SPX implied-volatility surface

25Δ risk reversal by expiry, every stored session

RR25 = 25Δ call vol − 25Δ put vol of the SVI-cal surface per listed expiry: 2026-10-28 solid, the other stored sessions grey, the session chosen on the comparison tabs dashed. A wing extrapolated beyond the quoted strikes leaves a gap.

SVI-cal surface, listed expiries; 2026-10-28 solid, the others grey, the session chosen under “Against an earlier session” dashed. A gap is a wing extrapolated beyond the quoted strikes.

The stored sessions are irregular (the surface report runs on sessions with a stored chain), so a change spans the gap shown, not a day. “—”: a session without that tenor among its listed expiries (never extrapolated), or an RR / BF with a wing beyond the quoted strikes. The tabs also set the dashed lines of the ATM and RR25 charts.

ATM implied vol per quoted expiry, from the mids at the parity forward and from the eSSVI and SVI-cal surfaces, and the forward ATM vol between the constant maturities 1M, 2M, 3M, 6M and 1Y as dotted steps; the dashed line is the SVI-cal ATM of the earlier session chosen on the comparison tabs, named with its date.

Forward ATM vol of the SVI-cal surface between its constant maturities, √((w₂ − w₁)/(T₂ − T₁)) with w = ATMF²·T (dotted steps): 1M → 2M 14.77% · 2M → 3M 14.98% · 3M → 6M 16.41% · 6M → 1Y 17.75%. Each pair spans at least 7 days, so no AM/PM pair of one date is a forward period.

Grey lines: the 5th, 25th, median, 75th and 95th percentiles of SPX realised vol RV_h = 100·√(252/h · Σ r²) over every window of h = 10, 21, 42, 63, 126 and 252 sessions (close-to-close log returns, zero mean, the site’s RV21 convention), in the sample the tab picks: since 2010 2010-01-04 → 2026-10-28, last 3 years 2023-10-30 → 2026-10-28. Every sample ends on the surface’s session, so no later close enters. An expiry of T calendar days sits at T·252/365 sessions. The windows overlap (one per session), so neighbouring windows share most of their returns: at 252 sessions the last 3 years hold about three independent years. Since 2010 includes 2020; the last 3 years does not.

The listed expiry nearest 30 days, 2026-11-27 (30 d), has an SVI-cal ATMF of 13.44% (the 1M tile’s 13.40% is the surface interpolated to exactly 30 days), above 57% of the 20-session RV windows since 2010 and 65% of those in the last 3 years. The 21-session cone since 2010 runs 6.5–29.0% (5th–95th percentile), median 12.4%; the 21 sessions to 2026-10-28 realised 9.7%.

Implied vol carries the variance risk premium, so an ATMF above the realised median is the usual state, not by itself rich. Close-to-close RV leaves out intraday range (the database holds daily closes only). The ATMF is the SVI-cal slice at ln(K/F) = 0 of a listed expiry, not an interpolation; 1 listed expiry beyond 252 sessions is past the cone and left out.

Breeden–Litzenberger density of ln(S T/F) from each listed SVI-cal slice, p(k) = g(k)/√(2πw)·exp(−d₋²/2), drawn only inside the expiry's quoted strikes; risk-neutral, not a forecast.

Risk-neutral density of ln(S_T/F) under each expiry’s T-forward measure, read off one butterfly-certified SVI-cal slice: p(k) = g(k)/√(2πw)·exp(−d₋²/2), with w(k) the slice’s total variance and g Durrleman’s function: the call price’s second strike derivative (Breeden–Litzenberger) in closed form. Each curve is a listed expiry (2026-11-27 (30 d, nearest 30 d); 2027-01-29 (93 d, nearest 91 d); 2027-04-16 (169 d, nearest 182 d)), not an interpolation, and is drawn only across its quoted strikes; beyond them the slice is the SVI linear wing, an extrapolation.

P(S T < 0.80 F) and P(S T < 0.90 F), the downside tails, in one colour, and P(S T > 1.10 F), the upside tail, in another, of each listed SVI-cal slice; a gap is a threshold beyond the expiry's quoted strikes, where no probability is printed.

Risk-neutral, not a forecast: these probabilities price claims and carry the market’s risk premia, so they are not real-world odds. A move is measured from each expiry’s parity forward F, not from spot. A gap is a threshold beyond the expiry’s quoted strikes (4 of 17 expiries quote no strike as high as 1.10 F): no probability is printed there.

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Two-sided out-of-the-money quotes on each expiry's parity forward (identical to JointCalibrator.spx_surface_full; 0-6 DTE expiries are excluded, as in the calibrator). Median half-spread 0.04 vol pts. The outer columns hold every quote beyond ±0.30, so the 3,343 quotes in the cells are all of them; the colour scale spans the cells shown.

Every model is scored on the same quote universe (the calibrator's cleaned quotes, 3,343) with the same metrics and dense-grid checks. Each model's out_of_sample set is the quotes it was not fitted to, so its size differs by model (essvi 1,689, svi 1,689, svi_cal 1,689, heston_v2 3,290); metrics.common_out_of_sample scores every model on the same 1,664 quotes, those no model was fitted to.

Surfaces are fitted to alternate strikes, so their out-of-sample quotes are the strikes in between (an interpolation test); Heston's in-sample set is its calibration anchors, fitted jointly with the VIX index leg, and its out-of-sample set is every other quote.

svi_cal is the arbitrage-free surface on this session: every slice passes svi.butterfly_certificate (Durrleman g >= 0 on a 1e-4 grid over k in [−3, 2] and a 1/k grid out to |k| = 2,000-3,000, each local minimum refined, and the |k| -> inf limit; minimum g 1.62e-09) and every consecutive pair passes svi.ordering_certificate (total variance ordered to 1e-10 on the same grids, wing slopes and tied-wing intercepts included; minimum w_(i+1) - w_i −6.6e-11). Numerical certificates, not closed-form; fields surface.json surfaces.svi_cal.butterfly_certificate / calendar_ordering. svi has no calendar constraint and its calendar violations are counted in dense_grid.

Yahoo snapshots carry no quote time: part of every model's spread misses can be stale quotes rather than model error.

The surfaces interpolate between fitted strikes where Heston must extrapolate from its anchors: read them as the quote-noise floor, not as rival models.

Each model's out-of-sample quotes of each expiry, the scorecard's quote sets; 1 half-spread is the edge of the quoted bid–ask.

Out-of-sample quotes only: the quote sets of the scorecard beside it, split by expiry.