  We consider the path-dependent volatility (PDV) model of Guyon and Lekeufack
(2023), where the instantaneous volatility is a linear combination of a
weighted sum of past returns and the square root of a weighted sum of past
squared returns. We discuss the influence of an additional parameter that
unlocks enough volatility on the upside to reproduce the implied volatility
smiles of S&P 500 and VIX options. This PDV model, motivated by empirical
studies, comes with computational challenges, especially in relation to VIX
options pricing and calibration. We propose an accurate neural network
approximation of the VIX which leverages on the Markovianity of the 4-factor
version of the model. The VIX is learned as a function of the Markovian factors
and the model parameters. We use this approximation to tackle the joint
calibration of S&P 500 and VIX options.
