Paper: arXiv 2609.05047

Authors: Vladimir Lucic

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly mathematical and theoretical proof, offering a novel counterexample to Gatheral’s conjecture within the Heston model. While the mathematical derivations are rigorous, there is no empirical testing or data validation, placing it firmly in the ‘Lab Rats’ quadrant. The clarity is good for a highly technical paper, and the methodology is detailed enough for theoretical replication.

Research Flowchart

  flowchart TD
    A[Research Goal: Revisit Gatheral's Conjecture] --> B{Model Setup: Heston vs. LV}
    B --> C{Inputs: Perfect Negative Correlation, Integrated Variances}
    C --> D[Analysis: Compare Expected Integrated Variance Payoffs]
    D --> E{Result: E[(I_T^H - K)^+] < E[(I_T^LV - K)^+]}
    E --> F[Conclusion: Heston integrated variance < LV integrated variance (convex order)]
    F --> G[Outcome: Counterexample to Gatheral's Conjecture]