Gumbel Copula Fits U.S. Equity and Treasury Volatility Nearly Perfectly

When market turbulence hits, the size of price swings can rise together across assets, and that joint motion matters for risk and portfolio decisions. This paper introduces the realized copula of volatility, a nonparametric way to measure how the hidden ups and downs of volatility move together, using high-frequency returns and local volatility estimates. The authors show their estimator is consistent, whether the sample period is fixed or gets longer, and they also derive a functional central limit theorem for the measurement error in the time-invariant marginal copula of volatility. In simulations, the method tracks both the empirical and marginal copula of volatility well, even with only a moderate amount of high-frequency data over a relatively short sample. The accompanying goodness-of-fit test also shows strong size control and excellent power. Applied to high-frequency transaction data from futures contracts tied to the U.S. equity and treasury bond markets, the framework points to a Gumbel copula as a near-perfect fit for the realized variance processes.

Equity and bond futures can look calm in the price tape. Their swings can still jump together under stress. That hidden link is what this study targets. It names that link the realized copula of volatility. A copula is a rule for how two things move together. Here, the things are the size of price swings, not the prices themselves. Noisy trades still reveal how volatility sticks together. This study shows that a Gumbel shape can fit those links in U.S. equity and treasury bond futures. That matters for risk and hedging when markets get rough. The method uses high-frequency returns to read the link. It turns fast data into a map of shared risk.

The surprise sits in the swings

The method stays consistent when the time span stays fixed. It also stays consistent when the time span grows. This work derives a functional central limit theorem. That is a rule for the whole error curve. The law covers the error in the time-invariant, or same-over-time, link. The same framework adds a test for the shape. Simulations show the stand-in works well with only moderate high-frequency data over a short sample. It tracks both the data-based link and the time-invariant link. The test keeps its size in check and shows strong power. On futures tied to U.S. equity and treasury bonds, the fit is striking.

How the link is built

The engine starts with high-frequency returns. Those returns feed local volatility estimates. Local volatility means the short-run size of price swings. The copula then strips away the scale of each market. A copula is a link rule for how two things move together. What remains is the shape of the joint move. In-fill means more samples inside the same time span. The larger-sample case lets the data span grow too. A goodness-of-fit test checks whether a chosen shape matches the data. This work checks that shape against a Gumbel form.

Why this matters for markets

Risk models often watch return correlation alone. This work shifts attention to volatility itself. That matters when shocks spread across assets. The data pair U.S. equity futures with treasury bond futures. The best fit points to a Gumbel copula. That link shape puts more weight on joint extremes. So a plain correlation check leaves out part of the story. The new test gives portfolio and pricing work a sharper way to compare shapes.

What to test next

Noisy trades still reveal how volatility sticks together. That is the surprise this study leaves behind. For U.S. equity and treasury bond futures, the hidden link becomes a shape you can test. That turns a vague risk idea into a model choice. The clear next check is whether the same Gumbel fit holds in other futures pairs. If it does, the realized copula of volatility becomes a wider tool, not a one-off result.