Full-Distribution Evolutionary Replacement

Evolutionary replacement means specifying a successor primitive that inherits the VIX complex’s three jobs — public coordinate, tradable settlement object, and legal pointer — while carrying the mass of the surface the second-moment index compresses away. Chapter VIII writes those jobs for dated strips. Chapter IX writes them for a perpetual as versioned I_t. If either job-set is omitted, the market keeps the VIX for listings and treats the rest as research.

Why replacement is on the table

Three pressures now point the same way. First, the listed surface is richer than it was in 2003: weeklys, 0DTE, finer strikes, a VIX-options surface, and a family of tenor clones. The flagship 30-day average is no longer the unique sufficient summary of what the exchange already lists. Second, risk systems and capital rules have become distribution-aware (expected shortfall, stress scenarios, tail add-ons) while the public volatility coordinate remains a variance-swap square root. Third, packaged products demonstrated that a single compressed number plus a futures roll plus an indenture is a brittle retail object. The scientific response is a better estimator. The legal response is a better bundle.

What “full distribution” means here

Full-distribution does not mean publishing a million-bin histogram every fifteen seconds. It means replacing a single functional Φ_2 (the second moment, or a discrete variance strip) with a documented vector of functionals Φ = (Φ_2, Φ_skew, Φ_tail, Φ_term, …) of the same option surface, each with a replication or extraction argument, each with a settlement rule. The state-price density is the generating object. The published vector is a set of coordinates on that object. SKEW is already one such coordinate. VIX9D and VIX3M are tenor coordinates on the same second-moment functional. VVIX is a second-moment coordinate on a different surface. The replacement thesis is that these should be designed as one family with one settlement philosophy, not as a menagerie of marketing tickers.

Design principles

1. Auditable booklet. Every functional needs a public methodology versioned like Cboe’s February 2026 VIX booklet. Unpublished “AI vol” is not a legal object.
2. Tradable settlement. If the successor cannot be challenged in an opening auction or an equivalent executable mechanism, it will not support listed futures of institutional size.
3. Moment hygiene. Publish variance and volatility separately. Do not hide Jensen in a square root and then list products on the square root.
4. Tail functionals with named g. Expected shortfall of the risk-neutral law, probability of a two-sigma decline, or a log-unrecovered jump term should be separate tickers or separate fields, not adjectives in a press release.
5. Term structure as a first-class object. A curve of Φ_2(T) and Φ_g(T) is the primitive; a 30-day point is a slice.
6. Wrapper last. Do not issue retail notes on a new functional until the wholesale futures book has depth, position accountability, and a documented issuer-hedge channel.
7. Jurisdiction declared in the first filing. Decide whether the flagship listed claim is a futures contract, a securities option, or both, and write the dual-commission sentence before the launch press release.
8. License path. Either migrate inside the incumbent administrator or budget a LIBOR-style benchmark war. There is no third way that still uses the word VIX.

Candidate successors

Several research and product lines already approximate parts of this program. Listed variance futures restore Φ_2 without the square-root display. SKEW and related tail indexes restore a shape coordinate. Model-free implied-distribution papers extract q(K, T) daily and price arbitrary g. Volatility-of-volatility indexes restore a second surface. Rough-vol and path-dependent vol contracts attempt to restore dynamics rather than the terminal law. Conviction or lattice indexes that map a surface onto instrument-specific confidence bands are attempts to restore decision functionals—g that is not a moment but a policy.

The legal ranking of these candidates is not the same as the econometric ranking. A functional that is estimable only by a proprietary neural net will lose a methodology contest at a DCM. A functional that requires the entire wing will fail in the exact crises when the wing’s bid is zero. A functional that cannot be hedged in listed SPX options will not attract the dealer balance sheet that makes a futures book tight. The feasible successor is therefore conservative: a published vector of strips, each a piecewise-linear g on the listed grid, each settled by an auction in the same options that enter the strip.

Full-distribution claims that should not be listed

Not every functional of q(K, T) should become a futures underlier. High-order moments are unstable on a truncated strike grid. Functionals that switch definition when liquidity dries up will produce settlement disputes. Functionals that embed a trading policy (leverage caps, “conviction,” regime labels) mix an index with an advisor and will be pulled toward investment-company or CTA analysis. The book’s advice is severe: listed underliers should be mechanical strips. Policy functionals can be published as research indexes or as advisory signals; they should not silently become the settlement price of a DCM contract.

Transitional law

ISDA equity-derivatives definitions, futures rulebooks, and note indentures need fallback language now, not after a methodology surprise. A sober fallback stack is: (1) administrator-announced successor in the same family; (2) specified alternate tenor or alternate functional already published; (3) dealer poll only as last resort, and never as the first line of a retail note. LIBOR taught the cost of “polled expert judgment” as a daily underlier. Volatility should not relearn it.