WSVI can fit an earnings-week dip at the forward in closed form, but the three-parameter model's single butterfly inequality disappears.
Shape coordinates and the added degrees of freedom
Fitting one expiry is a small problem with an unforgiving constraint. A handful of listed strikes becomes Black implied vols, after which the curve must remain low-dimensional, exactly differentiable and free of butterfly, spread and calendar arbitrage. Per-slice eSSVI achieves that with three parameters: total variance ϑ, correlation ρ*, and curvature ϕ. Clevenger and Wan make a sharper criticism than usual. Corollary 3.10 gives f''_base(z) = c/(8B(z)³) > 0 for every z when c > 0. At m = 0, the three-parameter slice is therefore strictly convex at every strike, including the forward. For m > 0, the same corollary says that f''(z) < 0 forces P''(z) < 0. The three-parameter family cannot produce a W-shaped smile.
WSVI sets total variance to w(k) = ς² f(z), where ς = σ0√T is the scale and z = k/ς is normalized strike. Its shape is f = P/2 + sqrt(P²/4 + cz²/2), with P(z) = 1 + sz + Σ a_j φ_j(z/λ_j). Each φ_j is a smooth one-sided step bounded in [0,1), with scale λ_j, integer order n_j ≥ 1 and a designated side. A slice with m amplitudes has 3 + m parameters. Set a_m = 0 and the construction returns the (m−1)-amplitude member exactly, making eSSVI the m = 0 rung of a true ladder.
Three exact results support the construction. For any m and any selection of members, the real-valued domain is exactly c ≥ 0. The wing slopes are C± = √χ ± s/2, where χ = s²/4 + c/2, and amplitudes cannot alter them. Meanwhile, f''(0) = c + Σ_{n_j=1} a_j/λ_j². So c no longer equals curvature at the forward, and negative order-one amplitudes can produce negative curvature while c > 0.
The paper contains no market data. Its only illustration is a hand-picked slice with σ0 = 0.60, T = 5/252, s = 0 and c = 0.3. Two order-one members use λ = 1 on opposite sides, both with amplitudes −0.35. The resulting values are f''(0) = −0.40 and g(0) = 0.80, with a bimodal implied density whose modes lie near k ≈ ±0.04. In the wings, g → 0.2499.
The wings stay put
The extension preserves its most useful design feature. Since every basis member is bounded in [0,1), Lee's moment bound remains ς(√χ + |s|/2) ≤ 2 for every amplitude count when c > 0. Proposition 3.13 states this as necessary for the absence of butterfly arbitrage. Under the illustration's settings, ςC± = 0.033, far inside the constraint.
Corollary 4.4 supplies the exact curvature budget at the forward: f''(0) ≥ (s²/2)(1 + ς²/4) − 2. Curvature may be negative without violating the butterfly condition provided it stays above that bound. Larger skew or total variance lifts the floor. Interior freedom with unchanged extrapolation has real value, and WSVI obtains it directly.
What replaces the cheap butterfly check?
Standard eSSVI has two diagnostics: an asymptotic necessary wing-slope condition, plus χ + |s|√χ/2 ≤ 1 as a sufficient condition for no butterfly arbitrage. The paper explicitly says this sufficient condition is neither necessary nor sufficient for m > 0. Its conclusion leaves an exact parameter-domain butterfly characterization for m > 0 as an open problem. Corollary 4.4 covers only the at-the-forward portion.
Klassen's necessary and sufficient conditions apply to the three-parameter curve in normalized-strike coordinates. Martini and Mingone derive the exact butterfly domain for the five-parameter SVI slice and a global arbitrage-free parametrization of eSSVI surfaces. Once WSVI adds amplitudes, the density factor g must be swept pointwise for g ≥ 0. This factor carries the sign of the implied density and is non-negative exactly when butterfly arbitrage is absent.
The authors state the price plainly. Their conclusion says that staying in volatility space leaves the absence of butterfly arbitrage as a condition that must be imposed rather than a property inherited from the construction. The introduction likewise promises additional shapes at the cost of arbitrage conditions that still require characterization and enforcement. In return, the family represents shapes unavailable to the three-parameter slice while remaining strictly inside the domain. The paper establishes both claims. Their practical balance depends on how often listed smiles need the dip, which the paper does not measure.
Two proved mitigations deserve credit. First, g = g0(z) − (θ/16) f'(z)² is affine and non-increasing in the level. A shape checked at the largest level where it will be used remains verified at every smaller level. This result applies over a bounded interval of log-moneyness: verify g ≥ 0 at the largest level θ_N for the relevant normalized strikes, and those slices carry no butterfly arbitrage on that interval.
