Sticky Strike vs Sticky Delta¶

Introduction¶

When the underlying price moves, the implied volatility surface changes in complex ways. Understanding these dynamics is critical for delta hedging, P&L attribution, and model selection. Two stylized assumptions—sticky strike and sticky delta—describe idealized behaviors of how implied volatility responds to spot movements and serve as polar benchmarks for analyzing real market behavior.

This section develops the mathematical framework for these assumptions, derives their implications for Greeks and hedging, and examines empirical evidence on actual smile dynamics.

Theoretical Framework¶

1. The Central Question¶

Consider an option with strike \(K\) and maturity \(T\). At time \(t\), with spot price \(S_t\), it has implied volatility \(\sigma_{\text{IV}}(K, T, S_t)\).

Question: When the spot moves from \(S_t\) to \(S_t + \Delta S\), how does the implied volatility change?

Three quantities characterize the answer: 1. The new implied volatility at strike \(K\): \(\sigma_{\text{IV}}(K, T, S_t + \Delta S)\) 2. The new implied volatility at the new ATM forward 3. The relationship between strike-space and delta-space

2. Coordinate Systems for the Smile¶

Strike space: Implied volatility as a function of absolute strike \(K\):

\[ \sigma_{\text{IV}} = \sigma(K, T) \]

Moneyness space: Implied volatility as a function of moneyness \(m = K/F\):

\[ \sigma_{\text{IV}} = \sigma(m, T) \]

Log-moneyness space: Implied volatility as a function of \(k = \ln(K/F)\):

\[ \sigma_{\text{IV}} = \sigma(k, T) \]

Delta space: Implied volatility as a function of option delta \(\Delta\):

\[ \sigma_{\text{IV}} = \sigma(\Delta, T) \]

The choice of coordinate system determines how we parametrize the smile and affects the interpretation of smile dynamics.

Sticky Strike Assumption¶

1. Definition¶

Under the sticky strike assumption, implied volatility is a function of absolute strike only, independent of spot:

\[ \sigma_{\text{IV}}(K, T; S) = \sigma(K, T) \quad \text{(independent of } S \text{)} \]

Consequence: When spot moves, the implied volatility at a fixed strike \(K\) remains unchanged:

\[ \frac{\partial \sigma_{\text{IV}}(K)}{\partial S} = 0 \]

2. Graphical Interpretation¶

Under sticky strike: - The smile "stays in place" in \((K, \sigma)\) space - The forward price \(F = S e^{(r-q)T}\) shifts with spot - The ATM point moves along the existing smile curve

Example: If spot moves from \(S = 100\) to \(S = 102\): - The strike \(K = 100\) (was ATM) now has the same IV as before - The new ATM forward (\(K \approx 102\)) picks up the IV from the old OTM call region

3. Mathematical Formulation¶

Let \(\sigma(K)\) be the smile function. Under sticky strike:

\[ \sigma_{\text{IV}}(K; S_t) = \sigma(K) \quad \forall S_t \]

The total derivative of IV with respect to spot is:

\[ \frac{d\sigma_{\text{IV}}}{dS} = \frac{\partial \sigma}{\partial K} \cdot \underbrace{\frac{\partial K}{\partial S}}_{=0} = 0 \]

since strike is held constant.

4. Implications for Delta¶

The Black-Scholes delta is:

\[ \Delta_{\text{BS}} = e^{-qT} \Phi(d_1) \]

Under sticky strike, the total delta includes the smile effect:

\[ \Delta_{\text{total}} = \frac{\partial C}{\partial S} = \Delta_{\text{BS}} + \mathcal{V} \cdot \frac{\partial \sigma_{\text{IV}}}{\partial S} \]

Since \(\frac{\partial \sigma_{\text{IV}}}{\partial S} = 0\) under sticky strike:

\[ \Delta_{\text{sticky-strike}} = \Delta_{\text{BS}} \]

Result: Under sticky strike, the Black-Scholes delta (computed at the current implied volatility) is the correct hedge ratio.

5. Local Volatility Connection¶

Theorem: The sticky strike assumption is consistent with local volatility models.

