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An extensive empirical study of the class of Volterra Bergomi models usingSPX options data between 2011 and 2022 reveals the following fact-check on twofundamental claims echoed in the rough volatility literature:  Do rough volatility models with Hurst index $H \in (0,1/2)$ really capturewell SPX implied volatility surface with very few parameters? No, roughvolatility models are inconsistent with the global shape of SPX smiles. Theysuffer 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-lawshape generated by rough volatility models. The skew of rough volatility modelsincreases too fast on the short end, and decays too slow on the longer endwhere "negative" $H$ is sometimes needed.  Do rough volatility models really outperform consistently their classicalMarkovian counterparts? No, for short maturities they underperform theirone-factor Markovian counterpart with the same number of parameters. For longermaturities, they do not systematically outperform the one-factor model andsignificantly underperform when compared to an under-parametrized two-factorMarkovian model with only one additional calibratable parameter.  On the positive side: our study identifies a (non-rough) path-dependentBergomi model and an under-parametrized two-factor Markovian Bergomi model thatconsistently outperform their rough counterpart in capturing SPX smiles betweenone week and three years with only 3 to 4 calibratable parameters.\end{abstract}

Volatility models in practice: Rough, Path-dependent or Markovian?