A GMM approach to estimate the roughness of stochastic volatility

TL;DR

This paper tackles the problem of estimating a log-normal stochastic volatility model driven by fractional Brownian motion, where the Hurst index $H$ captures roughness. It formulates a GMM estimation approach that leverages the integrated variance process and a bias-corrected realized variance proxy to achieve consistency and asymptotic normality across the memory spectrum. The key contributions include deriving low-order moment structure for integrated variance, embedding measurement-error bias corrections without additional nuisance parameters, and establishing CLTs under a long-span setting plus extensions to double-asymptotics with high-frequency data. Simulation studies show accurate finite-sample performance across rough, standard, and long-memory regimes, while empirical analysis on major equity indices reveals very rough volatility with $\hat{H}$ near zero, robust to proxy noise and sampling frequency. Overall, the work provides a practical, theoretically grounded method for detecting and quantifying roughness in stochastic volatility with real-world high-frequency data.

Abstract

We develop a GMM approach for estimation of log-normal stochastic volatility models driven by a fractional Brownian motion with unrestricted Hurst exponent. We show that a parameter estimator based on the integrated variance is consistent and, under stronger conditions, asymptotically normally distributed. We inspect the behavior of our procedure when integrated variance is replaced with a noisy measure of volatility calculated from discrete high-frequency data. The realized estimator contains sampling error, which skews the fractal coefficient toward "illusive roughness." We construct an analytical approach to control the impact of measurement error without introducing nuisance parameters. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show the bias correction attenuates any systematic deviance in the parameter estimates. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in stochastic volatility.

Figures (5)

• Figure 1: Sample path of spot and integrated variance.
• Figure 2: Kernel density estimate of standardized $H$.
• Figure 3: Distribution of the J-test of overidentifying restrictions.
• Figure 4: Properties of the roughness estimate.
• Figure 5: Properties of .SPX realized variance.

Theorems & Definitions (19)

• Theorem 2.1
• Example 3.1: Realized variance, CLT approximation
• Example 3.2: Realized variance, no drift or leverage effect
• Example 3.3: Bipower variation, CLT approximation
• Theorem 3.4
• Theorem 3.5
• Proposition 3.6
• Theorem 3.7
• Theorem 3.8
• Theorem 3.9