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Reconciling rough volatility with jumps

Author

Listed:
  • Eduardo Abi Jaber

    (CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Nathan de Carvalho

    (UPCité - Université Paris Cité)

Abstract

We reconcile rough volatility models and jump models using a class of reversionary Heston models with fast mean reversions and large vol-of-vols. Starting from hyper-rough Heston models with a Hurst index H ∈ (−1/2, 1/2), we derive a Markovian approximating class of one dimensional reversionary Hestontype models. Such proxies encode a trade-off between an exploding vol-of-vol and a fast mean-reversion speed controlled by a reversionary timescale ϵ > 0 and an unconstrained parameter H ∈ R. Sending ϵ to 0 yields convergence of the reversionary Heston model towards different explicit asymptotic regimes based on the value of the parameter H. In particular, for H ≤ −1/2, the reversionary Heston model converges to a class of Lévy jump processes of Normal Inverse Gaussian type. Numerical illustrations show that the reversionary Heston model is capable of generating at-the-money skews similar to the ones generated by rough, hyper-rough and jump models.

Suggested Citation

  • Eduardo Abi Jaber & Nathan de Carvalho, 2024. "Reconciling rough volatility with jumps," Post-Print hal-04295416, HAL.
    • Handle: RePEc:hal:journl:hal-04295416
    Note: View the original document on HAL open archive server: https://hal.science/hal-04295416v2
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    References listed on IDEAS

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    1. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. "Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    2. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
    3. 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.
    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    5. Eduardo Abi Jaber, 2021. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Post-Print hal-02412741, HAL.
    6. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    7. Ryan McCrickerd, 2019. "On spatially irregular ordinary differential equations and a pathwise volatility modelling framework," Papers 1902.01673, arXiv.org, revised Sep 2021.
    8. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    9. Eduardo Abi Jaber, 2018. "Lifting the Heston model," Papers 1810.04868, arXiv.org, revised Nov 2019.
    10. Martin Forde & Antoine Jacquier, 2011. "The large-maturity smile for the Heston model," Finance and Stochastics, Springer, vol. 15(4), pages 755-780, December.
    11. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 19(12), pages 1995-2013, December.
    12. 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.
    13. Eduardo Abi Jaber & Omar El Euch, 2019. "Multi-factor approximation of rough volatility models," Post-Print hal-01697117, HAL.
    14. Sassan Alizadeh & Michael W. Brandt & Francis X. Diebold, 2002. "Range‐Based Estimation of Stochastic Volatility Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1047-1091, June.
    15. Matthieu Garcin & Martino Grasselli, 2022. "Long versus short time scales: the rough dilemma and beyond," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 257-278, June.
    16. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2016. "Decoupling the short- and long-term behavior of stochastic volatility," Papers 1610.00332, arXiv.org, revised Jan 2021.
    17. 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.
    18. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    19. Eduardo Abi Jaber, 2020. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Papers 2009.10972, arXiv.org, revised May 2022.
    20. Callegaro Giorgia & Grasselli Martino & Pag`es Gilles, 2018. "Fast Hybrid Schemes for Fractional Riccati Equations (Rough is not so Tough)," Papers 1805.12587, arXiv.org, revised Feb 2020.
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    1. Xiaoyu Shen & Fang Fang & Chengguang Liu, 2024. "The Fourier Cosine Method for Discrete Probability Distributions," Papers 2410.04487, arXiv.org, revised Oct 2024.

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