Volatility: Skew Stickiness Ratio

An implied volatility surface is a function of strike and maturity, \(\sigma = \sigma(K, T)\). Tomorrow the underlying spot (or forward) moves. What happens to the surface? The answer is an assumption, not an observable, and it feeds directly into:

• delta and gamma, hence hedging
• P&L attribution
• scenario and stress testing
• spot–volatility dynamics in models

In Black-Scholes, volatility is assumed independent of spot, so the standard delta is \(\Delta_{BS} = \partial V / \partial S\). In reality the implied volatility surface may itself move when spot moves, so the total spot sensitivity of an option's value \(V\) is

\[\frac{dV}{dS} = \frac{\partial V}{\partial S} + \frac{\partial V}{\partial \sigma}\frac{d\sigma}{dS} = \Delta_{BS} + \text{Vega}\cdot\frac{d\sigma}{dS}\]

The first term is the conventional model delta. The second term is new: it captures the effect of the volatility surface moving with spot, scaled by the option's vega, \(\text{Vega} = \partial V/\partial\sigma\), its sensitivity to implied volatility. The value of \(d\sigma/dS\) — how much vol moves per unit spot move — depends entirely on the assumed surface dynamics.

Definition. Under sticky strike, implied volatility at a given strike does not change when spot moves:

\[\sigma_1(K, T) = \sigma_0(K, T) \quad \text{for every fixed } K, T\]

Equivalently, the derivative of the surface with respect to spot at fixed strike and tenor is zero:

\[\left.\frac{\partial\sigma(K, T; S)}{\partial S}\right|_{K,T} = 0\]

The volatility smile is fixed in strike space: a spot move does not shift the curve horizontally. However, the ATM point moves along the existing smile, because the ATM strike is tied to the forward, \(K_{ATM} \approx F\). Therefore ATM volatility does change when spot/forward changes — even though every fixed-strike vol is unchanged. The terminology is standard practitioner vocabulary, dating back at least to Derman's "Regimes of Volatility".

Definition. Under sticky delta, implied volatility is attached to a fixed option delta rather than a fixed strike. When spot moves, the strike associated with a given delta changes.

A common simplified representation parameterises the surface by forward log-moneyness,

\[k = \ln\left(\frac{K}{F}\right), \qquad \sigma = \sigma(k, T)\]

so that a fixed delta remains associated with approximately the same location on the smile. The smile is therefore translated in strike space as the forward moves. Since ATM corresponds to \(k = 0\), ATM volatility remains approximately unchanged under a pure sticky-delta/moneyness rule.

Strike must move with spot. Suppose spot moves from \(S_0\) to \(S_1\) and the forward from \(F_0\) to \(F_1\). Keeping log-moneyness fixed means

\[\ln\left(\frac{K_1}{F_1}\right) = \ln\left(\frac{K_0}{F_0}\right), \qquad \text{so} \quad K_1 = K_0\,\frac{F_1}{F_0}\]

and, approximately, \(K_1 \approx K_0\,(S_1/S_0)\) when \(F \propto S\). Thus a 10% rise in spot shifts the smile approximately 10% higher in strike space.

Derman's original paper describes three regimes, not two. The third — sticky local volatility (also called the sticky implied tree) — treats the local volatility \(\sigma_{loc}(S, t)\) as a static function, so the implied surface depends on spot and strike together, \(\Sigma = \Sigma(K, S)\). Since implied vol is approximately the average of local vols between spot and strike, its behaviour differs sharply from both sticky rules:

• Volatility is anti-correlated with the index: fixed-strike vol falls as the index rises, and ATM vol falls at roughly twice that rate.
• In SSR language this is a regime above sticky strike: \(\text{SSR} \approx 2\) for local volatility with weak skew — which is also the short-maturity limit of classical stochastic-volatility models (see the SSR section below).

Empirically, no pure regime holds for long. Derman's own observation: calm, rising markets behave closer to sticky strike; fearful markets approach sticky local volatility. And on S&P 500 OTC data, Daglish, Hull and Suo (2007) found sticky strike the worst-performing of the three rules. A theoretical caveat for sticky strike: Balland (2002) shows the only arbitrage-free sticky-strike model is plain Black-Scholes — the regimes are practical conventions, not exact dynamics.

