SIAM Journal on Financial Mathematics, Volume 11, Issue 1, Page SC1-SC13, January 2020.
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 our 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 S&P 500 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.

中文翻译：

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简短的交流：凸顺序的反转：局部波动不会使VIX期货的价格最大化

SIAM金融数学杂志，第11卷，第1期，SC1-SC13页，2020年1月。
经常有人指出，在根据香草校准的连续随机波动率模型中，通过Dupire局部波动率模型使VIX期货的价格最大化。在本文中，我们证明了这种说法是不正确的：我们建立了一个连续的随机波动率模型，在该模型中，VIX的期货价格远比其相关的局部波动率模型昂贵。更一般而言，在我们的模型中，平方VIX上的严格凸收益要比相关的本地波动率模型便宜得多。当从瞬时方差转移到平方VIX时，这对应于局部方差和随机方差之间的凸顺序的反转，因为在局部波动率模型中瞬时方差的凸现收益总是更便宜。因此，我们证明了凸序的这种反演，在S＆P 500市场中观察到的短期VIX期限可以通过连续随机波动率模型产生。我们还证明了该模型可以扩展，以便如市场数据所建议的那样，可以保留凸序以实现较长的到期期限。