RangeBased Volatility Estimators: Overview and Examples of Usage
Volatility estimation and forecasting plays a crucial role in many areas of finance.
For example, standard riskbased portfolio allocation methods (minimum variance, equal risk contributions, hierarchical risk parity…) critically depend on the ability to build accurate volatility forecasts^{1}.
Multiple methods for estimating volatility have been proposed over the past several decades, and in this blog post I will focus on rangebased volatility estimators.
These estimators, the first of which introduced by Parkinson^{2} as a way to compute the true variance of the rate of return of a common stock^{2}, rely on the highest and lowest prices of an asset over a given time period to estimate its volatility, hence their name^{3}.
After describing the four most well known rangebased volatility estimators, I will reproduce the analysis of Arthur Sepp in his presentation Volatility Modelling and Trading^{4} made at Global Derivatives Conference 2016 and test the predictive power of the naive volatility forecasts produced by these estimators for various ETFs.
Notes:
A very accessible series of papers about rangebased volatility estimators has recently^{5} been released by people at Lombard Odier, c.f. here, here and here.
Mathematical preliminaries
Volatility modelling
One of the main^{6} assumptions made when working with rangebased volatility estimators^{7} is that the price movements $S_t$ of the asset under consideration follow a geometric Brownian motion with unknown volatility coefficient^{8} $\sigma$ and unknown drift coefficient $\mu$, that is
\[d S_t = \mu S_t dt + \sigma S_t dW_t\], where $W_t$ is a standard Brownian motion.
Under this working assumption, $\sigma$ represents the volatility of the asset.
Volatility and variance estimators
Although anyone can empirically observe the impact of “volatility” on the prices of a given asset, the volatility coefficient $\sigma$ of this asset is not directly observable^{9} and must be estimated using stock market information.
A statistical estimator of $\sigma$ is then called a volatility estimator, and a statistical estimator of $\sigma^2$ is called a variance estimator.
Efficiency of a volatility estimator
In order to determine the quality of a volatility estimator, two measures are commonly used:

The bias of a volatility estimator measures whether this estimator produces, on average, too high or too low volatility estimates.
More formally, a volatility estimator $\sigma_A$ is said to be unbiased when $\mathbb{E}[\sigma_A] = \sigma$ and biased otherwise.

