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Correction to Black-Scholes formula due to fractional stochastic volatility

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  • Josselin Garnier
  • Knut Solna

Abstract

Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is shown how the associated implied volatility has a term structure that is a function of maturity to a fractional power.

Suggested Citation

  • Josselin Garnier & Knut Solna, 2015. "Correction to Black-Scholes formula due to fractional stochastic volatility," Papers 1509.01175, arXiv.org, revised Mar 2017.
    • Handle: RePEc:arx:papers:1509.01175
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    References listed on IDEAS

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    Cited by:

    1. Josselin Garnier & Knut Sølna, 2018. "Option pricing under fast-varying and rough stochastic volatility," Annals of Finance, Springer, vol. 14(4), pages 489-516, November.
    2. Chinonso Nwankwo & Weizhong Dai & Ruihua Liu, 2019. "Compact Finite Difference Scheme with Hermite Interpolation for Pricing American Put Options Based on Regime Switching Model," Papers 1908.04900, arXiv.org, revised Jun 2020.
    3. Archil Gulisashvili, 2020. "Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness," Papers 2002.05143, arXiv.org, revised Dec 2020.
    4. Archil Gulisashvili, 2018. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Papers 1808.00421, arXiv.org, revised Jun 2019.
    5. Masaaki Fukasawa, 2020. "Volatility has to be rough," Papers 2002.09215, arXiv.org.
    6. Gulisashvili, Archil, 2020. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3648-3686.
    7. Gerhold, Stefan & Gerstenecker, Christoph & Gulisashvili, Archil, 2021. "Large deviations for fractional volatility models with non-Gaussian volatility driver," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 580-600.
    8. Qi Zhao & Alexandra Chronopoulou, 2024. "A New Proxy for Estimating the Roughness of Volatility," JRFM, MDPI, vol. 17(4), pages 1-15, March.
    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. Ranieri Dugo & Giacomo Giorgio & Paolo Pigato, 2024. "The Multivariate Fractional Ornstein-Uhlenbeck Process," CEIS Research Paper 581, Tor Vergata University, CEIS, revised 28 Aug 2024.
    11. Chinonso I. Nwankwo & Weizhong Dai & Ruihua Liu, 2023. "Compact Finite Difference Scheme with Hermite Interpolation for Pricing American Put Options Based on Regime Switching Model," Computational Economics, Springer;Society for Computational Economics, vol. 62(3), pages 817-854, October.
    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. Josselin Garnier & Knut Solna, 2018. "Optimal hedging under fast-varying stochastic volatility," Papers 1810.08337, arXiv.org, revised Mar 2020.

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