IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1410.3394.html
   My bibliography  Save this paper

Volatility is rough

Author

Listed:
  • Jim Gatheral
  • Thibault Jaisson
  • Mathieu Rosenbaum

Abstract

Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H

Suggested Citation

  • Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2014. "Volatility is rough," Papers 1410.3394, arXiv.org.
    • Handle: RePEc:arx:papers:1410.3394
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1410.3394
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Stephen J. Hardiman & Nicolas Bercot & Jean-Philippe Bouchaud, 2013. "Critical reflexivity in financial markets: a Hawkes process analysis," Papers 1302.1405, arXiv.org, revised Jun 2013.
    2. Mehdi Lallouache & Damien Challet, 2016. "The limits of statistical significance of Hawkes processes fitted to financial data," Quantitative Finance, Taylor & Francis Journals, vol. 16(1), pages 1-11, January.
    3. Novikov, Alexander & Valkeila, Esko, 1999. "On some maximal inequalities for fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 47-54, August.
    4. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    5. Fulvio Corsi, 2009. "A Simple Approximate Long-Memory Model of Realized Volatility," Journal of Financial Econometrics, Oxford University Press, vol. 7(2), pages 174-196, Spring.
    6. Vladimir Filimonov & Didier Sornette, 2013. "Apparent Criticality and Calibration Issues in the Hawkes Self-Excited Point Process Model: Application to High-Frequency Financial Data," Swiss Finance Institute Research Paper Series 13-60, Swiss Finance Institute.
    7. Emmanuel Bacry & Jean-Fran�ois Muzy, 2014. "Hawkes model for price and trades high-frequency dynamics," Quantitative Finance, Taylor & Francis Journals, vol. 14(7), pages 1147-1166, July.
    8. Peter Carr & Liuren Wu, 2003. "What Type of Process Underlies Options? A Simple Robust Test," Journal of Finance, American Finance Association, vol. 58(6), pages 2581-2610, December.
    9. Vladimir Filimonov & Didier Sornette, 2012. "Quantifying reflexivity in financial markets: towards a prediction of flash crashes," Papers 1201.3572, arXiv.org, revised Apr 2012.
    10. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    11. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2003. "Modeling and Forecasting Realized Volatility," Econometrica, Econometric Society, vol. 71(2), pages 579-625, March.
    12. Jim Gatheral & Roel Oomen, 2010. "Zero-intelligence realized variance estimation," Finance and Stochastics, Springer, vol. 14(2), pages 249-283, April.
    13. Andersen, Torben G. & Bollerslev, Tim & Diebold, Francis X. & Ebens, Heiko, 2001. "The distribution of realized stock return volatility," Journal of Financial Economics, Elsevier, vol. 61(1), pages 43-76, July.
    14. Hull, John & White, Alan, 1993. "One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(2), pages 235-254, June.
    15. Gilles Teyssière & Alan P. Kirman (ed.), 2007. "Long Memory in Economics," Springer Books, Springer, number 978-3-540-34625-8, December.
    16. Andersen T. G & Bollerslev T. & Diebold F. X & Labys P., 2001. "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 42-55, March.
    17. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    18. Rosenbaum, Mathieu, 2009. "First order p-variations and Besov spaces," Statistics & Probability Letters, Elsevier, vol. 79(1), pages 55-62, January.
    19. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    20. Christian Y. Robert & Mathieu Rosenbaum, 2011. "A New Approach for the Dynamics of Ultra-High-Frequency Data: The Model with Uncertainty Zones," Journal of Financial Econometrics, Oxford University Press, vol. 9(2), pages 344-366, Spring.
    21. Alexandra Chronopoulou & Frederi Viens, 2012. "Estimation and pricing under long-memory stochastic volatility," Annals of Finance, Springer, vol. 8(2), pages 379-403, May.
    22. Andersen, Torben G. & Bollerslev, Tim, 1997. "Intraday periodicity and volatility persistence in financial markets," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 115-158, June.
    23. Rama Cont, 2007. "Volatility Clustering in Financial Markets: Empirical Facts and Agent-Based Models," Springer Books, in: Gilles Teyssière & Alan P. Kirman (ed.), Long Memory in Economics, pages 289-309, Springer.
    24. F. Comte & L. Coutin & E. Renault, 2012. "Affine fractional stochastic volatility models," Annals of Finance, Springer, vol. 8(2), pages 337-378, May.
    25. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    26. Rosenbaum, Mathieu, 2008. "Estimation of the volatility persistence in a discretely observed diffusion model," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1434-1462, August.
    27. Stephen Hardiman & Nicolas Bercot & Jean-Philippe Bouchaud, 2013. "Critical reflexivity in financial markets: a Hawkes process analysis," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 86(10), pages 1-9, October.
    28. Bouchaud,Jean-Philippe & Potters,Marc, 2003. "Theory of Financial Risk and Derivative Pricing," Cambridge Books, Cambridge University Press, number 9780521819169.
    29. Khalil Dayri & Mathieu Rosenbaum, 2012. "Large tick assets: implicit spread and optimal tick size," Papers 1207.6325, arXiv.org, revised Jan 2013.
    30. Mathieu Rosenbaum, 2011. "A new microstructure noise index," Quantitative Finance, Taylor & Francis Journals, vol. 11(6), pages 883-899.
    Full references (including those not matched with items on IDEAS)

