Joint SPX & VIX calibration with Gaussian polynomial volatility models: Deep pricing with quantization hints

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 2011 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. To ensure an efficient calibration and fair comparison between the models, we develop a generic unified method in our class of models for fast and accurate pricing of SPX & VIX derivatives based on functional quantization and neural networks. For the first time, we identify a conventional one-factor Markovian continuous stochastic volatility model that can achieve remarkable fits of the implied volatility surfaces of the SPX & 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.