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Realized variance

Realized variance is a non-parametric estimator used in financial econometrics to measure the ex-post quadratic variation of an asset's price process over a specified time interval, typically computed as the sum of squared intraday returns sampled at high frequencies, such as five-minute intervals.^[1] This approach provides a direct, model-free approximation of the integrated variance, which represents the accumulated squared volatility contributions from both continuous price movements and jumps in the underlying diffusion process.^[2] As the sampling frequency increases, realized variance converges in probability to the true quadratic variation under standard semimartingale assumptions, making it a consistent and efficient estimator superior to traditional low-frequency measures like daily squared returns.^[1] Theoretically, realized variance is grounded in continuous-time stochastic processes, where the price dynamics follow an Itô semimartingale: dp(t) = [\mu(t)](/page/MU) dt + [\sigma(t)](/page/Sigma) [dW(t)](/page/DW) + jump components, and the quadratic variation [p,p]_t = \int_0^t [\sigma^2(s)](/page/Sigma) ds + \sum squared jumps.^[2] For a day divided into M intraday periods, it is calculated as RV_t = \sum_{j=1}^M r_{t,j}^2, where r_{t,j} = \log(p_{t,j}) - \log(p_{t,j-1}) are the log returns.^[1] This formulation accounts for market microstructure noise through bias-corrected variants, such as two-scale or kernel-based estimators, which mitigate the effects of bid-ask bounce and other trading frictions at ultra-high frequencies.^[3] The concept traces its origins to Robert Merton's 1980 proposal for using high-frequency data to estimate integrated variance, but it gained prominence in the late 1990s through empirical applications in foreign exchange markets.^[3] Seminal work by Torben Andersen and Tim Bollerslev in 1998 demonstrated that aggregating 288 five-minute squared returns yields accurate daily volatility measures, validating ARCH/GARCH models' forecasting performance when evaluated against this benchmark.^[1] Subsequent advancements, including extensions by Andersen, Bollerslev, Diebold, and Labys in 2003, incorporated long-memory dynamics via vector autoregressive models on log realized variances, enhancing multi-period forecasts.^[2] In practice, realized variance underpins volatility forecasting, risk management, and derivative pricing, with models like the heterogeneous autoregressive (HAR) framework capturing persistent patterns in volatility persistence.^[3] It has been applied to equities, currencies, and commodities, enabling precise Value-at-Risk (VaR) calculations and portfolio optimization by providing superior proxies for latent volatility compared to parametric alternatives.^[2] Despite challenges from noise and jumps, ongoing refinements ensure its robustness in high-frequency trading environments.^[3]

Core Concepts

Definition and Motivation

Realized variance (RV), also known as realized volatility squared, serves as a nonparametric estimator of the integrated variance of an asset's log-price process over a fixed time interval, such as a trading day, by summing the squared intraday returns sampled at high frequencies.^[4] This approach approximates the quadratic variation of the underlying continuous semimartingale price process, which captures the true economic volatility arising from continuous price fluctuations in efficient markets. The motivation for realized variance stems from the shortcomings of traditional low-frequency volatility estimators, such as those based on daily close-to-close returns or parametric models like GARCH, which often suffer from measurement errors, model misspecification, and inability to precisely quantify ex-post volatility without imposing strong distributional assumptions.^[5] By exploiting the availability of high-frequency transaction data, RV provides a model-free, consistent ex-post measure that directly leverages the information content in intraday price movements to estimate the latent integrated variance more accurately, enabling better risk management, option pricing, and forecasting in financial applications.^[6] This nonparametric framework originated from the practical need to discern genuine market volatility from noise in increasingly data-rich environments, particularly as electronic trading expanded access to tick-by-tick observations. Conceptually, the roots of realized variance trace back to the mathematical theory of stochastic processes, where quadratic variation quantifies the pathwise variability of semimartingales, a class encompassing most financial price models.^[7] Empirically, its development gained traction in the 1990s with the proliferation of high-frequency data, building on earlier explorations of summed squared returns for variance decomposition in low-frequency settings, such as daily data analyses of stock return persistence. Initial applications focused on foreign exchange markets, where 5-minute intraday sampling demonstrated RV's superiority in volatility measurement over coarser alternatives.^[6] As an illustrative example, consider a stock's price observed at 5-minute intervals throughout the trading day. The intraday returns are computed as r_{t,i} = \log(P_{t,i}) - \log(P_{t,i-1}) for i = 1, \dots, M, where P_{t,i} denotes the price at the i-th interval on day t, and M is the total number of such intervals (e.g., 78 for a 6.5-hour day). The realized variance for that day is then RV_t = \sum_{i=1}^M r_{t,i}^2, which converges to the integrated variance as the sampling frequency increases under ideal conditions.^[8]

