FX Volatility Surface
Table of Contents
The FX volatility surface is the primary market data input for pricing and risk-managing FX options in ORE Studio. It is a two-dimensional function of implied Black-Scholes volatility across strike (or delta) and expiry tenor. This document covers pillar structure, delta conventions, smile models, and surface construction. For the generic concept of volatility and implied vol this page assumes, see Volatility. For how vol surfaces interact with pricing configuration see Pricing Configuration. Return to Knowledge.
1. Layperson's mental model
Think of the price of insurance against a currency moving, written as a landscape. One direction is how long the cover lasts, the expiry; the other is how far the currency must move before the cover pays, the strike, which FX measures in delta. The height at each spot is the volatility the market charges for that cover. The landscape is the surface.
Nobody surveys the whole landscape. The market plants a few stakes at each expiry and reads three numbers off them. The ATM volatility is the height in the middle of the valley. The risk reversal (RR) says which side of the valley rises faster, the side where the currency falls or the side where it rises. The butterfly or strangle (BF) says how steeply both sides rise compared with the middle. Everything between the stakes is filled in by a smile model.
Two things make this harder than it sounds. Surveyors disagree about where a stake stands: the same "25 delta" is a slightly different strike depending on whether the delta is measured against spot or forward and whether it counts the premium, so a number means nothing without its convention. And brokers quote the butterfly as the price of a whole strangle rather than as one height, so turning their butterfly into heights takes a calculation, not just arithmetic.
// the 25-delta call and put vols from the three quotes, in the smile convention double call_vol = atm + bf + 0.5 * rr; double put_vol = atm + bf - 0.5 * rr;
2. Overview
An FX volatility surface is a two-dimensional function of implied Black-Scholes volatility across strike (or delta) and expiry tenor. Unlike equity vol surfaces — which are quoted strike-by-strike — FX vol surfaces are quoted using a set of market pillars expressed in terms of option structure and delta: the ATM straddle, the Risk Reversal (RR), and the Strangle (STR). These three pillars, at each tenor, fully parameterise the surface under standard smile models such as SABR.
The surface must be constructed consistently across both dimensions: across strikes at a fixed maturity (the smile) and across maturities at a fixed strike or delta (the term structure).
3. Core Pillar Structures
3.1. ATM Volatility
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.
3.2. Risk Reversal (RR)
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.
3.3. Strangle (STR) / Butterfly (BF)
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.
The formula above is the smile butterfly, and with it the call and put volatilities follow from the quotes directly:
\[\sigma_{call} = \sigma_{ATM} + \sigma_{BF} + \tfrac{1}{2}\sigma_{RR}, \qquad \sigma_{put} = \sigma_{ATM} + \sigma_{BF} - \tfrac{1}{2}\sigma_{RR}\]
Brokers usually quote the market (or broker) strangle instead. Its quote \(\tilde{\sigma}_{BF}\) prices a strangle at the single volatility \(\sigma_{ATM} + \tilde{\sigma}_{BF}\), struck where that volatility puts the 25-delta call and put, and those strikes are not the strikes of the vanilla 25-delta options the risk reversal refers to. Recovering the call and put volatilities from a market strangle therefore needs a smile and a numerical solve: the smile must reproduce the ATM and risk reversal volatilities and the market strangle's price at once (Healy, 2025, equations 5 to 12; Clark, 2011). The two conventions give different numbers for the same surface, so a quote is read only together with the pair's butterfly style.
3.4. The Three-Pillar Parameterisation
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:
| Delta | Structures quoted |
|---|---|
| 25Δ | ATM, 25Δ RR, 25Δ STR |
| 10Δ | 10Δ RR, 10Δ STR (wings) |
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.
4. Delta Conventions
The definition of delta used to locate option strikes is not universal in FX and must be agreed between counterparties and within systems.
4.1. Spot Delta vs. Forward Delta
- 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).
4.2. Premium-Adjusted Delta
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.
4.3. Strike from Delta
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.
5. ATM Conventions
The definition of "ATM" determines the anchor of the smile. Two conventions are in common use:
- Delta-Neutral Straddle (DNS / Zero-Delta Straddle): The strike at which a
straddle has zero delta. For a currency pair with significant interest rate
differential, this differs from the forward. This is the standard convention
for tenors up to
10Y. - ATM Forward (ATMF): The strike equal to the forward rate. Simpler to
compute; standard for tenors
12Yand beyond (12Y--30Y).
Systems must implement the DNS/ATMF switch cleanly, as the transition creates a discontinuity in delta space at the switch tenor if not handled carefully.
6. Smile Models
6.1. SABR
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.
6.2. SVI and SSVI
SVI writes the total implied variance of one expiry as a five-parameter function of log-moneyness, and SSVI restricts it to a surface described by the ATM total variance of each expiry, one skew parameter \(\rho\) and a curvature function. The three FX pillars map onto SSVI as they map onto SABR: ATM onto the ATM total variance, RR onto \(\rho\), and STR onto the curvature. What SSVI adds is that the conditions excluding calendar spread and butterfly arbitrage across the whole surface are explicit inequalities on its parameters, where SABR fitted expiry by expiry carries no condition between expiries. The forms and the conditions are in SVI Volatility Parameterisation.
6.3. Black-Scholes (Flat Vol)
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.
6.4. Five-Point Spline
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.
