The rough Hawkes Heston stochastic volatility model

TL;DR

This paper introduces the rough Hawkes Heston stochastic volatility model, combining rough volatility and jump clustering, with explicit formulas and calibration demonstrating improved fit to S&P 500 and VIX options data.

Contribution

It extends the Heston model by incorporating rough volatility and Hawkes-type jumps, providing explicit solutions and a calibration method that captures market features more accurately.

Findings

Model accurately fits implied volatility smiles for S&P 500 and VIX options.

Low kernel power explains the VIX implied volatility shift.

Efficient approximation of solutions via multi-factor techniques.

Abstract

We study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes-type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier-Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati-Volterra equations, which can be efficiently approximated using a multi-factor approximation technique. We calibrate a parsimonious specification of our model characterized by a power kernel and an exponential law for the jumps. We show that our parsimonious setup is able to simultaneously capture, with a high precision, the behavior of the implied volatility smile for both S&P 500 and VIX options. In particular, we observe that in our setting the usual shift in the implied volatility of VIX options is explained by a very low value of the power in the kernel. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self-exciting jumps in order to capture the joint evolution of the S&P 500 and VIX.