Pricing and calibration in the 4-factor path-dependent volatility model

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 \emph{pathwise} neural network approximation of the VIX which leverages on the Markovianity of the 4-factor version of the model. The VIX is learned pathwise 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, quickly sample VIX paths, and price derivatives that jointly depend on S\&P 500 and VIX. As an interesting aside, we also show that this \emph{time-homogeneous}, low-parametric, Markovian PDV model is able to fit the whole surface of S\&P 500 implied volatilities remarkably well.