The efficiency of a volatility estimator measures the uncertainty of the volatility estimates produced by this estimator, with the greater the efficiency of the estimator, the more accurate the volatility estimates.
More formally, the relative efficiency $Eff \left( \sigma_A \right)$ of a volatility estimator $\sigma_A$ compared to a reference volatility estimator $\sigma_B$ is defined as the ratio of the variance of the estimator $\sigma_B^2$ over the variance of the estimator $\sigma_A^2$, that is,
\[Eff \left( \sigma_A \right) = \frac{Var \left( \sigma_B^2 \right)}{Var \left( \sigma_A^2 \right)}\]
To be noted that bias and efficiency are sometimes conflicting, which is more generally known in statistics as the biasvariance tradeoff.
Closetoclose volatility estimators
Let $C_1,…,C_T$ be the closing prices of an asset for $T$ time periods $t=1..T$^{10}.
Then,
\[\sigma_{cc,0} \left( T \right) = \sqrt{ \frac{1}{T1} \sum_{i=2}^T \ln{\frac{C_i}{C_{i1}}}^2 }\]is a biased^{11} estimator of the asset volatility $\sigma$ over the $T$ time periods, assuming zero drift (i.e., $\mu = 0$), c.f. Parkinson^{2}.
In addition,
\[\sigma_{cc} \left( T \right) = \sqrt{ \frac{1}{T2} \sum_{i=2}^T \left( \ln \frac{C_i}{C_{i1}}  \mu_{cc} \right)^2 }\], with $\mu_{cc} = \frac{1}{T1} \sum_{i=2}^T \ln \frac{C_i}{C_{i1}} $, is a biased^{11} estimator of the asset volatility $\sigma$ over the $T$ time periods, assuming nonzero drift (i.e., $\mu \ne 0$), c.f. Yang and Zhang^{12}.
These two estimators are known as closetoclose volatility estimators.
Rangebased volatility estimators
Let be:
 $t=1..T$, $T$ time periods^{10}
 $\left( O_1,H_1,L_1,C_1 \right), …, \left( O_T,H_T,L_T,C_T \right)$, the opening, highest, lowest and closing prices of an asset for time periods $t=1..T$
As mentioned in the introduction, a volatility estimator fully or partially relying on the highest prices $H_t, t=1..T$ and on the lowest prices $L_t, t=1..T$ is called a rangebased volatility estimator.
The underlying idea behind such estimators is that information contained in the asset highlow price ranges $H_t  L_t, t=1..T$ should allow to build volatility estimators that are more efficient than the closetoclose volatility estimators, which use only one price inside this range^{13}.
This quest for efficiency is important because, contrary to one of the working assumptions^{6}, the volatility of an asset is known to be timevarying^{14}, so that the less the number of time periods required to estimate its volatility, the more chances that its volatility is constant(ish) over the time periods under consideration.
As Rogers et al.^{15} put it:
[…] volatility may change over long periods of time; a highly efficient procedure will allow researchers to estimate volatility with a small number of observations.
Parkinson volatility estimator
Parkinson^{2} introduces an estimator for the diffusion coefficient of a Brownian motion without drift that relies on the highest and lowest observed values of this Brownian motion over a given time period.
When applied to the estimation of an asset volatility, this gives the Parkinson volatility estimator $\sigma_{P} \left( T \right)$ defined over $T$ time periods by
\[\sigma_{P} \left( T \right) = \sqrt{\frac{1}{T}} \sqrt{\frac{1}{4 \ln 2} \sum_{i=1}^T \left( \ln \frac{H_i}{L_i} \right) ^2}\]Intuitively, the Parkinson estimator should be “better” than the closetoclose estimators because large price movements impacting the highlow price range $H_t  L_t$ but leaving the closing price $C_t$ unchanged might occur within any time period $t$.
This is confirmed by the efficiency of this estimator, up to 5.2 times higher than the efficiency of the closetoclose estimators^{16}.
GarmanKlass volatility estimator
Garman and Klass^{17} propose to improve the Parkinson estimator by taking into account the opening prices $O_t, t=1..T$ and the closing prices $C_t, t=1..T$.
This leads to the GarmanKlass volatility estimator $\sigma_{GK} \left( T \right)$, defined over $T$ time periods by
\[\sigma_{GK} \left( T \right) = \sqrt{\frac{1}{T}} \sqrt{ \sum_{i=1}^T \frac{1}{2} \left( \ln\frac{H_i}{L_i} \right) ^2  \left( 2 \ln2  1 \right) \left( \ln\frac{C_i}{O_i} \right )^2 }\]For the historical comment, Garman and Klass^{17} establish in their paper that $\sigma_{GK}$ is the “best reasonable”^{18} volatility estimator that depends only on the highopen price range $H_t  O_t$, the lowopen price range $L_t  O_t$ and the closeopen price range $C_t  O_t$, $t=1..T$.
The GarmanKlass estimator is up to 7.4 times more efficient than the closetoclose estimators^{16}.
RogersSatchell volatility estimator
The Parkinson and the GarmanKlass estimators have both been derived under a zero drift assumption.
When this assumption is not verified for an asset, for example because of a strong upward or downward trend in the asset prices or because of the usage of large time periods (monthly, yearly…), these estimators should in theory not be used because the quality of their volatility estimates is negatively impacted by the presence of a nonzero drift^{19}^{15}.
In order to solve this problem, Rogers and Satchell^{19} devise the RogersSatchell volatility estimator $\sigma_{RS} \left( T \right)$, defined over $T$ time periods by
\[\sigma_{RS} \left( T \right) = \sqrt{\frac{1}{T}} \sqrt{ \sum_{i=1}^T \ln\frac{H_i}{C_i} \ln\frac{H_i}{O_i}  \ln\frac{L_i}{C_i} \ln\frac{L_i}{O_i} }\]The RogersSatchell estimator is up to 6 times more efficient than the closetoclose estimators^{19}, which is less than the GarmanKlass estimator^{20}.
YangZhang volatility estimator
The rangebased volatility estimators discussed so far do not take into account opening jumps in an asset prices^{21}, that is, the potential difference between an asset opening price $O_t$ and its closing price $C_{t1}$ for a time period $t$^{22}.
This limitation causes a systematic underestimation of the true volatility^{12}.
When trying to integrate opening jumps into the Parkinson, the GarmanKlass and the RogersSatchell estimators, Yang and Zhang^{12} discover that it is unfortunately not possible for any “reasonable” singleperiod^{23} volatility estimator to properly handle both a nonzero drift and opening jumps.
This leads them to introduce the multiperiod^{23} YangZhang volatility estimator $\sigma_{YZ} \left( T \right)$, defined over $T$ time periods by
\[\sigma_{YZ} \left( T \right) = \sqrt{ \sigma_{ov}^2+ k \sigma_{oc}^2 + (1k ) \sigma_{RS}^2 ) }\], where:

