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Realized range-based estimation of integrated variance
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\setcounter{page}{1} The volatility of asset prices is a key ingredient in several areas of financial economics. Not long ago, academic studies routinely used constant volatility models (e.g., \citeasnoun{black-scholes:73a}), despite empirical evidence in the data suggesting that the conditional variance is both time-varying and highly persistent. These facts were uncovered by the development and application of parametric models, such as ARCH (see, e.g., \citeasnoun{bollerslev-engle-nelson:94a}), through stochastic volatility models (e.g., \citeasnoun{ghysels-harvey-renault:96a}), and more recently non-parametric methods based on high-frequency data, the most conspicuous idea being realized variance ($RV$), see, e.g., \citeasnoun{andersen-bollerslev-diebold-labys:01a} or \citeasnoun{barndorff-nielsen-shephard:02a}; henceforth ABDL and BN-S.
$RV$ is the sum of squared returns over non-overlapping intervals within a sampling period. Given weak regularity conditions, $RV$ converges in probability to the quadratic variation ($QV$) of all semimartingales as the sampling frequency tends to infinity.
In practice, the consistency of $RV$ breaks down as data limitations prevent the sampling frequency from rising without bound. Most notably, market microstructure noise contaminates high-frequency asset prices. This invalidates the asymptotic theory, and $RV$ is known to be inconsistent in the presence of noise (e.g., \citeasnoun[2006]{bandi-russell:05a} or \citeasnoun{hansen-lunde:06a}). Therefore, it is common in applied work to construct $RV$ at a moderate frequency, where the impact of noise is small enough to be ignored, but this leads to loss of information. Though current research seeks to make $RV$ robust against microstructure noise (e.g., \citeasnoun{zhang-mykland-ait-sahalia:05a} or \citeasnoun{barndorff-nielsen-hansen-lunde-shephard:08a}), the most accurate estimator of $QV$ remains unknown. Set against this backdrop, we suggest the realized range-based variance ($RRV$).
Range-based estimation of volatility (developed in, e.g., \citeasnoun{feller:51a}, \citeasnoun{garman-klass:80a}, \citeasnoun{parkinson:80a}, \citeasnoun{rogers-satchell:91a}, \citeasnoun{kunitomo:92a}, and \citeasnoun{alizadeh-brandt-diebold:02a}) reveals more information than returns sampled at fixed intervals, because the extremes are formed from the entire price process. The daily squared range, for example, is about five times more efficient at estimating the scale of Brownian motion than the daily squared return. But, as noted in \citeasnoun{andersen-bollerslev:98a}, the accuracy of the high-low estimator is only around that afforded by $RV$ based on two- or three-hour returns, and the range has largely been neglected in the recent literature.
Intraday range-based estimation of volatility, however, has the potential of achieving smaller sampling errors than a sparsely sampled $RV$, because we can replace every squared return of $RV$ with a squared range and extract most of the information about volatility contained in the intermediate data points. No prior studies have explored the properties of such an estimator. Indeed, it is not clear what to expect from sampling, properly transformed, high-frequency ranges. Extrapolating from the daily interval would suggest that hourly ranges, say, achieve the accuracy of $RV$ based on five- or ten-minute returns, but the comparison is more complicated as each intraday range is constructed from less data.
We propose to sample and sum intraday price ranges to construct more efficient estimates of $QV$. Our contributions are four-fold. First, we develop a non-parametric method for measuring $QV$ with $RRV$. Second, and unlike the existing time-invariant theory for the high-low, we deal with estimation of time-varying volatility, when the driving terms of the price process are (possibly) continuously evolving random functions. Third, we derive a set of probabilistic laws for sampling intraday high-lows. Fourth, we remove the problems with downward bias reported in the previous range-based literature. The new estimator is defined as:
where $s_{p_{i \Delta, \Delta}, m} = \max_{0 \leq s,t \leq m} \left\{ p_{ \left(i - 1 \right)/n + t/mn} - p_{\left( i - 1 \right)/n + s/mn} \right\}$ is the observed range of a price process $p$ over the interval $\left[ \left( i - 1 \right) / n, i / n \right]$, $i = 1, \ldots, n$. $m$ is the number of high-frequency returns used to construct $s_{p_{i \Delta, \Delta}, m}$ and $\lambda_{2, m}$ is a constant. We prove that $RRV_{m}^{ \Delta}$ is consistent for the integrated variance ($IV$) and that $\sqrt{n} \left( RRV_{m}^{ \Delta} - IV \right)$ has a mixed Gaussian limit with a variance that can be much smaller relative to $RV$.
