Sticky Strike

Under the sticky strike rule, the implied volatility for a specific contractual strike price $K$ is invariant to movements in the spot price $S$. If the underlying asset rallies, the volatility assigned to that fixed strike does not change. This causes the volatility smile to shift horizontally along the strike axis, effectively changing the implied volatility for a given moneyness level. This behavior is the opposite of the sticky delta regime, where volatility is tied to the option's moneyness ($K/S$) rather than the absolute strike price.

The sticky strike dynamic is formally expressed as:

$\sigma_{imp}(K, t; S) = \sigma_{imp}(K)$

This states that implied volatility is a function of the strike $K$ only, not the spot price $S$. The partial derivative with respect to the underlying is zero: $\frac{\partial \sigma_{imp}}{\partial S} = 0$.

• Moneyness Shift: As $S$ increases, a fixed $K$ option becomes further out-of-the-money, but its implied volatility remains unchanged.
• Skew Persistence: The shape of the volatility skew relative to strike prices remains fixed, sliding along the price axis.

Sticky strike dynamics are most empirically valid in markets where volatility is driven by supply and demand for specific strikes rather than by broad market moves. This regime is often observed:

• Short-dated options: Near expiration, volatility is heavily influenced by gamma hedging flows at specific strikes.
• Commodity markets: Where volatility is tied to physical delivery levels and producer hedging at fixed price points.
• Fixed-income markets: Where specific strike levels correspond to key rate levels defended by central bank policy.

In equity index markets, sticky strike is generally rejected in favor of sticky delta or more complex dynamics.

Assuming sticky strike dynamics has profound consequences for delta hedging and profit and loss attribution:

• Delta Calculation: The Black-Scholes delta must be adjusted. Since volatility does not change with spot, the standard delta is correct, and no vanna adjustment is needed.
• P&L Decomposition: A long gamma position benefits fully from realized volatility without being offset by a decline in implied volatility (no 'volatility slide').
• Risk Management: Traders hedging exotic books must correctly identify the prevailing dynamics. Mistaking a sticky strike regime for sticky delta leads to systematic hedging errors and mispricing of path-dependent options.

These two regimes represent opposite ends of the volatility surface dynamics spectrum:

• Sticky Strike: $\sigma(K)$ is constant. The smile shifts horizontally. A fixed-strike option's implied volatility never changes. The at-the-money volatility changes as spot moves.
• Sticky Delta: $\sigma(\Delta)$ is constant. The smile is fixed in moneyness space. A fixed-strike option's implied volatility changes with spot to maintain constant moneyness volatility.

Most real-world markets exhibit a hybrid behavior, often modeled with a weighted combination or through stochastic volatility models that interpolate between these extremes.

The sticky strike assumption is closely related to local volatility models. In a pure local volatility framework, the instantaneous volatility is a deterministic function $\sigma(S, t)$. When the spot price moves, the volatility surface predicted by the local volatility model shifts in a manner that approximates sticky strike dynamics for small moves.

• Dupire Equation: The local volatility surface is calibrated to match the initial implied volatility surface exactly.
• Forward Smile: The local volatility model predicts how the smile evolves, which often resembles sticky strike behavior over short time horizons.
• Limitation: Pure local volatility models typically underpredict the movement of the smile, leading to the development of stochastic-local volatility hybrids.

The primary alternative to sticky strike. Under sticky delta, the implied volatility for a specific moneyness (e.g., 25-delta call) remains constant as the underlying moves. This means the volatility surface shifts horizontally with the spot price. It is the standard assumption for many equity derivatives desks because it preserves the shape of the skew relative to the forward price.

A refined version of sticky delta that accounts for the change in delta itself as the underlying moves. Under sticky true delta, the implied volatility for a specific delta value is preserved, but the delta is recalculated using the new spot price. This creates a more complex surface evolution that sits between pure sticky strike and pure sticky delta behavior.

A deterministic function σ(S,t) calibrated to exactly reproduce the current market prices of vanilla options. In a local volatility model, the instantaneous volatility is a function of the spot price and time. This framework inherently defines how the implied volatility surface moves: as the spot changes, the local vol at the new spot level dictates the new implied vol, creating a unique surface dynamic.

Models where volatility itself follows a random process, such as the Heston model. In these frameworks, the volatility surface dynamics are driven by changes in the latent variance state variable. A shock to the spot price can be correlated with a shock to variance via the spot-vol correlation (ρ) parameter, producing realistic joint dynamics that neither pure sticky strike nor sticky delta can capture alone.

The broader study of how the entire implied volatility surface evolves over time. Empirical analysis using Principal Component Analysis (PCA) typically identifies three dominant modes of surface movement:

• Parallel shift: The entire surface moves up or down (level factor)
• Twist: Short-dated vols move opposite to long-dated vols (term structure factor)
• Skew steepening/flattening: The slope across strikes changes (skew factor) Sticky strike is a specific, simplified rule within this broader field.

A trading strategy exploiting discrepancies between the implied volatility surface dynamics and forecasted realized volatility. If a trader believes the market is pricing options using a sticky strike assumption but the true dynamics follow sticky delta, they can construct delta-hedged portfolios to profit from the mispricing. This involves buying undervalued options and selling overvalued ones relative to the expected surface evolution.