In a local volatility model:

\[ dS_t = (r - q) S_t dt + \sigma_{\text{loc}}(S_t, t) S_t dW_t \]

The implied volatility surface is determined by the local volatility function via the Dupire equation. When spot moves: - The local volatility function \(\sigma_{\text{loc}}(S, t)\) is fixed - The implied volatility at each strike remains unchanged - This is precisely sticky strike behavior

Caveat: This is an idealization. Real local vol dynamics exhibit more complex behavior over short horizons.

Sticky Delta Assumption¶

1. Definition¶

Under the sticky delta assumption, implied volatility is a function of option delta only, independent of spot:

\[ \sigma_{\text{IV}}(\Delta, T; S) = \sigma(\Delta, T) \quad \text{(independent of } S \text{)} \]

Consequence: When spot moves, the implied volatility at a fixed delta level remains unchanged:

\[ \frac{\partial \sigma_{\text{IV}}(\Delta)}{\partial S} = 0 \]

2. Graphical Interpretation¶

Under sticky delta: - The smile "shifts" with spot in \((K, \sigma)\) space - A fixed-delta point (e.g., 25-delta put) always has the same IV - The strike corresponding to that delta changes with spot

Example: If spot moves from \(S = 100\) to \(S = 102\): - The 25-delta put strike shifts from ~\(K_1\) to ~\(K_2\) - Both \(K_1\) (at old spot) and \(K_2\) (at new spot) have the same IV - In strike space, the entire smile has "shifted right"

3. Mathematical Formulation¶

Let \(\sigma(\Delta)\) be the smile function in delta space. The strike corresponding to delta \(\Delta\) depends on spot:

\[ K(\Delta, S) = S \cdot f(\Delta, \sigma, T, r, q) \]

where \(f\) is the inverse delta function.

Under sticky delta, the IV at a fixed strike \(K_0\) changes as spot moves because the delta at \(K_0\) changes:

\[ \frac{d\sigma_{\text{IV}}(K_0)}{dS} = \frac{\partial \sigma}{\partial \Delta} \cdot \frac{\partial \Delta(K_0)}{\partial S} \]

Since \(\frac{\partial \Delta}{\partial S} > 0\) (delta increases with spot for calls), and \(\frac{\partial \sigma}{\partial \Delta}\) is typically negative (OTM puts have higher IV), we have:

\[ \frac{d\sigma_{\text{IV}}(K_0)}{dS} < 0 \quad \text{(for typical equity skew)} \]

Result: Under sticky delta, implied volatility at a fixed strike decreases when spot increases (for negatively skewed markets).

4. Implications for Delta¶

The total delta under sticky delta includes the smile effect:

\[ \Delta_{\text{total}} = \Delta_{\text{BS}} + \mathcal{V} \cdot \frac{\partial \sigma_{\text{IV}}}{\partial S} \]

Since \(\frac{\partial \sigma_{\text{IV}}}{\partial S} < 0\) for typical equity skew:

\[ \Delta_{\text{sticky-delta}} = \Delta_{\text{BS}} + \underbrace{\mathcal{V} \cdot \frac{\partial \sigma_{\text{IV}}}{\partial S}}_{< 0} \]

Result: Under sticky delta, the total delta is smaller than the Black-Scholes delta.

Intuition: When spot rises, IV falls, which partially offsets the direct price increase. The hedge ratio should account for this.

5. FX Market Convention¶

The sticky delta assumption is closely aligned with FX market conventions:

FX options are quoted in delta terms (25-delta call, 10-delta put, ATM)
Market makers quote IV at fixed delta levels
When spot moves, the strikes adjust but delta-quoted IVs stay similar

This makes sticky delta a natural assumption for FX volatility models.

Quantitative Comparison¶

1. Smile Parametrization¶

Consider a simple linear skew model:

\[ \sigma(k) = \sigma_0 + \beta k \]

where \(k = \ln(K/F)\) is log-moneyness and \(\beta < 0\) (downward skew).

Under sticky strike: \(\sigma(K)\) is fixed. When \(F\) changes (spot moves), log-moneyness \(k\) changes, so the IV at ATM changes.

Under sticky delta: The smile in \((k, \sigma)\) space shifts so that the delta-equivalent points have constant IV.

