Realized Volatility

Definition

Realized volatility (RV) is a nonparametric, model-free estimate of the latent integrated variance $IV_t = \int_0^1 \sigma_{t+s}^2\,ds$ constructed by summing squared intraday returns sampled at frequency $m$ :

$RV_t^{(m)} = \sum_{j=1}^{m} r_{t,j/m}^2$

By the quadratic variation theorem (Karatzas-Shreve 1988), $RV_t^{(m)} \xrightarrow{p} \int_0^1 \sigma_{t+s}^2\,ds$ as $m \to \infty$ under a continuous semimartingale price process. At 5-minute intervals ( $m = 288$ per trading day), the measurement noise is reduced by a factor of $m$ relative to squared daily returns, making RV the standard benchmark for evaluating and comparing volatility models.

Key Ideas

Squared daily returns are nearly pure noise. Under generalized autoregressive conditional heteroskedasticity (GARCH)(1,1) fitted to DM-$ exchange rates, measurement noise accounts for $\approx 93\%$ of the residual variance in Mincer-Zarnowitz forecast evaluation regressions — the low $R^2$ is a property of the criterion, not a flaw in the model (Andersen-Bollerslev 1998).
RV dramatically improves forecast evaluation. Replacing $r_t^2$ with 5-minute RV raises $R^2$ from $\approx 0.05$ to $\approx 0.48$ , close to the population maximum predicted by GARCH(1,1).
Returns standardized by $\sqrt{RV_t}$ are approximately $\mathcal{N}(0,1)$ . Andersen, Bollerslev, Diebold, and Labys (ABDL, 2001) document this across equity, foreign exchange (FX), and bond markets, implying the heavy-tailed unconditional distribution of returns is driven entirely by time-varying volatility — not by a fixed fat-tailed innovation distribution.
Microstructure noise limits sampling frequency. Bid-ask bounce, price discreteness, and asynchronous trading contaminate very-high-frequency returns. The optimal frequency trades off noise reduction against microstructure contamination, typically yielding 5–15 minute sampling for equity and FX data.
RV is the most precise of several daily volatility proxies. In the measurement hierarchy: $RV_t$ (5-minute) $\succ$ daily high-low range $\succ$ $|r_t|$ . RV Granger-causes the noisier proxies but not vice versa.
Model realized volatility directly (ABDL 2003): because RV is (nearly) observable, log realized volatilities can be modeled with standard linear Gaussian time series rather than latent-variable filtering. A simple trivariate long-memory (fractionally-integrated) Gaussian VAR for the log realized variances and covariance out-forecasts daily GARCH and RiskMetrics, and — combined with a lognormal-normal mixture — yields well-calibrated density and VaR forecasts.

How It Works

Quadratic Variation Foundation

Under a continuous semimartingale $dp_t = \mu_t\,dt + \sigma_t\,dW_t$ (no jumps), the quadratic variation over $[t-1, t]$ equals the integrated variance:

$[p,p]_t = \int_{t-1}^{t} \sigma_s^2\,ds$

and RV converges to it:

$RV_t^{(m)} = \sum_{j=1}^{m} \bigl(p_{t-1+j/m} - p_{t-1+(j-1)/m}\bigr)^2 \xrightarrow{p} \int_{t-1}^{t} \sigma_s^2\,ds \quad \text{as } m \to \infty$

This also establishes the mixture-of-normals representation of returns: conditioning on the full volatility path, the daily return is

$r_t \mid \{\sigma_{t+s}\} \sim \mathcal{N}\!\left(0,\; \int_0^1 \sigma_{t+s}^2\,ds\right)$

so that the marginal fat-tailed distribution of $r_t$ is a scale mixture of normals with mixing weight $IV_t$ .

Measurement-Error Decomposition (Andersen-Bollerslev 1998)

The standard volatility forecast evaluation regression uses squared daily returns as the proxy:

$r_t^2 = a + b\, h_t + u_t$

The residual mean squared error (MSE) decomposes as:

$\text{MSE}(h_t \to r_t^2) = \underbrace{\text{Var}(IV_t - h_t)}_{\text{model imperfection}} + \underbrace{\text{Var}(r_t^2 - IV_t)}_{\text{measurement noise}}$

For DM-$ data: model imperfection $= 0.084$ , measurement noise $= 1.137$ , total $= 1.221$ — noise accounts for 93% of the residual. The population $R^2$ under GARCH(1,1) is determined entirely by the signal-to-noise ratio in the criterion variable, not by the model's forecasting accuracy.

