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Functional central limit theorems for rough volatility

Author

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  • Horvath, Blanka
  • Jacquier, Antoine
  • Muguruza, Aitor
  • Søjmark, Andreas

Abstract

The non-Markovian nature of rough volatility makes Monte Carlo methods challenging, and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker’s approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent H∈(0,1). Some of the most relevant consequences of this ‘rough Donsker (rDonsker) theorem’ are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme and find remarkable agreement (for a large range of values of H). Our rDonsker theorem further provides a weak convergence proof for the hybrid scheme itself and allows constructing binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options.

Suggested Citation

  • Horvath, Blanka & Jacquier, Antoine & Muguruza, Aitor & Søjmark, Andreas, 2024. "Functional central limit theorems for rough volatility," LSE Research Online Documents on Economics 122848, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:122848
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    References listed on IDEAS

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    1. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
    2. 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.
    3. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    4. Maurizio Pratelli & Sabrina Mulinacci, 1998. "Functional convergence of Snell envelopes: Applications to American options approximations," Finance and Stochastics, Springer, vol. 2(3), pages 311-327.
    5. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    6. Giulia Livieri & Saad Mouti & Andrea Pallavicini & Mathieu Rosenbaum, 2018. "Rough volatility: Evidence from option prices," IISE Transactions, Taylor & Francis Journals, vol. 50(9), pages 767-776, September.
    7. Eyal Neuman & Mathieu Rosenbaum, 2017. "Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint," Papers 1711.00427, arXiv.org, revised May 2018.
    8. Archil Gulisashvili, 2017. "Large deviation principle for Volterra type fractional stochastic volatility models," Papers 1710.10711, arXiv.org, revised Aug 2018.
    9. Antoine Jacquier & Mikko S. Pakkanen & Henry Stone, 2017. "Pathwise large deviations for the Rough Bergomi model," Papers 1706.05291, arXiv.org, revised Dec 2018.
    10. Paul Gassiat, 2018. "On the martingale property in the rough Bergomi model," Papers 1811.10935, arXiv.org, revised Apr 2019.
    11. Marc Romano & Nizar Touzi, 1997. "Contingent Claims and Market Completeness in a Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 399-412, October.
    12. 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.
    13. Nieminen, Ari, 2004. "Fractional Brownian motion and Martingale-differences," Statistics & Probability Letters, Elsevier, vol. 70(1), pages 1-10, October.
    14. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    15. Antoine Jacquier & Claude Martini & Aitor Muguruza, 2018. "On VIX futures in the rough Bergomi model," Quantitative Finance, Taylor & Francis Journals, vol. 18(1), pages 45-61, January.
    16. Alfredas Račkauskas & Charles Suquet, 2004. "Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces," Journal of Theoretical Probability, Springer, vol. 17(1), pages 221-243, January.
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    Cited by:

    1. Bo Yuan & Damiano Brigo & Antoine Jacquier & Nicola Pede, 2024. "Deep learning interpretability for rough volatility," Papers 2411.19317, arXiv.org.

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    More about this item

    Keywords

    binomial trees; fractional brownian motion; functional limit theorems; rough volatility; Early Postdoc.Mobility grant 165248; Imperial CDT in Financial Computing & Analytics; T032146 grant;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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