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Perfect hedging in rough Heston models

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  • Omar El Euch
  • Mathieu Rosenbaum

Abstract

Rough volatility models are known to reproduce the behavior of historical volatility data while at the same time fitting the volatility surface remarkably well, with very few parameters. However, managing the risks of derivatives under rough volatility can be intricate since the dynamics involve fractional Brownian motion. We show in this paper that surprisingly enough, explicit hedging strategies can be obtained in the case of rough Heston models. The replicating portfolios contain the underlying asset and the forward variance curve, and lead to perfect hedging (at least theoretically). From a probabilistic point of view, our study enables us to disentangle the infinite-dimensional Markovian structure associated to rough volatility models.

Suggested Citation

  • Omar El Euch & Mathieu Rosenbaum, 2017. "Perfect hedging in rough Heston models," Papers 1703.05049, arXiv.org.
    • Handle: RePEc:arx:papers:1703.05049
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    References listed on IDEAS

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    1. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2015. "Hybrid scheme for Brownian semistationary processes," CREATES Research Papers 2015-43, Department of Economics and Business Economics, Aarhus University.
    2. Omar El Euch & Mathieu Rosenbaum, 2016. "The characteristic function of rough Heston models," Papers 1609.02108, arXiv.org.
    3. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2014. "Volatility is rough," Papers 1410.3394, arXiv.org.
    4. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
    5. 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.
    6. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    7. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    8. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    9. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Post-Print hal-01313994, HAL.
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    Citations

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    Cited by:

    1. Jim Gatheral & Martin Keller-Ressel, 2018. "Affine forward variance models," Papers 1801.06416, arXiv.org, revised Oct 2018.
    2. Mesias Alfeus & Ludger Overbeck & Erik Schlögl, 2019. "Regime switching rough Heston model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(5), pages 538-552, May.
    3. Eduardo Abi Jaber & Omar El Euch, 2018. "Multi-factor approximation of rough volatility models," Papers 1801.10359, arXiv.org, revised Apr 2018.
    4. Christian Bayer & Peter K. Friz & Paul Gassiat & Joerg Martin & Benjamin Stemper, 2017. "A regularity structure for rough volatility," Papers 1710.07481, arXiv.org.
    5. repec:uts:finphd:41 is not listed on IDEAS
    6. Eduardo Abi Jaber & Omar El Euch, 2018. "Multi-factor approximation of rough volatility models," Working Papers hal-01697117, HAL.
    7. Mesias Alfeus, 2019. "Stochastic Modelling of New Phenomena in Financial Markets," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2019, January-A.
    8. Eduardo Abi Jaber & Omar El Euch, 2019. "Multi-factor approximation of rough volatility models," Post-Print hal-01697117, HAL.
    9. Paul Jusselin & Mathieu Rosenbaum, 2018. "No-arbitrage implies power-law market impact and rough volatility," Papers 1805.07134, arXiv.org.
    10. Eduardo Abi Jaber & Omar El Euch, 2018. "Markovian structure of the Volterra Heston model," Working Papers hal-01716696, HAL.
    11. Eduardo Abi Jaber & Omar El Euch, 2019. "Markovian structure of the Volterra Heston model," Post-Print hal-01716696, HAL.
    12. Blanka Horvath & Antoine Jacquier & Peter Tankov, 2018. "Volatility options in rough volatility models," Papers 1802.01641, arXiv.org, revised Jan 2019.

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