2. Delta Comparison¶

Proposition: For a call option with negative skew (\(\beta < 0\)):

\[ \Delta_{\text{sticky-delta}} < \Delta_{\text{BS}} < \Delta_{\text{sticky-strike}} \cdot \text{(adjusted)} \]

Actually, under sticky strike:

\[ \Delta_{\text{sticky-strike}} = \Delta_{\text{BS}} \]

And under sticky delta:

\[ \Delta_{\text{sticky-delta}} = \Delta_{\text{BS}} + \mathcal{V} \cdot \frac{\partial \sigma}{\partial S} \]

For equity skew with \(\frac{\partial \sigma}{\partial S} < 0\):

\[ \Delta_{\text{sticky-delta}} < \Delta_{\text{sticky-strike}} \]

3. Numerical Example¶

Parameters: - \(S_0 = 100\), \(K = 100\) (ATM), \(T = 0.25\), \(r = 5\%\), \(q = 0\) - \(\sigma_{\text{ATM}} = 20\%\) - Skew: \(\frac{\partial \sigma}{\partial k} = -20\%\) per unit log-moneyness

Black-Scholes delta:

\[ \Delta_{\text{BS}} = \Phi(d_1) = \Phi(0.175) \approx 0.569 \]

Smile adjustment:

\[ \frac{\partial \sigma}{\partial S} = \frac{\partial \sigma}{\partial k} \cdot \frac{\partial k}{\partial S} = -0.20 \times \left(-\frac{1}{S}\right) = \frac{0.20}{100} = 0.002 \]

Vega:

\[ \mathcal{V} \approx 19.73 \]

Sticky delta adjustment:

\[ \Delta_{\text{sticky-delta}} = 0.569 + 19.73 \times 0.002 = 0.569 + 0.039 = 0.608 \]

Wait, this is larger, not smaller. Let me recalculate.

The sign depends on the direction of the skew effect. With \(\frac{\partial \sigma}{\partial k} < 0\):

\[ \frac{\partial \sigma}{\partial S} = \frac{\partial \sigma}{\partial k} \cdot \frac{\partial k}{\partial S} = \frac{\partial \sigma}{\partial k} \cdot \frac{-1}{S} \]

So if \(\frac{\partial \sigma}{\partial k} = -0.20\):

\[ \frac{\partial \sigma}{\partial S} = (-0.20) \times \left(\frac{-1}{100}\right) = +0.002 \]

This gives a positive adjustment, making \(\Delta_{\text{sticky-delta}} > \Delta_{\text{BS}}\).

Correction: The relationship depends on how skew is defined and whether we're looking at OTM puts or calls. For a negatively skewed smile where OTM puts have higher IV: - Spot up → fixed strike becomes more OTM put-like → IV increases - This is not sticky delta behavior

Let me reformulate more carefully.

4. Correct Formulation¶

Under sticky moneyness (log-moneyness constant):

\[ \sigma(k, T) \text{ fixed}, \quad k = \ln(K/F) \]

When \(S\) increases, \(F\) increases, so \(k = \ln(K/F)\) decreases for fixed \(K\).

\[ \frac{\partial \sigma}{\partial S}\bigg|_{K \text{ fixed}} = \frac{\partial \sigma}{\partial k} \cdot \frac{\partial k}{\partial S} = \frac{\partial \sigma}{\partial k} \cdot \left(-\frac{1}{S}\right) \]

For downward skew (\(\frac{\partial \sigma}{\partial k} < 0\)):

\[ \frac{\partial \sigma}{\partial S}\bigg|_{K \text{ fixed}} > 0 \]

Interpretation: Under sticky moneyness, when spot rises, the IV at a fixed strike increases because that strike becomes relatively more OTM (lower moneyness), moving into higher-IV territory.

Under sticky strike: \(\frac{\partial \sigma}{\partial S}\big|_{K \text{ fixed}} = 0\).

Under sticky delta: The IV at a fixed delta stays constant, meaning the smile in strike space shifts.

