Research

In the wings of an implied-vol smile, no-butterfly-arbitrage is more than a floor on the implied PDF (or on the convexity): it is a cap on the skew. Concretely, the no-butterfly-arbitrage condition at any point reduces to a quadratic in the skew $s$. Where this quadratic admits two real roots $s_{-}<s_{+}$ in the regime $d_1{d}_2>1$, the cap reads $s\le s_{-}$ at any right-wing point, with mirror $s\ge s_{+}$ at the left; we call this the maximum-skew constraint. Under strict Lee [Lee 2004], this regime holds asymptotically in either wing. This note studies that quadratic, first in the linear-in-variance edge-knot case introduced by the CVI paper [Deschâtres 2026], then in full generality.

At the edge knot of CVI's cubic spline, linear-in-variance wing extrapolation forces $c=0$. The CVI paper selected the smaller-magnitude root on practical grounds, leaving open whether any strike-arbitrage-free smile could lie on the larger-magnitude root. We close that gap via Lucic's pointwise strike-arbitrage slope bound [Lucic 2021], which sits strictly between $s_{-}$ and $s_{+}$: the larger-magnitude root carries a vertical-spread arbitrage incompatible with strike-arbitrage-freeness. We further identify the constraint as the finite-strike form of Lee's asymptotic large-strike bound [Lee 2004]: strictly tighter than Lee at any finite knot, approaching Lee only as the knot moves deep into the tail, and propagating along the linear-extrapolated wing to give necessary-and-sufficient strike-arbitrage-freeness at and beyond the knot. Furthermore, an equivalent form free of the $0/0$ pathology at $d_1{d}_2=1$ is given. The argument applies to any implied-volatility parametrization with linear-in-variance wing extrapolation.

Beyond CVI, the $c=0$ assumption is relaxed. Strict Lee gives the wing regime asymptotically; for asymptotically linear smooth tails with $c(z)\to 0$, including SVI/SSVI, the remaining real-root condition $c\le c^{\ast }(z)$ also holds asymptotically in either wing. The larger-magnitude root is excluded wherever such roots exist. Under Black-Scholes pricing, $s\le s_{-}$ alone is pointwise (i.e., at each finite strike of the slice) necessary and sufficient for strike-arbitrage-freeness (mirror $s\ge s_{+}$ at the left); the name maximum-skew constraint thus becomes literal: in either wing, wherever the smaller-magnitude root exists, it is the maximum admissible skew. Without Black-Scholes, the combined form $s\le \min \{s_{-},s_{\text{VS}}^{\text{call}}\}$ takes over, with butterfly binding for $c<c_{\text{cross}}$ and the call-spread Mills threshold for $c>c_{\text{cross}}$, where $c_{\text{cross}}$ is the convexity at which $s_{-}$ and $s_{\text{VS}}^{\text{call}}$ coincide. Moving out along an asymptotic linear-in-variance wing, $c_{\text{cross}}(z)\sim 1/z$; under the further assumption of eventually monotone convexity, $c(z)=o(1/z)$, so the Mills-ratio bound is asymptotically non-binding. For parametrizations with positive fast-decaying convexity ($c(z)\ge 0$ eventually and $c(z)=o(1/z^2)$; covering SVI and SSVI with $c(z)=O(1/z^3)$), the pointwise picture above extends asymptotically to a sufficient (not necessary) at-and-beyond form: a single-strike check against the CVI $c=0$ bound $s_{-}(z_b,0)$ (sufficient since $c\ge 0$ implies $s_{-}(z_b,0)\le s_{-}(z_b,c)$) propagates strike arbitrage-freeness along the entire wing past the strike, with the closed form worked out for raw SVI.

Across the smile, the picture extends into the interior as a band $s_{+}\le s\le s_{-}$ on the skew when butterfly roots are real (automatic for any $c\ge 0$, since the interior $c^{\ast }$ is negative). At very marked W shapes near the money, the butterfly quadratic has no real roots and the constraint becomes a convexity floor $c\ge c^{\ast }$. In short, butterfly arbitrage is best read as a cap or floor on the skew across the smile when real roots exist.