How vol surfaces work

An option's price depends on one number the market doesn't observe directly: its implied volatility. Desks quote that number on a surface, a grid of vols across expiries and, for most products, strikes. This page explains how an FX surface and a rate surface are quoted, why the nodes sit where they do, and how a vol is read between and beyond them. The FX vol surface and rate vol surface pages list the calls.

The FX smile

For each expiry, the interdealer FX market quotes three numbers:

the at-the-money (ATM) vol, for the delta-neutral straddle;
the 25-delta risk reversal (RR), the call wing's vol less the put wing's;
the 25-delta butterfly (BF), how far the average wing sits above ATM.

The two wings follow from them:

\sigma_{25P} \;=\; \sigma_{ATM} + BF - \tfrac{1}{2}RR, \qquad \sigma_{25C} \;=\; \sigma_{ATM} + BF + \tfrac{1}{2}RR.

A positive risk reversal means calls on the base currency are dearer than puts, and a positive butterfly means both wings are dearer than at-the-money, which is the smile's curvature. Helixio keys each node by the quote the market spells: 25P, ATM and 25C, from the lowest strike to the highest.

Placing a node on the strike axis

A quote names an option, not a strike. Each node sits at the strike where an option of its kind, priced at the node's own vol $\sigma_j$ , has the quoted delta. With $F$ the forward to delivery, $DF_b$ the base currency's discount factor and $\tau$ the time to expiry, under an unadjusted convention:

K_j \;=\; F\,\exp\!\left(\tfrac{1}{2}\sigma_j^2\tau - d_1\,\sigma_j\sqrt{\tau}\right), \qquad d_1 \;=\; \omega\,N^{-1}\!\left(\frac{|\Delta|}{s}\right),

where $\omega$ is $+1$ for a call and $-1$ for a put, and $s$ is $DF_b$ under a spot delta and $1$ under a forward delta. A premium-adjusted convention has no closed form, so its strike is solved numerically. The ATM node is the delta-neutral straddle, which sets $d_1 = 0$ :

K_{DNS} \;=\; F\,e^{\sigma_{ATM}^2\tau/2}.

Because each strike depends on the forward, a curve move shifts where the nodes sit: the smile moves with the forward, which is called sticky delta.

Delta conventions

Four delta conventions are in use: spot or forward, each plain or premium-adjusted (net of a premium paid in the base currency). Helixio spells them Spot, Fwd, PaSpot and PaFwd. A surface carries its convention, and every node is placed under it. The Garman–Kohlhagen delta of a call struck at $K$ is $s\,N(d_1)$ , and of a put $-s\,N(-d_1)$ , with $d_1 = \bigl(\ln(F/K) + \tfrac{1}{2}\sigma^2\tau\bigr)/(\sigma\sqrt{\tau})$ . The delta-neutral straddle's call therefore has a forward delta of exactly one half, and a spot delta of $DF_b/2$ .

Reading within a smile

Between two node strikes $K_j < K < K_{j+1}$ , the FX vol is linear in log-moneyness $x = \ln(K/F)$ :

\sigma(K) \;=\; \sigma_j + (\sigma_{j+1} - \sigma_j)\,\frac{x - x_j}{x_{j+1} - x_j}, \qquad x_j \;=\; \ln(K_j/F).

Beyond the outer nodes the vol is held at the wing's value: the surface doesn't extrapolate the skew into the tails.

Across expiries: total variance

Total variance is $\sigma^2\tau$ . Between two stored expiries $\tau_0 < \tau < \tau_1$ , every surface here, FX or rates, is linear in it:

\sigma^2(\tau)\,\tau \;=\; (1-w)\,\sigma^2(\tau_0)\,\tau_0 + w\,\sigma^2(\tau_1)\,\tau_1, \qquad w \;=\; \frac{\tau - \tau_0}{\tau_1 - \tau_0}.

Total variance can only grow with time, so this rule can't create calendar arbitrage between two arbitrage-free smiles, where interpolating the vol itself can. When vols rise with expiry, the interpolated vol sits above the straight line between the two node vols. Before the first expiry and after the last, the vol is held flat.

Because $w$ is a ratio of times, a day count that scales every $\tau$ alike leaves an interpolated vol unchanged. The day count still matters: an option counts its own life on it.

