Realized Range-Based Variance

• Realized range-based variance is a high-frequency estimator that aggregates squared high–low ranges to infer integrated variance more accurately.
• It offers higher efficiency and robustness than traditional realized variance by mitigating microstructure noise and bias.
• RRV is effectively integrated into advanced volatility models like realized GARCH and EGARCH, enhancing forecasting and tail-risk management.

Realized range-based variance (RRV) is a high-frequency statistical estimator designed to infer integrated variance (IV) of asset prices by aggregating the squared high–low range information over a partitioned trading period. Unlike realized variance, which relies on sums of squared returns at selected sampling intervals, RRV exploits the supremum–infimum path structure within each subinterval. Theoretical and empirical research demonstrates that RRV attains higher efficiency, exhibits superior robustness to market microstructure noise, and produces sharper volatility forecasts when integrated into advanced volatility modeling frameworks such as realized GARCH, EGARCH, and CARE models.

1. Mathematical Definition and Asymptotics

Let $p_t$ denote a log-price process given by a continuous Itô semimartingale: $$p_t = p_0 + \int_0^t \mu_u\,du + \int_0^t \sigma_u\,dW_u,$$ where $\sigma_u$ is spot volatility and $W_u$ is standard Brownian motion. Integrated variance on $[0,T]$, typically a single trading day, is defined as

$IV = \int_0^T \sigma_u^2 \, du.$

Partition $[0,T]$ into $n$ intervals $\{[t_{i-1}, t_i]\}_{i=1}^n$, each with $m$ tick-level quotes. The observed high and low over interval $[t_{i-1},t_i]$ are: $$H_{t_i} = \max_{0 \leq k \leq m} p_{t_{i-1}+k\Delta_i/m},\quad L_{t_i} = \min_{0 \leq k \leq m} p_{t_{i-1}+k\Delta_i/m}.$$ The realized range-based variance is then

$$RRV_{n,m} = \sum_{i=1}^n \frac{(H_{t_i} - L_{t_i})^2}{\lambda_{2,m}},$$

where $\lambda_{2,m} = \mathbb{E}[s_{W,m}^2]$ is the second moment of the $m$-grid range for a standard Brownian motion.

Under ideal continuous sampling ($m \to \infty$),

$$RRV_{n,\infty} = \frac{1}{\lambda_2} \sum_{i=1}^n (H_{t_i} - L_{t_i})^2,$$

with $$\lambda_2 = \mathbb{E}[\sup_{s,t \in [0,1]} (W_t - W_s)^2]$$.

Asymptotic mixed-normality is achieved: $$\sqrt{n}(RRV_{n,\infty} - IV) \xrightarrow{d} \mathcal{MN}(0,\,\Lambda\,IQ),$$ where $IQ = \int_0^T \sigma_u^4\,du$ and $$\Lambda = (\lambda_4 - \lambda_2^2)/\lambda_2^2 \approx 0.4073$$. This variance constant confirms roughly fivefold greater efficiency than realized variance (Christensen et al., 28 Jan 2026, Chang, 4 Feb 2025).

2. Discretization, Bias Correction, and Noise Robustness

In empirical practice, the true path supremum/infimum is unobservable; only discrete ticks are available. This introduces downward bias, controlled by the normalizing constant $\lambda_{2,m}$, which depends on the number of observations $m$ per interval: $$RRV_{n,m} = \sum_{i=1}^n \frac{s_{p,i,m}^2}{\lambda_{2,m}},$$ where $$s_{p,i,m} = \max_{k, \ell} \{ p_{t_{i-1}+k\Delta_i/m} - p_{t_{i-1}+\ell\Delta_i/m} \}$$.

Bias due to coarse sampling rates can therefore be explicitly corrected by Monte Carlo or infinite-series computation of $\lambda_{2,m}$. Sub-sampling schemes further mitigate microstructure effects: the "sub-sampled realized range" averages range statistics from multiple non-overlapping grids, phase-shifting the sampling start within each interval (Tendenan et al., 2020, Gerlach et al., 2016, Wang et al., 2017). This provides robustness to bid–ask bounce and transaction price irregularities.

Alternative range-based estimators—Parkinson, Garman–Klass, Rogers–Satchell—also clarify the spectrum of robustness to noise and drift (Mouti, 2023):

• Parkinson: $$\sigma^2_{P, t} = \frac{1}{4 \ln 2} (\ln H_t/L_t)^2$$.
• Garman–Klass: $$\sigma^2_{GK, t} = \frac{1}{2} (\ln H_t/L_t)^2 - (2 \ln 2 - 1)(\ln C_t/O_t)^2$$.
• Rogers–Satchell: $$\sigma^2_{RS, t} = u_t (u_t - r_t) + d_t (d_t - r_t)$$.

Range statistics are empirically validated as more noise-robust than return-based realized variance (Mouti, 2023).

3. Integration into Volatility Modeling Frameworks

RRV is incorporated as the realized measure $x_t$ in measurement equations of various volatility models.

