A 0.1 curvature gap at a T^-0.4 skew

A T^-0.4 ATM skew leaves a trader with two plausible models and a hard measurement problem. Rough volatility with H near 0.1 can produce it; so can infinite-variation jumps with Y near 1.8. Burés and Vives show that ATM curvature separates the two in theory. Whether a real short-dated surface can resolve that separation is the question for pricing.

Where the exponents come from

The paper places a general stochastic volatility process, including rough volatility, beneath the log-price. An independent Lévy jump component supplies small jumps that behave like a stable law, with activity index Y between 1 and 2. In that range, infinitely many small jumps have infinite variation; CGMY is the standard example. Using Malliavin calculus, the authors split ATM skew and curvature into contributions and take their short-maturity limits.

For skew, the exponent is T^min(H-1/2, (1-Y)/2). For curvature, it is T^min(2H-1, -Y/2). The more negative power dominates. Ownership of the skew changes at Y = 2-2H; ownership of the curvature changes at Y = 2-4H.

Those powers give each pure model a different relation between slopes. Pure rough volatility yields β_c = 2β_s, heuristically, under the paper's hypotheses. Pure constant-vol Lévy jumps yield β_c = β_s - 1/2. A mixed model can depart from both. One apparent combination is impossible: jump-driven skew with rough-driven curvature would require 2-2H < Y < 2-4H, which cannot hold for H > 0. A fitted β_c ≠ β_s - 1/2 therefore identifies rough-driven skew.

There is also a useful asymmetry in what the jumps measure. Their skew contribution depends on C+ - C-, while their curvature contribution depends on C+ + C-. Symmetric small jumps can set the curvature power law without contributing to the leading skew term.

The general theorems assume bounded volatility and three orders of Malliavin regularity. Rough Heston lacks those properties, so the authors use a truncated variance process. They show that implied vol, skew and curvature for that process differ from the original by o(T^m) for every m. The third-derivative bound in Proposition 5.12 has no proof in the paper, and the leading-term expansion for the second derivative in Proposition 5.11 also lacks details. There are no market data. Numerical results use rough Heston and CGMY prices obtained by Fourier inversion.

How much separation does curvature buy?

Set H = 0.1 and Y = 1.8, the two explanations for the T^-0.4 skew. Rough volatility gives β_c = 2H-1 = -0.8; jumps give β_c = -Y/2 = -0.9. On a log-log curvature plot, the slopes differ by 0.1.

The paper's mixed example narrows the comparison. With H = 0.2 and Y = 1.5, it gives β_s = -0.3 and β_c = -0.75. The curvature slope of -0.75 lies 0.15 from the -0.6 implied by 2β_s, yet only 0.05 from the pure-jump -0.8. Distinguishing "mixed" from "pure jumps with a slightly different Y" asks the shortest expiries to deliver a curvature exponent accurate to a few hundredths.

A boundary remains unresolved. On Y = 2-4H, the curvature coefficient may be C_vol or C_vol + C_jumps. The authors say their paper alone cannot determine which mechanism produced it.

Clean fits at unusually short expiries

Table 1 gives the results at the smallest maturity in each run. For rough-only, theoretical (β_s, β_c) values of (-0.40, -0.80) are estimated as (-0.4001, -0.7978). For jump-only, (-0.25, -0.75) become (-0.2514, -0.7520). For mixed, (-0.3, -0.75) become (-0.2969, -0.7531). Every error is under 0.004, within the 0.05 gap.

For the skew-exponent check, the authors "work with very small maturities that are not usually found in real markets". Table 1 likewise reports the smallest maturity of the curvature runs. Two sentences later, they warn: "Greater maturities, as seen in the plots, may generate ambiguity on the validity of our result." As a mathematical check, the exercise succeeds on its own terms. Its prices are noise-free model prices at expiries shorter than markets usually list, so errors under 0.004 say nothing about noise in real quotes. We did not find a test that adds noise to prices. The skew-exponent and skew-coefficient checks appear only in plots, without values in the text.

Several entries in the numerical section also appear to be transcription slips. The rough-only curvature run gives H = -0.5 and ρ = 0.1, apparently swapped. The mixed curvature run gives H = -0.7 and ρ = 0.2, also apparently swapped; H = 0.2 is the reading consistent with β_s = -0.3 above. Its CGMY tuple contains six values for five parameters. The regime summary puts the curvature critical line at Y = 4-2H, although the theorem gives 2-4H. And C_vol, the rough-volatility curvature coefficient, appears in three different forms across Theorem 2.5, Proposition 4.2 and the rough Heston section; the last has no separate Θ term. The exponents in Theorems 2.2 and 2.5 survive these issues. The coefficients a calibrator would fit do not.

Can listed weeklies clear 0.05?

Curvature is the binding measurement. Fitting its slope requires near-ATM strikes at the shortest listed expiries.

A steady curvature exponent, estimated from listed weeklies day to day to well inside 0.05, would change my view. Until that is shown on a real surface, this remains a sharp calibration constraint supported numerically only at maturities shorter than markets usually list.