Impact on Greeks¶

1. Adjusted Delta¶

The general formula for total delta is:

\[ \Delta_{\text{total}} = \frac{\partial C}{\partial S} = \Delta_{\text{BS}} + \frac{\partial C}{\partial \sigma} \cdot \frac{\partial \sigma}{\partial S} = \Delta_{\text{BS}} + \mathcal{V} \cdot \frac{\partial \sigma}{\partial S} \]

Assumption	\(\frac{\partial \sigma}{\partial S}\)	Delta Adjustment
Sticky strike	0	None
Sticky moneyness	\(-\frac{1}{S} \frac{\partial \sigma}{\partial k}\)	\(\mathcal{V} \cdot \frac{\partial \sigma}{\partial S}\)
Sticky delta	Depends on \(\frac{\partial \sigma}{\partial \Delta}\)	Complex

2. Vanna and Smile Dynamics¶

Vanna measures the sensitivity of delta to volatility, or equivalently, of vega to spot:

\[ \text{Vanna} = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial \mathcal{V}}{\partial S} \]

Under different smile dynamics:

Sticky strike:

\[ \text{Total Vanna} = \text{Vanna}_{\text{BS}} \]

Sticky delta:

\[ \text{Total Vanna} = \text{Vanna}_{\text{BS}} + \text{Volga} \cdot \frac{\partial \sigma}{\partial S} + \mathcal{V} \cdot \frac{\partial^2 \sigma}{\partial S \partial S} \]

The smile dynamics introduce additional vanna-like effects.

3. Gamma Adjustment¶

Similarly, the total gamma depends on smile dynamics:

\[ \Gamma_{\text{total}} = \Gamma_{\text{BS}} + 2 \cdot \text{Vanna} \cdot \frac{\partial \sigma}{\partial S} + \mathcal{V} \cdot \frac{\partial^2 \sigma}{\partial S^2} \]

Under sticky strike, the correction terms vanish.

P&L Attribution Under Different Dynamics¶

1. Delta-Hedged P&L¶

For a delta-hedged option position:

\[ \text{P\&L} = \frac{1}{2} \Gamma (\Delta S)^2 + \mathcal{V} \Delta\sigma + \text{Vanna} \cdot \Delta S \cdot \Delta\sigma + \Theta \Delta t \]

The vanna term captures the cross-effect between spot and volatility moves.

2. Sticky Strike P&L¶

Under sticky strike, \(\Delta\sigma|_{K \text{ fixed}} = 0\), so:

\[ \text{P\&L}_{\text{sticky-strike}} = \frac{1}{2} \Gamma (\Delta S)^2 + \Theta \Delta t \]

No volatility P&L (at fixed strike).

3. Sticky Delta P&L¶

Under sticky delta, \(\Delta\sigma|_{\Delta \text{ fixed}} = 0\), but \(\Delta\sigma|_{K \text{ fixed}} \neq 0\):

\[ \text{P\&L}_{\text{sticky-delta}} = \frac{1}{2} \Gamma (\Delta S)^2 + \mathcal{V} \Delta\sigma_K + \text{Vanna} \cdot \Delta S \cdot \Delta\sigma_K + \Theta \Delta t \]

where \(\Delta\sigma_K\) is the change in IV at the fixed strike \(K\).

Empirical Evidence¶

1. Equity Markets¶

Empirical studies of equity index options (S&P 500, EURO STOXX) show:

Short-term behavior: - Closer to sticky strike than sticky delta - IV at fixed strikes relatively stable over short horizons - Skew steepens after large down moves

Medium-term behavior: - Neither assumption holds perfectly - Spot-vol correlation (leverage effect) dominates - Smile dynamics are asymmetric: faster reaction to down moves

Quantitative finding: The "skew stickiness ratio" (SSR):

\[ \text{SSR} = \frac{\text{ATM IV change}}{\text{Predicted change under sticky strike}} \]

Empirical SSR for SPX is typically 0.3-0.6, indicating behavior between sticky strike and sticky moneyness.

2. FX Markets¶

Short-term: Closer to sticky delta - Market makers quote at fixed delta levels - IV at 25-delta put/call relatively stable

Medium-term: Mean reversion in both spot and volatility complicates the picture

Risk reversals: The 25D risk reversal (IV of 25D call minus 25D put) is more stable than individual strike IVs.

3. Single-Stock Options¶

Idiosyncratic behavior: - More noise, less clear pattern - Event-driven (earnings, M&A) dominates - Sector effects matter

General tendency: Between sticky strike and sticky delta, with significant variation.

Model Implications¶

1. Local Volatility Models¶

Local volatility models produce sticky strike behavior by construction: - The local vol surface \(\sigma_{\text{loc}}(S, t)\) is fixed - When spot moves, the path through local vol space changes - But implied volatility at each strike is determined by the same integral

Problem: This implies unrealistic forward smiles and dynamics.

