Barndorff-Nielsen-Shephard (2002) Econometric Analysis of Realized Volatility and Its Use in Estimating Stochastic Volatility Models

Summary

Barndorff-Nielsen and Shephard develop the econometric distribution theory of realized volatility as an estimator of volatility in a general continuous-time stochastic volatility (SV) model. Under an SV model where the log-price is a Brownian motion with a stationary, independent spot-variance process, they derive the exact moments and the asymptotic distribution of the realized-volatility error — the difference between realized volatility (the sum of intraday squared returns) and the (discretized) integrated variance they call actual volatility. The error is asymptotically mixed Gaussian with variance governed by the integrated quarticity. Because this gives realized volatility a known measurement-error structure, SV-model parameters can be estimated by treating realized volatility as a noisy observation in a state-space / Kalman-filter framework, avoiding simulation-intensive methods.

Key Claims

• Realized volatility estimates integrated variance. Over a day split into $M$ intraday intervals, realized volatility $[y_\delta]=\sum_{j}y_{\delta,j}^2$ converges (as $M\to\infty$, i.e. $\delta\to 0$) to the quadratic variation of the log-price, which under the SV model equals the integrated variance $\sigma^{2*}=\int_0^1\sigma^2(u)\,du$ ("actual volatility").
• A CLT for the realized-volatility error. The estimation error is asymptotically mixed Gaussian: $$\sqrt{M}\,\big([y_\delta] - \sigma^{2*}\big) \;\Rightarrow\; \sqrt{2\textstyle\int_0^1\sigma^4(u)\,du}\;\; N(0,1),$$ so the precision is controlled by the integrated quarticity $\int\sigma^4$. Conditional on the volatility path the limit is normal; unconditionally it is a variance mixture.
• Feasible inference via realized quarticity. The unknown integrated quarticity is itself consistently estimated by the (scaled) sum of intraday fourth powers of returns, giving a feasible standard error for realized volatility.
• Exact moments. Beyond the asymptotics, the paper derives the exact mean and variance of the realized-volatility error for finite $M$ under the SV model.
• Estimation without simulation. Realized volatility is an unbiased, known-variance proxy for actual volatility, so it can enter a linear state-space model as a measurement equation; the Kalman filter then delivers quasi-likelihood estimates of SV parameters (e.g. for superpositions of Ornstein–Uhlenbeck variance processes), sidestepping MCMC / indirect inference.
• Generality. The theory accommodates leverage, non-Gaussian (Lévy / OU-type) volatility, superposition of factors, and connects to power-variation measures.

Concepts Introduced or Extended

• Realized Volatility
• Quadratic Variation
• Stochastic Volatility

Entities Mentioned

• Ole E. Barndorff-Nielsen
• Neil Shephard

Quotes

"We derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility)."

"These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods."

My Take

This is the probabilist's foundation of the realized-volatility program — where ABDL (2003) supplied the modeling-and-forecasting case, Barndorff-Nielsen and Shephard supplied the asymptotic distribution theory that makes realized volatility a measured quantity with a known error. The two ideas that echo through everything downstream are the mixed-Gaussian CLT and the role of quarticity: once you know the measurement-error variance is $2\int\sigma^4/M$, realized volatility stops being a black-box proxy and becomes an observation you can Kalman-filter, which is exactly why SV estimation no longer required MCMC. The natural next step — separating the continuous (integrated variance) part from jumps via bipower variation — is the sequel (BNS 2004), and the whole apparatus underlies modern high-frequency volatility inference.