Gatheral's Conjecture Revisited

Abstract

We consider the Heston model with perfect negative spot--variance correlation and its one-dimensional local-volatility projection. Let $I_T^{\mathrm{H}}$ and $I_T^{\mathrm{LV}}$ denote their respective integrated variances over $[0,T]$. We establish the inequality \[ \mathbb{E}\bigl[(I_T^{\mathrm H}-K)^+\bigr] < \mathbb{E}\bigl[(I_T^{\mathrm{LV}}-K)^+\bigr] \] for every maturity $T>0$ and every strike $K>0$. Consequently, Heston integrated variance is strictly smaller in convex order than the integrated variance of the calibrated local-volatility model. This strict ordering gives a Heston-model counterexample to the convex-order inequality conjectured by J. Gatheral.