Quoted Vol

Raw option chains — every quoted strike, bid and ask, untouched. Each smile is read out of them in your browser and reduced to a scale and up to five shape numbers, as many as its strikes can actually identify; the term structure is carried by two factors running on activity time. Nothing is simulated and nothing heavy is fitted, and the whole reading lands in a few hundred milliseconds.

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Reading

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State

Explore by hand

1 · VIX futures?

The curve, and the two factors that build it

 

2 · VIX options?

Implied vol by strike

 

3 · SPX options?

Implied vol by strike

 

4 · The curve of scales?

What the model actually carries — one dot per marginal read

 

5 · The SPX–VIX basis?

Two ways of buying the same thing, and what separates them

 

6 · Priced crisis intensity?

λ(t) — regime-jump arrival rate read daily from long-dated smiles

 

Found this useful — or wrong? Every number here is measured against real quotes, so a counter-measurement, a maturity where it breaks, or a reference worth reading is the most useful thing you can leave.

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Where this comes from

Almost none of the machinery is new, and it is worth being plain about which part is which. Every brick below is standard; what is less usual is the assembly — a principal-component smile carried by a two-factor forward curve, with the activity clock governing both the reversion and the accumulation of shocks — and the choice to read everything daily rather than calibrate constants.

Breeden & Litzenberger (1978) The risk-neutral density as the second strike-derivative of the call price. Every marginal here starts there.
Skiadopoulos, Hodges & Clewlow (1999) · Cont & da Fonseca (2002) Principal components of implied volatility surfaces, and the finding that three of them — level, slope, curvature — carry more than 95%. Chart 3's shape numbers are exactly these.
Gatheral & Jacquier (2014) Arbitrage-free SVI volatility surfaces. The surface written as ATM total variance times a shape function — the structure this model converges to, and the reference that does impose no-arbitrage where this one does not.
Bergomi (2005, 2008) Smile Dynamics II and III. The two-factor forward-variance curve, with its short factor and its long anchor. Chart 1 is its decomposition, laid on marginals rather than on an instantaneous volatility.
Demeterfi, Derman, Kamal & Zou (1999) The log-contract, and the fact that the VIX is one over thirty days. This is why the futures curve costs no extra model.
Lee (2004) The moment formula for implied volatility at extreme strikes. Wings may grow like the square root of log-moneyness and no faster — extrapolating the edge slope in a straight line put the SPX forward variance 22% above the VIX future at one year.
Clark (1973) · Ané & Geman (2000) · Carr, Geman, Madan & Yor (2003) Subordination: price time runs on market activity, not the calendar. Here the same clock is applied to the factors' reversion and to their shocks.
Bakshi, Kapadia & Madan (2003) Model-free cumulants from option prices. Chart 5's λ(t) is read from the third one at long maturities.
Bacry, Delour & Muzy (2001) The multifractal random walk. Its intermittency coefficient sits at 0.02–0.05 on equities; measured independently on these VIX densities it comes out at 0.039.
Schönbucher (1999) · Brace, Goldys, Klebaner & Womersley (2001) Market models of implied volatility — the family this belongs to, and the source of its known weakness, below.
Papanicolaou (2021) Consistent time-homogeneous diffusion models. A model-free result on what the VIX right tail must be paid for out of.