Sticky Delta

Under the Sticky Delta rule, the implied volatility for a given moneyness (e.g., a 25-Delta call) remains constant as the underlying asset price fluctuates. This implies that the volatility surface shifts horizontally along the strike axis to maintain the same smile shape relative to the new spot price. It is also known as the floating-skew or sticky-moneyness model.

The sticky delta condition is expressed as:

• σ(K, S, t) = σ(K/S, t) This means implied volatility is a function of the ratio of strike to spot (moneyness), not the absolute strike K. When the spot price S moves, the volatility assigned to a specific strike changes because its moneyness has changed. This contrasts sharply with the Sticky Strike rule where σ(K, S, t) = σ(K, t).

This regime typically dominates in markets where:

• Spot moves are perceived as proportional shocks rather than absolute price level shifts.
• Foreign Exchange (FX) markets are the canonical example, where a 1% move in EUR/USD is structurally similar regardless of the absolute exchange rate.
• Equity indices with low volatility risk premium often exhibit sticky delta behavior during range-bound, non-crash periods.
• It reflects a belief that the relative risk distribution is stationary.

Sticky delta dynamics critically alter the minimum variance delta calculation. Because the volatility surface moves with the spot, the effective delta of an option includes a vanna component:

• Total Delta = Black-Scholes Delta + Vanna × (∂σ/∂S)
• Under sticky delta, ∂σ/∂S is non-zero for options away from the at-the-money strike.
• Hedging purely with the Black-Scholes delta will leave a residual vanna exposure, requiring a volatility-adjusted hedge ratio.

These two rules represent opposite ends of the volatility dynamics spectrum:

• Sticky Strike: Volatility for a fixed strike K remains constant. The surface does not shift with spot. Common in equity markets during crash regimes where investors focus on absolute downside protection levels.
• Sticky Delta: Volatility for a fixed moneyness K/S remains constant. The surface shifts horizontally with spot. Common in FX and normal equity conditions.
• In reality, markets often exhibit a hybrid behavior, interpolating between these two extremes.

To test if a market follows sticky delta dynamics, practitioners perform a floating regression:

• Regress changes in implied volatility against changes in the underlying spot price for a fixed delta bucket.
• Under pure sticky delta, the regression coefficient for a fixed delta bucket should be statistically insignificant (zero).
• Conversely, regressing volatility changes for a fixed strike bucket will show a significant negative coefficient, as the strike's moneyness changes with spot.

The alternative volatility dynamics rule where the implied volatility for a fixed strike price remains constant as the underlying asset price moves. Under this regime, a shift in the spot price causes the volatility surface to slide along the strike axis, effectively changing the implied volatility for a given moneyness level. This behavior is often observed during short-term, idiosyncratic stock moves where the market perceives the event as a one-off shock rather than a regime change.

A dimensionless measure defining an option's strike price relative to the current underlying price, typically expressed as K/S (simple moneyness) or log(K/S) (log-moneyness). It is the critical invariant in the Sticky Delta rule. Key categories include:

• At-the-Money (ATM): Strike equals spot.
• In-the-Money (ITM): Option has intrinsic value.
• Out-of-the-Money (OTM): Option has no intrinsic value. Sticky Delta postulates that an OTM put with a delta of -0.25 retains that same volatility even if the spot price rallies.

The quantitative process of fitting a parametric or non-parametric model to market-quoted option prices to construct a smooth, arbitrage-free implied volatility surface. Calibration engines must reconcile the Sticky Delta or Sticky Strike assumptions with observable market dynamics. The choice of dynamics rule directly impacts the Greeks calculated for risk management, particularly Vanna (sensitivity of Delta to volatility) and Volga (sensitivity of Vega to volatility).

A deterministic volatility function σ(S, t) that is exactly calibrated to the current market prices of vanilla options via the Dupire Equation. In a strict local volatility framework, the instantaneous volatility is a function of the future spot price and time, which inherently implies a Sticky Delta behavior. As the spot price moves, the volatility assigned to a given strike changes according to the pre-calibrated local volatility surface, preserving the volatility-by-moneyness relationship.

A class of models, such as the Heston Model or SABR Model, where the variance follows its own random diffusion process. These models introduce volatility of volatility and spot-vol correlation parameters. The dynamics of the surface under these models can blend Sticky Delta and Sticky Strike behaviors depending on the correlation structure. A negative spot-vol correlation generates the characteristic equity skew and causes the surface to shift in a manner that partially resembles Sticky Delta for short expirations.

A statistical dimensionality reduction technique applied to historical volatility surface movements. Principal Component Analysis typically decomposes surface shifts into three orthogonal factors:

• PC1 (Level): A parallel shift affecting all strikes and maturities.
• PC2 (Skew/Tilt): A change in the steepness of the skew.
• PC3 (Curvature): A change in the smile convexity. Empirical PCA often reveals that the dominant real-world dynamics are a hybrid of Sticky Delta and Sticky Strike, informing more accurate risk factor models.