2. Stochastic Volatility Models¶

Stochastic volatility models (Heston, SABR) produce dynamics between sticky strike and sticky delta: - The spot-vol correlation (\(\rho\)) controls the leverage effect - Negative \(\rho\) (typical for equities) produces skew that steepens when spot falls - This is neither pure sticky strike nor pure sticky delta

SABR: The SABR model with backbone parameter \(\beta\) interpolates: - \(\beta = 1\): Normal SABR, sticky strike-like - \(\beta = 0\): Lognormal SABR, sticky delta-like

3. Bergomi's Variance Curve Models¶

Bergomi's framework models the forward variance curve directly:

\[ d\xi_t^T = \xi_t^T \cdot \omega(T-t) \cdot dZ_t \]

The resulting smile dynamics are: - More realistic than local vol - Capture term structure of skew - Can match empirical SSR observations

Practical Hedging Implications¶

1. Delta Hedging Choices¶

Assumption	Hedge Ratio	When to Use
Sticky strike	\(\Delta_{\text{BS}}\)	Local vol models, short-term
Sticky delta	Adjusted \(\Delta\)	FX markets, longer horizons
Empirical	Blend or estimated	Most realistic

2. Estimating Smile Dynamics¶

Regression approach:

\[ \Delta\sigma_{\text{ATM}} = \alpha + \beta \cdot \frac{\Delta S}{S} + \epsilon \]

The coefficient \(\beta\) estimates the spot-vol sensitivity: - \(\beta = 0\): Sticky strike - \(\beta = -\text{skew}\): Sticky moneyness - Intermediate: Empirical dynamics

3. Robust Hedging¶

Given uncertainty about smile dynamics:

Conservative approach: Use the hedge ratio that performs better in adverse scenarios
Scenario analysis: Test P&L under sticky strike, sticky delta, and historical dynamics
Dynamic adjustment: Update hedge ratio estimates as market conditions change

Summary¶

Sticky strike and sticky delta represent idealized extremes of smile dynamics:

1. Sticky Strike¶

\[ \frac{\partial \sigma_{\text{IV}}(K)}{\partial S} = 0 \]

IV at fixed strike unchanged when spot moves
Consistent with local volatility models
Delta = Black-Scholes delta

2. Sticky Delta¶

\[ \frac{\partial \sigma_{\text{IV}}(\Delta)}{\partial S} = 0 \]

IV at fixed delta unchanged when spot moves
Smile shifts in strike space
Common in FX markets
Delta requires adjustment

3. Reality¶

Neither assumption holds perfectly
Short-term equity: closer to sticky strike
FX markets: closer to sticky delta
Leverage effect creates asymmetric dynamics
Empirical "skew stickiness ratio" typically 0.3-0.6

4. Implications¶

Aspect	Sticky Strike	Sticky Delta
Delta	\(\Delta_{\text{BS}}\)	\(\Delta_{\text{BS}} + \mathcal{V} \frac{\partial\sigma}{\partial S}\)
Vanna effect	Standard	Enhanced
Gamma	Standard	Modified
P&L attribution	No vol P&L at fixed \(K\)	Vol P&L present

Understanding these benchmarks enables more nuanced analysis of actual market dynamics and more robust hedging strategies.

Exercises¶

Exercise 1. Under the sticky strike assumption, the implied volatility at a fixed strike \(K\) does not change when the spot moves. If an ATM call option (\(K = S_0 = 100\)) has \(\sigma_{\text{IV}} = 22\%\) and the spot rises to \(S = 103\), what is the implied volatility at \(K = 100\) under sticky strike? What is the implied volatility at the new ATM strike (\(K = 103\)) if the smile has skew \(\frac{\partial \sigma}{\partial k} = -15\%\)?

Solution to Exercise 1

Under sticky strike, \(\partial\sigma_{\text{IV}}(K)/\partial S = 0\), so the implied volatility at any fixed strike is unchanged when spot moves.

When spot rises from \(S_0 = 100\) to \(S = 103\), the IV at \(K = 100\) remains \(\sigma_{\text{IV}}(K=100) = 22\%\).

