The realized copula of volatility

2026-04-17

AI Summary

This study addresses the nonparametric characterization of second-moment dependence structures in continuous-time multivariate asset price processes. Building on high-frequency returns, the authors construct local volatility estimators and introduce, for the first time, the “realized volatility copula” statistic to nonparametrically estimate the empirical copula of latent stochastic volatilities. They establish infill asymptotic consistency under both fixed and expanding time horizons and derive a functional central limit theorem for the empirical process of time-invariant marginal copulas subject to measurement error. Simulations demonstrate that the method accurately approximates the true volatility copula even with moderate sampling frequencies and short sample periods, while goodness-of-fit tests exhibit well-controlled size and high power. Empirically, the analysis reveals that the dependence between U.S. equity and Treasury futures volatilities is well captured by a Gumbel copula.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study a new measure of codependency in the second moment of a continuous-time multivariate asset price process, which we name the realized copula of volatility. The statistic is based on local volatility estimates constructed from high-frequency asset returns and affords a nonparametric estimator of the empirical copula of the latent stochastic volatility. We show consistency of our estimator with in-fill asymptotic theory, either with a fixed or increasing time span. In the latter setting, we derive a functional central limit theorem for the empirical process associated with the measurement error of the time-invariant marginal copula of volatility. We also develop a goodness-of-fit test to evaluate hypothesis about the shape of the latter. In a simulation study, we demonstrate that our estimator is a good proxy of both the empirical and marginal copula of volatility, even with a moderate amount of high-frequency data recorded over a relatively short sample. The goodness-of-fit test is found to exhibit size control and excellent power. We implement our framework on high-frequency transaction data from futures contracts that track the U.S. equity and treasury bond market. A Gumbel copula is found to offer a near-perfect bind between the realized variance processes in these data.

Problem

• realized copula
• volatility
• stochastic volatility
• empirical copula
• high-frequency data

Innovation

• realized copula
• stochastic volatility
• high-frequency data
• nonparametric estimation
• goodness-of-fit test