The paper is structured as follows. In the next section, we unfold the necessary diffusion theory, present various ways of measuring volatility and advance our methodological contribution by suggesting $RRV$ and a version thereof that handles non-trading effects. Under mild conditions, we prove consistency for the estimation method and derive a mixed Gaussian central limit theorem (CLT). Section 3 illustrates the approach through Monte Carlo analysis to uncover the finite sample properties, and we present some empirical results in section 4. Rounding up, section 5 offers conclusions and sketches several directions for future research.
In this section, we propose a new method for consistently estimating quadratic variation ($QV$) based on the price range. The theory is developed for the log-price of a univariate asset evolving in continuous time over some interval, say $p = \left( p_{t} \right)_{t \geq 0}$. $p$ is defined on a filtered probability space $\bigl( \Omega, \mathcal{F}, \left( \mathcal{F}_{t} \right)_{t \geq 0}, \mathbb{P} \bigr)$ and adapted to the filtration $\left( \mathcal{F}_{t} \right)_{t \geq 0}$, i.e. a collection of $\sigma$-fields with $\mathcal{F}_{u} \subseteq \mathcal{F}_{t} \subseteq \mathcal{F}$ for all $u \leq t < \infty$.
The basic building block is that $p$ constitutes a continuous sample path semimartingale.\footnote{We adopt the continuity assumption as a starting point only. In subsequent work, we have been analyzing the properties of our estimator, when $p$ exhibits jumps (see \citeasnoun{christensen-podolskij:12a}).} Hence, we write the time $t$ log-price in the generic form:
where $\mu = \left( \mu_{t} \right)_{t \geq 0}$ (the drift) is locally bounded and predictable, $\sigma = \left( \sigma_{t} \right)_{t \geq 0}$ (the volatility) is c\`{a}dl\`{a}g, and $W = \left( W_{t} \right)_{t \geq 0}$ is a standard Brownian motion.
Much work in financial econometrics is cast within this setting (see, e.g., \citeasnoun{andersen-bollerslev-diebold:10a} or BN-S (2007) for reviews and references). Except for the continuity of the local martingale, we impose little structure on the model. In fact, for semimartingales with a continuous martingale component as above, the form $\bigl( \int_{0}^{t} \mu_{u} \text{d}u \bigr)_{t \geq 0}$ is implicit, when the drift term is predictable (in the absence of arbitrage).\footnote{Moreover, all continuous local martingales, whose $QV$ (to be defined in a moment) is absolutely continuous, has the martingale representation of the second term in Equation (ref), e.g., \citeasnoun{doob:53a}. We refer to BN-S (2004, footnote 6) for further details.}
The objective is to estimate a suitable measure of the return variation over a subinterval $\left[ a, b \right] \subseteq \left[ 0, \infty \right)$, labeled the sampling period or measurement horizon. We assume $\left[ a, b \right] = \left[ 0, 1 \right]$; this will be thought of as representing a trading day, but the choice is arbitrary and can serve as a normalization. At any two sampling times $t_{i - 1}$ and $t_{i}$, with $0 \leq t_{i - 1} \leq t_{i} \leq 1$, the intraday return over $\left[ t_{i - 1}, t_{i} \right]$ is denoted by:
where $\Delta_{i} = t_{i} - t_{i - 1}$.
From the theory of stochastic integration, it is well-known that $QV$ is a natural measure of sample path variability for the class of semimartingales. $QV$ is defined by:
for any sequence of partitions, $0 = t_{0} < t_{1} < \ldots < t_{n} = 1$, such that $\max_{1 \leq i \leq n} \left\{ \Delta_{i} \right\} \to 0$ as $n \to \infty$ (e.g., \citeasnoun{protter:04a}).
In our framework, $QV$ is entirely induced by innovations to the local martingale and coincides with the integrated variance ($IV$), which is the object of interest:
$IV$ is central to financial economics, whether in asset and derivatives pricing, portfolio selection or risk management (e.g., \citeasnoun{andersen-bollerslev-diebold:10a}). The econometric problem is that $IV$ is latent, which complicates the empirical estimation of this quantity. We briefly review the literature on existing methods for measuring $IV$, before suggesting a new approach.
