Foreign Exchange Option Interpolation

• Foreign exchange option interpolation is a method that maps sparse FX option quotes to control points, ensuring a continuous, arbitrage-free implied volatility surface.
• It leverages parametric models (e.g., SABR, Heston, SVI) and non-parametric approaches (e.g., cubic splines) to align with market-standard delta conventions and enforce dynamic arbitrage constraints.
• Effective implementation demands precise delta-to-strike inversions, rigorous arbitrage checks, and cross-currency consistency to reliably price and hedge a range of FX derivatives.

Foreign exchange option interpolation is the collection of quantitative methodologies by which practitioners build a continuous, arbitrage-free implied volatility surface from sparse broker or market-quoted FX option data. Accurate interpolation is essential for consistent pricing, risk management, and hedging of vanilla and path-dependent FX derivatives. This field has a distinct technical landscape due to the FX market’s market-standard delta conventions, cross-currency symmetry requirements, and the mandate of rigorous static and dynamic arbitrage constraints.

1. Market Conventions and Smile Construction

The FX options market typically quotes option volatility smiles at key standard deltas (commonly 10Δ, 25Δ puts/calls, and ATM), supplied as ATM volatility (σATM), risk reversals (RRΔ), and butterflies (BF_Δ). These quotes imply a sparse grid for each maturity, with the actual strikes K_i computed via Black-Scholes delta equations and the Garman–Kohlhagen framework. The convention-specific definitions for delta (spot, forward, premium-included) and ATM strikes must be consistently applied, as mis-specification can lead to misaligned surfaces and arbitrage (Bossens et al., 2009).

A crucial pre-step is "delta-to-strike inversion," mapping these quotes to their precise K_i, which then serve as control points for the implied volatility smile σ(K) or σ(Δ). For cross-asset consistency and arbitrage prevention, the interpolated smile must be constructed so that it fits these anchor points exactly and is amenable to robust extrapolation in the wings.

2. Interpolation Schemes: Structural Overview

Multiple interpolation methodologies are used in practice, each with specific properties and risks:

Interpolation Family	Variable	Key Formula / Approach
Linear/Polynomial in Strike	K, log K	Linear, spline, or polynomial in K, log K
Spline in Moneyness (log(K/F))	ln(K/F)	Natural cubic spline / interpolation
Spline/Polynomial in Delta	Δ, Δ_F	Spline/poly on {Δ, σ(Δ)}
Parametric (SABR, SVI, LVG, etc.)	K, log(K/F), T	Closed/parametric volatility function
Vanna–Volga	K	Linear replication using market greeks

Each method’s arbitrage characteristics—specifically density positivity, monotonicity, and convexity—depend strongly on both interpolation variables and practical implementation choices (Healy, 22 Dec 2025).

3. Parametric and Model-based Approaches

The most robust interpolation strategies are based on parametric models that guarantee arbitrage-freeness under suitable constraints.

SABR Parameterisation. The SABR model, with parameters (α, β, ρ, ν), provides a closed-form approximation to the implied volatility smile, with the forward FX rate F and strike K inputs. Market anchors (σ_ATM, RR, BF) are used with fixed β (common choices: ½ or 1); calibration proceeds by minimizing squared differences at anchor strikes. The Hagan et al. formula yields a smooth, monotone, and typically arbitrage-free smile for reasonable parameters (Dellaportas et al., 2014).

Heston Stochastic Volatility Model. Here, the implied volatility surface is constructed by calibrating the Heston parameters (κ, θ, σ, ρ, v_0) to quoted smile data, performing pricing via semi-analytical characteristic function inversion (FFT or numerical quadrature), and recovering implied vols by inverting prices. This approach is notable for matching global wing shapes and ensuring positive density for well-chosen parameters (Janek et al., 2010), with calibration and parameter interpolation in maturity for time-dependent surfaces.

SSVI/xSSVI and LVG Models. SVI-type models explicitly impose static arbitrage constraints (via parameter and Lee’s moment conditions). xSSVI in particular enforces butterfly and calendar-spread arbitrage absence per slice. The Quadratic Local Variance-Gamma (LVG) provides explicit control over the local variance shape, guaranteeing positivity (Healy, 22 Dec 2025).