$\sigma_{co} \left( T \right)$ is the closetoopen volatility, defined as
\[\sigma_{co} = \sqrt{\frac{1}{T2} \sum_{i=2}^T \left( \ln \frac{O_i}{C_{i1}}  \mu_{co} \right)^2}\], with $\mu_{co} = \frac{1}{T1} \sum_{i=2}^T \ln \frac{O_i}{C_{i1}}$

$\sigma_{oc} $ is the opentoclose volatility, defined as
\[\sigma_{oc} \left( T \right) = \sqrt{\frac{1}{T2} \sum_{i=2}^T \left( \ln \frac{O_i}{C_{i}}  \mu_{oc} \right)^2}\], with $\mu_{oc} = \frac{1}{T1} \sum_{i=2}^T \ln \frac{C_i}{O_{i}}$

$\sigma_{RS}$ is the RogersSatchell volatility estimator over the time periods $t=2..T$

$k = \frac{0.34}{1.34 + \frac{T}{T2}}$
In addition to the new estimator $\sigma_{YZ}$, Yang and Zhang^{12} also provide multiperiod versions of the Parkinson, the GarmanKlass and the RogersSatchell estimators that support opening jumps^{24}.
The YangZhang estimator is up to 14 times more efficient than the closetoclose estimators^{12}, a result that Yang and Zhang^{12} comment as follows
The improvement of accuracy over the classical closetoclose estimator is dramatic for reallife time series
Other estimators
The family of rangebased volatility estimators has many other members:
 The Kunitomo^{25} volatility estimator
 The AlizadehBrandtDiebold^{26} volatility estimator
 The Meilijson^{27} volatility estimator
 …
Still, the Parkinson, the GarmanKlass, the RogersSatchell and the YangZhang volatility estimators are representative of this family, so that I will not detail any other rangebased volatility estimator in this blog post.
From volatility estimation to volatility forecasting
Rangebased volatility estimators are based on the assumption of independent sample and observations within the sample^{4}, so that the corresponding volatility forecasts are simply naive forecasts under a random walk model.
In other words, with such volatility estimators, the “natural” forecast of an asset volatility over the next $T$ time periods is the (past) estimate of the asset volatility over the last $T$ time periods.
That being said, it is perfectly possible to use rangebased volatility estimates together with any volatility forecasting model such as:
 A time series forecasting model (simple moving average, exponentially weighted moving average…), as detailed for example in Jacob and Vipul^{28}
 An econometric forecasting model (GARCH model…), c.f. Mapa^{29}
 A specific rangebased forecasting model (Chou’s^{30} Conditional AutoRegressive Range model, Harris and Yilmaz’s^{31} hybrid multivariate exponentially weighted moving average model…)
Performance of rangebased volatility estimators
Theoretical and practical performances of rangebased volatility estimators are studied in several papers, for example Shu and Zhang^{32}, Jacob and Vipul^{28} and Brandt and Kinlay^{33}, among others.
Most of these studies agree that rangebased volatility estimators are biased^{11}, but other conclusions differ depending on the exact methodology used.
In particular, as highlighted by Brandt and Kinlay^{33}, the results from empirical research differ significantly from those seen in simulation studies in a number of respects^{33}.
One perfect example of these differences is Shu and Zhang^{32} concluding, using a Monte Carlo simulation, that
If the drift term is large, the Parkinson estimator and the [GarmanKlass] estimator will significantly overestimate the true variance […]
, while Jacob and Vipul^{28} concluding, using real stock market data, that
Overall, the [GarmanKlass] estimator, which indirectly adjusts for the drift, performs better for the highdrift stocks.
Motivated by such inconsistencies, Lyocsa et al.^{34}, building on Patton and Sheppard^{35}, introduced what I will call the LyocsaPlihalVyrost volatility estimator $\sigma_{LPV}$, defined as the arithmetic average of the Parkinson, the GarmanKlass and the RogersSatchell volatility estimators^{36}