Not long ago, the daily squared return was employed as a non-parametric estimator of $IV$. With the advent of high-frequency data, however, more recent work has computed realized variance ($RV$), which is the sum of squared intraday returns sampled over non-overlapping intervals (see, e.g., ABDL (2001) or BN-S (2002)). More formally, consider an equidistant partition $0 = t_{0} < t_{1} < \dots < t_{n} = 1$, where $t_{i} = i / n$.\footnote{Though an irregular partition of the sampling period suffices for consistency, it is standard to compute an equidistant time series of intraday returns by various approaches, such as linear interpolation in, e.g., \citeasnoun{andersen-bollerslev:97b} or the previous-tick method suggested in \citeasnoun{wasserfallen-zimmermann:85a}. A side-effect of linear interpolation is that $RV^{ \Delta} \overset{p}{ \to} 0$ as $n \to \infty$, because the interpolated process is of continuous bounded variation, see \citeasnoun[Lemma 1]{hansen-lunde:06a}. Intuitively, a straight line is the minimum variance path between two points. \citeasnoun{oomen:05a} characterizes $RV$ under alternative sampling schemes.} Then, adopting the notation of \citeasnoun{hansen-lunde:05a}, we define $RV$ at sampling frequency $n$ by setting:
$RV$ builds directly on the theory of $QV$. From Equation (ref) and (ref), it follows that
as $n \to \infty$.
BN-S (2002) derived a distribution theory for $RV^{ \Delta}$ in relation to $IV$. The law of the scaled difference between $RV^{ \Delta}$ and $IV$ has a mixed Gaussian limit,
where
is the integrated quarticity ($IQ$). Thus, the size of the error bounds for $RV^{ \Delta}$ is positively related to $\sigma$, so $RV$ is a less precise estimator of $IV$ when $\sigma$ is high. BN-S (2002) also derived a feasible CLT, where all quantities except $IV$ can be computed directly from the data. This was done by simply replacing $IQ$ by a consistent estimator, such as realized quarticity ($RQ$):
making it possible to construct confidence bands for $RV^{ \Delta}$ to measure the size of the estimation error involved with finite sampling.
The choice of volatility proxy is not obvious in practice, since microstructure bias affects $RV$ if $n$ is too large. With noisy prices, $RV$ is both biased and inconsistent, see, e.g., \citeasnoun{zhou:96a}, \citeasnoun[2006]{bandi-russell:05a}, or \citeasnoun{hansen-lunde:06a}.\footnote{With IID noise, for instance, $RV$ diverges to infinity, i.e. $RV^{ \Delta} \overset{p}{ \to} \infty$ as $n \to \infty$.} Previous studies have recognized this by developing bias reducing techniques (e.g., pre-whitening of the high-frequency return series with moving average or autoregressive filters as in \citeasnoun{andersen-bollerslev-diebold-ebens:01a} and \citeasnoun{bollen-inder:02a}, or kernel-based estimation as in \citeasnoun{zhou:96a} and \citeasnoun{hansen-lunde:06a}). \citeasnoun{zhang-mykland-ait-sahalia:05a} also suggest a subsample estimator that is robust to noise in some situations. In empirical work, the benefits of more frequent sampling is traded off against the damage caused by cumulating noise, and - using various criteria to pick the optimal sampling frequency - the result is often sampling at a moderate frequency, e.g., every 5-, 10-, or 30-minutes, whereby data are discarded.
This pitfall of $RV$ motivates our choice of another proxy with a long history in finance: the price range or high-low. Using the terminology from above, we define the intraday range at sampling times $t_{i - 1}$ and $t_{i}$ as:
The subscript $p$ indicates that we use the range of the price process. Below, we also need the range of a standard Brownian motion over $\left[ t_{i - 1}, t_{i} \right]$, which is denoted by:
The foundations of the range go back to \citeasnoun{feller:51a}, who found its distribution by using the theory of Brownian motion.\footnote{There are two types of range-based volatility estimators: The first relies purely on the high-low, while the second combines the high-low with the open-close, e.g., \citeasnoun{garman-klass:80a} or \citeasnoun{rogers-satchell:91a}. Throughout, we only consider the high-low estimator.} According to his work, the density of $s_{W_{t_{i}, \Delta_{i}}}$ is given by:
with $\phi \left( y \right) = \exp \left(- y^{2}/2 \right) / \sqrt{2 \pi}$. The infinite series is evaluated by a suitable truncation. In Figure (ref), we plot the density function of $s_{W_{t_{i}, \Delta_{i}}}$ by taking $t_{i} = \Delta_{i} = 1$ (We use the shorthand notation $s_{W}$ for this random variable in the rest of the paper).
The figure also displays the distribution of the absolute return. By comparing these proxies, it is suggestive that the efficiency of the range is higher, or in other words that its variance vis-\`{a}-vis the return is lower.
