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Heston Stochastic Vol-of-Vol Model for Joint Calibration of VIX and S&P 500 Options

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  • Jean-Pierre Fouque
  • Yuri 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 (2011, CUP) 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 very 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

  • Jean-Pierre Fouque & Yuri F. Saporito, 2017. "Heston Stochastic Vol-of-Vol Model for Joint Calibration of VIX and S&P 500 Options," Papers 1706.00873, arXiv.org.
  • Handle: RePEc:arx:papers:1706.00873
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    References listed on IDEAS

    as
    1. 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.
    2. repec:cup:cbooks:9780521843584 is not listed on IDEAS
    3. 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.
    4. repec:wsi:ijtafx:v:17:y:2014:i:07:n:s0219024914500435 is not listed on IDEAS
    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. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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