Realized EGARCH Model

For returns $r_t$ and latent log-volatility $h_t$: $$\begin{aligned} r_t &= \mu + \sqrt{h_t}\,\epsilon_t,\ \log h_t &= \omega + \beta \log h_{t-1} + \tau_1 \epsilon_{t-1} + \tau_2 (\epsilon_{t-1}^2 - 1) + \gamma u_{t-1},\ \log x_t &= \xi + \phi \log h_t + \delta_1 \epsilon_t + \delta_2 (\epsilon_t^2 - 1) + u_t, \end{aligned}$$ where $x_t$ can be sub-sampled RRV $RRV^{sub}_t$, and $u_t$ is i.i.d. Gaussian noise (Tendenan et al., 2020).

Joint Realized EGARCH (REGARCH)

For multiple realized measures $x_{k,t}$ (e.g., both RV and RRV): $$\begin{aligned} r_t &= \sqrt{h_t} z_t,\ \log h_t &= \omega + \beta (\log h_{t-1} - \omega) + \tau_1 z_{t-1} + \tau_2(z_{t-1}^2 - 1) + \gamma_1 u_{1, t-1} + \gamma_2 u_{2, t-1},\ \log x_{k, t} &= \xi_k + \log h_t + \delta_{k, 1} z_t + \delta_{k, 2}(z_t^2 - 1) + u_{k, t}, \end{aligned}$$ with $z_t \sim \text{N}(0,1)$ (Chang, 4 Feb 2025).

Realized-CARE (Expectile)

Measurement equation uses (sub-)sampled range to drive expectile dynamics: $$x_t = \xi + \phi |\mu_t| + \tau_1 \epsilon_t + \tau_2 (\epsilon_t^2 - \overline{\epsilon}^2) + u_t, \quad u_t \sim N(0, \sigma_u^2),$$ where $\mu_t$ is the latent expectile (Gerlach et al., 2016).

Tail-Risk Frameworks

Bayesian estimation via adaptive Markov Chain Monte Carlo (MCMC), with block-wise mixtures and robust acceptance rate targeting, provides efficient inference for RRV-driven volatility models (Wang et al., 2017).

4. Statistical Efficiency and Comparative Performance

RRV achieves lower mean-squared error than classical realized variance in estimating IV. Under continuous sampling, the key efficiency constant is $\Lambda \approx 0.4073$, corresponding to a variance reduction by roughly a factor of five; practical gains remain significant for finite $m$ (Christensen et al., 28 Jan 2026, Chang, 4 Feb 2025).

Empirical studies on equity indices and large-cap assets consistently find:

• Measurement error variance $\sigma_{RRV}^2 < \sigma_{RV}^2$ (e.g., $0.183$ vs $0.234$ for Nikkei 225) (Chang, 4 Feb 2025).
• In-sample fit improvements (lower AIC/SBIC, higher log-likelihood for REGARCH models with RRV vs. RV).
• Superior out-of-sample forecasting: lower QLIKE loss, higher realized kernel proxy log-likelihood, and improved Model Confidence Set inclusion rates at 90% (Wang et al., 2017, Gerlach et al., 2016).
• More accurate tail-risk forecasts (VaR, Expected Shortfall), with sub-sampled RRV yielding nominal violation rates and lower capital buffer requirements.

5. Applications to Rough Volatility and Long-Memory Modeling

Recent evidence demonstrates that range-based volatility estimators validate fractional Brownian motion (fBm) models for log-volatility dynamics, with observed Hurst exponent $H \approx 0.07$–$0.1$, even lower than that found using return-based realized volatility (Mouti, 2023). Range-based realized volatility strongly supports the core tenets of the rough volatility paradigm, indicating that observed market volatility is intrinsically rough, not an artifact of microstructure noise.

Specification of the Rough Fractional Stochastic Volatility (RFSV) model using RRV as input confirms competitive or superior forecasting performance relative to AR, HAR, and traditional GARCH frameworks, particularly at multi-day horizons. RRV validates the scaling law $m(q, \Delta) \approx K_q \Delta^{qH}$ for log-variance increments, providing direct empirical support for the fBm hypothesis.

6. Empirical Implementation and Monte Carlo Studies

Extensive application to ultra-high-frequency TAQ data shows that feasible RRV estimators ($m \approx 13$ for 5-min midquotes) achieve approximately 58% of the variance of realized variance estimators, higher autocorrelation, and narrower confidence bands, all with negligible computational overhead once $\lambda_{2,m}, \lambda_{4,m}$ are known (Christensen et al., 28 Jan 2026). Across multiple market settings—large indices, individual equities, long time frames—RRV-driven volatility models dominate alternatives in statistical accuracy and model selection tests.

7. Summary of Properties and Model Integration

Estimator	Asymptotic Variance Factor	Microstructure Robustness	Empirical Forecast Performance
Realized Variance	2	Sensitive	Lower
RRV (full path)	0.4073	Robust	Higher
Sub-sampled RRV	$\leqslant$ RRV	Maximally Robust	Highest

RRV is an unbiased, statistically-efficient, bias-correctable high-frequency estimator of integrated variance, exhibiting robust performance in volatility prediction, tail risk management, and rough volatility model validation. Its superiority holds across diverse GARCH-type, EGARCH-type, and expectile-based frameworks and under differing sample periods, market indices, and asset classes (Christensen et al., 28 Jan 2026, Tendenan et al., 2020, Chang, 4 Feb 2025, Wang et al., 2017, Gerlach et al., 2016, Mouti, 2023).