For the new ATM strike \(K = 103\), we use the smile. The log-moneyness of \(K = 103\) relative to the original forward is:

\[ k = \ln(K/F) = \ln(103/F) \]

Under sticky strike, the smile function \(\sigma(K)\) is fixed. The new ATM strike \(K \approx 103\) was originally an OTM call. With skew \(\partial\sigma/\partial k = -15\%\) and log-moneyness \(k = \ln(103/100) \approx 0.03\):

\[ \sigma_{\text{IV}}(K=103) \approx 22\% + (-15\%)\times 0.03 = 22\% - 0.45\% = 21.55\% \]

The new ATM implied volatility is approximately 21.55%, lower than the original 22% because the smile is downward-sloping and the ATM point has shifted to higher strikes.

Exercise 2. Under the sticky delta assumption, the implied volatility at a fixed delta level is unchanged when spot moves. If the 25-delta put has \(\sigma_{\text{IV}} = 27\%\) when \(S_0 = 100\), what is its implied volatility when the spot moves to \(S = 105\)? In absolute strike space, does the \(K\) corresponding to the 25-delta put increase or decrease?

Solution to Exercise 2

Under sticky delta, the IV at a fixed delta level does not change when spot moves. The 25-delta put has \(\sigma_{\text{IV}} = 27\%\) at \(S_0 = 100\), and this remains \(27\%\) when spot moves to \(S = 105\).

In absolute strike space, the strike corresponding to the 25-delta put increases when spot rises. This is because the 25-delta put strike scales approximately with the forward price:

\[ K_{25\Delta P} \approx F \cdot e^{-c} \]

for some constant \(c > 0\) that depends on volatility and maturity. When \(S\) increases from 100 to 105, \(F\) increases proportionally, so \(K_{25\Delta P}\) also increases. The entire smile has effectively shifted to the right in strike space, maintaining the same shape in delta space.

Exercise 3. For a call option with \(S_0 = 100\), \(K = 100\), \(T = 0.25\), \(r = 5\%\), \(q = 0\), \(\sigma_{\text{ATM}} = 20\%\), and skew \(\frac{\partial \sigma}{\partial k} = -20\%\), compute the total delta under sticky moneyness. Use

\[ \Delta_{\text{total}} = \Delta_{\text{BS}} + \mathcal{V} \cdot \frac{\partial \sigma}{\partial S} \]

with \(\Delta_{\text{BS}} = 0.569\) and \(\mathcal{V} = 19.73\). Compare with the Black-Scholes delta (sticky strike delta).

Solution to Exercise 3

We are given \(\Delta_{\text{BS}} = 0.569\), \(\mathcal{V} = 19.73\), and \(\partial\sigma/\partial k = -0.20\).

Under sticky moneyness, the IV at a fixed strike changes when spot moves because log-moneyness \(k = \ln(K/F)\) changes. The sensitivity of IV to spot at fixed strike is:

\[ \frac{\partial \sigma}{\partial S}\bigg|_{K} = \frac{\partial \sigma}{\partial k}\cdot\frac{\partial k}{\partial S} = \frac{\partial \sigma}{\partial k}\cdot\left(-\frac{1}{S}\right) = (-0.20)\times\left(-\frac{1}{100}\right) = +0.002 \]

The total delta under sticky moneyness:

\[ \Delta_{\text{total}} = \Delta_{\text{BS}} + \mathcal{V}\cdot\frac{\partial\sigma}{\partial S} = 0.569 + 19.73\times 0.002 = 0.569 + 0.039 = 0.608 \]

Compared to the Black-Scholes delta (which equals the sticky strike delta, \(\Delta_{\text{SS}} = 0.569\)), the sticky moneyness delta is higher by 0.039, or about 6.9%. The positive adjustment arises because, under sticky moneyness, when spot rises the fixed strike becomes more in-the-money in moneyness terms, and the negative skew pushes IV higher at that fixed strike, amplifying the price increase.

Exercise 4. The skew stickiness ratio (SSR) is defined as the ratio of the actual ATM IV change to the predicted change under sticky strike. If the SSR for an equity index is 0.5, and the smile has \(\frac{\partial \sigma}{\partial k} = -18\%\), determine the actual delta adjustment relative to the full sticky-moneyness adjustment. Where does this place the true dynamics on the spectrum from sticky strike to sticky moneyness?

