  An extensive empirical study of the class of Volterra Bergomi models using
SPX options data between 2011 and 2022 reveals the following fact-check on two
fundamental claims echoed in the rough volatility literature:
  Do rough volatility models with Hurst index $H \in (0,1/2)$ really capture
well SPX implied volatility surface with very few parameters? No, rough
volatility models are inconsistent with the global shape of SPX smiles. They
suffer from severe structural limitations imposed by the roughness component,
with the Hurst parameter $H \in (0,1/2)$ controlling the smile in a poor way.
In particular, the SPX at-the-money skew is incompatible with the power-law
shape generated by rough volatility models. The skew of rough volatility models
increases too fast on the short end, and decays too slow on the longer end
where "negative" $H$ is sometimes needed.
  Do rough volatility models really outperform consistently their classical
Markovian counterparts? No, for short maturities they underperform their
one-factor Markovian counterpart with the same number of parameters. For longer
maturities, they do not systematically outperform the one-factor model and
significantly underperform when compared to an under-parametrized two-factor
Markovian model with only one additional calibratable parameter.
  On the positive side: our study identifies a (non-rough) path-dependent
Bergomi model and an under-parametrized two-factor Markovian Bergomi model that
consistently outperform their rough counterpart in capturing SPX smiles between
one week and three years with only 3 to 4 calibratable parameters.
\end{abstract}