\citeasnoun{parkinson:80a} used Feller's insights to derive the moment generating function of the range of a scaled Brownian motion, $p_{t} = \sigma W_{t}$.\footnote{Note, $\sigma$ does double-duty; representing either the process $\sigma = \left( \sigma_{t} \right)_{t \geq 0}$ or a constant diffusion parameter $\sigma_{t} = \sigma$. The meaning is clear from the context.} For the $r$th moment:
where $\lambda_{r} = \mathbb{E} \left[ s_{W}^{r} \right]$.\footnote{The explicit formula for $\lambda_{r}$ is: $\lambda_{r} = \frac{4}{ \sqrt{ \pi}} ( 1 - \frac{4}{2^{r}} ) 2^{r / 2} \Gamma ( \frac{r + 1}{2} ) \zeta \left( r - 1 \right)$, for $r \geq 1$; where $\Gamma \left( x \right)$ and $\zeta \left( x \right)$ denote the Gamma and Riemann's zeta function, respectively.}
Arguably, a process without drift and constant $\sigma$ is irrelevant from an empirical point of view. An overwhelming amount of research indicates that $\sigma$ is time-varying, see, e.g., \citeasnoun{ghysels-harvey-renault:96a}. Nonetheless, to our knowledge there exists little theory about range-based estimation of $IV$ in the presence of a continually evolving diffusion parameter.\footnote{A notable exception is \citeasnoun{gallant-hsu-tauchen:99a}, who estimate two-factor stochastic volatility models in a general continuous time framework. They derive the density function of the range in this setting, but do not otherwise explore its theoretical properties.} Previous work accounts for (randomly) changing volatility by holding $\sigma_{t}$ fixed within the trading day, while allowing for (stochastic) shifts between them (e.g., \citeasnoun{alizadeh-brandt-diebold:02a}). Still, there are strong intraday movements in $\sigma_{t}$ (e.g., \citeasnoun{andersen-bollerslev:97b}).
A major objective of this paper is therefore to extend the theoretical domain of the extreme value method to a more general class of stochastic processes. Contrary to extant research, we develop a statistical framework for the Brownian semimartingale in Equation (ref), featuring less restrictive dynamics for $\mu$ and $\sigma$.
As stated earlier, the (transformed) daily range is less efficient than $RV$ for moderate values of $n$; two- or three-hour returns suffice. But with tick-by-tick data at hand, we can construct more precise range-based estimates of $IV$ by sampling high-lows within the trading day. Curiously, a rigorous analysis of intraday ranges has been missing in the volatility literature.\footnote{In an independent, concurrent paper, \citeasnoun{martens-dijk:07a} have studied $RRV$ for homoscedastic diffusions, but they do not derive a general asymptotic theory.}
Accordingly, consider again the equidistant partition with $t_{i} = i / n$, for $i = 1, \ldots, n$.\footnote{We use equidistant estimation to ease notation. All our results generalize to an irregular subdivision of the sampling period, so long as $\max_{1 \leq i \leq n} \left\{ \Delta_{i} \right\} \to 0$ as $n \to \infty$, although the conditional variance in the CLT is modified slightly, as spelled out below.} We then propose a realized range-based variance ($RRV$) estimator of $IV$, which - at sampling frequency $n$ - is defined as:
$RRV^{ \Delta}$ has two advantages over the previous return- and range-based methods suggested in the literature on volatility estimation. First, $RRV^{ \Delta}$ inspects all data points (regardless of $n$), whereby we avoid neglecting information about $IV$. Second, the efficiency of $RRV^{ \Delta}$ is several times that of $RV^{ \Delta}$, leading to narrower confidence intervals for $IV$ (see below).
The properties of $RRV^{ \Delta}$ are trivial for the scaled Brownian motion, $p_{t} = \sigma W_{t}$. As the infill asymptotics start operating by letting $n \to \infty$, we achieve an increasing sequence of IID random variables, $\left\{ s_{p_{i \Delta, \Delta}} \right\}_{i = 1, \dots, n}$. Suitably transformed to unbiased measures of $\sigma^{2}$ using (ref), the consistency of $RRV^{ \Delta}$ follows from a standard law of large numbers by averaging. To see this, note that $\mathbb{E} \left[ RRV^{ \Delta} \right] = \sigma^{2}$ and $\text{var} \left[ RRV^{ \Delta} \right] = \Lambda n^{-1} \sigma^{4}$ with:
Hence, $RRV^{ \Delta} \overset{p}{ \to} \sigma^{2}$ as $n \to \infty$. Also, for this process a stardard CLT implies that:
If $\mu$ and $\sigma$ are stochastic, establishing the large sample properties of $RRV^{ \Delta}$ is more involved, but nonetheless possible. Overall, the basic idea extends to general Brownian semimartingales, given some regularity on $\mu$ and $\sigma$, as we next show.\footnote{Throughout the paper, proofs of the theorems are presented in the Appendix.}
No knowledge about the dynamics of $\sigma$ is needed for Theorem (ref) to hold, except for weak technical conditions, so it considerably extends the theory of range-based volatility estimation. We allow for very general continuous time processes, including, but not limited to, models with leverage, long-memory, diurnal effects or jumps (in $\sigma$). This is certainly not true in the previous range-based literature. Moreover, the theorem allows for drift due to the fact that the variation induced by the expected move in $p$, $\bigl( \int_{0}^{t} \mu_{u} \text{d}u \bigr)_{t \geq 0}$, is an order of magnitude lower than the variation induced by the continuous local martingale; comprised by $\bigl( \int_{0}^{t} \sigma_{u} \text{d}W_{u} \bigr)_{t \geq 0}$.
