Summary
This paper (ABDL 2003) provides a unified framework for folding high-frequency intraday data into the measurement, modeling, and forecasting of daily and lower-frequency return volatility and return distributions. Building on continuous-time arbitrage-free pricing and the theory of quadratic variation, it establishes that realized volatility — the sum of squared high-frequency intraperiod returns — is a model-free, consistent, and highly efficient estimator of the latent integrated volatility (the day's quadratic variation), turning volatility into an effectively observable series. Using continuously recorded Deutschemark/Dollar and Yen/Dollar spot rates (1986–1999), the authors show that a simple long-memory Gaussian vector autoregression (VAR) for the logarithmic daily realized volatilities out-forecasts standard GARCH and high-frequency competitors, and — coupled with a lognormal-normal mixture — delivers well-calibrated density forecasts and accurate value-at-risk (VaR) quantiles.
Key Claims
- Realized volatility estimates quadratic variation. For an arbitrage-free (special semimartingale) log-price vector, the continuously-compounded return's conditional covariance matrix is governed by the quadratic variation. As the intraday sampling interval →0, realized volatility ∑jrt,jrt,j′ converges almost surely to the increment of the quadratic variation process (integrated volatility), giving a nonparametric, model-free volatility measure that requires no specific volatility model to be estimated.
- Volatility becomes observable — so model it directly. Because realized volatility is (nearly) observable, one can treat log realized volatilities as data and model them with standard time-series tools, sidestepping the latent-variable filtering that GARCH and stochastic volatility models require. This continues the observable-proxy tradition of Taylor (1986) but with theoretically-grounded, far more efficient measures.
- Three empirical regularities (established in ABDL 2000a/2001, and the foundation of this paper): (1) raw returns are leptokurtic, but returns standardized by realized volatility are approximately Gaussian; (2) realized volatilities are right-skewed, but their logarithms are approximately Gaussian; (3) the long-run dynamics of log realized volatilities exhibit long memory / fractional integration.
- A simple long-memory Gaussian VAR forecasts best. A trivariate fractionally-integrated Gaussian VAR for the log realized DM and Yen variances and their realized covariance — estimated allowing for realized-volatility measurement error — produces out-of-sample (2.5-year) volatility forecasts that generally beat RiskMetrics, daily GARCH, FIEGARCH, and short-memory alternatives, at horizons from 1 to 10+ days.
- Well-calibrated density and VaR forecasts. Under the empirically-supported assumption that returns are conditionally normal given realized volatility, the VAR volatility forecast implies a lognormal-normal mixture predictive return distribution. Diebold-Gunther-Tay density-forecast evaluation confirms the resulting one-step and multi-step density forecasts are conditionally well-calibrated, yielding accurate conditional quantiles (VaR).
- Scalability to large covariance matrices. Because the approach models observable realized (co)variances with linear Gaussian methods rather than filtering latent volatility, it escapes the curse of dimensionality that constrains multivariate ARCH/SV models, holding promise for the large covariance matrices in asset pricing, portfolio allocation, and risk management.
Concepts Introduced or Extended
- Realized Volatility — canonical reference; construction from high-frequency returns, consistency for integrated volatility, log-Gaussianity, long-memory dynamics
- Quadratic Variation — the theoretical target; integrated volatility as the quadratic variation of the continuous martingale component
- Long Memory and Fractional Integration — fractionally-integrated dynamics of log realized volatility; the FI-VAR specification
- Stochastic Volatility — realized volatility as a nonparametric alternative to latent-volatility filtering
- GARCH — the daily-return volatility models the VAR is benchmarked against
- Value-at-Risk — lognormal-normal mixture density forecasts and conditional quantiles
- Volatility Forecast Evaluation — realized volatility as the ex-post volatility proxy against which forecasts are scored
Entities Mentioned
Quotes
"The theory of quadratic variation suggests that, under suitable conditions, realized volatility is an unbiased and highly efficient estimator of return volatility."
"Although raw returns are clearly leptokurtic, returns standardized by realized volatilities are approximately Gaussian… although the distributions of realized volatilities are clearly right-skewed, the distributions of the logarithms of realized volatilities are approximately Gaussian."
"Forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably."
My Take
This is the paper that operationalized realized volatility and launched the modern high-frequency volatility-measurement literature. Its conceptual move — treat volatility as (almost) observable via quadratic-variation theory, then apply simple linear Gaussian time-series methods to log realized volatilities — is deceptively powerful: it converts the hard latent-variable filtering problem of GARCH/SV into an ordinary forecasting problem, and in doing so wins the forecast horse race with a strikingly simple model. The three distributional regularities (Gaussian standardized returns, log-Gaussian RV, long-memory log-RV) are the empirical bedrock, and the long-memory VAR + lognormal-normal mixture is an elegant, practically useful package for density and VaR forecasting. The wiki already carries the authors' later syntheses (ABCD 2004, ABCD 2006) and the precursor Andersen-Bollerslev (1998) "Answering the Skeptics"; this paper is the theoretical and empirical keystone they build on. The main caveats — market-microstructure noise biasing RV at the highest frequencies, and jumps contributing to quadratic variation — are precisely what the subsequent literature (Barndorff-Nielsen-Shephard bipower variation, two-scales estimators) went on to address.