Convex-order behavior of the Heston model away from perfect negative correlation

Determine whether the strict convex-order inequalities between the integrated variance of the Heston model and that of its one-dimensional local-volatility projection persist for perfect positive correlation [?] No, the paper specifically leaves open the cases \(\rho=+1\) and \(-1<\rho<1\), including whether strict inequalities persist away from the endpoint.

Background

The paper establishes that, when the Heston model has perfect negative pricevariance correlation ρ=−1\rho=-1, its integrated variance is strictly smaller in convex order than the integrated variance of the calibrated one-dimensional local-volatility projection. The result is proved only at this endpoint and relies structurally on the perfect negative correlation, which yields a pathwise relation between spot, instantaneous variance, and accumulated variance.

The unresolved extension concerns the remaining correlation parameters: perfect positive correlation ρ=+1\rho=+1 and intermediate correlations −1<ρ<1-1<\rho<1. Specifically, it is unknown whether the strict comparison established at ρ=−1\rho=-1 continues to hold when the correlation is moved away from that endpoint.

References

The present note treats only the endpoint $\rho=-1$; the cases $\rho=+1$ and $-1<\rho<1$, including whether strict inequalities persist away from the endpoint, are left open.

— Gatheral's Conjecture Revisited  (2609.05047 - Lucic, 4 Sep 2026) in Section 1, Introduction