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Bounds for VIX futures given S&P 500 smiles

Author

Listed:
  • Julien Guyon

    (Bloomberg L.P.)

  • Romain Menegaux

    (Bloomberg L.P.)

  • Marcel Nutz

    (Columbia University)

Abstract

We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a family of functionally generated portfolios which often improves the classical bounds while still being tractable; more precisely, they are determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

Suggested Citation

  • Julien Guyon & Romain Menegaux & Marcel Nutz, 2017. "Bounds for VIX futures given S&P 500 smiles," Finance and Stochastics, Springer, vol. 21(3), pages 593-630, July.
    • Handle: RePEc:spr:finsto:v:21:y:2017:i:3:d:10.1007_s00780-017-0334-6
    DOI: 10.1007/s00780-017-0334-6
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Sebastian Herrmann & Florian Stebegg, 2017. "Robust Pricing and Hedging around the Globe," Papers 1707.08545, arXiv.org, revised Apr 2019.
    2. Mathias Beiglbock & Benjamin Jourdain & William Margheriti & Gudmund Pammer, 2021. "Stability of the Weak Martingale Optimal Transport Problem," Papers 2109.06322, arXiv.org, revised Apr 2022.
    3. Benjamin Jourdain & Gudmund Pammer, 2023. "An extension of martingale transport and stability in robust finance," Papers 2304.09551, arXiv.org.
    4. Johannes Muhle-Karbe & Marcel Nutz, 2018. "A risk-neutral equilibrium leading to uncertain volatility pricing," Finance and Stochastics, Springer, vol. 22(2), pages 281-295, April.
    5. Antoine Jacquier & Aitor Muguruza & Alexandre Pannier, 2021. "Rough multifactor volatility for SPX and VIX options," Papers 2112.14310, arXiv.org, revised Nov 2023.
    6. Julio Backhoff-Veraguas & Gudmund Pammer, 2019. "Stability of martingale optimal transport and weak optimal transport," Papers 1904.04171, arXiv.org, revised Dec 2020.

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    More about this item

    Keywords

    VIX futures; Price bounds; Model-free pricing; Robust hedging;
    All these keywords.

    JEL classification:

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

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