Inversion of convex ordering in the VIX market

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 ${\mathbb E}[\sigma _t^2|S_t] = \sigma _{\mathrm {lv}}^2(t,S_t) $E[σt2|St]=σlv2(t,St). 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 $\sigma _{\mathrm {lv}}^2(t,S_t) $σlv2(t,St) is smaller than the instantaneous variance $\sigma _t^2 $σt2 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 ${\rm VIX}_T^2 $VIXT2 in the market local volatility model is larger than the market-implied distribution of ${\rm VIX}_T^2 $VIXT2 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.