The Discrete-Time 4-Factor Path-Dependent Volatility Model: Calibration under P and Q

This article examines calibration "under P" and "under Q" for a discrete-time version of the 4-factor path-dependent volatility (PDV) model introduced by Guyon and Lekeufack. First, the article demonstrates the model's ability to fit option data across multiple dimensions, including the VIX time series, the S&P 500 volatility surface, and S&P 500 and VIX smiles jointly. The article then focuses on estimating the historical measure model via maximum likelihood. It is then demonstrated that the considered PDV model outperforms existing models in the literature based on likelihood metrics. The paper subsequently evaluates the consistency between the probability measure implied by the model estimated under P and the option market data. The obtained results reveal that these measures are fairly close, which both validates the model's consistency with market data and supports the hypothesis of high endogeneity the P-and Q-dynamics, along with their tight interconnection. In light of this relationship, we propose a new model estimation approach combining P-and Q-information in order to enhance calibration robustness. The approach's effectiveness is then benchmarked against classical calibration methods.