\[\sigma_{LPV} = \frac{\sigma_{P} + \sigma_{GK} + \sigma_{RS}}{3}\]As Lyocsa et al.^{34} explain, the motivation behind using the (naive) equally weighted average is based on the assumption that we have no prior information on which estimator might be more accurate^{34}.
I personally like the idea of an averaged estimator, but at this point, I think it is safe to highlight that there is no “best” rangebased volatility estimator…
Implementation in Portfolio Optimizer
Portfolio Optimizer implements all the volatility estimators discussed in this blog post:
 The closetoclose volatility estimators, through the endpoint
/assets/volatility/estimation/closetoclose
 The Parkinson volatility estimator, through the endpoint
/assets/volatility/estimation/parkinson
 The GarmanKlass volatility estimator, through the endpoint
/assets/volatility/estimation/garmanklass
 The original GarmanKlass volatility estimator^{18}, through the endpoint
/assets/volatility/estimation/garmanklass/original
 The RogersSatchell volatility estimator, through the endpoint
/assets/volatility/estimation/rogerssatchell
 The YangZhang volatility estimator, through the endpoint
/assets/volatility/estimation/yangzhang
, as well as their jumpadjusted variations, whenever applicable.
Examples of usage
To illustrate possible uses of rangebased volatility estimators, I propose to reproduce a couple of results from Sepp^{4}:
 The estimation and the forecast of the SPY ETF monthly volatility
 The forecast of the monthly volatility of misc. ETFs representative of different asset classes (U.S. treasuries, international stock market, gold…)
Such examples will allow to compare the empirical behavior of the different volatility estimators and maybe reach a conclusion as to their relative performance in this specific setting.
Estimating SPY ETF volatility
I will estimate the SPY ETF monthly volatility using all the daily open/high/low/close prices^{37} observed during that month^{38}.
Figure 1, limited to 5 volatility estimators for readability purposes, illustrates the results obtained over the period 31 January 2005  29 February 2016^{39}.
Figure 1 is mostly identical to the figure on slide 22 from Sepp^{4}, on which it seems in particular that the closetoclose and the YangZhang volatility estimators provide higher estimates of volatility when the overall level of volatility is high^{4}.
Overall, though, the behavior of the different volatility estimators is essentially the same on this specific example, which is confirmed by their correlations displayed in Figure 2.
Forecasting misc. ETFs volatility
Using the same methodology as in Sepp^{4}, I will now evaluate the quality of the naive forecasts produced by all the rangebased volatility estimators implemented in Portfolio Optimizer against the next month’s closetoclose observed volatility^{40}, for 10 ETFs representative of misc. asset classes:
 U.S. stocks (SPY ETF)
 European stocks (EZU ETF)
 Japanese stocks (EWJ ETF)
 Emerging markets stocks (EEM ETF)
 U.S. REITs (VNQ ETF)
 International REITs (RWX ETF)
 U.S. 710 year Treasuries (IEF ETF)
 U.S. 20+ year Treasuries (TLT ETF)
 Commodities (DBC ETF)
 Gold (GLD ETF)
These ETFs are used in the Adaptative Asset Allocation strategy from ReSolve Asset Management, described in the paper Adaptive Asset Allocation: A Primer^{41}.
For each ETF, Sepps’s methodology is as follows:

At each month’s end, compute the volatility estimates $\sigma_{cc, t}$, $\sigma_{P, t}$, … using all the ETF daily open/high/low/close prices^{37} observed during that month^{38}
Under a random walk volatility model, each of these estimates represents the next month’s volatility forecast $\hat{\sigma}_{t+1}$

At each month’s end, also compute the next month’s closetoclose volatility estimate $\sigma_{cc, t+1}$ using all the ETF daily close prices^{37} observed during that month^{38}
This estimate is the volatility benchmark, which represents how the ETF “volatility” is perceived by an investor monitoring her portfolio daily.