When RV at frequency $m$ replaces $r_t^2$ , the measurement noise shrinks by the factor $m$ . The $R^2$ improvement is precisely predicted by theory:

Frequency	$m$	Theory $R^2$ (DM-$)	Empirical $R^2$ (DM-$)
Daily	1	0.063	0.047
8-hour	3	0.151	0.133
Hourly	24	0.383	0.331
5-minute	288	0.483	0.479

The convergence of empirical to theoretical $R^2$ across five frequencies is itself confirmatory evidence of correct GARCH specification.

Microstructure Noise and Optimal Sampling

The observed log-price is $\tilde p_{t,j} = p_{t,j} + e_{t,j}$ , where $e_{t,j}$ is i.i.d. microstructure noise with variance $\xi^2$ (bid-ask bounce, discreteness, asynchrony). The bias in RV is:

$E[RV_t^{(m)}] = IV_t + 2m\xi^2$

which grows without bound as $m \to \infty$ — the "signature plot" signature. The optimal frequency $m^*$ minimizes MSE and corresponds empirically to 5–15 minute intervals. Noise-robust alternatives include:

Bipower variation (Barndorff-Nielsen–Shephard 2004): $BV_t = \tfrac{\pi}{2}\sum_j |r_{t,j}|\,|r_{t,j-1}|$ , consistent for $IV_t$ even with finite-activity price jumps. The difference $RV_t - BV_t$ estimates total jump variation.
Kernel-based RV (Barndorff-Nielsen et al. 2008): applies a smooth weight function to realized autocovariances of intraday returns, removing microstructure bias while remaining consistent for $IV_t$ .
Pre-averaging (Jacod et al. 2009): averages returns over a local window before squaring; achieves the optimal $m^{1/4}$ convergence rate under noise.

Log-Normality of RV (ABDL 2001)

Across equity, FX, and bond markets, $\log RV_t$ is approximately Gaussian. This motivates the log-normal mixture for the conditional return distribution:

$\log(\sigma_t)\,|\,\mathcal{F}_{t-1} \sim \mathcal{N}(\mu_t, \tau^2), \qquad r_t\,|\,\sigma_t \sim \mathcal{N}(0,\sigma_t^2)$

where $\sigma_t^2 \approx RV_t$ . This model generates well-calibrated value at risk (VaR) coverage rates, outperforming GARCH with normal innovations by exploiting the observed near-normality of $r_t/\sqrt{RV_t}$ — see Value at Risk.

The autocorrelation function (ACF) of $\log RV_t$ decays hyperbolically, motivating autoregressive fractionally integrated moving average (ARFIMA) models for RV forecasting (fractional $d \approx 0.4$ ) — see Long Memory and Fractional Integration.

RV Sampling Frequency and Monthly Regression Bias (Bollerslev-Zhou 2006)

Bollerslev and Zhou (BZ, 2006) use a Monte Carlo study (Table 2) to quantify how the choice of intraday sampling frequency affects the three return-volatility regression slopes at monthly frequency. Key findings:

5-minute RV introduces negligible measurement error for monthly volatility feedback, leverage, and implied-vol forecasting regressions. Biases are near zero and comparable to those from the infeasible integrated variance itself.
Daily-squared-return RV introduces large biases. For the volatility feedback slope, the bias toward zero is ~0.08 at T=150 and barely shrinks to 0.078 at T=600 — the bias is not alleviated by larger samples. This is because daily-squared-return noise is a fixed fraction of the monthly integrated variance rather than a measurement-error component that shrinks with T.
Practical implication: monthly return-volatility regressions should use 5-minute (or finer) intraday realized vol; using daily returns severely understates the predictability of volatility by lagged returns.

This result complements Andersen-Bollerslev (AB, 1998): where AB (1998) showed that noise in the criterion variable (squared daily return as volatility proxy) inflates GARCH forecast evaluation residuals, BZ (2006) show that noise in the regressor (daily RV as integrated variance proxy) induces persistent slope biases.

RV as a Volatility Proxy in the MIM (Engle-Gallo 2006)

The Multiple Indicators Model (MIM) treats three daily volatility proxies as a joint vector multiplicative error model (MEM): absolute returns $|r_t|$ , daily high-low range $hl_t$ , and realized volatility $v_t$ . Bayesian information criterion (BIC)-selected cross-equation lags run from $v_t$ to the noisier proxies, but not in reverse — confirming the Granger-causality hierarchy that maps to the measurement precision ordering. See MEM and MIM for the full system specification and multi-step forecasting dynamics.