Mathematical Formulation

The log-price process X_t is modeled as a continuous Itô semimartingale over the time interval [0, T], satisfying the stochastic differential equation

dX_t = \mu_t \, dt + \sigma_t \, dW_t,

where W_t denotes a standard Brownian motion, \mu_t represents the drift process, and \sigma_t > 0 is the spot volatility process assumed to be càdlàg with locally square-integrable paths.^[9] The integrated variance, which quantifies the accumulated squared volatility over the horizon, is then given by

\int_0^T \sigma_t^2 \, dt.

This setup captures the continuous diffusion component of price dynamics without discontinuous jumps.^[9] The realized variance estimator RV_T is constructed from high-frequency observations of X_t as

RV_T = \sum_{i=1}^n (X_{t_i} - X_{t_{i-1}})^2,

where the sampling times are equidistant with t_i = iT/n for i = 1, \dots, n, and n \to \infty corresponds to the high-frequency sampling limit.^[10] This sum of squared intraday returns serves as a nonparametric measure of ex-post variation in log prices.^[10] The formulation relies on several key assumptions for its validity: the price process exhibits no jumps (ensuring continuity of paths), the time horizon T is fixed (e.g., corresponding to one trading day), and observations are sampled at increasingly fine intervals to achieve consistency in the limit.^[9] Under these ideal conditions—where the semimartingale is of finite variation in the drift and the volatility process is independent of the Brownian innovation—RV_T converges in probability to the quadratic variation process \langle X \rangle_T = \int_0^T \sigma_t^2 \, dt.^[10]

Estimation Methods

Under Ideal Conditions

Under ideal conditions, realized variance is estimated from high-frequency intraday price data assuming the underlying price process is an Itô semimartingale, with no measurement errors such as bid-ask bounce or other microstructure effects, and using equidistant sampling intervals throughout the trading day.^[11] These assumptions align with the price process being modeled as a semimartingale, where the log-price follows a stochastic differential equation driven by a Brownian motion, possibly with a jump component, without additive noise.^[10] In this setting, the estimator leverages the quadratic variation property of such processes to approximate the unobservable quadratic variation directly from observable returns. The computation proceeds in a straightforward step-by-step manner using intraday log-returns. First, collect observed log-prices p_{t_i} = \log P_{t_i} at n equidistant time points t_i = i [\Delta t](/page/Delta) over the interval [0, T], where [\Delta t](/page/Delta) = T/n is the sampling interval. Next, calculate the intraday log-returns r_{t_i} = p_{t_i} - p_{t_{i-1}} for i = 1, \dots, n. The realized variance RV is then obtained by summing the squared returns:

RV = \sum_{i=1}^n r_{t_i}^2.

This sum provides a nonparametric estimate of the period's quadratic variation without requiring parametric assumptions about the drift or volatility dynamics.^[11] The choice of sampling frequency significantly influences the precision; for instance, using 5-minute intervals (common in foreign exchange markets, yielding n \approx 288 per trading day) offers a robust approximation, while finer 1-minute sampling (n \approx 1440) further reduces discretization bias under these ideal conditions, as the estimator's efficiency improves with higher n.^[10] As the sampling frequency increases (n \to \infty, or equivalently \Delta t \to 0), the realized variance converges in probability to the quadratic variation, defined as

