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Asymptotic behaviour of the fractional Heston model

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  • Hamza Guennoun
  • Antoine Jacquier
  • Patrick Roome
  • Fangwei Shi

Abstract

We consider the fractional Heston model originally proposed by Comte, Coutin and Renault. Inspired by recent ground-breaking work on rough volatility, which showed that models with volatility driven by fractional Brownian motion with short memory allows for better calibration of the volatility surface and more robust estimation of time series of historical volatility, we provide a characterisation of the short- and long-maturity asymptotics of the implied volatility smile. Our analysis reveals that the short-memory property precisely provides a jump-type behaviour of the smile for short maturities, thereby fixing the well-known standard inability of classical stochastic volatility models to fit the short-end of the volatility smile.

Suggested Citation

  • Hamza Guennoun & Antoine Jacquier & Patrick Roome & Fangwei Shi, 2014. "Asymptotic behaviour of the fractional Heston model," Papers 1411.7653, arXiv.org, revised Aug 2017.
    • Handle: RePEc:arx:papers:1411.7653
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    References listed on IDEAS

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    1. Fouque,Jean-Pierre & Papanicolaou,George & Sircar,Ronnie & Sølna,Knut, 2011. "Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives," Cambridge Books, Cambridge University Press, number 9780521843584, October.
    2. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    3. 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.
    4. Martin Forde & Antoine Jacquier, 2011. "The large-maturity smile for the Heston model," Finance and Stochastics, Springer, vol. 15(4), pages 755-780, December.
    5. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    6. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    7. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    8. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    9. Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
    10. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 74(1), pages 3-30, September.
    11. Vilela Mendes, R. & Oliveira, M.J. & Rodrigues, A.M., 2015. "No-arbitrage, leverage and completeness in a fractional volatility model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 470-478.
    12. 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.
    13. F. Comte & L. Coutin & E. Renault, 2012. "Affine fractional stochastic volatility models," Annals of Finance, Springer, vol. 8(2), pages 337-378, May.
    14. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    15. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    16. Antoine Jacquier & Martin Keller-Ressel & Aleksandar Mijatovic, 2011. "Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models," Papers 1108.3998, arXiv.org.
    17. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2014. "Volatility is rough," Papers 1410.3394, arXiv.org.
    18. C. W. J. Granger & Roselyne Joyeux, 1980. "An Introduction To Long‐Memory Time Series Models And Fractional Differencing," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 15-29, January.
    19. Martin Forde & Antoine Jacquier, 2009. "Small-Time Asymptotics For Implied Volatility Under The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 861-876.
    20. Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
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    Cited by:

    1. Omar El Euch & Mathieu Rosenbaum, 2016. "The characteristic function of rough Heston models," Papers 1609.02108, arXiv.org.
    2. Antoine Jacquier & Patrick Roome, 2015. "Black-Scholes in a CEV random environment," Papers 1503.08082, arXiv.org, revised Nov 2017.
    3. Jiro Akahori & Xiaoming Song & Tai-Ho Wang, 2022. "Probability Density of Lognormal Fractional SABR Model," Risks, MDPI, vol. 10(8), pages 1-27, August.
    4. Antoine Jacquier & Fangwei Shi, 2016. "The randomised Heston model," Papers 1608.07158, arXiv.org, revised Dec 2018.
    5. Blanka Horvath & Antoine Jacquier & Aitor Muguruza & Andreas Sojmark, 2017. "Functional central limit theorems for rough volatility," Papers 1711.03078, arXiv.org, revised Nov 2023.
    6. Christian Bayer & Peter K. Friz & Archil Gulisashvili & Blanka Horvath & Benjamin Stemper, 2017. "Short-time near-the-money skew in rough fractional volatility models," Papers 1703.05132, arXiv.org, revised Mar 2018.
    7. 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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