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Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint

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  • Eyal Neuman
  • Mathieu Rosenbaum

Abstract

Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when $H$ goes to zero. We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.

Suggested Citation

  • Eyal Neuman & Mathieu Rosenbaum, 2017. "Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint," Papers 1711.00427, arXiv.org, revised May 2018.
    • Handle: RePEc:arx:papers:1711.00427
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    2. Christian Bayer & Fabian Andsem Harang & Paolo Pigato, 2020. "Log-modulated rough stochastic volatility models," Papers 2008.03204, arXiv.org, revised May 2021.
    3. Takaishi, Tetsuya, 2020. "Rough volatility of Bitcoin," Finance Research Letters, Elsevier, vol. 32(C).
    4. Paul Hager & Eyal Neuman, 2020. "The Multiplicative Chaos of $H=0$ Fractional Brownian Fields," Papers 2008.01385, arXiv.org.
    5. Forde, Martin & Fukasawa, Masaaki & Gerhold, Stefan & Smith, Benjamin, 2022. "The Riemann–Liouville field and its GMC as H→0, and skew flattening for the rough Bergomi model," Statistics & Probability Letters, Elsevier, vol. 181(C).
    6. 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.
    7. Blanka Horvath & Antoine Jacquier & Aitor Muguruza & Andreas Søjmark, 2024. "Functional central limit theorems for rough volatility," Finance and Stochastics, Springer, vol. 28(3), pages 615-661, July.
    8. 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.
    9. Wu, Peng & Muzy, Jean-François & Bacry, Emmanuel, 2022. "From rough to multifractal volatility: The log S-fBM model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    10. Tetsuya Takaishi, 2019. "Rough volatility of Bitcoin," Papers 1904.12346, arXiv.org.
    11. Blanka Horvath & Antoine Jacquier & Aitor Muguruza & Andreas Sojmark, 2017. "Functional central limit theorems for rough volatility," Papers 1711.03078, arXiv.org, revised Nov 2023.
    12. Ran Wei & Jinjiong Yu, 2024. "The critical disordered pinning measure," Papers 2402.17642, arXiv.org, revised Mar 2024.
    13. Christian Bayer & Paul Hager & Sebastian Riedel & John Schoenmakers, 2021. "Optimal stopping with signatures," Papers 2105.00778, arXiv.org.

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