How Wrong Is Your Volatility Number? Barndorff-Nielsen and Shephard Answer

Realized volatility is a lovely idea: chop the trading day into short intervals, square each interval's return, add them all up, and you have measured how volatile the day was. No model, no assumptions, no maximum likelihood. Just addition.

But there is a question lurking underneath, and it is the question a statistician always asks: how accurate is that number?

Because it is not exact. You did not sample continuously, you sampled every five minutes. Your sum is an estimate of the day's true volatility, not the thing itself. So how far off is it likely to be? Is realized volatility a precision instrument or a rough approximation? Nobody knew. Ole Barndorff-Nielsen and Neil Shephard worked it out.

The problem: a measurement without error bars is half a measurement

If your realized volatility for today comes out at 1.2%, and I ask "could the truth have been 1.5%?", you need to be able to answer. Without error bars you cannot:

• Say whether today was genuinely more volatile than yesterday, or whether the difference is measurement noise.
• Build a proper model of volatility, because you do not know how much of the wiggle in your realized series is real and how much is estimation error.
• Test any hypothesis about volatility at all, because every test needs a distribution.

Andersen and Bollerslev and their coauthors had shown that realized volatility converges to the truth as you sample more finely. That is a statement about the limit. It does not tell you how much error remains when you sample every five minutes, which is what you actually do.

The key idea via analogy: the noisy speedometer

Imagine estimating how far you drove by glancing at your speedometer every few minutes and multiplying by the elapsed time. Add up the segments and you get total distance. As you glance more often, your estimate gets better. But at any finite number of glances, you have error, and the size of that error depends on how erratically you were driving. On a steady motorway cruise, few glances suffice. In stop-start city traffic, you need far more.

Barndorff-Nielsen and Shephard derived exactly this for volatility. Their central result is a central limit theorem for realized volatility: they showed that the error in your realized volatility estimate is approximately bell-shaped, and, crucially, they worked out how big that error is.

The answer is illuminating and slightly alarming. The precision of your realized volatility depends on how volatile the volatility itself was. Specifically, the error is governed by a quantity called realized quarticity, which is essentially the sum of the fourth powers of your intraday returns. If the day contained a few enormous moves, your estimate is much less precise than the raw number suggests.

Put plainly: on the wild days you most want to measure accurately, your measurement is at its least accurate. The error bars widen exactly when you need them tight.

And they can be wide. Realized volatility computed from a reasonable number of intraday observations is not a precise instrument. It is a good one, far better than a squared daily return, but treating it as if it were the exact truth is a mistake, and this paper is what allowed people to stop making it.

The second contribution: fitting real models with it

Having established how accurate realized volatility is, they used it. Stochastic volatility models, the continuous-time cousins of GARCH where volatility follows its own random process, are notoriously painful to estimate. Volatility is hidden, and getting at the parameters usually requires heavy simulation-based methods.

Barndorff-Nielsen and Shephard showed you can sidestep much of that. Since realized volatility gives you a (noisy) direct look at the hidden state, you can use it to estimate stochastic volatility model parameters with far less machinery, as long as you correctly account for the measurement error, which is precisely what their theory tells you how to do.

Why it mattered

• It gave realized volatility a statistical foundation. Before this paper, realized volatility was a good idea supported by intuition and simulation. After it, it was a proper estimator with a known distribution, and the entire high-frequency econometrics field could be built on top.
• It made inference possible. Every subsequent test in this literature, including the jump tests that Barndorff-Nielsen and Shephard themselves later built, rests on knowing the sampling distribution of realized measures.
• It warned against false precision. The finding that estimation error blows up on turbulent days is a genuinely practical caution. If you feed realized volatility into a model as if it were exact, you will be overconfident precisely during crises.
• It made stochastic volatility models estimable. A whole class of elegant continuous-time models became far more usable.

The honest limitations

• The theory assumes no jumps. The clean results here hold for a price that moves continuously. Real prices jump on news. The authors themselves fixed this in later work with bipower variation, but this paper's framework does not handle jumps.
• It assumes no microstructure noise. The theory treats observed prices as the true efficient prices. In reality, at high frequencies, prices bounce between bid and ask, and that bounce inflates realized volatility. Sample too finely and the theory misleads you badly. The two-time-scales estimator and its relatives exist to fix this.
• The error bars themselves are estimated. The quantity that determines your precision, realized quarticity, must itself be estimated from the same data, and it is even noisier than realized volatility. Your error bar has its own error bar.
• The finite-sample approximation can be poor. The bell-curve approximation for the error works better if you take logarithms first, and better with more intraday observations. With coarse sampling it can be unreliable.

The one-line takeaway

Barndorff-Nielsen and Shephard turned realized volatility from a good idea into a proper statistical estimator by working out its error bars, and discovered the uncomfortable fact that your volatility measurement is least reliable on exactly the violent days you most need it to be right.