Once all months have been processed that way, regress the volatility forecasts on the volatility benchmarks by applying the MincerZarnowitz^{42} regression model:
\[\hat{\sigma}_{t+1} = \alpha + \beta \sigma_{cc, t+1} + \epsilon_{t+1}\], where $\epsilon_{t+1}$ is an error term.
Then, the estimator producing [the best] volatility forecast is indicated by [a] high explanatory power R^2, [a] small intercept $\alpha$ and [a] $\beta$ coefficient close to one^{4}.
Forecasting SPY ETF volatility
In the case of the SPY ETF, Figure 3 illustrates Sepps’s methodology for the LyocsaPlihalVyrost volatility estimator $\sigma_{LPV}$ over the period 31 January 2005  29 February 2016.
Detailed results for all regression models over the period 31 January 2005  29 February 2016:
Volatility estimator  $\alpha$  $\beta$  $R^2$ 

Closetoclose  4.1%  0.75  57% 
Closetoclose (zero drift)  3.9%  0.77  57% 
Parkinson  3.5%  0.95  58% 
Parkinson (jumpadjusted)  3.4%  0.79  58% 
GarmanKlass  3.7%  0.92  57% 
GarmanKlass (jumpadjusted)  3.6%  0.77  58% 
GarmanKlass (original)  3.7%  0.92  57% 
GarmanKlass (original, jumpadjusted)  3.6%  0.77  58% 
RogersSatchell  4.0%  0.88  56% 
RogersSatchell (jumpadjusted)  3.9%  0.74  57% 
YangZhang  3.8%  0.75  58% 
LyocsaPlihalVyrost  3.7%  0.92  57% 
While these figures are far^{43} from those on slide 42 from Sepp^{4}, with for example nearly no variation in terms of $R^2$ among the different volatility estimators, two observations are similar:
 All volatility estimators have comparable $\alpha$
 The Parkinson, the GarmanKlass and the RogersSatchell volatility estimators have a $\beta$ much closer to 1 than the closetoclose volatility estimator
Forecasting the other ETFs volatility
Going beyond the SPY ETF, averaged results for all ETFs/regression models over each ETF price history^{44} are the following:
Volatility estimator  $\bar{\alpha}$  $\bar{\beta}$  $\bar{R^2}$ 

Closetoclose  5.8%  0.66  44% 
Closetoclose (zero drift)  5.6%  0.67  45% 
Parkinson  5.6%  0.94  44% 
Parkinson (jumpadjusted)  4.9%  0.70  45% 
GarmanKlass  5.7%  0.93  43% 
GarmanKlass (jumpadjusted)  5.0%  0.70  44% 
GarmanKlass (original)  5.7%  0.93  43% 
GarmanKlass (original, jumpadjusted)  5.0%  0.70  44% 
RogersSatchell  6.1%  0.88  42% 
RogersSatchell (jumpadjusted)  5.2%  0.68  43% 
YangZhang  5.1%  0.69  44% 
LyocsaPlihalVyrost  5.7%  0.92  43% 
A couple of remarks:
 Forecasts produced by all the volatility estimators explain on average only ~45% of the variability of the ETFs monthly volatility
 Forecasts produced by the jumpadjusted volatility estimators seem to offer no improvement on average over the forecasts produced by the closetoclose volatility estimator
 Forecasts produced by the Parkinson, the GarmanKlass and the RogersSatchell volatility estimators seem to be much less biased on average than the forecasts produced by the closetoclose volatility estimator, a property inherited by the LyocsaPlihalVyrost volatility estimator
As an empirical conclusion, it is disappointing that the naive monthly volatility forecasts produced by rangebased volatility estimators have about the same predictive power as the forecasts produced by the closetoclose volatility estimator. Nevertheless, because these forecasts are much less biased than their closetoclose counterparts, they still represent an improvement for the many investors who currently rely on close prices only^{45}.
To also be noted, similar to one of the conclusions of Lyocsa et al.^{34}, that the LyocsaPlihalVyrost volatility estimator should probably be preferred to the Parkinson, the GarmanKlass or the RogersSatchell volatility estimators because using only one rangebased estimators has occasionally led to very inaccurate forecasts, which could successfully be avoided by using the average of the three rangebased estimators^{34}.
Conclusion
One aspect of rangebased volatility estimators not discussed in this blog post is their capability to capture important stylized facts about asset returns^{46}.
This, together with possible ways to incorporate them in more predictive volatility models than the random walk model, will be the subject of future blog posts.
Meanwhile, for more volatile discussions, feel free to connect with me on LinkedIn or to follow me on Twitter.
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As well as correlation forecasts. ↩