Moving a smile

A scenario states a smile move in vol points, the unit the quotes are made in: 0.002 is 0.20 vol points. A risk reversal move $s_{RR}$ and a butterfly move $s_{BF}$ change the wings by

\Delta\sigma_{25P} \;=\; s_{BF} - \tfrac{1}{2}s_{RR}, \qquad \Delta\sigma_{25C} \;=\; s_{BF} + \tfrac{1}{2}s_{RR}, \qquad \Delta\sigma_{ATM} \;=\; 0,

so the shocked smile is the one the moved quotes would build. A smile move is always absolute: a relative move is refused, because a risk reversal can be zero or negative.

Rate vol grids

Swaptions are quoted on a grid of option expiry by underlying swap tenor, one at-the-money vol per cell. Both axes are in years, so the same grid also serves a bond option, which reads the tenor axis at a duration. A rate vol is quoted in one of two conventions:

a normal vol, in basis points of rate per year, priced with the Bachelier formula;
a lognormal vol, in percent of the rate, priced with Black-76.

The surface carries its convention, and a pricer takes the formula from the surface rather than guessing from the level. A read at expiry $T$ , tenor $m$ and strike $K$ collapses one axis at a time: strike, then tenor, then expiry. Across strikes and tenors the vol is linear; between tenor nodes $m_j < m < m_{j+1}$ ,

\sigma(m) \;=\; \sigma_j + (\sigma_{j+1} - \sigma_j)\,\frac{m - m_j}{m_{j+1} - m_j},

and across expiries it is linear in total variance, as above. Outside every edge the read is the edge node's vol.

Strike axes

A cap market quotes a vol for each expiry and absolute strike on one index tenor. A swaption cube adds strikes to the expiry by tenor grid, quoted as spreads $s$ to the at-the-money forward $F$ :

K \;=\; F + s.

A strike of 3% and a spread of 300 bp look alike as numbers, so the surface records which axis it carries, and a read naming the wrong kind of coordinate is refused rather than misread.

From formula to API

Symbol	Meaning	Rust	Python
$\sigma_{ATM}$ , $RR$ , $BF$	an FX expiry's three quotes	the arguments of `FxVolSmile::from_quotes`	a smile tuple's last three entries, or `vols.FxVolSmile.from_quotes`
$\sigma_j$	an FX node's vol	`FxVolSmile::vols`; `vol_at_quote` on `FxVolSurface::view`	`vols.FxVolSmile.vols`; `vols.FxVolSurface.vol_at_quote`
$K_j$ , $K_{DNS}$	a node's strike	`strike_from_delta`, `atm_delta_neutral_strike`	`vols.strike_from_delta`, `vols.atm_delta_neutral_strike`
$\sigma(K)$	the FX vol at a strike	`vol_at_strike` on `FxVolSurface::view`	`vols.FxVolSurface.vol_at_strike`
$\Delta$	an option's delta	`delta`	`vols.fx_delta`
$s$	the spot or forward delta scale	`DeltaConvention::Spot`, `DeltaConvention::Fwd`	`vols.DeltaConvention.Spot`, `vols.DeltaConvention.Fwd`
$s_{RR}$ , $s_{BF}$	a smile move	a `Shift` on `Selector::FxSmile` with a `SmileComponent`	`Scenario.shift` on `"fx_rr(EUR/USD)"` or `"fx_bf(EUR/USD)"`
$\tau$ , $T$	time to an expiry	`FxVolSurface::day_count`; a rate read's `expiry_years`	`vols.FxVolSurface.day_count`; `vols.RateVolSurface.vol`'s `expiry_years`
$w$	the total-variance weight	computed inside every read	computed inside every read
$m$	an underlying tenor in years	a rate read's `tenor_years`; `RateVolSurface::tenors`	`vols.RateVolSurface.vol`'s `tenor_years`; `vols.RateVolSurface.tenors`
$K$ , $s$	a rate strike or spread	`vol_at_strike`, `vol_at_spread` on `RateVolSurface::view`	the `strike` argument of `vols.RateVolSurface.vol`
the convention	normal or lognormal	`RateVolSurface::convention`	`vols.RateVolSurface.vol_convention`