Solution to Exercise 4

The SSR of 0.5 means the actual ATM IV change is 50% of what sticky moneyness predicts.

Under sticky moneyness, the full delta adjustment is:

\[ \mathcal{V}\cdot\frac{\partial\sigma}{\partial S} = \mathcal{V}\cdot\left(-\frac{1}{S}\right)\cdot\frac{\partial\sigma}{\partial k} \]

The actual delta adjustment is:

\[ \text{Actual adjustment} = \text{SSR}\times\mathcal{V}\cdot\frac{\partial\sigma}{\partial S} = 0.5\times\mathcal{V}\cdot\frac{\partial\sigma}{\partial S} \]

This places the true dynamics exactly halfway between sticky strike (SSR = 0, no adjustment) and sticky moneyness (SSR = 1, full adjustment).

With \(\partial\sigma/\partial k = -0.18\), the full adjustment per unit spot move is \(\mathcal{V}\times 0.18/S\). The actual adjustment is \(0.5\times\mathcal{V}\times 0.18/S = \mathcal{V}\times 0.09/S\).

An SSR of 0.5 is in the middle of the empirical range (0.3--0.6 for SPX), consistent with the observation that equity index dynamics lie between the two extremes, influenced by both local volatility effects (sticky strike) and stochastic volatility effects (sticky moneyness/delta).

Exercise 5. Explain why FX markets naturally align with the sticky delta assumption while equity markets tend toward sticky strike behavior. Discuss the role of quoting conventions (delta-quoted vs. strike-quoted) in this distinction.

Solution to Exercise 5

FX markets naturally align with sticky delta because:

FX options are quoted at fixed delta levels (10-delta, 25-delta, ATM-delta). Market makers post bid/offer at these delta points, so the market convention itself anchors IV to delta rather than strike.
FX spot-vol correlation is often near zero (e.g., EUR/USD), meaning the smile shifts symmetrically with spot, consistent with sticky delta.
There is no strong directional leverage effect in major FX pairs, unlike equities.

Equity markets tend toward sticky strike because:

Equity options are listed at fixed absolute strikes (e.g., \(K = 90, 95, 100, \ldots\)). Traders monitor IV at these specific strikes.
The local volatility framework, which produces sticky strike behavior, captures short-term equity dynamics reasonably well.
However, the leverage effect (negative spot-vol correlation) means the actual behavior deviates from pure sticky strike over longer horizons.

The quoting convention plays a reinforcing role: when market makers anchor quotes to fixed deltas (FX) versus fixed strikes (equities), the respective sticky assumption becomes a self-fulfilling approximation over short horizons, because hedging and market-making activity tends to stabilize IV in whichever coordinate the market uses.

Exercise 6. In the SABR model, the backbone parameter \(\beta\) interpolates between sticky strike (\(\beta = 1\)) and sticky delta (\(\beta = 0\)) behavior. For intermediate \(\beta = 0.5\) with \(F = 100\), \(\alpha = 0.20\), \(\nu = 0.3\), and \(\rho = -0.4\), qualitatively describe how the smile shifts when the forward moves from 100 to 105. Is the behavior closer to sticky strike or sticky delta?

Solution to Exercise 6

In the SABR model with \(\beta = 0.5\), the dynamics are:

\[ dF_t = \alpha_t F_t^{0.5}\,dW_t^F \]

When \(F\) moves from 100 to 105, the effective local volatility depends on \(F^{0.5}\). The ATM volatility scales approximately as \(\alpha\,F^{\beta - 1} = \alpha\,F^{-0.5}\).

At \(F = 100\): effective vol \(\propto \alpha/\sqrt{100} = \alpha/10\).

At \(F = 105\): effective vol \(\propto \alpha/\sqrt{105} \approx \alpha/10.247\).

The ATM vol decreases slightly as \(F\) increases, which means the smile partially shifts in strike space. Under pure sticky strike (\(\beta = 1\)), ATM vol would be independent of \(F\). Under pure sticky delta (\(\beta = 0\), normal model), ATM vol would shift fully.

With \(\beta = 0.5\), the behavior is intermediate but closer to sticky strike than sticky delta. The negative \(\rho = -0.4\) adds a negative skew, so when \(F\) rises, the left wing steepens somewhat. The overall shift is moderate: the smile moves partially to the right but not as much as sticky delta would predict, placing the dynamics roughly midway between the two extremes but leaning toward sticky strike.

