Volatility is (mostly) path-dependent

Julien Guyon and Jordan Lekeufack
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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, MATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École nationale des ponts et chaussées - Centre Inria de Paris - Inria - Institut National de Recherche en Informatique et en Automatique, Bloomberg L.P.
Jordan Lekeufack: Department of Statistics [Berkeley] - UC Berkeley - University of California [Berkeley] - UC - University of California, Bloomberg L.P.

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Abstract: We learn from data that volatility is mostly path-dependent: up to 90% of the variance of the implied volatility of equity indexes is explained endogenously by past index returns, and up to 65% for (noisy estimates of) future daily realized volatility. The path-dependency that we uncover is remarkably simple: a linear combination of a weighted sum of past daily returns and the square root of a weighted sum of past daily squared returns with different time-shifted power-law weights capturing both short and long memory. This simple model, which is homogeneous in volatility, is shown to consistently outperform existing models across equity indexes and train/test sets for both implied and realized volatility. It suggests a simple continuous-time path-dependent volatility (PDV) model that may be fed historical or risk-neutral parameters. The weights can be approximated by superpositions of exponential kernels to produce Markovian models. In particular, we propose a 4-factor Markovian PDV model which captures all the important stylized facts of volatility, produces very realistic price and (rough-like) volatility paths, and jointly fits SPX and VIX smiles remarkably well. We thus show that a continuous-time Markovian parametric stochastic volatility (actually, PDV) model can practically solve the joint SPX/VIX smile calibration problem. This article is dedicated to the memory of Peter Carr whose works on volatility modeling have been so inspiring to us.

Keywords: Volatility modeling; Path-dependent volatility; Endogeneity; Empirical PDV model; 4-factor Markovian PDV model; Joint S&P 500/VIX smile calibration; Stochastic volatility; Spurious roughness (search for similar items in EconPapers)
Date: 2023-07-19
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Citations: View citations in EconPapers (9)

Published in Quantitative Finance, 2023, 23 (9), pp.1221-1258. ⟨10.1080/14697688.2023.2221281⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-04373380

DOI: 10.1080/14697688.2023.2221281

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