/VIX Smile Calibration," Post-Print hal-03932780, HAL.
    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. Federico M. Bandi & Nicola Fusari & Guido Gazzani & Roberto Ren`o, 2026. "Ultra-short-term volatility surfaces," Papers 2603.29430, arXiv.org.

    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. Guido Gazzani & Julien Guyon, 2024. "Pricing and calibration in the 4-factor path-dependent volatility model," Papers 2406.02319, arXiv.org, revised Feb 2025.
    2. Christa Cuchiero & Guido Gazzani & Janka Möller & Sara Svaluto‐Ferro, 2025. "Joint calibration to SPX and VIX options with signature‐based models," Mathematical Finance, Wiley Blackwell, vol. 35(1), pages 161-213, January.
    3. Fabio Baschetti & Giacomo Bormetti & Pietro Rossi, 2025. "Joint deep calibration of the 4-factor PDV model," Papers 2507.09412, arXiv.org.
    4. Julien Guyon & Jordan Lekeufack, 2023. "Volatility is (mostly) path-dependent," Post-Print hal-04373380, HAL.
    5. Ofelia Bonesini & Emilio Ferrucci & Ioannis Gasteratos & Antoine Jacquier, 2024. "Rough differential equations for volatility," Papers 2412.21192, arXiv.org, revised Mar 2026.
    6. Julien Guyon & Jordan Lekeufack, 2023. "Volatility is (mostly) path-dependent," Quantitative Finance, Taylor & Francis Journals, vol. 23(9), pages 1221-1258, September.
    7. Julien Guyon, 2024. "Dispersion-constrained martingale Schrödinger problems and the exact joint S&P 500/VIX smile calibration puzzle," Finance and Stochastics, Springer, vol. 28(1), pages 27-79, January.
    8. Eduardo Abi Jaber & Camille Illand & Shaun (Xiaoyuan) Li, 2025. "Joint SPX & VIX calibration with Gaussian polynomial volatility models: Deep pricing with quantization hints," Mathematical Finance, Wiley Blackwell, vol. 35(2), pages 470-519, April.
    9. Jim Gatheral & Paul Jusselin & Mathieu Rosenbaum, 2020. "The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem," Papers 2001.01789, arXiv.org.
    10. Christa Cuchiero & Guido Gazzani & Janka Moller & Sara Svaluto-Ferro, 2023. "Joint calibration to SPX and VIX options with signature-based models," Papers 2301.13235, arXiv.org, revised Jul 2024.
    11. Giulia Di Nunno & Kk{e}stutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From constant to rough: A survey of continuous volatility modeling," Papers 2309.01033, arXiv.org, revised Aug 2025.
    12. Yan, Tingjin & Yin, Jie & Wang, Ling & Wong, Hoi Ying, 2025. "4/2 rough and smooth," Journal of Banking & Finance, Elsevier, vol. 181(C).
    13. Eduardo Abi Jaber & Camille Illand & Shaun Xiaoyuan Li, 2024. "Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-03902513, HAL.
    14. Léo Parent, 2022. "The EWMA Heston model," Post-Print hal-04431111, HAL.
    15. R'emy Chicheportiche & Jean-Philippe Bouchaud, 2012. "The fine-structure of volatility feedback I: multi-scale self-reflexivity," Papers 1206.2153, arXiv.org, revised Sep 2013.
    16. Siu Hin Tang & Mathieu Rosenbaum & Chao Zhou, 2023. "Forecasting Volatility with Machine Learning and Rough Volatility: Example from the Crypto-Winter," Papers 2311.04727, arXiv.org, revised Feb 2024.
    17. Julien Guyon & Scander Mustapha, 2023. "Neural Joint S&P 500/VIX Smile Calibration," Post-Print hal-03932780, HAL.
    18. Federico M. Bandi & Nicola Fusari & Guido Gazzani & Roberto Ren`o, 2026. "Ultra-short-term volatility surfaces," Papers 2603.29430, arXiv.org.
    19. Eduardo Abi Jaber & Camille Illand & Shaun Xiaoyuan Li, 2024. "Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints," Post-Print hal-03902513, HAL.
    20. Antoine Jacquier & Aitor Muguruza & Alexandre Pannier, 2021. "Rough multifactor volatility for SPX and VIX options," Papers 2112.14310, arXiv.org, revised Nov 2023.

    More about this item

    Statistics

    Access and download statistics

    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:taf:quantf:v:25:y:2025:i:3:p:471-489. 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: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/RQUF20 .

    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.