A local-volatility engine should produce SSR equal to two as maturity collapses to the valuation date. Fukasawa proves that result even when local volatility depends on calendar time. The expansions carry o(√(T-t)) errors. The paper gives a limit for the ratio, with no rate.
We ran no test of this. The result is an asymptotic theorem. An empirical exercise would instead examine a local-volatility surface fitted to listed quotes, rather than test the proof.
What does SSR measure?
Bergomi introduced the skew stickiness ratio in Smile Dynamics IV (Risk, December 2009). It divides the response of at-the-money implied volatility to a spot move by the at-the-money log-strike skew. In Bergomi's framing, the ratio measures the associated cross-gamma risk. Two models may fit today's static smile and still give entirely different answers. Hedge a short-dated book on the assumption that the smile slides with spot, and a wrong assumption leaves the residual in gamma P&L.
Fukasawa assumes zero interest and dividend rates and uses a one-factor diffusion, dS = a(S,u) dB, with relative local volatility v(s,t) = a(s,t)/s. The ratio is defined through quadratic covariation, d⟨v̂,log S⟩ over d⟨log S⟩, normalised by the skew. Markovian dynamics reduce this to R = (v̂_s + v̂_K)/v̂_K, where the subscripts denote spot and strike derivatives of implied volatility. SSR equal to two therefore has a precise meaning: at-the-money implied volatility responds equally to a spot move and a strike move.
If local volatility has no calendar-time dependence, the backward pricing equation in spot and the Dupire forward equation in strike imply the exact symmetry v̂(t,s;K) = v̂(t,K;s). Differentiating at the money gives v̂_s = v̂_K. The ratio is exactly two.
Bergomi gave two arguments for the time-dependent case, and Fukasawa opens by explaining why neither is complete. The first expands around flat local volatility at fixed maturity. Its numerator and denominator vanish in the reference model, while "no remainder estimate uniform in maturity is provided." The second differentiates the pointwise harmonic-mean short-maturity formula. The problem is stated directly: "pointwise convergence does not imply convergence of the spot and strike derivatives entering the SSR."
One sign supplies the factor of two
The spot derivative of the put price becomes a digital probability exactly: P_s(t,s;K) = -P(S̃_T < K). Fukasawa changes measure using dQ = J_T dP. Here J is the derivative flow, the pathwise sensitivity of terminal spot to initial spot. Under this measure, spot acquires a drift and follows dS̃ = a a_s du + a dB̃.
The proof then localises the process to a box in spot and time. With ε = √(T-t), a first-order Watanabe expansion follows. Theorems 2.1 and 2.3 of Watanabe (1987) pull the indicator H_- back through the Wiener functional. Uniform nondegeneracy permits this step because the Malliavin covariance has bounded inverse moments of every order, uniformly in ε.
In the stochastic Taylor expansion of the original dynamics, one term contains (W² - 1). The drift a a_s changes it to (W² + 1). Everything rests on this sign flip. The original digital moves to 1/2 + (a_1/2)φ(0)√(T-t), while the auxiliary digital moves to 1/2 - (a_1/2)φ(0)√(T-t), with a_1 = a_s(s,t). The ratio's numerator and denominator become φ(0) s v_s √(T-t) and (φ(0)/2) s v_s √(T-t).
Ratio two.
The expansions also recover the one-half skew rule. The at-the-money log-strike skew tends to (1/2) s v_s(s,t), one half of the relative local volatility slope. Alòs and García-Lorite established this for bounded, uniformly positive local volatility with bounded derivatives. Fukasawa removes the uniform lower bound, along with global bounds on second and higher spatial derivatives.
The remaining assumptions are local smoothness, positivity, and the global bound sup|a_s| = L. Localisation carries the proof. Exit probabilities from the box are bounded by c_p h^{p/2}. Fukasawa takes p = 2 for digital payoffs and p = 4 for the put, producing a transfer error of O(h). It disappears relative to √h. Positivity of the solution remains an assumption, leaving absorbing specifications such as CEV with β below one outside the theorem.
The screen is still out of reach
Every result here is asymptotic. The theorem takes T to t and supplies no rate or maturity-uniform bound. Fukasawa makes the same criticism of Bergomi's first argument: "no remainder estimate uniform in maturity is provided." Our SSR work uses one-month and three-month options. A limit says nothing at either fixed maturity.
The paper offers no simulation of how quickly R(t,s;T) approaches two. It also contains no market data. The maturity range over which the limit becomes informative is therefore unestablished.
Fukasawa's theorem also assumes v_s(s,t) ≠ 0. If local volatility is locally flat in spot, both sides vanish and the claim has no content. Interest and dividend rates are Zero throughout. The model class is local volatility only; stochastic and rough volatility receive no treatment.
A useful consistency check survives these limits. At maturities short enough for the asymptotic result to matter, though the paper does not identify them, a local-volatility engine producing SSR far from two is revealing a problem in its own calculations. The same applies when a short-dated at-the-money skew misses half the calibrated slope s v_s. Blame the engine.
This is the third recent theory paper on smile dynamics we have examined that ends where numerical work would begin, following this one and this one. The omission matters less here. Bergomi had already identified the local-volatility value of two, and the proof is short enough to inspect line by line. The author discloses that ChatGPT was used to draft and edit the exposition and prepare the LaTeX manuscript. An eight-page argument can still be checked directly.
A numerical study of convergence speed at one and three months on calibrated surfaces would turn the theorem into a hedging benchmark. The paper contains none. Two is correct at zero maturity and remains an open question everywhere you trade.