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Local volatility of volatility for the VIX market

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  • Gabriel Drimus
  • Walter Farkas

Abstract

Following a trend of sustained and accelerated growth, the VIX futures and options market has become a closely followed, active and liquid market. The standard stochastic volatility models—which focus on the modeling of instantaneous variance—are unable to fit the entire term structure of VIX futures as well as the entire VIX options surface. In contrast, we propose to model directly the VIX index, in a mean-reverting local volatility-of-volatility model, which will provide a global fit to the VIX market. We then show how to construct the local volatility-of-volatility surface by adapting the ideas in Carr (Local variance gamma. Bloomberg Quant Research, New York, 2008 ) and Andreasen and Huge (Risk Mag 76–79, 2011 ) to a mean-reverting process. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Gabriel Drimus & Walter Farkas, 2013. "Local volatility of volatility for the VIX market," Review of Derivatives Research, Springer, vol. 16(3), pages 267-293, October.
    • Handle: RePEc:kap:revdev:v:16:y:2013:i:3:p:267-293
    DOI: 10.1007/s11147-012-9086-9
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    References listed on IDEAS

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    1. Leif Andersen & Jesper Andreasen, 2000. "Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing," Review of Derivatives Research, Springer, vol. 4(3), pages 231-262, October.
    2. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    3. 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.
    4. Matthias Fengler, 2009. "Arbitrage-free smoothing of the implied volatility surface," Quantitative Finance, Taylor & Francis Journals, vol. 9(4), pages 417-428.
    5. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    6. Ernst Eberlein & Dilip Madan, 2009. "Sato processes and the valuation of structured products," Quantitative Finance, Taylor & Francis Journals, vol. 9(1), pages 27-42.
    7. 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.
    8. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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    Citations

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    Cited by:

    1. 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.
    2. Martino Grasselli & Andrea Mazzoran & Andrea Pallavicini, 2024. "A general framework for a joint calibration of VIX and VXX options," Annals of Operations Research, Springer, vol. 336(1), pages 3-26, May.
    3. Felix Brinkmann & Olaf Korn, 2018. "Risk-adjusted option-implied moments," Review of Derivatives Research, Springer, vol. 21(2), pages 149-173, July.
    4. Emanuele Nastasi & Andrea Pallavicini & Giulio Sartorelli, 2020. "Smile Modeling In Commodity Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(03), pages 1-28, May.
    5. Carol Alexander & Julia Kapraun & Dimitris Korovilas, 2015. "Trading and Investing in Volatility Products," Financial Markets, Institutions & Instruments, John Wiley & Sons, vol. 24(4), pages 313-347, November.
    6. Ying-Li Wang & Cheng-Long Xu & Ping He, 2023. "A Markovian empirical model for the VIX index and the pricing of the corresponding derivatives," Papers 2309.08175, arXiv.org.
    7. Andrew Papanicolaou, 2022. "Consistent time‐homogeneous modeling of SPX and VIX derivatives," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 907-940, July.
    8. Bo Jing & Shenghong Li & Yong Ma, 2020. "Pricing VIX options with volatility clustering," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 40(6), pages 928-944, June.

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    More about this item

    Keywords

    VIX futures; VIX options; Volatility of volatility ; Volatility derivatives; G12; G13;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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