In empirical work, the consistency of $RRV^{ \Delta}$ becomes unreliable due to microstructure noise, if $n$ is too large. Theorem (ref) does not indicate the precision of $RRV^{ \Delta}$ if $n$ is fixed at a moderate level, and econometricians often compute confidence bands as a guide to the error made from estimation in finite samples. To strengthen the convergence in probability, we next develop a distribution theory for $RRV^{ \Delta}$.
The above weak assumptions on $\sigma$ are too general to prove a CLT, and we need slightly stronger conditions:\\[-0.25cm]
Assumption (V): $\sigma$ does not vanish (V$_{1}$) and satisfies:
where $\mu^{ \prime} = \left( \mu_{t}^{ \prime} \right)_{t \geq 0}$, $\sigma^{ \prime} = \left( \sigma_{t}^{ \prime} \right)_{t \geq 0}$ and $v^{ \prime} = \left( v_{t}^{ \prime} \right)_{t \geq 0}$ are c\`{a}dl\`{a}g, with $\mu^{ \prime}$ also being locally bounded and predictable, and $B^{ \prime} = \left( B_{t}^{ \prime} \right)_{t \geq 0}$ is a Brownian motion independent of $W$.}\\[-0.25cm]
We prove our result by invoking stable convergence in law. This is standard in the $RV$ literature. But to avoid any confusion about our terminology, we present the definition.
We use the symbol $X_{n} \overset{d_{s}}{ \to} X$ to denote stable convergence. Note that this implies weak convergence, which may be equivalently defined by taking $Y = 1$ (see, e.g., \citeasnoun{renyi:63a} or \citeasnoun{aldous-eagleson:78a} for more details).
We now state the main result, which is a (non-standard) CLT.
A critical feature of this theorem is that the left-hand side converges to a stochastic integral with respect to $B$, which is independent of the driving term $\sigma$. This implies $\sqrt{n} \left( RRV^{ \Delta} - IV \right)$ has a mixed normal limit, with $\sigma$ governing the mixture.\footnote{Earlier drafts of this paper had a non-mixed Gaussian CLT and the stronger conditions, $\mu = 0$ and $\sigma$ is H\"{o}lder continuous of order $\gamma > 1/2$, i.e. $\sigma_{t} - \sigma_{s} = \text{O}_{p} \left( \mid t - s \mid^{ \gamma} \right)$ for $t \to s$. We have substantially weakened these restrictions and also proved the mixed Gaussian CLT. Svend E. Graversen was helpful in pointing our attention to a result that enabled us to remove these assumptions (see Lemma (ref) in the Appendix).} In general, this introduces heavier tails in the unconditional distribution of $RRV^{ \Delta}$ than for Gaussian random variables. To summarize:
Hence, the sampling errors of $RRV^{ \Delta}$ are about one-fifth of those based on $RV^{ \Delta}$. This is not surprising: $RRV^{ \Delta}$ uses all the data, whereas $RV^{ \Delta}$ is based on high-frequency returns sampled at fixed points in time. As, for the moment, $p$ is assumed fully observed, $RV^{ \Delta}$ is neglecting a lot of information.
$IQ$ on the right-hand side in (ref) is infeasible, i.e. it cannot be computed directly from the data. We can estimate it with the realized range-based quarticity ($RRQ$):
With techniques similar to the proof of Theorem (ref), we can show that $RRQ^{ \Delta} \overset{p}{ \to} IQ$. Thus, by using the properties of stable convergence (e.g., \citeasnoun{jacod:97a}), we get the next corollary.
In practice, we draw inference about $IV$ from a finite data sample and cannot extract the true range, so the intraday high-low statistic will be progressively more downward biased as $n$ gets larger. Building on the simulation evidence of \citeasnoun{garman-klass:80a}, \citeasnoun{rogers-satchell:91a} proposed a technique for bias correcting the range that largely removed the error from a numerical perspective.
