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Short-term at-the-money asymptotics under stochastic volatility models

Author

Listed:
  • Omar El Euch
  • Masaaki Fukasawa
  • Jim Gatheral
  • Mathieu Rosenbaum

Abstract

A small-time Edgeworth expansion of the density of an asset price is given under a general stochastic volatility model, from which asymptotic expansions of put option prices and at-the-money implied volatilities follow. A limit theorem for at-the-money implied volatility skew and curvature is also given as a corollary. The rough Bergomi model is treated as an example.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:1801.08675
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    File URL: http://arxiv.org/pdf/1801.08675
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    References listed on IDEAS

    as
    1. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    2. Peter Friz & Stefan Gerhold & Arpad Pinter, 2018. "Option pricing in the moderate deviations regime," Mathematical Finance, Wiley Blackwell, vol. 28(3), pages 962-988, July.
    3. Alexey Medvedev & Olivier Scaillet, 2007. "Approximation and Calibration of Short-Term Implied Volatilities Under Jump-Diffusion Stochastic Volatility," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 427-459.
    4. 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.
    5. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    6. Josselin Garnier & Knut Solna, 2015. "Correction to Black-Scholes formula due to fractional stochastic volatility," Papers 1509.01175, arXiv.org, revised Mar 2017.
    7. Masaaki Fukasawa, 2017. "Short-time at-the-money skew and rough fractional volatility," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 189-198, February.
    8. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
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    Citations

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    Cited by:

    1. Liexin Cheng & Xue Cheng, 2024. "Approximating Smiles: A Time Change Approach," Papers 2401.03776, arXiv.org, revised Apr 2024.
    2. Marc Lagunas-Merino & Salvador Ortiz-Latorre, 2020. "A decomposition formula for fractional Heston jump diffusion models," Papers 2007.14328, arXiv.org.
    3. Christian Bayer & Fabian Andsem Harang & Paolo Pigato, 2020. "Log-modulated rough stochastic volatility models," Papers 2008.03204, arXiv.org, revised May 2021.
    4. 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.
    5. Masaaki Fukasawa, 2020. "Volatility has to be rough," Papers 2002.09215, arXiv.org.
    6. 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.
    7. Masaaki Fukasawa & Jim Gatheral, 2021. "A rough SABR formula," Papers 2105.05359, arXiv.org.
    8. Carsten Chong & Viktor Todorov, 2022. "Short-time expansion of characteristic functions in a rough volatility setting with applications," Papers 2208.00830, arXiv.org.
    9. Qinwen Zhu & Grégoire Loeper & Wen Chen & Nicolas Langrené, 2021. "Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing," Mathematics, MDPI, vol. 9(5), pages 1-21, March.
    10. Martin Forde & Stefan Gerhold & Benjamin Smith, 2021. "Small‐time, large‐time, and H→0 asymptotics for the Rough Heston model," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 203-241, January.
    11. Peter K. Friz & Paul Gassiat & Paolo Pigato, 2018. "Precise asymptotics: robust stochastic volatility models," Papers 1811.00267, arXiv.org, revised Nov 2020.
    12. Elisa Alos & Frido Rolloos & Kenichiro Shiraya, 2019. "On the difference between the volatility swap strike and the zero vanna implied volatility," Papers 1912.05383, arXiv.org, revised Dec 2020.
    13. 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.
    14. Qinwen Zhu & Gregoire Loeper & Wen Chen & Nicolas Langrené, 2021. "Markovian approximation of the rough Bergomi model for Monte Carlo option pricing," Post-Print hal-02910724, HAL.
    15. Carsten Chong & Marc Hoffmann & Yanghui Liu & Mathieu Rosenbaum & Gr'egoire Szymanski, 2022. "Statistical inference for rough volatility: Minimax Theory," Papers 2210.01214, arXiv.org, revised Feb 2024.
    16. Huy N. Chau & Duy Nguyen & Thai Nguyen, 2024. "On short-time behavior of implied volatility in a market model with indexes," Papers 2402.16509, arXiv.org, revised Apr 2024.

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