  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.