Nonetheless, it is misleading to think about ranges as downward biased. The source of the bias is $\lambda_{2}$, which is constructed on the presumption that $p$ is fully observed. Therefore, we will now develop an estimator that accounts for the number of high-frequency data points used in forming the high-low, in order to scale properly. To formalize this idea, additional notation is required. Assume, without loss of generality, that $mn + 1$ equidistant observations of the price process are available, giving $mn$ returns. These are split into $n$ intervals each with $m$ innovations. We denote the observed range over the $i$th interval by:
Also, we let:
and then define a new realized range-based estimator by setting:
where $\lambda_{r, m} = \mathbb{E} \bigl[ s_{W, m}^{r} \bigr]$. $\lambda_{r, m}$ is the $r$th moment of the range of a standard Brownian motion over a unit interval, when we only observe $m$ increments of the underlying continuous time process.
To our knowledge, there is no explicit formula for $\lambda_{r, m}$, but it is easily simulated to any degree of accuracy. Figure (ref) details this for $r = 2$ and all values of $m$ that integer divide 23,400.
Of course, $\lambda_{2, m} \to \lambda_{2}$ as $m \to \infty$, but note also that $\lambda_{2, 1} = 1$, which defines $RV^{ \Delta}$. The downward bias reported in simulation studies on the range-based estimator is a consequence of the fact that $1 / \lambda_{2}$ was applied in place of $1 / \lambda_{2, m}$, as the bias is in one-to-one correspondence with the difference.
Having completed these preliminaries, we prove consistency and asymptotic normality for the estimator in Equation (ref). Note that $m$ is not required to approach infinity for the CLT to work; convergence to any natural number is sufficient.
To provide an impression of the efficiency of $RRV^{ \Delta}_{m}$, Figure (ref) depicts $\Lambda_{m}$ on the y-axis, as a function of $m$ along the x-axis. The steep initial decline in $\Lambda_{m}$ renders the advantage of $RRV^{ \Delta}_{m}$ large compared to $RV^{ \Delta}$ even for moderate values of $m$. For $m = 10$, say, since the scalar appearing in front of $IQ$ in the CLT for $RRV^{ \Delta}_{m}$ is about 0.7, the confidence intervals for $IV$ are much narrower. In our experience $m = 10$, or higher values, is usually obtained for moderately liquid assets at empirically relevant frequencies, such as 5-minute sampling.\footnote{Under parametric assumptions and no microstructure noise, $RV$ is the maximum likelihood estimator. Thus, our efficiency comparison should be viewed as the potential reduction in variance that can be achieved with $RRV_{m}^{ \Delta}$ when microstructure noise is preventing $RV^{ \Delta}$ from being sampled at the maximum frequency ($mn$) and $n$ is set at a moderate level where the impact of noise is minimal.}
To study the finite sample properties of $RRV^{ \Delta}_{m}$, this section uses repeated samples from a stochastic volatility model. We simulate the following system of stochastic differential equations:
where $W$ and $B$ are independent Brownian motions, while $\left( \theta, \omega, \eta \right)$ are parameters. Thus, the log-variance of spot prices evolves as a mean reverting Ornstein-Uhlenbeck process with mean $\omega$, mean reversion parameter $\theta$ and volatility $\eta$ (see, e.g., \citeasnoun{gallant-hsu-tauchen:99a}, \citeasnoun{alizadeh-brandt-diebold:02a}, and \citeasnoun{andersen-benzoni-lund:02a}). The vector $\left( \theta, \omega, \eta \right) = \left( 0.032, -0.631, 0.115 \right)$ is taken from \citeasnoun{andersen-benzoni-lund:02a}, who apply Efficient Method of Moments (EMM) to calibrate numerous continuous time models.
Initial conditions are set at $p_{0} = 0$ and $\ln \sigma_{0}^{2} = \omega$, and we generate $T = 1,000,000$ daily replications from this model each with $mn$ returns, where $mn$ depends on the setting (see below). Throughout, we continue to ignore the irregular spacing of empirical high-frequency data and work with equidistant data.
The distributional result for $RRV^{ \Delta}_{m}$ is detailed by setting $m = 10$. The reported results are not very sensitive to specific choices of $m$, but in general higher values improve the coverage rates of the asymptotic confidence bands. We simulate $n = 10$, $50$, $100$ for a total of $mn = 100$, $500$, $1000$ increments each day, allowing us to show the convergence in distribution to the standard normal for high-frequency sample sizes that resemble those of moderately liquid assets.
Figure (ref) (upper panel) graphs kernel densities for the standardized errors of $RRV_{m}^{ \Delta}$; cf. the ratio in Equation (ref). For $n = 10$, the distribution is left-skewed with a poor approximation in both the center and tail areas compared to the N(0,1) reference density. The distortions are diminished by progressively increasing the sample. With $n = 100$ the tails are tracked quite closely.
