Volatility Surface Analytics

Overview

Implementation spec for the Volatility tab of SOPHIE's Options Viewer (VolatilityChartView) — the smile, term structure, risk-neutral density, and skew/kurtosis measures. The Volatility Surface and Volatility Smile & Skew cover the concepts; this page documents the computation.

Data Quality: SPX/SPXW Duplicate Listings

A finding that applies to every chart on this tab, so it's documented once here rather than repeated per section.

A standard SPX monthly expiration (3rd Friday) is frequently shared with a same-day SPXW contract — a genuinely different, separately-traded PM-settled product, with its own open interest pool, listed under the identical calendar date. On live data this duplicated 26–46% of strikes on every single monthly checked. Two real examples at strike 7275 on the same nominal expiration:

Symbol	Bid	Ask	Open Interest	Last Trade
`SPX260918C07275000`	435.70	438.20	855	2026-08-18 15:57
`SPXW260918C07275000`	434.50	440.40	302	2026-08-03 10:02

Two different instruments, two different OI pools, two different last-trade times — merged into one "expiration" bucket by whatever code groups contracts by calendar date alone. For any per-strike single-value view (smile, RND, skew, the term structure's ATM pick), interleaving both as if they were one instrument corrupts the shape entirely: iterating the strike-sorted array hits each strike twice with different quotes, and — critically for the Breeden-Litzenberger density further down this page, which differentiates a sequence of adjacent strikes — scrambles the neighbor relationships the second derivative depends on. On one affected monthly this alone was enough to make a straightforward thin-quote check (see below) flag 98% of the density curve, not because the market was that illiquid, but because half the "neighbors" in the sequence were the wrong instrument's quote.

The fix, scoped to this tab only: collapse duplicate strikes to a single contract per calendar date, preferring the standard AM-settled SPX root over SPXW when both are present (falling back to whichever has higher open interest if the root can't be parsed). Deliberately not applied to GEX — summing both instruments' gamma at a strike is the correct whole-book dealer-hedging reading there, not an error. Whether the platform should instead treat same-day SPX/SPXW as two genuinely separate, independently selectable expiration entries everywhere (the chain matrix, the payoff builder, the expiration selector itself) is a broader question this fix does not attempt to answer.

Per-Point Data Quality Flags

Excluding fully-expired cycles (below) catches the extreme case. The same underlying mechanism — a wide or untraded quote making price-to-IV inversion unreliable — shows up in milder form on individual strikes within otherwise-healthy, live cycles: a strike nobody has traded today, sitting on a wide resting quote. Rather than exclude those (they're still real, tradeable strikes), the term structure, RND, and skew charts mark them:

thin quote  =  spread% > 20%   OR   (volume == 0  AND  open interest < 10)

A flagged point renders as an amber ring instead of a solid dot, and its tooltip adds an explicit caution note. Nothing is hidden — the point (and the line connecting it) still renders normally — the flag just tells you not to over-read that specific number.

Checked against a live chain after fixing the SPX/SPXW duplication above, the flag rate rises smoothly with distance from spot, which is the shape you'd want from a genuine liquidity signal rather than noise: 0% within ±5% of spot (the true ATM core), 25% by ±10%, 28% by ±15% (the tab's default display range), 43% by ±25%. Before the duplication fix, an affected monthly's RND showed 98% flagged — the duplication bug, not real thinness.

Why each condition actually signals a problem

The two conditions in the flag catch two different failure modes, not one:

Wide spread (> 20%) means the quote itself is unreliable right now — the market maker isn't committing to a tight two-sided price, so the midpoint used to solve for IV could be far from where the contract would actually trade. A 20-point-wide market on a $2 option isn't "expensive," it's a market maker declining to make a real market.
Zero volume with under 10 open interest means nobody has traded this contract and almost nobody is holding it. The quoted price is theoretical — likely a market maker's own model output re-quoted, not something arrived at through actual price discovery. IV solved from a theoretical quote just gives you the market maker's model back, which tells you nothing about what the market as a whole thinks.

Either condition alone can produce a wrong number; a strike hitting both at once (common at the far wings) is the least trustworthy of all.

What to actually do when you see the amber ring

The flag is a caution, not a stop sign — these are real, tradeable strikes. What changes is how much weight to put on the number and how to approach the trade itself.

Before placing an order on a flagged strike: the displayed quote is a snapshot, possibly stale. Pull up a live quote for that specific contract before sizing a limit order — don't route off the chart's numbers directly. Expect the real fill to require more price improvement patience than an unflagged, tightly-quoted strike at the same delta.

When choosing between two similar strikes: if a strategy calls for, say, a ~30-delta short put and the strikes bracketing that delta differ in flag status, prefer the unflagged one. It isn't just about a cleaner-looking chart — a tight, actively-traded market is direct evidence you'll get filled near the displayed price rather than eating the full spread. All else equal, liquidity should win the tie.

Reading Term Structure: if the ATM point for one specific expiration is flagged, treat that single cycle's contribution to the curve skeptically — don't anchor a contango/backwardation call on one flagged point at either end. Look at the trend across several unflagged neighboring cycles instead; a slope that only appears because of one thin point isn't a real regime read (this is exactly the mechanism behind the stale-cycle bug documented above — a bad single point drove the entire Term Structure Slope card to the wrong regime).

