IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2002.09215.html
   My bibliography  Save this paper

Volatility has to be rough

Author

Listed:
  • Masaaki Fukasawa

Abstract

First, we give an asymptotic expansion of short-dated at-the-money implied volatility that refines the preceding works and proves in particular that non-rough volatility models are inconsistent to a power law of volatility skew. Second, we show that given a power law of volatility skew in an option market, a continuous price dynamics of the underlying asset with non-rough volatility admits an arbitrage opportunity. The volatility therefore has to be rough in a viable market of the underlying asset of which the volatility skew obeys a power law.

Suggested Citation

  • Masaaki Fukasawa, 2020. "Volatility has to be rough," Papers 2002.09215, arXiv.org.
    • Handle: RePEc:arx:papers:2002.09215
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2002.09215
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Antoine Jacquier & Mikko S. Pakkanen & Henry Stone, 2017. "Pathwise large deviations for the Rough Bergomi model," Papers 1706.05291, arXiv.org, revised Dec 2018.
    2. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-778, April.
    3. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    4. Paul Gassiat, 2018. "On the martingale property in the rough Bergomi model," Papers 1811.10935, arXiv.org, revised Apr 2019.
    5. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    6. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    7. Peter Friz & Stefan Gerhold & Arpad Pinter, 2018. "Option pricing in the moderate deviations regime," Mathematical Finance, Wiley Blackwell, vol. 28(3), pages 962-988, July.
    8. Elisa Alòs & Jorge León & Josep Vives, 2007. "On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility," Finance and Stochastics, Springer, vol. 11(4), pages 571-589, October.
    9. C. Bayer & P. K. Friz & A. Gulisashvili & B. Horvath & B. Stemper, 2019. "Short-time near-the-money skew in rough fractional volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 779-798, May.
    10. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    11. Josselin Garnier & Knut Solna, 2015. "Correction to Black-Scholes formula due to fractional stochastic volatility," Papers 1509.01175, arXiv.org, revised Mar 2017.
    12. Masaaki Fukasawa, 2017. "Short-time at-the-money skew and rough fractional volatility," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 189-198, February.
    13. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    14. Omar Euch & Masaaki Fukasawa & Mathieu Rosenbaum, 2018. "The microstructural foundations of leverage effect and rough volatility," Finance and Stochastics, Springer, vol. 22(2), pages 241-280, April.
    15. Omar El Euch & Masaaki Fukasawa & Jim Gatheral & Mathieu Rosenbaum, 2018. "Short-term at-the-money asymptotics under stochastic volatility models," Papers 1801.08675, arXiv.org, revised Mar 2019.
    16. Masaaki Fukasawa, 2014. "Volatility Derivatives And Model-Free Implied Leverage," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(01), pages 1-23.
    17. Masaaki Fukasawa, 2010. "Normalization for Implied Volatility," Papers 1008.5055, arXiv.org, revised Sep 2010.
    18. Elisa Alòs & Kenichiro Shiraya, 2019. "Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach," Finance and Stochastics, Springer, vol. 23(2), pages 423-447, April.
    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. Peter K. Friz & Paul Gassiat & Paolo Pigato, 2022. "Short-dated smile under rough volatility: asymptotics and numerics," Quantitative Finance, Taylor & Francis Journals, vol. 22(3), pages 463-480, March.
    2. Giacomo Giorgio & Barbara Pacchiarotti & Paolo Pigato, 2023. "Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 30(3), pages 123-152, May.
    3. Aur'elien Alfonsi & Ahmed Kebaier, 2021. "Approximation of Stochastic Volterra Equations with kernels of completely monotone type," Papers 2102.13505, arXiv.org, revised Mar 2022.

    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. Florian Bourgey & Stefano De Marco & Peter K. Friz & Paolo Pigato, 2023. "Local volatility under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1119-1145, October.
    2. Florian Bourgey & Stefano De Marco & Peter K. Friz & Paolo Pigato, 2023. "Local volatility under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1119-1145, October.
    3. Jacquier, Antoine & Pannier, Alexandre, 2022. "Large and moderate deviations for stochastic Volterra systems," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 142-187.
    4. Giacomo Giorgio & Barbara Pacchiarotti & Paolo Pigato, 2023. "Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 30(3), pages 123-152, May.
    5. Peter K. Friz & Paul Gassiat & Paolo Pigato, 2022. "Short-dated smile under rough volatility: asymptotics and numerics," Quantitative Finance, Taylor & Francis Journals, vol. 22(3), pages 463-480, March.
    6. Antoine Jacquier & Alexandre Pannier, 2020. "Large and moderate deviations for stochastic Volterra systems," Papers 2004.10571, arXiv.org, revised Apr 2022.
    7. Giacomo Giorgio & Barbara Pacchiarotti & Paolo Pigato, 2023. "Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 30(3), pages 123-152, May.
    8. Qinwen Zhu & Gregoire Loeper & Wen Chen & Nicolas Langrené, 2021. "Markovian approximation of the rough Bergomi model for Monte Carlo option pricing," Post-Print hal-02910724, HAL.
    9. Peter K. Friz & Paul Gassiat & Paolo Pigato, 2018. "Precise asymptotics: robust stochastic volatility models," Papers 1811.00267, arXiv.org, revised Nov 2020.
    10. Omar El Euch & Masaaki Fukasawa & Jim Gatheral & Mathieu Rosenbaum, 2018. "Short-term at-the-money asymptotics under stochastic volatility models," Papers 1801.08675, arXiv.org, revised Mar 2019.
    11. Qinwen Zhu & Grégoire Loeper & Wen Chen & Nicolas Langrené, 2021. "Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing," Mathematics, MDPI, vol. 9(5), pages 1-21, March.
    12. Christian Bayer & Fabian Andsem Harang & Paolo Pigato, 2020. "Log-modulated rough stochastic volatility models," Papers 2008.03204, arXiv.org, revised May 2021.
    13. Christian Bayer & Peter K. Friz & Paul Gassiat & Jorg Martin & Benjamin Stemper, 2020. "A regularity structure for rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 782-832, July.
    14. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
    15. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Papers 2210.12393, arXiv.org.
    16. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    17. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Working Papers hal-03827332, HAL.
    18. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Post-Print hal-02946146, HAL.
    19. Aït-Sahalia, Yacine & Li, Chenxu & Li, Chen Xu, 2021. "Closed-form implied volatility surfaces for stochastic volatility models with jumps," Journal of Econometrics, Elsevier, vol. 222(1), pages 364-392.
    20. Paul Gassiat, 2022. "Weak error rates of numerical schemes for rough volatility," Papers 2203.09298, arXiv.org, revised Feb 2023.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:arx:papers:2002.09215. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.