4. Non-parametric and Spline Interpolators: Risks and Counterexamples

Non-parametric approaches such as natural cubic splines or polynomials (in log-moneyness, delta, or variance) are widely used for their flexibility and exact fit to market points. However, practical experience and recent analyses demonstrate substantial risks:

• Spline in log-moneyness (ln(K/F)): Yields a smooth surface, but wing extrapolation may induce negative local variance if the slope at the boundary is not managed. Still, for five-point FX smiles, cubic spline in log-moneyness is found to be the most robust of the “exact” interpolants (Healy, 22 Dec 2025).
• Delta-based Splines/Polynomials: Spline or polynomial in delta (forward or simple) can produce oscillatory densities, non-monotonic mappings, or root-finding pathologies. The quartic-exponential in simple delta, though “exact” at control points, can interpolate below/above anchor levels and generate local arbitrage or even non-convergent inversions (Healy, 22 Dec 2025).
• Arbitrage Pathologies: Explicit counterexamples document spike-and-dip behavior in risk-neutral density, negative variance regions, multi-modality, and fixed-point failures, particularly in thinly quoted or extreme skew markets (Healy, 22 Dec 2025).

5. Cross-Currency and Copula Interpolation

For cross pairs (e.g., EUR/JPY constructed from EUR/USD and USD/JPY), cross-consistent interpolation must respect absence of triangular arbitrage and correct marginal recovery. The intermediate-currency (numéraire) approach ensures that calibration on one pair suffices to define the volatility smile for all inverse and cross pairs, by deriving all surfaces under a common risk-neutral measure Q_X (Maurer et al., 2019). This method applies both to parametric stochastic-volatility models (Heston, SABR) and non-parametric approaches.

Advanced copula-based constructions, such as the corrected Hermite polynomial expansion, build a nonnegative joint density for the underlying pairs, calibrated to the observed smile marginals. The Hermite copula consistently outperforms standard one-parameter copulas (Gaussian, Frank, etc.) for cross-asset smile synthesis in both fit and out-of-sample tracking (Shiraya et al., 2023).

6. Practical Implementation and Recommendations

Empirical evidence and theoretical pathology motivate several practical lessons:

• Preferred Schemes: Calibrate parametric models (SABR with β=1, SVI/xSSVI, LVG) to anchor points and use them for global interpolation and robust extrapolation (Dellaportas et al., 2014, Healy, 22 Dec 2025).
• Safety in Spline: If pure interpolation is required, use a cubic spline in log-moneyness over carefully delta-inverted control points; always inspect wing slopes and second derivatives (Healy, 22 Dec 2025).
• Delta-space Caution: Avoid simple-delta polynomial fits for long-dated, low-vol, or highly skewed markets. Avoid naive spline in delta except with explicit monitoring of monotonicity, bracketed inversions, and static arbitrage (Healy, 22 Dec 2025, Healy, 22 Dec 2025).
• Cross-Consistency: Employ intermediate-currency (Q_X) architectures or Hermite copula expansions for cross-pair smile synthesis, ensuring coherent risk-neutral densities and absence of triangular arbitrage (Maurer et al., 2019, Shiraya et al., 2023).
• Arbitrage Checks: After each interpolation, numerically verify ∂C/∂K≤0 (monotonicity) and ∂²C/∂K²≥0 (convexity), and in surface routines, check calendar-spread monotonicity in total variance (Dellaportas et al., 2014, Healy, 22 Dec 2025).
• Wing Extrapolation: Linear extrapolation in ln(K) vs. total variance, enforcing Lee’s formula asymptotics, prevents density negativity and preserves tail regularity. Parametric models typically satisfy this (Janek et al., 2010, Dellaportas et al., 2014).

7. Summary Table: Interpolation Schemes and Risks

Method	Arbitrage Robustness	Typical Pitfalls
SABR (β=1)	High	Far OTM breakdown on extreme ν, ρ
Heston / LVG	High	Weak wing shape (Heston), param link
SVI/xSSVI	Enforced (with constraint)	Param errors can cause local issues
Spline in log-moneyness	Medium-High (if slope-sane)	Negative variance in extrapolated wings
Spline/poly in Δ-space	Low-Medium	Oscillations, root-finding failures, local arbitrage
Hermite Copula (cross-pair)	High	Projection step required

Empirical market usage and extensive analysis corroborate that interpolation methodology choice must balance exact quote fit, computational tractability, and strict adherence to arbitrage constraints at every stage—from quote consumption, through model calibration, to out-of-sample pricing (Healy, 22 Dec 2025, Healy, 22 Dec 2025, Dellaportas et al., 2014, Janek et al., 2010, Shiraya et al., 2023).