This paper addresses computational challenges in the 4-factor path-dependent volatility (4FPDV) model of Guyon and Lekeufack (2023), which defines instantaneous volatility as a linear combination of weighted past returns and the square root of weighted past squared returns. The authors introduce an additional parameter β₁,₂ that adds volatility during upward trends, enabling the model to reproduce the implied volatility smiles of both SPX and VIX options, especially for short maturities. To overcome the difficulty of computing the VIX analytically, they develop a pathwise neural network approximation that leverages the Markovian structure of the 4FPDV model, learning the VIX as a function of the four factors and model parameters. This approximation allows for fast simulation of VIX paths and efficient joint calibration to SPX and VIX options.

The paper demonstrates that the 4FPDV model calibrates the SPX implied volatility surface remarkably well across maturities up to one year, and the joint calibration to SPX and VIX options yields accurate fits. The authors also analyze the stability of calibrated parameters over time and show that the neural network approximation can be used to price light exotic derivatives depending on both SPX and VIX. Overall, the work provides a practical and efficient framework for using the 4FPDV model in real-world applications, combining interpretability with computational tractability.