Joint deep calibration of the 4-factor PDV model

ArXiv ID: 2507.09412 “View on arXiv”

Authors: Fabio Baschetti, Giacomo Bormetti, Pietro Rossi

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.

Keywords: Joint Calibration, VIX, Neural Networks, Markov Path-Dependent Volatility, Implied Volatility, Volatility/Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced stochastic calculus, neural network theory, and regression techniques (high math), while also demonstrating practical implementation with specific calibration times, GPU usage, and a focus on making the model accessible for trading desks (high rigor).

  flowchart TD
    A["Research Goal<br>Joint SPX/VIX Calibration<br>to 4-Factor PDV Model"] --> B{"Data Inputs"}
    B --> B1["SPX Market Data"]
    B --> B2["VIX Market Data<br>Futures & Options"]
    
    B --> C["Methodology<br>Deep Calibration via Neural Networks"]
    
    C --> D["Computational Process<br>Forward Pass: Matrix-Vector Products"]
    D --> E["Outcome 1<br>SPX Implied Volatilities"]
    D --> F["Outcome 2<br>VIX Futures"]
    D --> G["Outcome 3<br>VIX Call Option Prices"]
    
    E & F & G --> H["Final Result<br>Joint Calibration in Seconds<br>vs. Nested Monte Carlo"]