Draw the smile as a curve on a chart with strike on the horizontal axis. Sticky strike nails the curve to the axis: when spot moves, the curve does not move — but the dot marking "at the money" slides along it, because the ATM strike follows the forward. Sticky delta does the opposite: the dot stays put (the delta is what the smile is attached to), so the whole curve slides horizontally with spot. The two rules differ on one question: does ATM vol stay fixed (sticky delta) or does it ride the existing smile (sticky strike)?

At a glance:

The two rules are also two practical ways to compute a delta. Desks call them "smile delta" conventions — the delta-quoting side of the same choice (see the delta conventions on the FX Volatility Surface page).

Sticky strike as a computation. Revalue the option with its own implied vol at its strike, then keep that vol fixed as spot moves: the delta is Black-Scholes delta at \(\sigma(K)\). This is the easiest convention to compute, hence the most widely used. Its weakness: it implies the strike is always repriced at the original vol.

Sticky delta as a computation. Keep the smile attached to the delta: an option at 15\(\Delta\) always trades at the same implied vol, but the strike behind that delta changes as spot moves. The convention mirrors the idea that, locally, ATM vol and the smile do not move. Its weakness: ATM and RR do not actually stay fixed when spot moves, so the convention is only a local approximation. Its practical virtue: for small spot moves it avoids "local headaches" — no per-strike revaluation is needed.

Neither rule is taken literally. Traders compensate for the flaws with a spot ladder — revaluing the book's vega at a set of spot levels — and adjust the delta from their experience of how spot, vol and RR actually co-move. That experience is the informal counterpart of the SSR: the SSR is the same spot/vol/RR co-movement, measured and expressed as a number (next section).

A third, cruder convention exists for context: the flat-smile delta — one single vol for all strikes, which is just Black-Scholes delta. It is acknowledged as wrong once a smile exists.

Real-world ATM volatility is not perfectly sticky-strike or perfectly sticky-delta. To describe where a market actually sits, define the ATM skew as the slope of the smile in log-moneyness space at the ATM point:

\[\text{ATM skew} = \left.\frac{\partial\sigma}{\partial \ln K}\right|_{K=F}\]

(In FX, the risk reversal RR is the market's headline measure of skew; the local ATM slope is the same phenomenon at the money. The exact conversion between RR and the ATM slope is smile-model dependent — see the FX Volatility Surface page's SABR section.)

The Skew Stickiness Ratio is the ratio of the observed ATM-volatility response to spot, over the ATM skew:

\[\text{SSR} = \frac{d\sigma_{ATM}/d\ln S}{\left.\partial\sigma/\partial\ln K\right|_{K=F}}\]

For a finite spot move, the corresponding approximation is

\[\text{SSR} \approx \frac{\Delta\sigma_{ATM}}{\Delta\ln S} \cdot \frac{1}{\left.\partial\sigma/\partial\ln K\right|_{K=F}}, \qquad \Delta\ln S = \ln\left(\frac{S_1}{S_0}\right) \approx \frac{\Delta S}{S}\]

Interpreting the number:

• SSR \(= 0\) — sticky delta: ATM volatility does not respond to spot.
• SSR \(= 1\) — sticky strike: ATM volatility moves by the amount the static ATM skew predicts.
• \(0 < \text{SSR} < 1\) — an intermediate response.
• SSR \(> 1\) — ATM volatility moves more strongly with spot than pure sticky strike predicts.

The SSR relation can be rearranged to give the local dynamics of the ATM volatility directly — every derivative below is evaluated at the ATM point:

\[\frac{d\sigma}{d\ln S} = \text{SSR}\cdot\frac{\partial\sigma}{\partial\ln K}, \qquad \text{hence} \quad \frac{d\sigma}{dS} = \frac{\text{SSR}}{S}\cdot\frac{\partial\sigma}{\partial\ln K}\]

under the approximation that the forward moves proportionally with spot. The limiting cases fall out immediately:

\[\text{SSR} = 0 \;\Rightarrow\; \frac{d\sigma}{dS} = 0 \quad \text{(sticky delta)}, \qquad \text{SSR} = 1 \;\Rightarrow\; \frac{d\sigma}{dS} = \frac{1}{S}\frac{\partial\sigma}{\partial\ln K} \quad \text{(sticky strike)}\]

SSR thus gives a continuous measure between the two classical assumptions — and permits values outside \([0, 1]\) when the market is more extreme than either.

