The Mathematical Form

Graduate work on the VIX begins by naming the functional. The published index is a particular map of a particular option surface. Name the map, or you cannot say what a successor conserves. Versioned I_t is that naming, extended: a density print, a realized clock, or dual F, each a published member.

The risk-neutral second moment

Let Q be a risk-neutral measure consistent with the observed SPX option surface at a fixed expiry T. Let F be the T-forward on the index. Breeden–Litzenberger identifies the risk-neutral density of S_T from the second derivative of the call price with respect to strike. Integrating K^{-2} against OTM option prices recovers the price of the log contract and therefore a model-light estimate of E^Q[∫_0^T (dS_u / S_u)^2] when paths are continuous. That expectation is the variance-swap rate. The VIX is one hundred times the square root of a 30-day interpolation of two such rates.

Discretization choices that are legal facts

Every operational choice in the Cboe booklet is a legal fact because settlement uses those choices, not the integral. The strike grid is the listed SPX grid. The contribution of each strike is ΔK / K². The option price is a midquote for the live index and an opening print for the SOQ. Series with a zero bid are dropped, which endogenously shortens the wing when liquidity withdraws. Only Friday-expiring SPX options inside a 23-to-37 calendar-day window enter the flagship VIX; other tenors have their own windows. Interest-rate inputs are specified by the administrator. These are not “implementation details.” They are the definition.

A replacement methodology that changes the wing rule, the midquote rule, or the expiry window has changed the claim, even if it keeps the 100√(·) display. Counsel should treat methodology amendments the way they treat index transition events in equity derivatives: as potential market-disruption and fallback problems.

Term structure and basis

The cash VIX is a 30-day constant-maturity statistic. VIX futures are claims on the then-prevailing cash VIX at listed expiries. The futures curve is therefore a curve of expected future 30-day variance rates (under a futures-implied measure, with risk premia), not a curve of 60-day or 90-day variance rates. VIX3M is the closer cousin of a three-month variance rate. Confusing the VX curve with the variance term structure is a standard error in both trading books and offering circulars.

In ordinary time the VX curve sits in contango: longer futures print above spot VIX. A long-only short-dated futures product then bleeds through roll. In stress the curve inverts (backwardation), short-dated futures rally toward or through spot, and the same product makes money until the roll normalizes. This path dependence is not a defect of “the VIX.” It is the geometry of a futures curve on a mean-reverting, uninvestable cash index.

Convexity, VVIX, and vol of vol

Because the listed products are written on √variance rather than on variance, a movement in the variance rate does not map linearly into VIX points. Options on VIX exist because the market wants convexity on that already-convex transform. VVIX applies the VIX estimator to the VIX option surface and is therefore a second-moment statistic of the VIX itself—vol of vol in the same compressed sense. A full-distribution view would replace both VIX and VVIX with explicit functionals of the joint law of (S, σ) or with a documented set of moments and tail probabilities.

What is discarded

• Skew and kurtosis of the 30-day risk-neutral law (partly recovered by SKEW, not by VIX).
• Pathwise vol-of-vol and rough-vol structure.
• Overnight versus regular-hours quadratic variation.
• Single-name and sector variance that does not survive index diversification.
• Realized-versus-implied premia; VIX is a Q-measure object, not a forecast of subsequent realized vol.
• Liquidity state of the wing, except insofar as zero bids drop out.

Variance risk premium

The difference between the VIX-style Q-variance rate and subsequent realized variance is the variance risk premium. It is usually positive: option markets price more variance than is later realized. That premium is why short-vol strategies have a long-run drift and why they die in clusters. A legal analysis of “VIX products as hedges” that does not mention this premium is incomplete. A hedge that collects the premium is an insurance underwriting book. A hedge that pays the premium is insurance. Most retail VIX products do neither cleanly, because they hold futures, not variance swaps, and because they roll.