Kim Christensen, Wenjing Liu, Zhi Liu, Yoann Potiron
arXiv 17 Apr 2026 · Econometrics
arXiv:2604.15811 · PDF · DOI · OpenAlex · Extracted main text
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 hypotheses 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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Christensen, Thyrsgaard, and Veliyev (2019) The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing | 0.961 | 9 | 5 | 89% |
| 2 | Jacod and Protter (2012) Discretization of Processes | 0.941 | 6 | 4 | 83% |
| 3 | Li, Todorov, and Tauchen (2013) Volatility occupation times | 0.807 | 19 | 7 | 53% |
| 4 | Black (1976) Studies of stock market volatility changes | 0.644 | 2 | 2 | 100% |
| 5 | Christie (1982) The stochastic behavior of common stock variances: Value, leverage and interest rate effects | 0.644 | 2 | 2 | 100% |
| 6 | Mancini (2009) Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps | 0.644 | 2 | 2 | 100% |
| 7 | Newey and West (1994) Automatic lag selection in covariance matrix estimation | 0.644 | 2 | 2 | 100% |
| 8 | Ibragimov (1975) Independent and stationary sequences of random variables | 0.511 | 4 | 2 | 25% |
| 9 | Fermanian, Radulović, and Wegkamp (2004) Weak convergence of empirical copula processes | 0.511 | 2 | 2 | 50% |
| 10 | Rosenblatt (1956) A central limit theorem and a strong mixing condition | 0.511 | 2 | 1 | 100% |
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