The Skew Stickiness Ratio was introduced by Lorenzo Bergomi (Stochastic Volatility Modeling, 2016, ch. 9.4) and is now standard vocabulary for the joint dynamics of an asset and its volatility (Fukasawa 2026). It is a statistic of that joint dynamics, and in practice it must be estimated:

• Estimation. Empirically, SSR is the regression coefficient of ATM implied-vol changes on underlying log returns, normalised by the ATM skew — the finite-move approximation above is its informal counterpart. The ratio is well-defined only when the ATM skew is non-zero: it is most reliable where the skew is deep and persistent (equity indices), and needs care where the smile is flat.
• Reference values. SSR \(= 0\) and \(1\) are the two classical regimes; sticky local volatility with weak skew corresponds to \(\text{SSR} \approx 2\), and classical (non-rough) stochastic-volatility models have short-maturity limit 2, declining toward 1 as maturity grows (Vargas, Dao and Bouchaud 2013). Empirically, index markets sit around \(\text{SSR} = 3/2\) for tenors from one month to a few years (Bergomi, Euro Stoxx 50 and S&P 500), and can exceed 2 at short maturities (Vargas, Dao and Bouchaud, S&P 500 and DAX). Under flat variance-swap curves and time homogeneity, the SSR is model-independently bounded in \([1, 2]\) (Bergomi).
• Why it matters. SSR measures the cross-gamma risk between spot and volatility. A model whose SSR does not match the market's will mis-hedge vega and misprice the spot-vol interaction in scenarios.

Substituting the SSR relation into the total spot sensitivity gives the adjusted delta. For an option at the money — the standard framing for quoting a delta hedge — the volatility that matters is the ATM volatility, whose dynamics the SSR parameterises:

\[\Delta_{adj} = \Delta_{BS} + \text{Vega}\cdot\frac{\text{SSR}}{S}\cdot\frac{\partial\sigma}{\partial\ln K}\]

For an ATM option, the limiting cases are:

• SSR \(= 0\) (sticky delta): ATM vol does not respond to spot, so \(\Delta_{adj} = \Delta_{BS}\) — the conventional delta is the complete answer.
• SSR \(= 1\) (sticky strike): the ATM point slides along the fixed smile, and the vega term contributes through the ATM skew.

For a fixed-strike option the picture inverts. Under sticky strike, the vol at the option's strike never moves, so the conventional Black-Scholes delta is the complete answer; under sticky delta the smile translates with spot, the vol at the fixed strike changes (by roughly \(-\text{skew}\times\Delta\ln S\) at the money), and the delta deviates from \(\Delta_{BS}\). Derman notes the sticky-delta rule is self-referential — the delta is itself a function of volatility. The two framings are consistent: the SSR formula above is the ATM case, and "the option is at the money" is the standard market-maker's assumption when quoting a delta hedge.

Once volatility depends on spot, gamma is affected too. Starting from \(\Delta_{adj} = dV/dS\), the total second derivative is

\[\Gamma_{adj} = \frac{d^2 V}{dS^2} = V_{SS} + 2\,V_{S\sigma}\frac{d\sigma}{dS} + V_{\sigma\sigma}\left(\frac{d\sigma}{dS}\right)^2 + V_\sigma\frac{d^2\sigma}{dS^2}\]

In Greek terminology:

\[\Gamma_{adj} = \Gamma_{BS} + 2\,\text{Vanna}\cdot\frac{d\sigma}{dS} + \text{Vomma}\cdot\left(\frac{d\sigma}{dS}\right)^2 + \text{Vega}\cdot\frac{d^2\sigma}{dS^2}\]

where vanna \(= \partial^2 V/\partial S\,\partial\sigma\) (the cross-sensitivity: how delta responds to vol, equivalently how vega responds to spot) and vomma \(= \partial^2 V/\partial\sigma^2\) (the curvature of vega in vol). For an ATM option under sticky delta (\(d\sigma/dS = 0\)) all the extra terms vanish and \(\Gamma_{adj} = \Gamma_{BS}\); under any non-zero spot-vol dynamics, gamma gains vanna, vomma, and vol-of-vol contributions.