Reading the Risk-Neutral Density: the second derivative amplifies noise in its inputs, so a cluster of flagged points can manufacture a bump or dip in the curve that isn't a real market view — it's numerical noise from a wide quote getting differentiated twice. Trust the overall shape (where the bulk of probability mass sits, whether the tails look fat) in regions with few flags; don't read fine local wiggles inside a flagged cluster as genuine skew or kurtosis.

Reading 25-Delta Skew: if a cycle's 25-delta call, put, or ATM contract is flagged, that cycle's risk-reversal or butterfly reading may reflect a stale quote rather than a real shift in demand for downside protection. Cross-check against that expiration's own smile chart directly — if the flagged 25-delta point sits far off the smooth curve traced by its unflagged neighbors, the skew number computed from it is not to be trusted; use a nearby unflagged expiration's skew reading instead when timing a skew-based trade (e.g. a risk reversal, or choosing which cycle to sell a put spread in based on relative put-side richness).

The general principle: a flag doesn't mean "avoid this strike" — plenty of legitimate trades happen in modestly thin names. It means "verify before you rely on this number," the same discipline any experienced options trader already applies before sizing an order in a name they don't usually trade.

IV Smile & Skew Curve

Plots strike $K$ against implied volatility $\sigma(K)$ for the selected cycle, restricted to within ±22% of spot. The band is deliberate: far-tail strikes quote wide and thin, and their implied vols are numerically unstable enough to visually dominate a curve they tell you nothing reliable about.

Term Structure & Forward Implied Volatility

Plots ATM implied volatility across all expirations, ordered by DTE.

Contango (normal): $\sigma_{\text{far}} > \sigma_{\text{near}}$ — upward sloping, calm market.
Backwardation (inverted): $\sigma_{\text{near}} > \sigma_{\text{far}}$ — downward sloping, near-term event risk or panic.

Genuine near-term richness vs. dead-contract noise

Short-dated options do legitimately carry a modest term-structure hump — a few vol points, from weekend theta, event risk, or a thin liquidity premium. That's real and common. What's not common, and not the same thing, is what a stale front cycle can do to this chart.

A cycle's listed daysToExpiration is computed relative to the data feed's own last-refreshed timestamp, not real wall-clock time. When the feed is frozen — markets closed, a weekend, an evening — a contract that has already expired can still be labeled 0 DTE. Its quotes are then frozen at end-of-day levels with near-zero remaining time value, and inverting a price back to an implied volatility is numerically unstable as $T \to 0$ : on one weekend snapshot this produced solved IVs of 78%, 193%, and 336% on SPX strikes just $5 apart on the same already-dead cycle — a spread that has nothing to do with a real smile, and swamped the entire chart. The Term Structure Slope card read Backwardation purely from that dead front point; the true, live-cycle reading was Contango.

The fix is to exclude any cycle whose expiration date has already passed in the exchange's own timezone (America/New_York) from every chart on this tab — term structure, forward vol, multi-expiration smile overlay, and the 25-delta skew term structure — regardless of what the feed's own DTE field claims. A small amber notice appears above the affected charts whenever a cycle was excluded this way, so it's visible rather than silently correcting the picture.

The forward implied volatility between two horizons $T_1 < T_2$ strips out the overlapping near-term vol to isolate what the market prices for the period between them:

\sigma_{\text{fwd}}(T_1, T_2) = \sqrt{\frac{T_2 \sigma_2^2 - T_1 \sigma_1^2}{T_2 - T_1}}

This follows from variance being additive in time while volatility is not — total variance to $T_2$ is the sum of variance to $T_1$ and forward variance across the gap. A steep front-month spike often leaves forward vol nearly flat, which tells you the event risk is genuinely localised rather than a repricing of the whole curve.

How it's used: term structure shape is a direct trading signal in its own right for volatility desks — backwardation (near-term vol above far-term) is the textbook setup for calendar spreads (sell the rich front month, buy the cheaper back month) and is closely watched around known binary events (earnings, FOMC, CPI) where the front-month IV bump is expected to collapse once the event passes. In index vol specifically, VIX futures traders watch the VIX/VIX3M ratio the same way — a ratio above roughly 1 signals backwardation and is one of the standard triggers cited for de-risking short-vol ETN/ETF positions (see the VRP methodology page for how this feeds into regime classification).

Breeden-Litzenberger Risk-Neutral Density

Extracts the market's implied probability density $f(K)$ for the underlying at expiration, directly from call prices, via the Breeden & Litzenberger (1978) result:

f(K) = e^{rT} \frac{\partial^2 C(K)}{\partial K^2}

Non-uniform strike grid

This is the subtle part. Real SPX chains alternate between 5-, 10-, and 25-point strike gaps, and the textbook uniform second-difference

\frac{C_{i+1} - 2C_i + C_{i-1}}{h^2}

is simply wrong when $\Delta K$ changes between the two sides — it silently assumes a single $h$ that does not exist. The error shows up exactly where the grid spacing changes, producing spurious density spikes.

