  We consider the joint SPX-VIX calibration within a general class of Gaussian
polynomial volatility models in which the volatility of the SPX is assumed to
be a polynomial function of a Gaussian Volterra process defined as a stochastic
convolution between a kernel and a Brownian motion. By performing joint
calibration to daily SPX-VIX implied volatility surface data between 2012 and
2022, we compare the empirical performance of different kernels and their
associated Markovian and non-Markovian models, such as rough and non-rough
path-dependent volatility models. In order to ensure an efficient calibration
and a fair comparison between the models, we develop a generic unified method
in our class of models for fast and accurate pricing of SPX and VIX derivatives
based on functional quantization and Neural Networks. For the first time, we
identify a \textit{conventional one-factor Markovian continuous stochastic
volatility model} that is able to achieve remarkable fits of the implied
volatility surfaces of the SPX and VIX together with the term structure of VIX
futures. What is even more remarkable is that our conventional one-factor
Markovian continuous stochastic volatility model outperforms, in all market
conditions, its rough and
  non-rough path-dependent counterparts with the same number of parameters.