Exercise 7. A hedger must choose between using \(\Delta_{\text{BS}}\) (sticky strike) and the adjusted delta (sticky delta) for an equity index call spread. The portfolio is long an ATM call and short a 110-strike call, both with \(T = 0.5\) years. Under which assumption is the net delta larger? Describe a scenario analysis comparing P&L under both assumptions when the index falls by 5% with a concurrent 3% rise in ATM implied volatility.

Solution to Exercise 7

The portfolio is long an ATM call (\(K = 100\)) and short a 110-strike call, both with \(T = 0.5\). The net delta of a call spread is positive since \(\Delta(K=100) > \Delta(K=110)\).

Under sticky strike (\(\partial\sigma/\partial S = 0\)):

\[ \Delta_{\text{net}}^{\text{SS}} = \Delta_{\text{BS}}(K=100) - \Delta_{\text{BS}}(K=110) \]

Under sticky delta, the total delta for each leg includes the smile adjustment \(\mathcal{V}\cdot\partial\sigma/\partial S\). For typical equity negative skew (\(\partial\sigma/\partial S < 0\)), the adjustment is negative for each option. The OTM call (\(K = 110\)) has smaller vega than the ATM call, so the magnitude of the negative adjustment is smaller for the short leg. Hence:

\[ \Delta_{\text{net}}^{\text{SD}} = [\Delta_{\text{BS}}(100) + \mathcal{V}_{100}\cdot\Sigma_S] - [\Delta_{\text{BS}}(110) + \mathcal{V}_{110}\cdot\Sigma_S] \]

Since \(\mathcal{V}_{100} > \mathcal{V}_{110}\) and \(\Sigma_S < 0\), the long leg is reduced more than the short leg. Therefore \(\Delta_{\text{net}}^{\text{SD}} < \Delta_{\text{net}}^{\text{SS}}\), meaning the net delta is larger under sticky strike.

Scenario analysis (index falls 5%, ATM IV rises 3%):

Under sticky strike: IV at both strikes is unchanged, so P&L is purely from delta and gamma effects. The call spread loses on the downside (\(\Delta > 0\)), partially offset by gamma.
Under sticky delta: IV at fixed strikes increases (spot down pushes up IV at each strike due to negative skew). The long ATM call gains from vega (higher IV increases its value), and the short 110-call also increases in value (a loss on the short position). Since \(\mathcal{V}_{100} > \mathcal{V}_{110}\), the net vega effect is positive, partially offsetting the directional loss. The P&L under sticky delta is less negative than under sticky strike because the vol increase provides a cushion via the net long vega of the call spread.

Sticky Strike vs Sticky Delta¶

Introduction¶

Theoretical Framework¶

1. The Central Question¶

2. Coordinate Systems for the Smile¶

Sticky Strike Assumption¶

1. Definition¶

2. Graphical Interpretation¶

3. Mathematical Formulation¶

4. Implications for Delta¶

5. Local Volatility Connection¶

Sticky Delta Assumption¶

1. Definition¶

2. Graphical Interpretation¶

3. Mathematical Formulation¶

4. Implications for Delta¶

5. FX Market Convention¶

Quantitative Comparison¶

1. Smile Parametrization¶

2. Delta Comparison¶

3. Numerical Example¶

4. Correct Formulation¶

Impact on Greeks¶

1. Adjusted Delta¶

2. Vanna and Smile Dynamics¶

3. Gamma Adjustment¶

P&L Attribution Under Different Dynamics¶

1. Delta-Hedged P&L¶

2. Sticky Strike P&L¶

3. Sticky Delta P&L¶

Empirical Evidence¶

1. Equity Markets¶

2. FX Markets¶

3. Single-Stock Options¶

Model Implications¶

1. Local Volatility Models¶

2. Stochastic Volatility Models¶

3. Bergomi's Variance Curve Models¶

Practical Hedging Implications¶

1. Delta Hedging Choices¶

2. Estimating Smile Dynamics¶

3. Robust Hedging¶

Summary¶

1. Sticky Strike¶

2. Sticky Delta¶

3. Reality¶

4. Implications¶

Further Reading¶

Exercises¶