Inversion of Convex Ordering: Local Volatility Does Not Maximize the Price of VIX Futures

SIAM Journal on Financial Mathematics, 11-1 (2020), pp. SC-1-SC-13

10 Pages Posted: 22 Oct 2019 Last revised: 2 Mar 2020

See all articles by B. Acciaio

B. Acciaio

affiliation not provided to SSRN

Julien Guyon

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

Date Written: October 11, 2019

Abstract

It has often been stated that, within the class of continuous stochastic volatility models calibrated to vanillas, the price of a VIX future is maximized by the Dupire local volatility model. In this article we prove that this statement is incorrect: we build a continuous stochastic volatility model in which a VIX future is strictly more expensive than in its associated local volatility model. More generally, in this model, strictly convex payoffs on a squared VIX are strictly cheaper than in the associated local volatility model. This corresponds to an inversion of convex ordering between local and stochastic variances, when moving from instantaneous variances to squared VIX, as convex payoffs on instantaneous variances are always cheaper in the local volatility model. We thus prove that this inversion of convex ordering, which is observed in the SPX market for short VIX maturities, can be produced by a continuous stochastic volatility model. We also prove that the model can be extended so that, as suggested by market data, the convex ordering is preserved for long maturities.

Keywords: VIX, VIX futures, stochastic volatility, local volatility, convex order, inversion of convex ordering

JEL Classification: G13

Suggested Citation

Acciaio, B. and Guyon, Julien, Inversion of Convex Ordering: Local Volatility Does Not Maximize the Price of VIX Futures (October 11, 2019). SIAM Journal on Financial Mathematics, 11-1 (2020), pp. SC-1-SC-13, Available at SSRN: https://ssrn.com/abstract=3468537 or http://dx.doi.org/10.2139/ssrn.3468537

B. Acciaio

affiliation not provided to SSRN

No Address Available

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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