FX Volatility Surface

The ATM (at-the-money) quote is the implied volatility of the option struck at the ATM level, quoted as a single number per tenor. The definition of "ATM" matters and is convention-dependent (see * ATM Conventions below).

A straddle is a long call and long put at the same strike and expiry. At-the-money, the two delta components cancel and the position has (approximately) zero delta. The straddle price is driven almost entirely by volatility level.

ATM vol captures the height of the smile at a given tenor.

A risk reversal is the difference in implied volatility between an out-of-the-money call and an OTM put at the same delta magnitude:

\[\sigma_{RR}^{25\Delta} = \sigma_{call}^{25\Delta} - \sigma_{put}^{25\Delta}\]

A positive RR means the call is more expensive than the put (market is paying for upside protection in the foreign currency). RR measures the skew of the smile — the asymmetry between the two wings.

At 25Δ the convention is to quote the 25-delta call vol minus the 25-delta put vol. At 10Δ the same structure gives a wider, more extreme read on the wings.

A strangle (or butterfly in some conventions) measures the average premium of the two OTM wings relative to ATM:

\[\sigma_{STR}^{25\Delta} = \frac{\sigma_{call}^{25\Delta} + \sigma_{put}^{25\Delta}}{2} - \sigma_{ATM}\]

The strangle captures the curvature of the smile — how "fat" the wings are relative to the centre. A positive strangle means OTM options are more expensive than ATM, as is typical in FX.

ATM, RR, and STR together encode the first three degrees of freedom of the smile at each tenor: level (ATM), skew (RR), and curvature (STR). Under SABR and similar models, these three numbers uniquely determine the smile.

Standard delta grids used in practice:

The full five-pillar quote (10Δ put, 25Δ put, ATM, 25Δ call, 10Δ call) gives a richer representation of the wings and is needed for more exotic pricing. SABR does not natively support five-point smiles; five-point spline interpolation is used for those cases.

• Spot Delta: \(\Delta_{spot} = e^{-r_f T} N(d_1)\) — measures sensitivity of option value to the spot rate. Standard in the short end.
• Forward Delta: \(\Delta_{fwd} = N(d_1)\) — measures sensitivity to the forward rate. Standard for longer maturities.

The convention switch from spot to forward delta typically occurs around 1Y for most currency pairs, but varies. Quoting and risk systems must flag which convention is in use (often denoted S/F on screens).

When the option premium is paid in the foreign currency (as is common for currency pairs where the foreign currency is the numeraire), the delta must be adjusted for the premium received. Premium-adjusted delta differs from unadjusted delta by the option value divided by spot; the difference is small for OTM options but significant for ITM ones. This affects where OTM strikes sit and must be handled consistently across the surface.

Given a delta convention and a vol, the corresponding strike is backed out from the Black-Scholes formula. The surface is therefore ultimately stored in strike space (or log-moneyness) internally, even though it is quoted in delta space in the market.

SABR (Stochastic Alpha Beta Rho) is the standard smile model for FX. With \(\beta = 1\) (log-normal backbone), SABR is a three-parameter model:

• \(\alpha\): initial volatility (maps to ATM)
• \(\nu\) (vol-of-vol): controls curvature (maps to STR)
• \(\rho\) (spot-vol correlation): controls skew (maps to RR)

The calibration procedure maps the three market pillars (ATM, RR, STR) to the three SABR parameters at each tenor. SABR produces smooth, arbitrage-free smiles in the body but can exhibit instability in the far wings (very low or very high deltas), requiring wing stabilisation.

Additional SABR term-structure parameters — per tenor — include correlation sensitivity (up/down), vol adjustments, RR infinite flag, vol swap adjustments, and standard deviation weighting. These are model parameters rather than market observables. In ORE these appear as entries in the EngineData XML configuration.

A flat Black-Scholes surface (one vol per tenor, no smile) is always maintained alongside the main smile model. It is used for re-parameterisation, as a fallback, and for ATM-only pricing. For EM currency pairs where smile data is sparse, a flat vol surface may be the primary representation.

For currency pairs that are well-quoted at 10Δ and 25Δ, a five-point spline gives a more detailed smile. SABR does not support five-point calibration; instead, spline interpolation is applied directly across the five strikes. The extra points give finer control over the wings but require more calibration data and introduce more model risk.

A split tenor (typically 5Y) separates the spline interpolation of the short end and long end of the surface. This prevents changes in long-dated vols from affecting short-dated smiles and vice versa. For curves that extend only to a short maturity, the last tenor is used as the split tenor.