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Joint Implied Willow Tree: An Approach for Joint S&P 500/VIX Calibration

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  • Bing Dong
  • Wei Xu
  • Zhenyu Cui

Abstract

Since the inception of Volatility Index (VIX) options trading, academic literature has persistently sought accurate methods for jointly calibrating the prices of the S&P 500 index (SPX) and VIX options. This study introduces a novel nonparametric approach, called the joint implied willow tree (JIWT) method, aimed at resolving this joint calibration challenge. The resulting willow tree adheres to the martingale constraint for the SPX and ensures that the VIX is derived as the implied volatility of a 30‐day log contract on the SPX. A notable advantage of our method is its ability to recover not only the unconditional probabilities for a fixed maturity but also the conditional probabilities across different maturities. Consequently, we reconstruct the entire term structure of the SPX, aligning it with market information from both SPX and VIX options. Numerical and empirical analyses demonstrate that the JIWT method excels in accurately capturing the volatility smile of SPX and VIX across various maturities.

Suggested Citation

  • Bing Dong & Wei Xu & Zhenyu Cui, 2025. "Joint Implied Willow Tree: An Approach for Joint S&P 500/VIX Calibration," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 45(6), pages 547-568, June.
    • Handle: RePEc:wly:jfutmk:v:45:y:2025:i:6:p:547-568
    DOI: 10.1002/fut.22572
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    References listed on IDEAS

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    1. Stéphane Goutte & Amine Ismail & Huyên Pham, 2017. "Regime-switching stochastic volatility model: estimation and calibration to VIX options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(1), pages 38-75, January.
    2. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    3. Bardgett, Chris & Gourier, Elise & Leippold, Markus, 2019. "Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets," Journal of Financial Economics, Elsevier, vol. 131(3), pages 593-618.
    4. Julien Guyon & Jordan Lekeufack, 2023. "Volatility is (mostly) path-dependent," Quantitative Finance, Taylor & Francis Journals, vol. 23(9), pages 1221-1258, September.
    5. Hadrien de March & Pierre Henry-Labordere, 2019. "Building Arbitrage-Free Implied Volatility: Sinkhorn'S Algorithm And Variants," Working Papers hal-02011533, HAL.
    6. Stéphane Goutte & Amine Ismail & Huyên Pham, 2017. "Regime-switching Stochastic Volatility Model : Estimation and Calibration to VIX options," Working Papers hal-01212018, HAL.
    7. 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.
    8. Pacati, Claudio & Pompa, Gabriele & Renò, Roberto, 2018. "Smiling twice: The Heston++ model," Journal of Banking & Finance, Elsevier, vol. 96(C), pages 185-206.
    9. Julien Guyon & Jordan Lekeufack, 2023. "Volatility is (mostly) path-dependent," Post-Print hal-04373380, HAL.
    10. Bjørn Eraker & Aoxiang Yang, 2022. "The Price of Higher Order Catastrophe Insurance: The Case of VIX Options," Journal of Finance, American Finance Association, vol. 77(6), pages 3289-3337, December.
    11. Song, Zhaogang & Xiu, Dacheng, 2016. "A tale of two option markets: Pricing kernels and volatility risk," Journal of Econometrics, Elsevier, vol. 190(1), pages 176-196.
    12. Duan, Jin-Chuan & Yeh, Chung-Ying, 2010. "Jump and volatility risk premiums implied by VIX," Journal of Economic Dynamics and Control, Elsevier, vol. 34(11), pages 2232-2244, November.
    13. Changfu Ma & Wei Xu & Yue Kuen Kwok, 2020. "Willow tree algorithms for pricing VIX derivatives under stochastic volatility models," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 1-28, March.
    14. Wei Xu & Zhiwu Hong & Chenxiang Qin, 2013. "A new sampling strategy willow tree method with application to path-dependent option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 861-872, May.
    15. 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.
    16. Kanniainen, Juho & Lin, Binghuan & Yang, Hanxue, 2014. "Estimating and using GARCH models with VIX data for option valuation," Journal of Banking & Finance, Elsevier, vol. 43(C), pages 200-211.
    17. Kaeck, Andreas & Alexander, Carol, 2012. "Volatility dynamics for the S&P 500: Further evidence from non-affine, multi-factor jump diffusions," Journal of Banking & Finance, Elsevier, vol. 36(11), pages 3110-3121.
    18. Lin, Yueh-Neng, 2013. "VIX option pricing and CBOE VIX Term Structure: A new methodology for volatility derivatives valuation," Journal of Banking & Finance, Elsevier, vol. 37(11), pages 4432-4446.
    19. Wang, Qi & Wang, Zerong, 2020. "VIX valuation and its futures pricing through a generalized affine realized volatility model with hidden components and jump," Journal of Banking & Finance, Elsevier, vol. 116(C).
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