See Parkinson, Michael H., The Extreme Value Method for Estimating the Variance of the Rate of Return, The Journal of Business 53 (1980), 6165, which is the final version of the working paper The random walk problem: extreme value method for estimating the variance of the displacement (diffusion constant) started 4 years before. ↩ ↩^{2} ↩^{3} ↩^{4}

Because the range of prices of an asset over a given time period is contained, by definition, within its highest and its lowest price. ↩

See Sepp, Artur, Volatility Modelling and Trading. Global Derivatives Workshop Global Derivatives Trading & Risk Management, Budapest, 2016. ↩ ↩^{2} ↩^{3} ↩^{4} ↩^{5} ↩^{6} ↩^{7} ↩^{8}

At the date of publication of this post. ↩

Other working assumptions are also commonly made, like assuming that the asset does not pay dividends, assuming that the volatility coefficient $\sigma$ remains constant, assuming that the geometric Brownian motion model also applies during time periods with no trading activity (e.g., stock market closure), etc. ↩ ↩^{2}

In details, the geometric Brownian motion assumption slightly differs between authors; for example, Garman and Klass^{17} assume that asset prices follow a more generic diffusion process, which includes the geometric Brownian motion as a specific case. ↩

$\sigma$ is also called the diffusion coefficient of the geometric Brownian motion, but in the context of this blog post, I think it is clearer to explicitly call it the volatility coefficient. ↩

See Andersen, T., Bollerslev, T., Diebold, F., & Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica, 71, 579–625. ↩

In practice, a time period $t$ usually corresponds to a trading day, a week or a month, so that the closing prices $C_t, t=1..T$ are simply the daily, weekly or monthly closing prices of the asset. ↩ ↩^{2}

These estimators are biased, due to Jensen’s inequality; c.f. also Molnar^{46}. ↩ ↩^{2} ↩^{3}

See Yang, D., and Q. Zhang, 2000, DriftIndependent Volatility Estimation Based on High, Low, Open, and Close Prices, Journal of Business 73:477–491. ↩ ↩^{2} ↩^{3} ↩^{4} ↩^{5} ↩^{6}

The asset closing price $C_t, t=1..T$. ↩

See French, K. R., Schwert, G. W., & Stambaugh, R. F. (1987). Expected stock returns and volatility. Journal of Financial Economics, 19, 3–29. ↩

See L. C. G. Rogers, S. E. Satchell & Y. Yoon (1994) Estimating the volatility of stock prices: a comparison of methods that use high and low prices, Applied Financial Economics, 4:3, 241247. ↩ ↩^{2}

See Colin Bennett, Trading Volatility, Correlation, Term Structure and Skew. ↩ ↩^{2}

See Garman, M. B., and M. J. Klass, 1980, On the Estimation of Security Price Volatilities from Historical Data, Journal of Business 53:67–78. ↩ ↩^{2} ↩^{3}

More precisely, Garman and Klass^{17} establish that a variation of $\sigma_{GK}$ is the “best” reasonable estimator but note that $\sigma_{GK}$ is 1) more practical and 2) as efficient as this variation, which I will call the original GarmanKlass volatility estimator $\sigma_{GKo}$. ↩ ↩^{2}

See L. C. G. Rogers and S. E. Satchell, Estimating Variance From High, Low and Closing Prices, The Annals of Applied Probability, Vol. 1, No. 4 (Nov., 1991), pp. 504512. ↩ ↩^{2} ↩^{3}

Such a decrease in efficiency cannot be avoided because the RogersSatchell estimator belongs to class of estimators studied in Garman and Klass^{17}, so that its efficiency is necessarily smaller than the efficiency of the GarmanKlass estimator (maximal by definition). ↩

