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Consistent Modeling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model


  • Jan Baldeaux
  • Alexander Badran


The paper demonstrates that a pure-diffusion 3/2 model is able to capture the observed upward-sloping implied volatility skew in VIX options. This observation contradicts a common perception in the literature that jumps are required for the consistent modelling of equity and VIX derivatives. The pure-diffusion model, however, struggles to reproduce the smile in the implied volatilities of short-term index options. One remedy to this problem is to augment the model by introducing jumps in the index. The resulting 3/2 plus jumps model turns out to be as tractable as its pure-diffusion counterpart when it comes to pricing equity, realized variance and VIX derivatives, but accurately captures the smile in implied volatilities of short-term index options.

Suggested Citation

  • Jan Baldeaux & Alexander Badran, 2012. "Consistent Modeling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model," Papers 1203.5903,, revised Aug 2012.
  • Handle: RePEc:arx:papers:1203.5903

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    References listed on IDEAS

    1. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    2. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    3. Peter Carr & Jian Sun, 2007. "A new approach for option pricing under stochastic volatility," Review of Derivatives Research, Springer, vol. 10(2), pages 87-150, May.
    4. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, June.
    5. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    6. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    7. Alexey Medvedev & Olivier Scaillet, "undated". "Approximation and Calibration of Short-Term Implied Volatilities under Jump-Diffusion Stochastic Volatility," Swiss Finance Institute Research Paper Series 06-08, Swiss Finance Institute, revised Jan 2006.
    8. Peter Carr & Hélyette Geman & Dilip Madan & Marc Yor, 2005. "Pricing options on realized variance," Finance and Stochastics, Springer, vol. 9(4), pages 453-475, October.
    9. Andrey Itkin & Peter Carr, 2010. "Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case," Review of Derivatives Research, Springer, vol. 13(2), pages 141-176, July.
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    Cited by:

    1. Wei Lin & Shenghong Li & Shane Chern, 2017. "Pricing VIX Derivatives With Free Stochastic Volatility Model," Papers 1703.06020,
    2. Ma, Jingtang & Li, Wenyuan & Han, Xu, 2015. "Stochastic lattice models for valuation of volatility options," Economic Modelling, Elsevier, vol. 47(C), pages 93-104.
    3. Thomas Kokholm & Martin Stisen, 2015. "Joint pricing of VIX and SPX options with stochastic volatility and jump models," Journal of Risk Finance, Emerald Group Publishing, vol. 16(1), pages 27-48, January.
    4. repec:ids:ijbder:v:3:y:2017:i:2:p:153-175 is not listed on IDEAS
    5. Alexander Badran & Beniamin Goldys, 2015. "A Market Model for VIX Futures," Papers 1504.00428,
    6. Andrew Papanicolaou & Ronnie Sircar, 2014. "A regime-switching Heston model for VIX and S&P 500 implied volatilities," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1811-1827, October.
    7. Ivan Guo & Gregoire Loeper, 2016. "Pricing Bounds for VIX Derivatives via Least Squares Monte Carlo," Papers 1611.00464,
    8. repec:spr:joptap:v::y::i::d:10.1007_s10957-017-1168-2 is not listed on IDEAS
    9. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2017. "Explicit Implied Volatilities For Multifactor Local-Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 926-960, July.
    10. Wei Lin & Shenghong Li & Xingguo Luo & Shane Chern, 2015. "Consistent Pricing of VIX and Equity Derivatives with the 4/2 Stochastic Volatility Plus Jumps Model," Papers 1510.01172,, revised Nov 2015.
    11. Wendong Zheng & Pingping Zeng, 2015. "Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model," Papers 1504.08136,
    12. Li, Jing & Li, Lingfei & Zhang, Gongqiu, 2017. "Pure jump models for pricing and hedging VIX derivatives," Journal of Economic Dynamics and Control, Elsevier, vol. 74(C), pages 28-55.
    13. Stéphane Goutte & Amine Ismail & Huyên Pham, 2017. "Regime-switching Stochastic Volatility Model : Estimation and Calibration to VIX options," Working Papers hal-01212018, HAL.
    14. Chi Hung Yuen & Wendong Zheng & Yue Kuen Kwok, 2015. "Pricing Exotic Discrete Variance Swaps under the 3/2-Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(5), pages 421-449, November.
    15. 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,

    More about this item

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G1 - Financial Economics - - General Financial Markets
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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