Bounds for VIX Futures Given S&P 500 Smiles

24 Pages Posted: 20 Sep 2016 Last revised: 23 Jun 2017

See all articles by Julien Guyon

Julien Guyon

Ecole des Ponts ParisTech

Romain Menegaux

Bloomberg L.P.

Marcel Nutz

Columbia University

Date Written: September 19, 2016


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, 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.

Keywords: VIX Futures, Price bounds, Superreplication, Subreplication, Model-free, Robust hedging, S&P 500 smile

JEL Classification: G12

Suggested Citation

Guyon, Julien and Menegaux, Romain and Nutz, Marcel, Bounds for VIX Futures Given S&P 500 Smiles (September 19, 2016). Available at SSRN: or

Julien Guyon (Contact Author)

Ecole des Ponts ParisTech ( email )


Romain Menegaux

Bloomberg L.P. ( email )

731 Lexington Avenue
New York, NY 10022
United States

Marcel Nutz

Columbia University ( email )

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