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

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  • Jan Baldeaux
  • Alexander Badran

Abstract

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, arXiv.org, revised Aug 2012.
    • Handle: RePEc:arx:papers:1203.5903
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. Alexander Badran & Beniamin Goldys, 2015. "A Market Model for VIX Futures," Papers 1504.00428, arXiv.org.
    3. Elisa Al`os & David Garc'ia-Lorite & Aitor Muguruza, 2018. "On smile properties of volatility derivatives and exotic products: understanding the VIX skew," Papers 1808.03610, arXiv.org.
    4. Andrew Papanicolaou, 2021. "Extreme-Strike Comparisons and Structural Bounds for SPX and VIX Options," Papers 2101.00299, arXiv.org, revised Mar 2021.
    5. Andrea Barletta & Elisa Nicolato & Stefano Pagliarani, 2019. "The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 928-966, July.
    6. Ivan Guo & Gregoire Loeper, 2018. "Pricing Bounds for Volatility Derivatives via Duality and Least Squares Monte Carlo," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 598-617, November.
    7. Wendong Zheng & Pingping Zeng, 2015. "Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model," Papers 1504.08136, arXiv.org.
    8. Anupam Dutta & Elie Bouri & David Roubaud, 2021. "Modelling the volatility of crude oil returns: Jumps and volatility forecasts," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(1), pages 889-897, January.
    9. 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.
    10. Andrew Papanicolaou, 2018. "Consistent Time-Homogeneous Modeling of SPX and VIX Derivatives," Papers 1812.05859, arXiv.org, revised Mar 2022.
    11. Antoine Jacquier & Aitor Muguruza & Alexandre Pannier, 2021. "Rough multifactor volatility for SPX and VIX options," Papers 2112.14310, arXiv.org, revised Nov 2023.
    12. 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.
    13. Ma, Jingtang & Li, Wenyuan & Han, Xu, 2015. "Stochastic lattice models for valuation of volatility options," Economic Modelling, Elsevier, vol. 47(C), pages 93-104.
    14. Ben Hambly & Juozas Vaicenavicius, 2015. "The 3/2 Model As A Stochastic Volatility Approximation For A Large-Basket Price-Weighted Index," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(06), pages 1-25.
    15. 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.
    16. Julien Guyon, 2020. "Inversion of convex ordering in the VIX market," Quantitative Finance, Taylor & Francis Journals, vol. 20(10), pages 1597-1623, October.
    17. Daniel Guterding, 2020. "Inventory effects on the price dynamics of VSTOXX futures quantified via machine learning," Papers 2002.08207, arXiv.org.

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    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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