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Log-modulated rough stochastic volatility models

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Listed:
  • Christian Bayer
  • Fabian Andsem Harang
  • Paolo Pigato

Abstract

We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index $H$. The so-obtained log-modulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for $H = 0$. As a consequence, the resulting super-rough stochastic volatility models can be analysed over the whole range $0 \le H

Suggested Citation

  • Christian Bayer & Fabian Andsem Harang & Paolo Pigato, 2020. "Log-modulated rough stochastic volatility models," Papers 2008.03204, arXiv.org, revised May 2021.
    • Handle: RePEc:arx:papers:2008.03204
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    References listed on IDEAS

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    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. Martin Keller-Ressel & Martin Larsson & Sergio Pulido, 2018. "Affine Rough Models," Papers 1812.08486, arXiv.org.
    3. Eduardo Abi Jaber, 2021. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02412741, HAL.
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    6. 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.
    7. Archil Gulisashvili, 2020. "Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness," Papers 2002.05143, arXiv.org, revised Dec 2020.
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    9. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2019. "Is Volatility Rough ?," Papers 1905.04852, arXiv.org, revised May 2019.
    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. Eduardo Abi Jaber, 2020. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Working Papers hal-02412741, HAL.
    12. 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.
    13. Paul Hager & Eyal Neuman, 2020. "The Multiplicative Chaos of $H=0$ Fractional Brownian Fields," Papers 2008.01385, arXiv.org.
    14. Eduardo Abi Jaber, 2019. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Papers 1912.07445, arXiv.org, revised Jun 2020.
    15. Masaaki Fukasawa, 2010. "Normalization for Implied Volatility," Papers 1008.5055, arXiv.org, revised Sep 2010.
    16. Eyal Neuman & Mathieu Rosenbaum, 2017. "Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint," Papers 1711.00427, arXiv.org, revised May 2018.
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    Cited by:

    1. Eduardo Abi Jaber & Camille Illand & Shaun Xiaoyuan Li, 2022. "Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints," Working Papers hal-03902513, HAL.
    2. 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).
    3. 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.
    4. Archil Gulisashvili, 2022. "Multivariate Stochastic Volatility Models and Large Deviation Principles," Papers 2203.09015, arXiv.org, revised Nov 2022.
    5. 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.
    6. Eduardo Abi Jaber & Camille Illand & Shaun & Li, 2022. "Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints," Papers 2212.08297, arXiv.org, revised Dec 2024.
    7. Alexandre Pannier, 2023. "Path-dependent PDEs for volatility derivatives," Papers 2311.08289, arXiv.org, revised Jan 2024.
    8. Christian Bayer & Paul Hager & Sebastian Riedel & John Schoenmakers, 2021. "Optimal stopping with signatures," Papers 2105.00778, arXiv.org.
    9. 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.
    10. Peng Wu & Jean-Franc{c}ois Muzy & Emmanuel Bacry, 2022. "From Rough to Multifractal volatility: the log S-fBM model," Papers 2201.09516, arXiv.org, revised Jul 2022.

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