Joint deep calibration of the 4-factor PDV model

Joint calibration to SPX and VIX market data is a delicate task that requires sophisticated modeling and incurs high computational costs. The latter is especially true when the pricing of volatility derivatives hinges on nested Monte Carlo simulation. One such example is the 4-factor Markov Path-Dependent Volatility (PDV) model of Guyon and Lekeufack (2023). Nonetheless, its realism has earned it considerable attention in recent years. Gazzani and Guyon (2025) marked a relevant contribution by learning the VIX as a random variable, i.e., a measurable function of the model parameters and the Markovian factors. A neural network replaces the inner simulation, making the joint calibration problem accessible. However, the minimization loop remains slow due to the expensive outer simulation. The present paper overcomes this limitation by learning SPX implied volatilities, VIX futures, and VIX call option prices. The pricing functions reduce to simple matrix–vector products that can be evaluated on the fly, shrinking calibration times to just a few seconds. Notably, we provide standard errors for the optimal calibration parameters.