The general non-uniform three-point second derivative is used instead:

\frac{\partial^2 C}{\partial K^2} \approx \frac{2 \left[ \Delta K_1 \cdot C(K_{i+1}) - (\Delta K_1 + \Delta K_2) \cdot C(K_i) + \Delta K_2 \cdot C(K_{i-1}) \right]}{\Delta K_1 \cdot \Delta K_2 \cdot (\Delta K_1 + \Delta K_2)}

where $\Delta K_1 = K_i - K_{i-1}$ and $\Delta K_2 = K_{i+1} - K_i$ .

Post-processing: densities are floored at zero ( $\max(0, \partial^2 C / \partial K^2)$ , since a negative probability density is a numerical artefact rather than a signal) and normalised so $\sum f(K_i)\,\Delta K_i = 100\%$ .

How it's used: the RND is how risk desks convert an entire options chain into "what does the market actually think the probability distribution of the underlying looks like" — genuinely richer than a single implied vol number, because it captures the market's skew and kurtosis views directly rather than assuming a lognormal shape. It's used to compare the option-implied (risk-neutral) distribution against a physical-measure forecast to find where the market's pricing of tail risk looks rich or cheap, and it underlies more sophisticated hedge construction than a flat-vol confidence interval can — see the Implied Price Ranges section of the Multi-Leg Payoff & Probability of Profit Methodology page for the simpler symmetric/skew-adjusted alternative most retail platforms use instead.

25-Delta Skew & Kurtosis

Two standard summary measures of surface shape, both quoted off the 25-delta wings:

Risk reversal (skew) — the price of downside protection relative to upside:

\text{RR}_{25\Delta} = \text{IV}(\text{Put}_{25\Delta}) - \text{IV}(\text{Call}_{25\Delta})

Positive means put skew, i.e. demand for downside protection exceeds upside calls. For index options this is nearly always positive; the informative signal is its level against its own history, not its sign.

Butterfly (convexity / kurtosis) — how much the wings are bid relative to the at-the-money base:

\text{Fly}_{25\Delta} = \frac{\text{IV}(\text{Put}_{25\Delta}) + \text{IV}(\text{Call}_{25\Delta})}{2} - \text{IV}_{\text{ATM}}

A rising butterfly means the market is paying up for tail outcomes on both sides — fat-tail probability being repriced independently of direction.

How they're used: risk reversal is a standard sentiment and positioning gauge — a sharp rise in put skew (independent of the VIX level itself) is read as crash-hedging demand building, and is watched by both directional traders (as a contrarian fear gauge) and vol desks pricing their own skew risk. Butterfly is the more specialised of the two: convexity/vol-of-vol traders express views on it directly via iron butterflies and condors, and a persistently elevated butterfly across a chain is one of the standard reasons a "sell premium everywhere" strategy underperforms a delta-neutral one that specifically targets skew.

Limitations

RND is only as good as the quotes. The second derivative amplifies quote noise; a single stale mid mid-chain produces a visible artefact.
Delta-based wings are model-dependent. The 25-delta strike is itself derived from a model, so risk reversal and butterfly inherit that model's assumptions.
The ±22% smile band hides genuine tail information — deliberately, but it does mean the smile view is not the place to study far-tail pricing.
Live-cycle filtering trusts the exchange calendar, not the feed's own DTE label. If the feed's clock is wrong in the other direction — reporting a cycle as already expired when it's actually still live — that cycle would be dropped instead. This hasn't been observed, but the filter is only as reliable as Intl.DateTimeFormat's America/New_York conversion and the browser's system clock.
SPX/SPXW dedup keeps one instrument per strike, not both. Preferring the standard SPX root discards the SPXW quote entirely for that strike on this tab — correct for a single-instrument IV curve, but it means the SPXW-specific liquidity at that strike (its own OI, its own spread) is invisible here. It's still fully visible in the Options Chain matrix and GEX, which are unaffected by this fix.
The thin-quote threshold (20% spread, or zero volume with under 10 OI) is a judgment call, not derived from a model. It was tuned by checking that the resulting flag rate rises smoothly with distance from spot on real data rather than being arbitrary, but a different threshold would flag a different set of points.

Key Takeaways

Forward vol isolates the period between two expirations, because variance (not volatility) adds in time.
SPX strike grids are non-uniform, so the RND must use the general three-point second derivative — the uniform formula produces artefacts at every spacing change.
Densities are floored at zero and normalised to 100%.
Risk reversal measures directional skew; butterfly measures tail convexity.
Cycles that have already expired in wall-clock time are excluded from every chart on this tab — a stale front cycle can flip the Term Structure Slope's regime call entirely, not just shift a number.
Standard monthlies routinely list a same-day SPXW contract as a separate instrument under the same calendar date — deduping to one contract per strike is what makes the per-point quality flag (and the smile/RND/skew shape generally) meaningful rather than noise.
Individual thin or wide-spread points are flagged with an amber ring rather than hidden — the flag rate should rise smoothly with distance from spot; a flat, high rate near the money is itself a sign something upstream (like the duplication above) needs checking.