7. Surface Construction: Across Time
The smile at each tenor is calibrated independently; the surface is then stitched together across maturities using one of two interpolation conventions:
- Market Interpolation: Interpolate directly in market parameter space. For
example, if RR is 2% at
1Yand 2.5% at2Y, then at18Mit is interpolated to ~2.25%. Simple and produces surfaces consistent with market quotes at standard tenors. - Physical Interpolation: Convert market parameters to model (physical) parameters first (e.g. RR → \(\rho\), STR → \(\nu\)), interpolate in model space, then convert back. Produces smoother model dynamics across time but can deviate from quoted market levels at off-pillar tenors.
The choice of interpolation convention (market vs. physical) is a system-level or currency-pair-level configuration. EM currency pairs often prefer linear vol or linear variance interpolation when ATM curve data is unreliable due to missing holidays or events.
Standard vol tenors for consensus submission and surface construction:
1W, 2W, 1M, 2M, 3M, 6M, 9M, 12M, 18M, 2Y, 3Y, 4Y,
5Y, 7Y, 10Y, 12Y, 15Y, 20Y, 25Y, 30Y.
7.1. Curve Split Tenor
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.
8. Volatility Kinds
Different volatility concepts are in use simultaneously; they must not be conflated:
| Kind | Description |
|---|---|
| Implied Vol | ATM implied vol of the spot-delta-neutral straddle at a given expiry |
| Forward-Forward Vol | Implied vol for a forward-starting option between two future dates |
| Normalised Vol | Volatility with a standardised day count (e.g. 252 or 365) |
| Historic Vol | Realised volatility computed from historical spot moves |
| Decreasing-Maturity Forward Vol | Historic vol computed over a window that ends at a fixed future date |
9. Volatility Masters and Slaves
In a multi-currency system, some currency pairs may not have independent vol surfaces. Instead, a slave pair derives its ATM vol, RR, and STR from one or more master pairs via a defined relationship:
- Correlation slaving: The slave surface is implied from a triangular relationship (e.g. EUR/GBP from EUR/USD and GBP/USD vols and their correlation).
- Basis slaving: The slave ATM vol is the master ATM plus a spread; RR and STR may be independently set or also slaved.
This mechanism reduces the number of independently managed surfaces while maintaining consistency across crosses.
10. Related Research
10.1. DYSANOS: Generative Dynamic Smooth Arbitrage-free Non-parametric Option Surfaces
DYSANOS (Hans Buehler, University of Oxford — Mathematical Institute; Blanka Horvath, University of Oxford; Anastasis Kratsios, McMaster University) proposes generating paths of option price surfaces (not just a single static snapshot) that are strictly free of static arbitrage by construction, rather than smile-model-parametric (SABR, spline) or checked-after-the-fact. Any option at any strike can be priced off the model efficiently — cheap enough for a GPU training loop. The model is trained under the statistical (real-world) measure and the paper shows how to change to the risk-neutral measure, yielding what the authors call an Option Market Model: a joint generative model of the whole surface's evolution through time, rather than a per-tenor calibration.
This is a different axis from the smile models above (SABR, flat BS, five-point spline), which parameterise a single surface at a point in time from ATM/RR/STR pillars. DYSANOS instead targets dynamics: sampling plausible, arbitrage-free future surface paths, non-parametrically, without committing to a specific smile-model functional form. Two places this is directly relevant to ORE Studio:
- Synthetic market data generation (
ores.marketdata.synthetic, cf. the GMM (Gaussian Mixture Model) components already in the codebase): a generative, arbitrage-free-by-construction path model is a natural fit for producing realistic, internally-consistent synthetic vol surface time series, rather than independently sampling static pillars per date and risking arbitrage violations or unrealistic day-to-day dynamics. - Scenario generation for risk: arbitrage-free surface paths are exactly what's needed for historical-simulation-style or Monte Carlo vol scenario generation, as an alternative/complement to bumping SABR parameters directly.
Not yet evaluated for implementation feasibility (GPU training loop, model architecture, calibration to ORE Studio's own pillar/tenor conventions) — flagged here as a candidate direction, not a committed design.
11. ORE Studio Notes
ORE uses the FxVolatility market datum type for FX vol surfaces. Relevant
ORE configuration files:
market.xml— declares which vol surface type (FxBlackVolatilitySurface,FxBlackVolatilitySurfaceDelta, etc.) is used per currency pair.pricingengines.xml— maps product types to pricers; the pricer selects which vol surface to consume.
ORE Studio will need to model vol surface configurations as domain entities (see ORE Model Configuration for the wider configuration landscape).
12. See also
- Volatility — the structure note that orders this cluster, and where to read this page in it.
- SVI Volatility Parameterisation — completes: a smile model whose arbitrage conditions are stated on its parameters.
- Paper summary: accounting for biases in Black-Scholes — completes: the ATM, RR and STR pillars expressed through the skewness and kurtosis of the return distribution.
- Constructing a volatility surface: from quotes to a published set — exemplifies: how the system carries a surface from quotes to a published set.
12.1. Further reading
- Healy, J. (2025). "Counterexamples for FX Options Interpolations - Part II." arXiv:2512.19625. The smile and broker conventions for the risk reversal and butterfly, and the solve for the market strangle.
- Reiswich, D. and Wystup, U. (2009). "FX volatility smile construction." CPQF Working Paper 20. EconPapers.
- Clark, I. J. (2011). Foreign Exchange Option Pricing: A Practitioner's Guide. Wiley. The market strangle solve; cited through Healy (2025).