Duration-Based Volatility Estimation
Torben G. Andersen, Dobrislav Dobrev, Ernst Schaumburg∗
First version: March 10, 2008
This version: July 2, 2008
Preliminary Draft: Comments Welcome

Abstract
We develop a novel approach to estimating the integrated variance of a general
jump-diffusion with stochastic volatility. Our approach exploits the relationship between the speed (distance traveled per fixed time unit) and passage time (time taken
to travel a fixed distance) of the Brownian motion. The new class of duration-based
IV estimators derived in this paper is shown to be robust to both jumps and market
microstructure noise. Moreover, their asymptotic and finite sample properties compare
favorably to those of commonly used robust IV estimators.

∗

Torben G. Andersen: Northwestern University, Kellogg School, t-andersen@northwestern.edu
Dobrislav Dobrev: Federal Reserve Board of Governors, Dobrislav.P.Dobrev@frb.gov
Ernst Schaumburg: Northwestern University, Kellogg School, e-schaumburg@northwestern.edu
We are grateful to the participants of the 2008 SITE-CREATES workshop at Stanford University and seminar
participants at the Federal Reserve Board of Governors for their comments.
The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting
the views of the Board of Governors of the Federal Reserve System or of any other person associated with
the Federal Reserve System.
Andersen gratefully acknowledges financial support from the NSF through a grant to the NBER and by
CREATES funded by the Danish National Research Foundation.

1

Introduction

The importance of return volatility for asset pricing theory as well as practical financial
management has led to a rich literature on volatility estimation, mostly based on time series
observations of asset prices. Information regarding asset values has historically been available
to the research community only through selective transaction and settlement prices such as
daily opening and closing prices and recorded high and low prices during the trading day.
However, within the last decade tick-by-tick data for many liquid financial markets, detailingall transactions and quotes with corresponding time stamps, have become commonplace. The
price history is available as a sequence of observations on a two-dimensional vector whose
coordinates take values on a price and time grid dictated by the institutional set-up of themarket.1 For example, for stocks traded on the New York Stock Exchange (NYSE) prices
are recorded in increments of one cent and time in increments of one second.
The task of volatility estimation is therefore to convert the sequence of recorded highfrequency price-time data into a measure of return variation for a given period of interest
such as a trading day. A common trait of the prevailing approaches to measuring the realized
return variation is to directly measure the size of the price changes over an exogenously given
time grid, either in the form of equidistant sampling (calendar time) or over a certain numberof transactions or quotes (event or tick time). Such measures readily map into corresponding
volatility measures expressed in terms of the observed price variation per time unit.
In this paper we introduce a natural dual approach to volatility estimation which focusesinstead on measuring the time duration per unit of price change and develop a new familyof robust volatility estimators based on the theory of passage times for Brownian excursions.
While the concept of duration-based volatility estimation is not new to financial econometrics
it has, to our knowledge, not been used for measuring the realized return variation in models
with stochastic volatility and jumps in a systematic way, nor have its robustness properties
been fully explored. The closest precursor to our work is arguably Cho and Frees (1988)
who also consider duration based volatility estimation. However, they limit attention to
the unduly restrictive case of constant volatility and do not consider the significant biases
1

Often additional information may be available, such as the depth of the order book and size of transactions, etc. We do not consider procedures which draw on such auxiliary information. The expanded
information set can be analyzed using the theory of marked point processes, see, e.g., the surveys by Engle
and Russell (2004), Bauwens and Hautsch (2008) and Pacurar (2008).

