Volatility Is (Mostly) Path-Dependent

47 Pages Posted: 3 Aug 2022 Last revised: 30 May 2023

See all articles by Julien Guyon

Julien Guyon

Ecole des Ponts ParisTech

Jordan Lekeufack

University of California, Berkeley

Date Written: July 27, 2022

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

JEL Classification: G13

Suggested Citation

Guyon, Julien and Lekeufack Sopze, Jordan, Volatility Is (Mostly) Path-Dependent (July 27, 2022). Available at SSRN: https://ssrn.com/abstract=4174589 or http://dx.doi.org/10.2139/ssrn.4174589

Julien Guyon (Contact Author)

Ecole des Ponts ParisTech ( email )

Paris
France

Jordan Lekeufack Sopze

University of California, Berkeley ( email )

CA
United States

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