> ## Documentation Index > Fetch the complete documentation index at: https://docs.backquant.com/llms.txt > Use this file to discover all available pages before exploring further. # Vol surface methodology > How the IV surface, smiles and constant-maturity vols are built, what is checked, and what is not Every implied-vol number in v2 starts from one surface per coin, rebuilt about every 30 seconds from the multi-venue options chain. This page is the full recipe, so you can reproduce it or decide where you disagree with it. ## Inputs * **The chain.** One row per contract across every venue that lists it (see [Venues](/concepts/venues)). A contract quoted on several venues carries their implied vols blended by open interest. * **Out of the money only.** Puts below the forward, calls above, the average of the two at the money. ITM options add no information that the OTM side and parity do not already give, and their marks are noisier. * **Coordinates.** Log-moneyness `k = ln(K / F)` against each expiry's own forward, and total variance `w = sigma^2 T`, with `T` in years (calendar days / 365). ## Forwards Each expiry gets its own forward. One forward for the whole surface is only right when basis is zero, and crypto basis is not (BTC's long-dated expiries sit several percent over spot). 1. **Put-call parity.** `F = K + (C - P)` per strike in the strike currency, undiscounted, then the median across strikes. Needs at least two call/put pairs; a per-strike solve more than 50% from spot is a bad mark and is dropped first. Against Deribit's listed BTC futures the parity forward came within 0.12% at every tenor inside 180 days. 2. **Delta.** Where parity has too few pairs, the strike where the call-equivalent delta crosses 0.5. 3. **Spot.** A forward from either method must sit within `0.5% + 30% x T` of spot (the floor is mark noise, the rate is carry); outside it the next method is tried, and if neither passes the expiry is read against spot. `expiry_forwards[].method` (`parity`, `delta` or `anchor`) on [`/v2/options/iv/surface`](/api/v2/options/iv/surface) says which one each expiry used. ## Smiles Each expiry's smile is fitted with raw SVI: ``` w(k) = a + b ( rho (k - m) + sqrt((k - m)^2 + sigma^2) ) ``` * **Window.** Strikes within four ATM standard deviations (`4 x ATM vol x sqrt(T)`, at least 0.10 in log-moneyness). The deep wings carry no liquidity, and their marks are what wreck a naive fit. * **Weights.** Square root of open interest, so a dead wing mark cannot drag the fit. * **Honesty gate.** A fit is used only if it lands within 2 vol points RMSE of the quotes it was fitted to. Otherwise the expiry is published as `interp`: its quotes joined in total variance, linear in `k`. In practice that is same-day expiries, whose smile is a kink SVI cannot draw. The fitted parameters, the RMSE, the quote count and the log-moneyness span of the expiry's quotes (`svi`, `smile_rmse`, `n_quotes`, `k_range`) are on every expiry of the surface. Rebuild any smile yourself from them. Note that `k_range` can be wider than the fit window: between the window and the outermost quote the curve is the fit's continuation, which `smile_rmse` does not measure. ## The grid `z_grid[i][j]` is the vol in percent at tenor `y_grid[i]` (days) and strike `x_grid[j]`. Rows are real listed expiries out to 180 days; no tenor is invented between them. * **Strike window.** Forward plus or minus 40%, tightened to the strikes most expiries actually quote. * **No invented wings.** A fitted row is drawn at most half its own quoted span (and never more than 0.25 in log-moneyness) past its outermost quote, and never above 1.5 times its highest quoted vol. An `interp` row reaches 0.10 past its quotes. Past that, the cell is null. * **Unpriceable cells.** A cell whose out-of-the-money option is worth less than a millionth of the forward has no vol worth reporting and is null. A row left less than 35% filled is dropped whole. * **Range.** Vols outside 5% to 250% are dropped as broken marks, never clipped to the limit. Expiries past 180 days are fitted the same way and published in `long_expiries` with the same fields, but not drawn on the grid. They are there for the 6-month to 1-year constant-maturity tenors. ## Static arbitrage Two conditions make a surface arbitrage-free: no calendar arbitrage (total variance at a fixed moneyness never falls as maturity rises) and no butterfly arbitrage (every smile implies a non-negative density). **On the grid**, calendar arbitrage is removed: each cell is raised to the running maximum of total variance down the maturity axis, the smallest change that restores the condition. Where that would push a cell past what its own expiry could plausibly be worth (1.5 times its highest quoted vol), the cell becomes null instead, so a same-day wing mark is never carried forward into a later expiry. `n_calendar_lifted` counts the cells raised. **On the fitted smiles**, both conditions are checked on every response of [`/v2/options/iv/constant-maturity`](/api/v2/options/iv/constant-maturity) and [`/v2/options/iv/smile`](/api/v2/options/iv/smile), across each expiry's quoted strikes (`k_range`): * `butterfly_free`: Durrleman's condition `g(k) >= 0` everywhere in that range, where ``` g(k) = (1 - k w' / (2 w))^2 - (w'^2 / 4) (1 / w + 1 / 4) + w'' / 2 ``` The risk-neutral density is proportional to `g`, so a negative `g` is a butterfly that pays you to own it. `min_density_g` is the lowest value found. * `calendar_free`: the smile's total variance is at or above the previous fitted expiry's at every strike both quote. Failures are listed in `arbitrage.violations`; nothing is hidden or quietly repaired. On live BTC on 2026-10-11, all 13 fitted expiries (1 day to 474 days) passed both. **What is not done.** Each expiry is fitted on its own (raw SVI, not a joint SSVI surface), so the checks above are tests, not constraints of the fit. Butterfly repair is not applied to `interp` rows. A projection onto the nearest convex smile exists, and was measured on the live chains: on SOL it doubled the 90th-percentile repricing error to remove butterflies worth fractions of a cent, so it stays off until it can be bounded by each option's value. ## Constant maturity [`/v2/options/iv/constant-maturity`](/api/v2/options/iv/constant-maturity) reports ATM and the 25- and 10-delta wings at 7, 14, 30, 60, 90, 120, 180, 270 and 365 days. Per expiry: * **ATM** is the vol at the forward (`k = 0`). * **A wing** is the vol at the strike with that Black-76 forward delta, `N(d1) = delta` with `d1 = (-k + w(k) / 2) / sqrt(w(k))`. Puts are read on the call-equivalent axis, so the 25-delta put sits at call delta 0.75. * **The vol is read off the quoted smile**, not the SVI fit: total variance linear in `k` between the neighbouring quotes. Checked on live BTC, SVI matched the quotes at both 25-delta wings on most expiries but missed the December put wing by about 0.7 vol, which is most of a risk reversal. * **A wing the quotes do not reach is null.** That vol was never quoted. Per tenor, each point's total variance is interpolated linearly in time between the two fitted expiries that bracket it (sticky delta), then turned back into vol. There is no extrapolation: a tenor before the first fitted expiry or after the last is null. `bracket_dte` names the two expiries behind each tenor, and `expiries[].n_quotes` shows how deep each one's book is, so you can see when a long tenor leans on a thin expiry. Conventions, in vol points: | Field | Definition | | - | - | | `rr_25`, `rr_10` | call vol minus put vol at that delta (negative when puts are bid) | | `bf_25`, `bf_10` | average of the call and put vols minus ATM | [`/v2/options/iv/skew`](/api/v2/options/iv/skew) reports skew as put minus call. It is the same quantity with the opposite sign. ## Freshness The surface, smiles and constant-maturity vols share one build, about every 30 seconds; `meta.computed_at` is that build's time. These are current values: there is no constant-maturity history. ## See also This documentation is built and hosted on [Mintlify](https://mintlify.com), a developer documentation platform.