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Heston stochastic vol-of-vol model for joint calibration of VIX and S&P 500 options

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  • J.-P. Fouque
  • Y. F. Saporito

Abstract

A parsimonious generalization of the Heston model is proposed where the volatility-of-volatility is assumed to be stochastic. We follow the perturbation technique of Fouque et al [Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, 2011, Cambridge University Press] to derive a first-order approximation of the price of options on a stock and its volatility index. This approximation is given by Heston’s quasi-closed formula and some of its Greeks. It can be efficiently calculated since it requires to compute only Fourier integrals and the solution of simple ODE systems. We exemplify the calibration of the model with S&P 500 and VIX data.

Suggested Citation

  • J.-P. Fouque & Y. F. Saporito, 2018. "Heston stochastic vol-of-vol model for joint calibration of VIX and S&P 500 options," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 1003-1016, June.
    • Handle: RePEc:taf:quantf:v:18:y:2018:i:6:p:1003-1016
    DOI: 10.1080/14697688.2017.1412493
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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, November.
    2. Jean-Pierre Fouque & George Papanicolaou & Ronnie Sircar & Knut Solna, 2004. "Maturity cycles in implied volatility," Finance and Stochastics, Springer, vol. 8(4), pages 451-477, November.
    3. Jean-Pierre Fouque & Yuri F. Saporito & Jorge P. Zubelli, 2014. "Multiscale Stochastic Volatility Model For Derivatives On Futures," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(07), pages 1-31.
    4. Jan Baldeaux & Alexander Badran, 2014. "Consistent Modelling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(4), pages 299-312, September.
    5. Han, Chuan-Hsiang & Molina, German & Fouque, Jean-Pierre, 2014. "McMC estimation of multiscale stochastic volatility models with applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 103(C), pages 1-11.
    6. Peter Carr & Dilip B. Madan, 2014. "Joint modeling of VIX and SPX options at a single and common maturity with risk management applications," IISE Transactions, Taylor & Francis Journals, vol. 46(11), pages 1125-1131, November.
    7. 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.
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