Garman and Klass^{17} provide a volatility estimator that takes into account opening jumps, but this estimator has a dependency on an unknown $f$ parameter which makes it unusable in practice; Yang and Zhang^{12} show that this dependency is actually spurious and provide a usable form of this estimator. ↩

When the time periods $t$ are measured in trading days, opening jumps are called overnight jumps. ↩

A singleperiod volatility estimator is a volatility estimator that can be used to estimate the volatility of an asset over a single time period $t$ using price data for this time period only; for example, the Parkinson, the GarmanKlass and the RogersSatchell estimators are singleperiod estimators while the closetoclose estimators are multiperiod estimators. ↩ ↩^{2}

C.f. also Molnar^{46} on this subject. ↩

See Kunitomo, N. (1992). Improving the Parkinson method of estimating security price volatilities. Journal of Business, 65, 295–302. ↩

See Alizadeh ,S., Brandt, W. M., and Diebold, X.F., 2002. Rangebased estimation of stochastic volatility models. Journal of Finance 57: 10471091. ↩

See Meilijson , I. (2011). The Garman–Klass Volatility Estimator Revisited. REVSTATStatistical Journal, 9(3), 199–212. ↩

See Jacob, J. and Vipul, (2008), Estimation and forecasting of stock volatility with rangebased estimators. J. Fut. Mark., 28: 561581. ↩ ↩^{2} ↩^{3}

See Mapa, Dennis S., 2003. A RangeBased GARCH Model for Forecasting Volatility, MPRA Paper 21323, University Library of Munich, Germany. ↩

See Chou, R.Y. (2005). Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range (CARR) Model. Journal of Money Credit and Banking, 37(3): 561582. ↩

See Harris, R. D. F., & Yilmaz, F. (2010). Estimation of the conditional variance–covariance matrix of returns using the intraday range. International Journal of Forecasting, 26, 180–194. ↩

See Shu, J. and Zhang, J.E. (2006), Testing range estimators of historical volatility. J. Fut. Mark., 26: 297313. ↩ ↩^{2}

See Brandt, Michael W. and Kinlay, J, Estimating Historical Volatility (March 10, 2005). ↩ ↩^{2} ↩^{3}

See Lyocsa S, Plihal T, Vyrost T. FX market volatility modelling: Can we use lowfrequency data? Financ Res Lett. 2021 May;40:101776. doi: 10.1016/j.frl.2020.101776. Epub 2020 Sep 30. ↩ ↩^{2} ↩^{3} ↩^{4} ↩^{5}

See Patton A.J., Sheppard K. Optimal combinations of realised volatility estimators. Int. J. Forecast. 2009;25(2):218–238. ↩

The YangZhang volatility estimator is excluded to avoid mixing jumpadjusted volatility estimators with nonjumpadjusted ones. ↩

(Adjusted) prices have have been retrieved using Tiingo. ↩ ↩^{2} ↩^{3}

The jumpadjusted YangZhang volatility estimator, as well as the closetoclose volatility estimators, require the closing price of the last day of the previous month as an additional price. ↩ ↩^{2} ↩^{3}

This period more or less matches with the period used in Sepp^{4}. ↩

The next month’s closetoclose volatility is then taken as a proxy for the next month’s realized volatility; this choice is important, because different proxies might result in different conclusions as to the outofsample forecast performances. ↩

See Butler, Adam and Philbrick, Mike and Gordillo, Rodrigo and Varadi, David, Adaptive Asset Allocation: A Primer. ↩

See Mincer, J. and V. Zarnowitz (1969). The evaluation of economic forecasts. In J. Mincer (Ed.), Economic Forecasts and Expectations. ↩

This is due to slight differences in methodology, with mainly 1) the definition of “monthly volatility” in Sepp^{4} taken to be the volatility from the 3rd Friday of a month to the 3rd Friday of the next month and 2) the usage in Sepp^{4} of a linear regression model robust to outliers. ↩

The common ending price history of all the ETFs is 31 August 2023, but there is no common starting price history, as all ETFs started trading on different dates. ↩

For example, for all investors running some kind of monthly tactical asset allocation strategy. ↩

See Peter Molnar, Properties of rangebased volatility estimators, International Review of Financial Analysis, Volume 23, 2012, Pages 2029,. ↩ ↩^{2} ↩^{3}