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From rough to multifractal volatility: The log S-fBM model

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  • Wu, Peng
  • Muzy, Jean-François
  • Bacry, Emmanuel

Abstract

We introduce a family of random measures MH,T(dt), namely log S-fBM, such that, for H>0, MH,T(dt)=eωH,T(t)dt where ωH,T(t) is a Gaussian process that can be considered as a stationary version of an H-fractional Brownian motion. Moreover, when H→0, one has MH,T(dt)→M˜T(dt) (in the weak sense) where M˜T(dt) is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal (H=0) and rough volatility (0

Suggested Citation

  • 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).
    • Handle: RePEc:eee:phsmap:v:604:y:2022:i:c:s0378437122005866
    DOI: 10.1016/j.physa.2022.127919
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    References listed on IDEAS

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    Cited by:

    1. Paolo Dai Pra & Paolo Pigato, 2025. "A stochastic volatility approximation for a tick-by-tick price model with mean-field interaction," Papers 2504.03445, arXiv.org, revised Mar 2026.
    2. 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.
    3. Mathieu Rosenbaum & Jianfei Zhang, 2022. "On the universality of the volatility formation process: when machine learning and rough volatility agree," Papers 2206.14114, arXiv.org.
    4. Yoshioka, Hidekazu & Yoshioka, Yumi, 2025. "Stochastic volatility model with long memory for water quantity-quality dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 195(C).
    5. Ofelia Bonesini & Antoine Jacquier & Alexandre Pannier, 2023. "Rough volatility, path-dependent PDEs and weak rates of convergence," Papers 2304.03042, arXiv.org, revised Feb 2026.
    6. Alexandre Pannier, 2023. "Path-dependent PDEs for volatility derivatives," Papers 2311.08289, arXiv.org, revised Jul 2025.
    7. Siu Hin Tang & Mathieu Rosenbaum & Chao Zhou, 2024. "Forecasting volatility with machine learning and rough volatility: example from the crypto-winter," Digital Finance, Springer, vol. 6(4), pages 639-655, December.
    8. Othmane Zarhali & Cecilia Aubrun & Emmanuel Bacry & Jean-Philippe Bouchaud & Jean-Franc{c}ois Muzy, 2025. "Why is the volatility of single stocks so much rougher than that of the S&P500?," Papers 2505.02678, arXiv.org, revised Dec 2025.
    9. Rudy Morel & St'ephane Mallat & Jean-Philippe Bouchaud, 2023. "Path Shadowing Monte-Carlo," Papers 2308.01486, arXiv.org.
    10. Othmane Zarhali & Nicolas Langren'e, 2026. "Fast simulation of Volterra processes using random Fourier features with application to the log-stationary fractional Brownian motion," Papers 2603.02946, arXiv.org.

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