BN-S (2005) showed that log-based inference via standard linearization methods improved the raw distribution theory for $RV^{ \Delta}$. They reported better finite sample behavior for the errors of the log-transform than those extracted with the feasible version of the CLT outlined in Equation (ref). The shape of the actual densities for $RRV_{m}^{ \Delta}$ suggests that this also applies to our setting. By the delta method, the log-version of the CLT for $RRV_{m}^{ \Delta}$ takes the form:
In the lower panel of Figure (ref), we plot the density functions of the feasible log-based t-statistics. The coverage probabilities of Equation (ref) are a much better guide for small values of $n$, with $n = 100$ providing a near perfect fit to the N(0,1) distribution. Hence, the results for $RRV_{m}^{ \Delta}$ are consistent with the findings for $RV^{ \Delta}$.
This technique is also applicable to study other (differentiable) functions of $RRV_{m}^{ \Delta}$. For convenience, we state the CLT of a particularly useful transformation, obtained by taking square roots:
We investigate the empirical properties of intraday ranges by analyzing a major stock from the Dow Jones Industrial Average, General Motors (GM).
High-frequency data were extracted from the TAQ database, which is a recording of trades and quotes from the securities listed on New York Stock Exchange (NYSE), American Stock Exchange (AMEX), and National Association of Securities Dealers Automated Quotation (NASDAQ). The sample period covers January 3, 2000 through December 31, 2004; a total of 1,255 trading days. We restrict attention to NYSE updates and only report the results of the quotation data, for which the midquote is used.\footnote{The analysis of transaction data is available upon request.} All raw data were filtered for irregularities (e.g., prices of zero, entries posted outside the NYSE opening hours, or quotes with negative spreads), and a second algorithm handled outliers in the price series.
The average number of data points after filtering is given in Table (ref). The column \#$r_{ \tau_{i}} \neq 0$, where $r_{ \tau_{i}} = p_{ \tau_{i}} - p_{ \tau_{i - 1}}$ and $\tau_{i}$ is the arrival time of the $i$th tick, counts the number of price changes relative to the previous posting. \#$\Delta r_{ \tau_{i}} \neq 0$ does the same for second differences, but after having removed updates with $r_{ \tau_{i}} = 0$. These numbers are important to calculate $\lambda_{2, m}$ and $\lambda_{4, m}$ that are required to estimate $RRV_{m}^{ \Delta}$ and construct confidence bands. Initially, we found $mn$ on the basis of all non-zero returns; i.e. the \#$r_{ \tau_{i}} \neq 0$ numbers. This meant $mn$ was too high, because of instantaneous reversals (e.g., bid-ask bounce behavior). We assessed that a proper method to determine $mn$ was to only count repeated reversals once. Thus, to compute $mn$ we use the \#$\Delta r_{ \tau_{i}} \neq 0$ numbers.
The estimation of $RV^{ \Delta}$ and $RRV_{m}^{ \Delta}$ proceeds with 5-minute sampling through the trading session starting 9:30{\scriptsize AM} EST until 4:00{\scriptsize PM} EST; i.e. by setting $n = 78$ or $\Delta = 300$ seconds.\footnote{This choice was guided by signature plots, i.e. sample averages of the estimators across different sampling frequencies $n$. We found increasing signs of microstructure noise by moving below the 5-minute frequency.} We use the previous-tick method to compute returns for $RV^{ \Delta}$. Note that since the empirical high-frequency data are irregularly distributed, there are, in general, different values of $m$ in the 5-minute intervals. This does not cause any problems, however, for the theory extends directly to this setting, provided we use the individual values of $m$ in the estimation.
Sample statistics for the resulting time series are printed in Table (ref). $RV^{ \Delta}$ has a lower minimum and a higher maximum than $RRV_{m}^{ \Delta}$, while its overall mean is higher. Both kurtosis figures are consistent with a mixed Gaussian limit. The variance of $RRV_{m}^{ \Delta}$ is only 58% that of $RV^{ \Delta}$. This is much lower, but as expected still somewhat higher than predicted by the theory (relative to $RV^{ \Delta}$). First off, here we are looking at a time series variance for the whole sample, so the CLT factors are not directly applicable. Second, the data from the empirical price process are, in all likelihood, not drawn from a Brownian semimartingale (e.g., there are jumps and microstructure frictions). $RRV_{m}^{ \Delta}$, in turn, behaves differently for other specifications, which we address elsewhere.