[p, p]_T = \int_0^T \sigma_s^2 \, ds + \sum_{0 < s \leq T} (\Delta p_s)^2,

where \sigma_s^2 denotes the spot variance at time s, the integral is the continuous integrated variance, and the sum captures squared jumps. This consistency holds under the stated assumptions, with the estimator exhibiting an asymptotic mixed Gaussian distribution scaled by \sqrt{n}, enabling inference on the latent quadratic variation.^[11] To illustrate, simulations of standard Brownian motion paths with constant volatility \sigma^2 = 1 over a unit interval demonstrate that RV tracks the true integrated variance \sigma^2 T = 1 closely, with mean squared error decreasing monotonically as n rises from 100 to 10,000 observations, confirming the theoretical convergence in a controlled noiseless environment.^[12]

Handling Microstructure Noise

Market microstructure noise arises primarily from bid-ask bounce, where transaction prices alternate between bid and ask quotes, inducing negative autocorrelation in observed returns.^[13] Inventory costs faced by market makers, who adjust prices to manage their holdings of securities, contribute additional noise by creating temporary price deviations from the efficient price. Episodic liquidity effects, such as varying trading intensity or order imbalances, further introduce time-dependent noise that correlates with the underlying price process.^[14] This noise contaminates high-frequency price observations, modeled as Y_t = X_t + \varepsilon_t, where Y_t is the observed log-price, X_t the efficient log-price, and \varepsilon_t the i.i.d. microstructure noise with variance \mathbb{E}[\varepsilon_t^2].^[15] Consequently, the naive realized variance estimator \mathrm{RV} = \sum_{i=1}^n (\Delta Y_i)^2 exhibits upward bias, with \mathbb{E}[\mathrm{RV}] \approx \int_0^T \sigma^2(t) \, dt + \frac{2n}{T} \mathrm{Var}(\varepsilon), where the second term inflates the estimate proportionally to the sampling frequency n/T.^[15] At ultra-high frequencies, such as tick-by-tick sampling, this noise term dominates, rendering standard realized variance unreliable for estimating integrated variance.^[14] To address this bias, the two-scale realized variance (TSRV) estimator combines high-frequency data with auxiliary low-frequency subsamples (e.g., every 5 minutes) to isolate and subtract the noise contribution, achieving consistency as the sampling frequency increases.^[16] Specifically, TSRV averages multiple subsampled realized variances and adjusts by the ratio of scales: \widehat{[X,X]}_T = \overline{\mathrm{RV}}_K - \frac{\bar{n}}{n} \mathrm{RV}_n, where K is the number of subsamples and optimal K \propto n^{2/3}.^[16] Realized kernel estimators mitigate noise by weighting autocovariances of returns with a positive definite kernel, such as the Parzen kernel, which truncates higher-order terms to reduce noise variance while preserving the quadratic variation signal; the bandwidth is selected to minimize mean squared error.^[17] Empirical studies using tick-by-tick transaction data from the Dow Jones Industrial Average stocks over 2000–2002 demonstrate that microstructure noise bias severely distorts realized variance at frequencies finer than 5 minutes, with signature plots showing declining estimates due to noise dominance.^[14] Post-2000s availability of high-frequency datasets like TAQ has enabled these corrections, such as TSRV and kernels, to substantially improve estimation accuracy, reducing bias by orders of magnitude compared to naive methods.^[14]

Theoretical Properties

Asymptotic Behavior

Under ideal sampling conditions with increasing intraday frequency n, the realized variance RV_n converges in probability to the integrated variance \int_0^T \sigma_t^2 \, dt, where \sigma_t^2 denotes the instantaneous variance process of the asset returns.^[18] More precisely, the centered and scaled error exhibits asymptotic mixed normality:

\sqrt{n} \left( RV_n - \int_0^T \sigma_t^2 \, dt \right) \xrightarrow{d} \mathrm{MN}\left(0, 2 \int_0^T \sigma_t^4 \, dt \right),

where \mathrm{MN}(0, V) denotes a mixed normal distribution with mean zero and random variance V conditional on the path of the volatility process \{\sigma_t\}.^[18] This limiting distribution arises under the assumption of a continuous semimartingale model for log-prices, with the mixing reflecting the stochastic nature of volatility.^[4] The rate of convergence is O_p(1/\sqrt{n}), reflecting the \sqrt{n}-consistency of RV_n as an estimator of integrated variance, which facilitates the construction of standard errors for inference.^[4] This rate stems from the central limit theorem applied to the sum of squared intraday returns, treating them as martingale increments in the high-frequency limit.^[18] The foundational result on this asymptotic mixed normality for continuous processes is established in the seminal work of Barndorff-Nielsen and Shephard (2002), which derives the limit using stochastic volatility models satisfying mild regularity conditions on the drift and volatility processes.^[4] These properties enable practical inference, such as hypothesis testing for volatility persistence through tests based on the scaled RV_n statistics or detection of jumps by comparing RV_n to alternative quadratic variation estimators.^[4]

Bias and Consistency

Realized variance (RV), defined as the sum of squared intraday returns, serves as a nonparametric estimator of the quadratic variation of a price process. Early theoretical foundations for its use in volatility modeling were established by Andersen and Bollerslev, who demonstrated through empirical analysis of high-frequency exchange rate data that RV provides an efficient and approximately unbiased measure of daily return volatility under standard diffusion assumptions.^[8] Subsequent formal proofs confirmed that, under the semimartingale assumption for the log-price process, RV converges in probability to the quadratic variation as the sampling interval Δt approaches zero, irrespective of the presence of a drift component. This consistency holds because the quadratic variation captures the cumulative effect of diffusive movements and finite-activity jumps, with the drift contributing negligibly in the high-frequency limit. In finite samples, however, RV exhibits biases arising from the discrete nature of sampling. Additionally, the choice of sampling scheme influences these biases; for instance, calendar time sampling (fixed intervals) can lead to higher variance and potential underestimation compared to business time sampling (proportional to trading volume), which better aligns with economic activity and reduces inefficiency in the estimator.^[19] Regarding robustness, standard RV remains consistent for the total quadratic variation even in the presence of jumps, as it incorporates both continuous and discontinuous components. However, when the objective is to estimate the continuous integrated variance (excluding jumps), standard RV becomes inconsistent for discontinuous processes, as jumps inflate the measure; jump-robust variants, such as bipower variation, restore consistency by filtering out jump contributions under mild regularity conditions on jump activity.

Realized Volatility

Realized volatility is defined as the square root of realized variance, providing a nonparametric estimate of the integrated standard deviation of returns over a given period.^[2] Specifically, if RV_t = \sum_{i=1}^M r_{t,i}^2 denotes the realized variance based on M intraday returns r_{t,i}, then realized volatility is RVOL_t = \sqrt{RV_t}, which converges in probability to \sqrt{\int_{t-1}^t \sigma_s^2 ds} as the sampling frequency increases, approximating the true latent volatility path. This measure offers advantages over realized variance for practical applications in finance, as it expresses volatility in percentage terms that align more intuitively with risk assessments, such as Value-at-Risk calculations or performance benchmarks.^[20] However, taking the square root introduces a bias due to Jensen's inequality, since the square root function is concave, leading to E[\sqrt{RV_t}] < \sqrt{E[RV_t]}, which underestimates the expected integrated volatility unless corrected.^[21] To obtain annualized realized volatility from daily estimates, RVOL_t is typically scaled by \sqrt{252}, reflecting the approximate number of trading days in a year and assuming independence across days under a random walk model for returns. This scaling yields figures comparable to yearly risk metrics, such as an annual volatility of approximately 15.8% from a daily realized volatility of 1%.^[2] Empirically, in high-liquidity assets like major foreign exchange rates (e.g., DM/ and ¥/), realized volatility derived from high-frequency data exhibits stronger correlation with option-implied volatility than low-frequency standard deviations computed from daily returns, with out-of-sample R^2 values up to 0.25 versus 0.10 for the latter, highlighting its superior ability to capture current market conditions.^[22]

