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Affine forward variance models

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  • Jim Gatheral
  • Martin Keller-Ressel

Abstract

We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterized by the affine form of their cumulant generating function, which can be obtained as solution of a convolution Riccati equation. We further introduce the class of affine forward order flow intensity (AFI) models, which are structurally similar to AFV models, but driven by jump processes, and which include Hawkes-type models. We show that the cumulant generating function of an AFI model satisfies a generalized convolution Riccati equation and that a high-frequency limit of AFI models converges in distribution to the AFV model.

Suggested Citation

  • Jim Gatheral & Martin Keller-Ressel, 2018. "Affine forward variance models," Papers 1801.06416, arXiv.org, revised Oct 2018.
    • Handle: RePEc:arx:papers:1801.06416
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    References listed on IDEAS

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    1. Omar El Euch & Mathieu Rosenbaum, 2016. "The characteristic function of rough Heston models," Papers 1609.02108, arXiv.org.
    2. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    3. Thibault Jaisson & Mathieu Rosenbaum, 2013. "Limit theorems for nearly unstable Hawkes processes," Papers 1310.2033, arXiv.org, revised Mar 2015.
    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. Omar El Euch & Mathieu Rosenbaum, 2017. "Perfect hedging in rough Heston models," Papers 1703.05049, arXiv.org.
    6. Gallant, A Ronald & Rossi, Peter E & Tauchen, George, 1992. "Stock Prices and Volume," The Review of Financial Studies, Society for Financial Studies, vol. 5(2), pages 199-242.
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    Cited by:

    1. Abi Jaber, Eduardo & El Euch, Omar, 2019. "Markovian structure of the Volterra Heston model," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 63-72.
    2. Bingyan Han & Hoi Ying Wong, 2019. "Merton's portfolio problem under Volterra Heston model," Papers 1905.05371, arXiv.org, revised Nov 2019.
    3. Siow Woon Jeng & Adem Kiliçman, 2021. "On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model," Mathematics, MDPI, vol. 9(22), pages 1-32, November.
    4. repec:hal:wpaper:hal-02265210 is not listed on IDEAS
    5. Eduardo Abi Jaber & Omar El Euch, 2018. "Markovian structure of the Volterra Heston model," Working Papers hal-01716696, HAL.
    6. Martin Keller-Ressel & Martin Larsson & Sergio Pulido, 2018. "Affine Rough Models," Papers 1812.08486, arXiv.org.
    7. Martin Keller-Ressel & Martin Larsson & Sergio Pulido, 2023. "Rough affine models," Post-Print hal-02265210, HAL.
    8. Bingyan Han & Hoi Ying Wong, 2019. "Mean-variance portfolio selection under Volterra Heston model," Papers 1904.12442, arXiv.org, revised Jan 2020.
    9. Jim Gatheral & Radoš Radoičić, 2019. "Rational Approximation Of The Rough Heston Solution," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-19, May.
    10. Eduardo Abi Jaber & Omar El Euch, 2019. "Markovian structure of the Volterra Heston model," Post-Print hal-01716696, HAL.
    11. Siow Woon Jeng & Adem Kiliçman, 2021. "SPX Calibration of Option Approximations under Rough Heston Model," Mathematics, MDPI, vol. 9(21), pages 1-11, October.
    12. Blanka Horvath & Antoine Jacquier & Peter Tankov, 2018. "Volatility options in rough volatility models," Papers 1802.01641, arXiv.org, revised Jan 2019.
    13. Benjamin James Duthie, 2019. "Portfolio optimisation under rough Heston models," Papers 1909.02972, arXiv.org.

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