• Hedging. Delta depends on the assumed spot-volatility dynamics. A surface-aware hedge can differ from the conventional Black-Scholes delta.
• P&L attribution. Spot moves can induce changes in implied volatility. Ignoring this interaction shifts P&L between the delta and vega buckets (see P&L attribution).
• Risk and scenarios. Spot and volatility should not necessarily be shocked independently. SSR provides a convenient way to specify their local co-movement in scenario and stress tests.
• Modeling. Local- and stochastic-volatility models generate their own spot-volatility dynamics; comparing a model's response with an observed SSR helps assess its realism (the Vol surface no-arbitrage conditions page covers how a static surface is constrained, a complementary problem to the dynamics here).

Nothing in the sticky-strike / sticky-delta / SSR machinery is specific to FX. The terminology originated in equity option markets, and the same question — what happens to the smile when spot moves — is asked in every market that trades a vol surface: equity indices and single names, rates (swaption markets), commodities, and crypto.

• Equity (index options). The classic home of the terminology: Derman's "Regimes of Volatility" framed sticky strike vs sticky delta for equity index options. Equity surfaces are quoted strike-by-strike, so they are sticky strike by construction in the same sense as a strike-space FX surface — the smile does not move when spot moves unless the desk re-quotes it. Empirically, no pure regime holds: measured index SSRs sit around 3/2 (see the SSR section), and at short maturities they can exceed 2 — more ATM-vol response than any classical regime predicts (Vargas, Dao and Bouchaud, S&P 500 and DAX). The regime also flips with market conditions: in a sharp sell-off, ATM vol typically rises by more than the static skew predicts, an SSR above 1 (the 2008 and 2020 crash vol spikes are extreme examples). The SSR framing is standard in the equity vol literature; Bergomi's Stochastic Volatility Modeling develops the index version in detail.
• Rates (swaption markets). Swaption vol surfaces are anchored to ATM forward swap rates, and the standard models — SABR on forward rates, the (L)MM family — attach volatility to the forward rate level, which is sticky delta in spirit. Rate vol desks face the same spot-vol co-movement question whenever the underlying swap rate moves.
• Commodities and crypto. Surfaces exist per underlying and are quoted in strike, moneyness, or delta space depending on the venue; the framework above applies unchanged once the parameterisation is known.

The practical takeaway: the framework is asset-class agnostic. For any market you need two things — (1) the parameterisation convention (strike vs delta/moneyness space; the FX Volatility Surface page's delta and ATM conventions are the FX instance of a general question), and (2) the empirical SSR, which you can estimate from a time series of ATM vols and spot moves via the finite-move approximation above.

Suppose \(S = 100\), \(\text{Vega} = 0.20\), and the ATM skew is \(\partial\sigma/\partial\ln K = -5\% = -0.05\). Assume SSR \(= 1.5\). Then

\[\frac{d\sigma}{dS} = \frac{1.5}{100}\,(-0.05) = -0.00075\]

so the surface-induced delta adjustment is

\[\text{Vega}\cdot\frac{d\sigma}{dS} = 0.20 \times (-0.00075) = -0.00015\]

The option's total spot sensitivity is 0.00015 lower than the conventional Black-Scholes delta — small here, but of exactly the same order as the everyday hedge adjustments a desk applies, and it scales with vega and skew.

ORE stores FX volatility surfaces either in strike space (FxBlackVolatilitySurface) or in delta space (FxBlackVolatilitySurfaceDelta) — see the FX Volatility Surface page's market.xml notes. The storage choice is a surface-dynamics assumption: a strike-space surface is sticky strike by construction (reprice after a spot move and every fixed-strike vol is unchanged), while a delta-space surface shifts its strikes as the forward moves — sticky-delta-like. The same phenomenon appears in Volatility surface driving: a derived surface must be rebased as spot moves, and the page flags large driver/derived spot divergence as "a known sticky-strike limitation" — the limitation this page's mechanics describe. The same construction holds beyond FX: QuantLib's BlackVarianceSurface, which ORE uses for rates and equity vol, stores vols in strike space, so it is sticky strike by construction — see the Vol surface no-arbitrage conditions page's class table.