Why It Matters

Resolves the GARCH forecast-evaluation paradox. $R^2 \approx 0.02$ – $0.05$ in squared-return regressions is exactly what a well-specified GARCH(1,1) predicts. The Andersen-Bollerslev (1998) measurement-error analysis closed a decade of incorrect skepticism about GARCH forecasting ability.
Provides a model-free variance proxy. RV requires no parametric model; it is a direct observation of $IV_t$ up to sampling error. This makes it usable as a dependent variable in forecasting regressions, as a real-time risk signal, and as a neutral benchmark for model comparison.
Enables distributional regularities. The near-normality of $r_t/\sqrt{RV_t}$ is a sharp empirical regularity that strongly constrains the class of admissible models: the non-Gaussian behavior of raw returns must come from time-varying volatility alone.
Identifies jump risk. The gap $RV_t - BV_t$ consistently estimates the jump component of quadratic variation, enabling model-free tests for jump activity and jump timing.

Realized Power and MIDAS Volatility Forecasting (Ghysels-Santa-Clara-Valkanov 2006)

Ghysels, Santa-Clara, and Valkanov (2006) compare five daily volatility predictors — realized variance $\tilde{Q}^{(m)}$ , squared returns $r^2$ , absolute returns $|r|$ , daily range $[hi\text{-}lo]$ , and realized power $\tilde{P}^{(m)} = \sum_{j=1}^m |r_{t-(j-1)/m}|$ — within the Mixed Data Sampling (MIDAS) distributed-lag regression, benchmarked against the ABDL ARFI(5,d) long-memory model (see MIDAS Regression).

Realized power strictly dominates all other predictors at every forecast horizon $H=1,5,10,20$ days and for every asset examined (DJ index + DIS, GE, JPM, XOM, MCD, HON). In-sample MSE ratios vs. ABDL range 0.606–0.912 in levels; out-of-sample 0.714–0.897. The predictor ranking is: $\tilde{P}^{(m)} \succ [hi\text{-}lo] \succ \tilde{Q}^{(m)} \succ |r| \succ r^2$

Why realized power outperforms realized variance: Under Barndorff-Nielsen–Shephard (2004) bipower theory, $\tilde{P}^{(m)}$ converges to the continuous component of quadratic variation (integrated variance without jumps), while $\tilde{Q}^{(m)}$ includes both continuous and jump parts. Because jumps are transient, their inclusion in the predictor reduces persistence and degrades multi-step forecasts. $\log\tilde{P}^{(m)}$ has higher ACF persistence and less measurement noise than $\log\tilde{Q}^{(m)}$ .

No gain from direct intraday use: Using raw 5-min returns as MIDAS regressors (rather than daily realized aggregates) does not improve MSE at any horizon — the relevant information in high-frequency data is fully captured by daily realized power.

RV Inconsistency under Lévy Time-Change (Barndorff-Nielsen-Shephard 2006)

When log-prices follow a time-changed Lévy process $Y_t = mt + Z_{\tau_t}$ with $Z$ non-Brownian, RV is an inconsistent estimator of the time-change $\tau_i$ . The quadratic variation of $Y$ includes a jump contribution, so $[Y]_{d,i} \xrightarrow{p} [Y]_i \neq k_2\tau_i$ regardless of sampling frequency. Specifically:

The variance of the RV error $[Y]_{d,i} - k_2\tau_i$ contains an irreducible term $k_4\Delta\xi$ driven by the 4th cumulant $k_4$ of the Lévy increments, which does not vanish as $M\to\infty$
The bias $E([Y]_{d,i} - k_2\tau_i) = O(M^{-1})$ is small; squared bias is $O(M^{-2})$ and dominated by the $O(1)$ variance in MSE
The ACF inequality: $\text{Cor}([Y]_{d,i}, [Y]_{d,i+s}) \leq \text{Cor}(\tau_i, \tau_{i+s})$ when $k_4 > 0$ — RV underestimates the autocorrelation of the true variance process, so actual volatility predictability exceeds what RV-based studies show
For a symmetric Lévy process, $k_4 = 0 \iff Z$ is Brownian motion (by Lévy-Khintchine); thus testing $k_4 = 0$ is a jump test

The practical implication is that the high predictability of RV found in Andersen-Bollerslev (1998) and subsequent work is a lower bound on the true predictability of $\tau_i$ . See Time-Changed Lévy Process for the full framework.

Open Questions

Multivariate realized covariance raises the Epps effect: non-synchronous trading across assets causes realized covariances to shrink toward zero at high sampling frequency. Refresh-time sampling and Hayashi-Yoshida estimators partially address this, but scalability to large cross-sections remains challenging.
Long memory vs. structural breaks in $\log RV_t$ . The hyperbolic ACF decay could reflect genuine fractional integration ( $d \approx 0.4$ ) or aggregation of short-memory components across volatility regimes. Disentangling these is unresolved in finite samples.
Jumps vs. continuous variation. Finite-sample power of $RV - BV$ jump tests is limited, especially for small jumps. Jump-robust estimators (quantile-based, truncated RV) offer complementary approaches.