Bipower and Other Nonparametric Measures

Bipower variation (BV) extends the standard realized variance by providing a jump-robust estimator of the integrated variance in jump-diffusion models. Defined as

\text{BV} = \frac{\pi}{2} \sum_{i=2}^{M} |r_{i}| \, |r_{i-1}|,

where r_i are high-frequency log-returns and M is the number of intraday observations, BV converges in probability to the quadratic variation of the continuous component of the price process, \int_0^T \sigma_t^2 \, dt, even in the presence of finite-activity jumps, as the contribution from jumps becomes negligible asymptotically. This measure was introduced to address the sensitivity of standard realized variance to jumps, which can inflate estimates in volatile markets. Building on BV, jump-robust realized variance decomposes total variation into continuous and jump components. The continuous variation is estimated using BV, while the jump variation \text{[RJ](/page/RJ)} is obtained as \text{[RJ](/page/RJ)} = \text{RV} - \text{[BV](/page/.bv)}, where RV is the standard realized variance; jumps are implicitly detected through this difference, with significant jumps contributing disproportionately to RV. This approach allows for consistent estimation of the continuous quadratic variation under jump-diffusion dynamics, enabling separate analysis of diffusive and discontinuous price movements. A related test for the presence of jumps compares RV and BV via a standardized statistic, rejecting the null of no jumps when the difference exceeds a threshold derived from the asymptotic variance of BV. Other nonparametric measures complement these by targeting higher-order properties or adjustments. Realized quarticity (RQ), defined as

\text{RQ} = \frac{M}{3} \sum_{i=1}^{M} r_i^4,

estimates the integrated quarticity \int_0^T \sigma_t^4 \, dt, which is crucial for inferring the asymptotic variance of realized variance, approximately \frac{2}{M} \int_0^T \sigma_t^4 \, dt.^[23] Multipower variation generalizes bipower to arbitrary powers and lags, such as

\text{MPV}(\mu_1, \dots, \mu_k) = \frac{1}{k \prod_{j=1}^k \mu_j} \sum_{i=1}^{M-k+1} \prod_{l=0}^{k-1} |r_{i+l}|^{\mu_{l+1}},

providing consistent estimators for powers of integrated volatility or higher moments in the presence of jumps when the powers satisfy certain conditions (e.g., \sum \mu_j < 2); the normalizing constant is adjusted using Gamma functions for exact consistency. Realized kernels adjust realized variance for autocorrelation in returns by incorporating lagged products weighted by a kernel function, yielding a bias-corrected estimator of quadratic variation that accounts for serial dependence without assuming specific noise structures. These measures, developed primarily by Barndorff-Nielsen and Shephard between 2004 and 2006, form the foundation for robust nonparametric estimation in models with jumps and stochastic volatility.^[24]

Applications in Finance

Risk Management and Forecasting

Realized variance serves as a key input in Value-at-Risk (VaR) models by providing an ex-post measure of actual market volatility, which can scale parametric VaR estimates or enhance historical simulation approaches for capturing tail risk. In parametric VaR, realized variance replaces or augments implied volatility assumptions, allowing for more accurate scaling of standard deviations under normal distributions, particularly when intraday data reveals microstructure effects not captured by daily returns. For historical simulation VaR, realized variance enables the construction of empirical distributions from high-frequency returns, improving the assessment of extreme losses by incorporating realized jumps and continuous components. This integration is especially beneficial in emerging markets, where jumps in realized volatility contribute significantly to tail events, leading to better VaR coverage rates.^[25] In volatility forecasting, the Heterogeneous Autoregressive (HAR) model has become a cornerstone for predicting future realized variance, leveraging its ability to capture long-memory properties in volatility dynamics without assuming fractional integration. The HAR model specifies realized variance at horizon h as:

