Joint deep calibration of the 4-factor PDV model

TL;DR

This paper introduces a neural network-based approach to efficiently calibrate a complex 4-factor PDV model by learning key market prices, significantly reducing calibration time from hours to seconds.

Contribution

It develops a novel calibration method that replaces nested simulations with learned pricing functions, enabling rapid and accurate joint calibration of SPX and VIX market data.

Findings

Calibration time reduced to a few seconds

Accurate joint calibration of SPX and VIX derivatives

Neural network-based pricing functions outperform traditional methods

Abstract

Joint calibration to SPX and VIX market data is a delicate task that requires sophisticated modeling and incurs significant computational costs. The latter is especially true when 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 and makes the joint calibration problem accessible. However, the minimization loop remains slow due to 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.