2

induced by finite samples and duration censoring.2,3
The main complication is that return volatility is stochastically evolving throughout
the trading day: it displays a pronounced intraday pattern (diurnal effect), is subject tolarge short-lived shocks (e.g. news release effect), and features a strong degree of temporalpersistence at the inter-daily horizon (volatility clustering effect). This raises a number ofquestions regarding the underlying concept of volatility. First, should we think of volatility
in continuous time or as a discrete-time process updated at equidistant points (in calendartime) or for each new transaction or quote (in tick time)? Second, do we estimate volatility
at each point in time within the trading day and then subsequently aggregate to obtaina measure covering the interval of interest, or should we aim more directly for an overall
daily measure in one step? Third, how do we formally interpret the volatility measure if
it represents return variation over a period where the underlying high-frequency volatility
potentially is fluctuating strongly?
We adopt the coherent framework for addressing these issues which has been developed
within the recent literature on realized volatility. We posit an underlying continuous-time
setting and a frictionless market in which the observed (equilibrium) price dynamics does
not allow for risk-free arbitrage opportunities and expected returns are finite. This implies
that the asset price process constitutes a special semimartingale for which the relevant asset
return variation is given by the associated quadratic variation process, see, e.g., Back (1991)
and Andersen, Bollerslev, and Diebold (2008) for discussion. A general representation ofa continuous-time semimartingale in financial economics is the stochastic volatility (SV)jump-diffusion. The standard “plain vanilla” realized volatility (RV) estimator is given by
the cumulative sum of equidistant high-frequency intraday squared returns and, under the2This early application of the price duration approach to volatility estimation appears to have generated
little, if any, subsequent research. We have identified one actual application, namely Park (1993), whileless directly related papers, which fail to cite Cho and Frees (1988), include Zumbach (1998) who considers
the scaling properties of price durations across different thresholds and Kamstra and Milevsky (2005) who
develop a test for the geometric Brownian motion with drift for the S&P 500 index based on the closed-form
expression for the first hitting time distribution.
3
A more fertile literature with applications exploiting price durations is the discrete-time parametric approach to high-frequency trade dynamics using point processes. These are extensions of the Autoregressive
Conditional Duration (ACD) models, pioneered by Engle and Russell (1998). Later work develops Stochastic
Duration and Stochastic Intensity models, but all rely on discrete time ARCH or SV style specifications, seee.g., Lunde and Timmermann (2004) and the survey by Bauwens and Hautsch (2008)). The conceptual differences between our approach and this literature mirror the distinction between the realized volatility measuresand high-frequency intraday ARCH and SV models. The realized volatility approach is nonparametric andmodel-free, based on a general continuous-time setting and focused on ex-post volatility measurement overa non-trivial interval, while ACD type models are parametric, cast in discrete time and focus on estimationof the spot volatility or price intensity at each transaction or quote, that is, an ex-ante forecast of return
variation over the next instant. As such, they represent distinct methodologies, even if they explore similar
quantities.

3

given (ideal) conditions, it is consistent for the underlying quadratic return variation as theintraday return horizon shrinks to zero. The attraction of the standard RV approach is thusthat it constitutes a valid volatility measure for a very general class of underlying processes
while the implementation is model independent. In practice, however, one must contend
with the fact that market imperfections introduce significant microstructure noise in ultra
high-frequency data. Consequently, most implementations of RV estimators are limited to
exploit the data at relatively low frequencies, such as 1-5 minute returns.
A similar natural limitation applies to the duration-based volatility (DV) estimatorsdeveloped in this paper. However, the rule of thumb for limiting the impact of noise on the
DV estimators is expressed explicitly in the domain of price thresholds rather than implicitly
in the time domain. In particular, we show that our DV estimators require price thresholds
larger than 2-3 bid-ask log-spreads for the measured price durations to be sufficiently robust
to market microstructure features reflected in real data quotes. Moreover, the DV estimators
are inherently robust to consecutive price duplicates, frequently encountered in real data
due to price discreteness, since such duplicates do not affect the point in time at which theprice is observed to exceed a chosen threshold. These features of the dual duration-based
approach render the DV estimators viable and interesting alternatives for robust volatility
measurement in the presence of microstructure noise without any need for delicate noise
filtering techniques. In particular, we show that our duration-based estimators have excellent
asymptotic efficiency relative to existing comparable noise-robust IV estimators.
When price jumps are present, the standard RV measures include the squared jumpcomponents. A number of extensions of RV have been developed to estimate the jump andcontinuous components of volatility separately. In this context, an important feature ofour duration-based approach is that it explicitly limits the impact of jumps exceeding thechosen price change threshold. Thus, we show that our DV estimators are robust to jumps
in the sense that they are consistent for the continuous part of the quadratic variation as
the threshold shrinks to zero.4 We confirm that these properties carry over to practical settings with actual equity price quote data and further illustrate the point through extensiveMonte Carlo simulation studies. Specifically, analyzing tick-by-tick data for the full sample
of 30 Dow Jones stocks over January 2005 through May 2007, we find that our DV estimators perform well compared with leading sub-sampled RV and bi-power variation estimatorsof the integrated variance. Moreover, our duration based estimators are not substantiallyharder to implement and are not sensitive to the choice of threshold in the range of 3 to 6
bid-ask log-spreads. Since the duration approach extracts information from the same un4This feature is akin to jump truncation in the RV context, see e.g. Mancini (2006), but is a natural featureof the basic estimation technique rather than necessitating the choice of a separate bandwidth parameter.

