Identify the ATM-IV curvature mechanism on the critical exponent line

Determine which mechanism—rough volatility, infinite-variation Lévy jumps, or both—generates the leading short-time behavior of the at-the-money implied-volatility curvature when the critical relation between the roughness and jump-activity parameters holds, namely when 2H - 1 = -Y/2 (equivalently, Y = 2 - 4H), including whether the leading coefficient is C_vol or C_vol + C_jumps.

Background

The paper derives short-time power-law asymptotics for the at-the-money implied-volatility skew and curvature in a model combining stochastic volatility with a stable-like infinite-variation Lévy process. The dominant contribution to the curvature changes at the threshold 2H - 1 = -Y/2, where H is the volatility roughness parameter and Y is the Lévy jump-activity index.

On the critical line, the rough-volatility and jump contributions have the same power of maturity and may combine in the leading coefficient. The paper states that its results do not determine whether the observed curvature is generated by the volatility component alone or by the mixed contribution, leaving the mechanism-identification problem unresolved in this regime.

References

If the exponent of the curvature lies on the critical line Y = 2-4H, our paper alone does not fully give an answer on which mechanism generated the ATM-IV curvature, since the coefficient of the curvature may be either C_{vol} or C_{vol}+C_{jumps}.

— On the skew-curvature power law relationship in a stochastic volatility model with infinite variation Lévy jumps  (2610.23488 - Burés et al., 16 Oct 2026) in Section 2.3, subsection “Consequences of the main results”