Inversion of Convex Ordering in the VIX Market

Quantitative Finance, Forthcoming

32 Pages Posted: 1 Jan 2020 Last revised: 2 Apr 2020

See all articles by Julien Guyon

Julien Guyon

Bloomberg L.P.; Columbia University - Department of Mathematics; New York University - Courant Institute of Mathematical Sciences

Date Written: December 14, 2019


We investigate conditions for the existence of a continuous model on the S&P 500 index (SPX) that jointly calibrates to a full surface of SPX implied volatilities and to the VIX smiles. We present a novel approach based on the SPX smile calibration condition (the fact that the conditional expectation of the instantaneous variance given the spot equals the market local variance). In the limiting case of instantaneous VIX, a novel application of martingale transport to finance shows that such model exists if and only if, for each time t, the local variance is smaller than the instantaneous variance in convex order. The real case of a 30 day VIX is more involved, as averaging over 30 days and projecting onto a filtration can undo convex ordering.

We show that in usual market conditions, and for reasonable smile extrapolations, the distribution of the VIX squared in the market local volatility model is larger than the market-implied distribution of the VIX squared in convex order for short maturities T, and that the two distributions are not rankable in convex order for intermediate maturities. In particular, a necessary condition for continuous models to jointly calibrate to the SPX and VIX markets is the inversion of convex ordering property: the fact that, even though associated local variances are smaller than instantaneous variances in convex order, the VIX squared is larger in convex order in the associated local volatility model than in the original model for short maturities. We argue and numerically demonstrate that, when the (typically negative) spot-vol correlation is large enough in absolute value, (a) traditional stochastic volatility models with large mean reversion, and (b) rough volatility models with small Hurst exponent, satisfy the inversion of convex ordering property, and more generally can reproduce the market term-structure of convex ordering of the local and stochastic squared VIX.

Keywords: VIX, convex order, inversion of convex ordering, martingale transport, local volatility, stochastic volatility, mean reversion, rough volatility, smile calibration

JEL Classification: G13

Suggested Citation

Guyon, Julien, Inversion of Convex Ordering in the VIX Market (December 14, 2019). Quantitative Finance, Forthcoming, Available at SSRN: or

Julien Guyon (Contact Author)

Bloomberg L.P. ( email )

731 Lexington Avenue
New York, NY 10022
United States

Columbia University - Department of Mathematics ( email )

3022 Broadway
New York, NY 10027
United States

New York University - Courant Institute of Mathematical Sciences ( email )

New York University
New York, NY 10012
United States

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