The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem

TL;DR

This paper introduces the quadratic rough Heston model, combining rough volatility and price-feedback effects, as a novel approach to jointly calibrate SPX and VIX smiles, challenging previous conjectures about the limitations of continuous-path models.

Contribution

The paper presents the quadratic rough Heston model as a counterexample to the conjecture that continuous-path models cannot jointly calibrate SPX and VIX smiles.

Findings

Successfully calibrates both SPX and VIX smiles simultaneously.

Demonstrates the effectiveness of combining rough volatility with price-feedback effects.

Challenges previous beliefs about the limitations of continuous-path models.

Abstract

Fitting simultaneously SPX and VIX smiles is known to be one of the most challenging problems in volatility modeling. A long-standing conjecture due to Julien Guyon is that it may not be possible to calibrate jointly these two quantities with a model with continuous sample-paths. We present the quadratic rough Heston model as a counterexample to this conjecture. The key idea is the combination of rough volatility together with a price-feedback (Zumbach) effect.