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

Julien Guyon () and Léo Parent
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Julien Guyon: CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École nationale des ponts et chaussées - IP Paris - Institut Polytechnique de Paris, MATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - Centre Inria de Paris - Inria - Institut National de Recherche en Informatique et en Automatique - ENPC - École nationale des ponts et chaussées - IP Paris - Institut Polytechnique de Paris, NYU Tandon School of Engineering
Léo Parent: CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École nationale des ponts et chaussées - IP Paris - Institut Polytechnique de Paris

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Abstract: 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.

Keywords: Historical Measure; Maximum Likelihood; Hedging; Pricing; Risk-Neutral Measure; VIX; S&P 500; Econometrics; Smile Calibration; Calibration; Path-Dependent Volatility; Volatility (search for similar items in EconPapers)
Date: 2025-08-07
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Persistent link: https://EconPapers.repec.org/RePEc:hal:wpaper:hal-05446613

DOI: 10.2139/ssrn.5382733

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