The second mitigation is global. If c > 0 and ςmax(C+,C−) < 2, then g ≥ 0 across all of R gives C(−∞) = 1 and C(+∞) = 0, forcing 0 ≤ C ≤ 1 everywhere. Those wing limits are unavailable on a bounded interval. Monotonicity of C then restricts variation without fixing its level: "Checking one condition on a finite interval does not remove the need to check the other."
WSVI also introduces a third check that m = 0 did not require. The ray factor R = 1 − kw'/(2w) determines whether total variance increases with the level because ∂w/∂θ = f R. Since the density factor contains R squared, g ≥ 0 cannot determine the sign of R. The base member receives R = 1/2 + 1/(4B) > 1/2 automatically; amplitudes alone can drive it negative. The domain floor requires a hard parameter bound as well. At c = 0, the shape degenerates to max(P,0), creating a kink and zeros wherever P ≤ 0.
Their counterexample at s = −0.55, c = −0.05 is worth memorizing.
Although χ = +0.0506 is positive and the wing slopes are real, the radicand turns negative on z ∈ (1.154, 4.278). Its vertex value is −0.1235, while the recovered implied correlation is |ρ*| = 1.222, outside the admissible range. Controlling the leading coefficient gives no control over the interior.
Five parameters on one weekly chain
The index set of scales, orders and sides is chosen exogenously for each underlying and shared across expiries. Calibration is deferred to a separate forthcoming manuscript by the authors. This paper does show that order and scale do not act as orthogonal knobs. In the paper's quantile table, the member's 50% point moves from 1.151 at n = 1 to 2.374 at n = 4 and 4.144 at n = 12. Absolute width u75 − u25 rises from 1.274 to 3.599. Relative to the member's own location, width contracts from 1.107 to 0.868. Higher order moves the feature outward and makes it wider. Two knobs, one direction.
With m = 2, the fit uses five parameters on near-money strikes from a chain only days from expiry. I did not find a penalization or selection criterion beyond the nested structure of the ladder. The paper also flags an implementation detail: its general second-derivative formula includes e^(n−2), which is undefined at u = 0 when n = 1. Order one therefore needs a special case.
Shared shapes move with √θ
Theorem 4.16 gives exact calendar-freeness for expiries that share one shape vector, provided c > 0, the ray factor is non-negative over the relevant normalized strikes, and levels are non-decreasing. The authors acknowledge the restriction. A common shape in z locates each feature at k = ±λ_j ς, so the feature moves with √θ from one expiry to another. A scheduled-event feature instead lies at log-moneyness determined by the expected move on both expiries around the announcement and does not scale that way.
Theorem 4.17 allows the shape to drift, though its condition is sufficient only. The bound uses f/(2B) ∈ [0,1] and z²/(4B) ≤ |z|/(2√(2c)). Those triangle-inequality steps are likely to bind well before the true condition.
Evidence that would change the verdict
The missing empirical question is how often listed near-expiry smiles truly require negative curvature at the forward. An out-of-sample comparison with eSSVI and a two-component mixture on identical chains would answer much of it. So would parameter stability through the announcement, together with evidence that amplitudes improve a hedge as well as the residual. The paper acknowledges the gap: "Empirical assessment of how often market smiles require this additional flexibility is outside the scope of this paper."
Its named alternative is substantive. Glasserman and Pirjol connect W-shaped implied volatility curves to the Gaussian mixture model, and the paper identifies two related routes: a mixture model, or the parameter-randomization framework of Zaugg, Perotti and Grzelak. Both produce the shape with a density that is non-negative by construction. Their stated cost is a move into price space. The implied volatility has no closed form and must be recovered through a Taylor expansion in log-moneyness with a radius of convergence that is not known a priori. The choice is closed form with pointwise checks versus a density that is non-negative by construction without closed-form implied volatility. Real chains are needed to judge that trade. This paper presents none, and calibration is deferred to a separate forthcoming manuscript by the authors.
We have previously covered a component justified structurally that never bound on out-of-sample quotes: the SPXW 0DTE ranker whose abstention gate did nothing on the hold-out (/articles/the-5-76-sharpe-lives-in-a-1-82-percent-denominator). I treat the amplitudes similarly until the fit appears. They do have one advantage: because the m = 0 rung is the incumbent, the ladder contains its own test.
Every proposition in the paper looks right to me. The identity f''(0) = c + Σ_{n_j = 1} a_j/λ_j² offers an elegant way to retain a positive domain while allowing the smile to dip at the forward. The amplitude still has no empirical price.