RV_{t+h} = \beta_0 + \beta_d RV_t + \beta_w \left( \frac{1}{5} \sum_{i=1}^{5} RV_{t-i+1} \right) + \beta_m \left( \frac{1}{22} \sum_{i=1}^{22} RV_{t-i+1} \right) + \epsilon_{t+h},

where RV_t denotes daily realized variance, the weekly term averages the prior five days, and the monthly term averages the prior 22 trading days, with coefficients \beta_d, \beta_w, \beta_m reflecting short-, medium-, and long-term persistence. This cascade structure approximates long-memory behavior through heterogeneous trader horizons, making it parsimonious and easy to estimate via ordinary least squares. Extensions like HAR-CJ further decompose realized variance into continuous and jump components to refine forecasts in turbulent periods.^[26] Empirically, HAR-based forecasts using realized variance outperform traditional GARCH models in out-of-sample predictions, particularly for intraday portfolio risk, by better accommodating the leverage effect and volatility clustering observed in high-frequency data. Studies across equity indices show HAR models reducing forecast errors by 10-20% relative to GARCH, with superior performance in multi-step horizons due to their explicit modeling of temporal aggregation. This advantage stems from realized variance's nonparametric nature, which avoids parametric misspecification common in GARCH, enabling more reliable risk assessments for dynamic portfolios.^[26]^[25] A notable application occurred during the 2008 financial crisis, where spikes in realized volatility (annualized square root of realized variance) for the S&P 500—reaching levels of 43.6% compared to 13.4% pre-crisis—signaled abrupt regime shifts from calm to turbulent markets, aiding risk managers in detecting heightened uncertainty post-Lehman Brothers' collapse. Analysis of high-frequency data revealed that these realized variance surges, often exceeding 300% of prior norms, correlated strongly with VIX elevations and equity drawdowns, providing early warnings of systemic stress that parametric models overlooked. Such insights underscored realized variance's role in real-time regime detection, informing capital allocation and stress testing during the crisis. Similar dynamics were evident during the COVID-19 market crash in March 2020, when the annualized realized volatility of the S&P 500 surged to approximately 76%, further demonstrating its utility in identifying extreme market stress.^[27]^[28]

Derivative Pricing and Hedging

Realized variance serves as the primary settlement variable in variance swaps, over-the-counter derivatives that enable speculation on or hedging against fluctuations in asset volatility. The fair strike K of a variance swap is set such that the expected payoff is zero, specifically \mathbb{E}[RV_T - K] = 0, where RV_T denotes the realized variance over the contract period [0, T]. This strike reflects the market's risk-neutral expectation of future realized variance, derived from a portfolio of European options on the underlying asset via static replication. At maturity, the payoff is the notional amount multiplied by (RV_T - K), which ex post reveals the variance risk premium (VRP) as the difference between the implied variance (embedded in K) and the observed realized variance.^[29]^[30] Options on realized variance extend this framework by providing nonlinear exposure to variance outcomes, with pricing achieved through replication strategies that combine dynamic trading in the underlying asset and its derivatives. These options can be replicated by dynamically adjusting positions in log contracts and vanilla options, leveraging the quadratic variation properties of realized variance. In the Heston stochastic volatility model, closed-form approximations for option prices on realized variance are obtainable by solving associated partial differential equations or using Fourier transform methods, allowing for efficient valuation under realistic dynamics that capture volatility clustering and leverage effects.^[31]^[32] Hedging strategies for variance swaps often involve delta-hedging with VIX futures, which track expectations of future volatility and provide a liquid instrument for offsetting variance risk exposure. By maintaining a dynamic position in VIX futures alongside the underlying asset, market makers can neutralize sensitivities to both price and volatility shocks. Furthermore, realized variance facilitates ex-post profit and loss (P&L) attribution in option portfolios, enabling the decomposition of returns into components driven by directional bets, volatility timing, and vega exposure, thus aiding in performance evaluation and risk adjustment.^[30]^[33] The integration of realized variance into derivative markets expanded rapidly in the post-1990s era, fueled by the launch of the CBOE Volatility Index (VIX) in 1993 and its 2003 revision to a model-free implied variance measure comparable to variance swap fair values. This evolution facilitated the proliferation of exchange-traded volatility products, including VIX futures and options, and underscored the empirical significance of the VRP in equities, where studies report average premiums of 5-10% in volatility terms across major indices.^[34]^[35]

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