4derlying data as RV estimators, but does so in a very different manner, the two procedures
are natural complements, even ignoring the rather compelling efficiency properties of theduration-based measures. Finally, we note that the same duration-based approach delivers
jump- and noise-robust estimators for the more general notion of integrated power-variation.
Such measures are useful for deriving feasible inference tools for the integrated volatility andfor the construction of formal jump tests among other things.
The remainder of the paper is organized as follows. Section 2 briefly reviews the related
literature and introduces the basic theory of Brownian excursions in the constant volatilitycase. Sections 3 and 4 develop the DV estimators following a duration-based dual approach
to RV and derive the asymptotic properties based on a localization argument. Section 5
applies the DV estimators to the set of 30 Dow Jones stocks and confirms that the desirable
robustness and efficiency properties hold when applied to real data. Section 6 separately
investigates the robustness of the DV estimators to noise and jumps in a Monte Carlosetting. Section 7 concludes.

2

Related Realized Volatility Literature

In the literature on continuous-time volatility estimation, the evolution of the log price of asingle financial asset is typically given in reduced form by a semi-martingale characterized
as the unique (weak) solution to the stochastic differential equation
dst = µt dt + σ t dWt + ∆Xt , s0 ≡ s(1)The solution to (1) is known as a jump-diffusion process with instantaneous drift µt , instantaneous volatility σ t and jumps given by the finite activity jump process Xt . Two key objectsof interest in financial applications are the integrated volatility (IV) and quadratic variation(QV) of the log price process, defined as

IVt =

Z t
0

σ 2u du ,QVt = IVt +

X

∆Xu2

(2)

0<u≤t

The quadratic variation has two separate components, namely, first, the return variation
generated by the diffusive volatility term and, second, the cumulative sum of squared jumpsizes. The latter component is important for a variety of reasons, including the interpretationof news and practical risk management. However, the former is often of key importance for5volatility forecasting as the continuous or diffusive component of return variation is themain driver of the pronounced persistence in return volatility. As such, methods have beendeveloped to separately identify these two components, as we discuss below.
Although the volatility process of asset returns is itself inherently unobservable, thevolatility path {σ t } and jumps {∆Xt } are de facto observable when the econometrician
observes the log price process {st } in continuous time. This is due to the convergence inprobability (see e.g. Karatzas and Shreve (1991, Theorem 1.5.8)):∀t : p lim 
m→∞

Z t
0

σ 2u du +X0<u≤t∆Xu2 −



m h
X

i2

s kt+1 − s kt  = 0

k=0

m

m

(3)

In practice, however, the choice of sampling frequency is limited by the liquidity of thegiven asset as well as the significant market microstructure noise found in the data frommost financial markets. The presence of such noise in high-frequency returns implies thatthe log price process itself is latent and the limit (3) cannot be expected to hold in practice.
Nonetheless, (3) motivates the use of realized volatility (RV), as an estimator of the quadratic
variation:5