The correlation between $RV^{ \Delta}$ and $RRV_{m}^{ \Delta}$ is 0.982, pointing towards little gain - at relevant frequencies - from taking linear combinations of the estimators to further reduce sampling variation. From the joint asymptotic distribution of $(RV^{ \Delta}, RRV_{m}^{ \Delta})$, the conditional covariance matrix at time $u$ is given by:
The covariance term appearing in $\Sigma_{u}$ is hard to tackle analytically. In unreported results, we used simulations to inspect the structure of the correlation coefficient around a grid of values for $m$ that matches our sample. Based on this, we found that the estimated empirical correlation is slightly higher than the theoretical level.
In Figure (ref), $IV$ estimates are drawn for the two methods, $RV^{ \Delta}$ and $RRV_{m}^{ \Delta}$. The time series agree on the level of $IV$. The key point is that the sample path of $RRV_{m}^{ \Delta}$ is less volatile compared to $RV^{ \Delta}$ (but still appears quite erratic). Again, this suggests that the sampling errors of $RV^{ \Delta}$ are larger compared to $RRV_{m}^{ \Delta}$, and that the theoretical gains of the realized range-based estimator also hold for the empirical identification of $\sigma$, at least for the 5-minute frequency.
To underscore these insights, we extracted data from July 1, 2002 to December 31, 2002 to plot the $IV$ estimates in Figure (ref) together with 95% confidence intervals, constructed from the log-based theory. The confidence bands widen as expected, when $\sigma$ goes up. Nonetheless, the stability of $RRV_{m}^{ \Delta}$ feeds into much smaller intervals, consistent with the theoretical relationship between the $m$ and $\Lambda_{m}$ scalars from Figure (ref). This implies that very few increments are required for $RRV_{m}^{ \Delta}$ to gain a significant advantage in efficiency over $RV^{ \Delta}$.\footnote{With $\# \Delta r_{ \tau_{i}} \neq 0$ equal to 1,017 on average for the midquote data, we have roughly $m = 13$ increments within each of the seventy eight 5-minute intervals during the trading day.}
These empirical findings translate into a more persistent time series behavior for $RRV_{m}^{ \Delta}$, as shown by the autocorrelation functions in Figure (ref). We included the first 75 lags and report Bartlett two standard error bands for testing a white noise null hypothesis. All autocorrelations are positive, starting at about 0.60 - 0.70 and ending around 0.10 - 0.15. The decay pattern in the series is identical but it evolves more smoothly and at higher levels for $RRV_{m}^{ \Delta}$. Combined, these observations might be put to work in a forecasting exercise, although we do not pursue this idea here.
All told, realized range-based estimation of $IV$ offers several advantages compared to $RV$, both from a theoretical and practical viewpoint. We acknowledge, however, that the probabilistic theory proposed in this paper needs further refinement at higher frequencies, where microstructure noise is more problematic. Statistical tools for controlling the impact of such noise is crucial for getting consistent estimates of $IV$. These techniques have already been developed for $RV$, see, e.g., \citeasnoun{zhang-mykland-ait-sahalia:05a} or \citeasnoun{barndorff-nielsen-hansen-lunde-shephard:08a}. It presents a topic for future research to verify if our method extends along these lines, and we are currently undertaking a formal analysis of $RRV$ and market microstructure noise.
Realized range-based variance ($RRV$) is an approach based on intraday price ranges for non-parametric measurement of the integrated variance ($IV$) of continuous semimartingales. Under weak regularity conditions, we have shown that it can extract $IV$ more accurately than previous methods, when microstructure noise is preventing realized variance ($RV$) from being sampled at the maximum frequency. Another contribution of this paper, particularly useful in empirical analysis, is the solution to the downward bias problem that has haunted the range-based literature for decades.
The finite sample distributions of the estimator were inspected with Monte Carlo analysis. For moderate samples, the coverage probabilities of the confidence bands for the t-statistics correspond with the limit theory, in particular for log-based inference.
We highlighted the empirical potential of $RRV$ vis-\`{a}-vis $RV$ by applying our method to a set of high-frequency data for General Motors. Consistent with the theory, $RRV$ has smaller confidence bands than $RV$. Although empirical price processes are very different from diffusion models and real data are noisy objects, we feel the results support our theory quite well and opens up alternative routes for estimating $IV$.
In future projects, we envision several extensions of the current framework. First, there is plenty of evidence against the continuous sample path diffusion. We are convinced that a range-based statistic can estimate quadratic variation ($QV$), when the price also exhibits jumps. This theory is being developed in \citeasnoun{christensen-podolskij:12a}, along with realized range-based bi-power variation. Second, with microstructure noise in observed asset prices, further comparisons of $RRV$ and $RV$ are needed. Finally, we can handle the bivariate case with the polarization identities, so multivariate range-based analysis constitutes a promising future application.