The Bachelier butterfly condition becomes a linear constraint once the smile is written in reciprocal implied scale.
Start with one expiry of listed calls. Invert each price to obtain the total normal implied standard deviation s(K), the Bachelier quantity rather than its lognormal counterpart. Normalize strike as y = (K - F)/s(K), then define q = 1/s in that coordinate. Sun claims that butterfly no-arbitrage across the slice is exactly
q''(y) + y q'(y) - q(y) ≤ 0,
provided the normalized coordinate remains increasing, q - y q' > 0. The operator is linear in q. On a discrete strike grid, it becomes a bank of linear inequalities on fitted values.
The Black-convention counterpart is v''(k) + v(k) a'(k) b'(k) ≥ 0, which is nonlinear in v. Our reading is that positivity must then be checked strike by strike on the fitted surface. We discussed this earlier in WSVI and the W-shaped smile.
The deformation η
The linear form follows from one definition. Express the risk-neutral CDF as h(z) in an increasing normalized coordinate z, and write
η(z) = (h(z) - Φ(z)) / φ(z),
which gives h = Φ + φη. Differentiation lets the Gaussian density supply the needed term: h'(z) = φ(z) m(z), where m(z) = 1 + η'(z) - zη(z). The quantity m is the implied density in that coordinate divided by the standard normal density. Static no-arbitrage reduces to m ≥ 0.
Digital bounds also become explicit. The condition 0 ≤ h ≤ 1 gives the Mills-ratio barriers -R(-z) ≤ η(z) ≤ R(z). The paper defines the Mills ratio by R(z) = Φ(-z)/φ(z).
For Bachelier coordinates, η(y) = s'(K(y)). Normal implied-volatility skew is therefore the normalized gap between the CDFs. The Mills bands become closed-form admissible slope bands for quoted normal vol, without any calibration between them.
Under the Black convention, the same slope η = v'(k) carries two interpretations. Fukasawa's coordinates are a = k/v - v/2 and b = k/v + v/2, the negatives of d1 and d2. Sun shows that η in b is the CDF deformation under the pricing measure. In a, η gives the deformation under the share measure. Translating m into log-moneyness recovers v'' + v a' b'. With constant Black volatility, a' = b' = 1/v₀ and m = 1, matching the Gaussian benchmark exactly.
There is no data here.
The analysis runs from start to finish without an empirical application. Its only worked examples are the two constant-volatility cases, one Bachelier and one Black. The conclusion states the limit plainly: "The present paper remains within Gaussian implied-volatility conventions and isolates the distributional meaning of their normalized slopes." The abstract describes the contribution in similar terms, saying it "interprets implied-volatility skew as a normalized deformation of the risk-neutral distribution rather than merely as a slope parameter."
The introduction attributes the slope bounds and one-coordinate parameterizations to Fukasawa and to Lucic. Most of Sun's claim is consequently a reading of known material. There is one exception with direct implementation value: q'' + yq' - q ≤ 0 is a constraint that can be handed to a solver.
Monotonicity carries the argument
Writing q(y) requires y(K) to be invertible. The paper says finite-strike monotonicity of y(K) under static no-arbitrage was established in Sun [9], and proceeds by assuming it. The cited item appears as "Preprint, 2026," without a venue. The Black-side monotonicity of a and b receives the same treatment, through references to Fukasawa, Lucic and the same companion. Any implementation of the linear inequality takes on that dependency. With crossed quotes and stale deep wings on a live chain, monotonicity remains an empirical question at the available strikes.
Smoothness is the second dependency. The function η is a strike derivative of implied volatility, while m contains a second derivative of q. Listed option chains offer discrete strikes and noisy quotes. The paper supplies no discretization scheme or error analysis for recovering s'(K) from a finite grid.
This gap is also where linearity becomes useful. A finite-difference version of the operator can constrain the fitted q values directly, avoiding differentiation of the observations themselves.
The local-volatility section limits itself to work "in forward units under zero rates and carry," and defers deterministic carry to "the usual forward normalization." Early exercise is absent from the text. The omission is survivable for SPY and index chains. For single-name US equity options carrying discrete dividends and American features, the stated local-variance formulas are different from those an implementation would use.
Can η produce a smile?
Section 5 gets closest to an actual construction. It takes b as the independent variable, makes η(b) primitive, and reconstructs the smile through v'(b) = ηv / (1 - (b - v)η). Sun's screens are the Mills band, density positivity M_b(η) ≥ 0, and the orientation condition 1 - (b - v(b))η(b) > 0. The third condition includes v, which emerges only after the ODE has been integrated. It must therefore be monitored during the solve.
This is the buildable part, and it remains unfinished. Proposition 2.2 characterizes admissible η under the CDF boundary conditions h(−∞) = 0, h(+∞) = 1. The result changes variables onto the set of CDFs and is complete in that sense. It supplies no finite-parameter family for calibration. SVI never appears, and the paper offers no comparison with an existing arbitrage-free parameterization on either stability or calibration cost.
What would settle it?
Nobody has yet fitted q on a real strike grid under the linear constraint, leaving the practical question open. One piece of evidence would change our view: the Bachelier coordinate remaining monotone across the full quoted grid on the days that matter. Every step of the Bachelier argument relies on that property. The Black half carries the same reliance on increasing a and b.