    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. Giulia Di Nunno & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2022. "Option pricing in Sandwiched Volterra Volatility model," Papers 2209.10688, arXiv.org, revised Dec 2023.
    2. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2017. "Decoupling the short- and long-term behavior of stochastic volatility," CREATES Research Papers 2017-26, Department of Economics and Business Economics, Aarhus University.
    3. Bollerslev, Tim & Osterrieder, Daniela & Sizova, Natalia & Tauchen, George, 2013. "Risk and return: Long-run relations, fractional cointegration, and return predictability," Journal of Financial Economics, Elsevier, vol. 108(2), pages 409-424.
    4. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2016. "Decoupling the short- and long-term behavior of stochastic volatility," Papers 1610.00332, arXiv.org, revised Jan 2021.
    5. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    6. de Truchis, Gilles & Keddad, Benjamin, 2016. "On the risk comovements between the crude oil market and U.S. dollar exchange rates," Economic Modelling, Elsevier, vol. 52(PA), pages 206-215.
    7. Abderrazak Ben Maatoug & Rim Lamouchi & Russell Davidson & Ibrahim Fatnassi, 2018. "Modelling Foreign Exchange Realized Volatility Using High Frequency Data: Long Memory versus Structural Breaks," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 10(1), pages 1-25, March.
    8. Corsi, Fulvio & Fusari, Nicola & La Vecchia, Davide, 2013. "Realizing smiles: Options pricing with realized volatility," Journal of Financial Economics, Elsevier, vol. 107(2), pages 284-304.
    9. Morelli, Giacomo & Santucci de Magistris, Paolo, 2019. "Volatility tail risk under fractionality," Journal of Banking & Finance, Elsevier, vol. 108(C).
    10. Proietti, Tommaso, 2014. "Exponential Smoothing, Long Memory and Volatility Prediction," MPRA Paper 57230, University Library of Munich, Germany.
    11. Sévi, Benoît, 2014. "Forecasting the volatility of crude oil futures using intraday data," European Journal of Operational Research, Elsevier, vol. 235(3), pages 643-659.
    12. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold, 2003. "Some Like it Smooth, and Some Like it Rough: Untangling Continuous and Jump Components in Measuring, Modeling, and Forecasting Asset Return Volatility," PIER Working Paper Archive 03-025, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 01 Sep 2003.
    13. Roger Martins & Dieter Hendricks, 2016. "The statistical significance of multivariate Hawkes processes fitted to limit order book data," Papers 1604.01824, arXiv.org, revised Apr 2016.
    14. Rossi, Eduardo & Santucci de Magistris, Paolo, 2013. "Long memory and tail dependence in trading volume and volatility," Journal of Empirical Finance, Elsevier, vol. 22(C), pages 94-112.
    15. Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," PIER Working Paper Archive 05-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    16. Patton, Andrew J., 2011. "Data-based ranking of realised volatility estimators," Journal of Econometrics, Elsevier, vol. 161(2), pages 284-303, April.
    17. Matthieu Garcin & Martino Grasselli, 2020. "Long vs Short Time Scales: the Rough Dilemma and Beyond," Papers 2008.07822, arXiv.org, revised Nov 2021.
    18. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2006. "Volatility and Correlation Forecasting," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 15, pages 777-878, Elsevier.
    19. Christophe Chorro & Florian Ielpo & Benoît Sévi, 2017. "The contribution of jumps to forecasting the density of returns," Post-Print halshs-01442618, HAL.
    20. 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.

    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:1410.3394. 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.