RVt =

N h
X

sti − sti−1

i2

(4)

i=1

The sampling frequency of the N observations {ti }i=1,...,N in (4) is often deliberately chosen
to be lower than the actual frequency of the data in order to ameliorate the potential bias
due to the aforementioned market microstructure effects. Improving the robustness andefficiency of the basic RV estimator (4) to market micro structure noise as well as devising
methods for estimating integrated volatility (i.e., filtering out jumps) has been an extremely
active area of recent research spurring the development of a new generation of volatility
estimators. These include:- Subsampled RV estimators in which (4) is applied to relatively low frequency intraday
returns but overlapping subsamples are used to produce a sequence of robust, albeit
individually inefficient, estimators of QV which can be averaged to produce the subsampled QV estimate, see, e.g., Zhang, Mykland, and Ait Sahalia (2005). The resulting
5

The term ‘realized volatility’ was coined by Fung and Hsieh (1991) to describe their ad-hoc daily volatility
estimate calculated as the sum of squared 15 min returns across a number of markets, including S&P500
futures, Bond futures and Currency futures. Such measures were only later put in a proper probabilistic
context as an approximation of the quadratic variation of a semi-martingale, see, e.g. Andersen and Bollerslev(1998) and Barndorff-Nielsen and Shephard (2002)).

6

estimator gains in robustness to market microstructure noise but it will estimate thefull QV and does not involve an attempt to correct for jumps.6- Kernel based RV estimators that are closely related to the subsampled RV estimatorsand can be constructed to have good robustness against micro structure noise as shownby Barndorff-Nielsen, Hansen, Lunde, and Shephard (2007).
- Bi-power variation estimators exploiting the finite activity of large jumps to produce
a jump-robust estimator of IV. Rather than squaring returns, Bi-power estimators
cumulate products of adjacent absolute returns, thereby ensuring that isolated jumps
will cease to matter asymptotically (see e.g. Barndorff-Nielsen and Shephard (2004)).7- Range based RV estimators that differ from the aforementioned methods which allconceptually rely on the relationship between squared or absolute returns and volatility(4). Instead, as a natural extension to Parkinson (1980), range based RV estimators
exploit the distribution of the maximum and minimum of the log-price process. Recent
contributions include Christensen and Podolskij (2007b), Christensen and Podolskij(2007a) and Dobrev (2007).
In this paper we take a conceptually different approach to robust IV estimation that
relies on the theory of durations for Brownian excursions. While duration based volatility
estimation is not new to financial econometrics, it has to our knowledge not been used forIV estimation in models of stochastic volatility in a systematic way nor have its robustness
properties been explored. We develop the duration analogues to the sub-sampled RV, bipower and range based estimators and show that these have better asymptotic properties
in the hypothetical no-noise continuous time limit and argue that they conceptually should
exhibit similar robustness properties to both market microstructure noise and jumps as theexisting estimators. These conjectures are substantiated by our Monte Carlo simulations
and further supported by our analysis of the tick-by-tick data of Dow Jones 30 index stocks.
6

Further refinements along these lines include the Two-scale and Multiscale RV estimators of Zhang,
Mykland, and Ait Sahalia (2005) and Zhang (2006).
7
Extensions of this approach include alternative transformations of the bipower variation statistic anddifferent lag choices between the nearby absolute returns whose product is cumulated. In addition, bipower
variation has been extended to the concept of multi-power variation, see, e.g., Barndorff-Nielsen, Graversen,
Jacod, and Shephard (2006) and Barndorff-Nielsen, Shephard, and Winkel (2006). In addition, many alternative jump detection procedures have also been proposed. For a recent overview of these developments, seeAndersen and Todorov (2007).

7

3

Constant Volatility Estimators Based on Passage Times

A great number of classic results exist on the properties of Brownian passage times. Althoughthese results are limited to the constant drift and volatility case, they can be extended to amore general class of diffusions by a suitable time-change or localization argument. In thissection we, for simplicity, introduce our estimators under the assumption of zero drift andconstant volatility, deferring the more general stochastic volatility case to Section 4 below.

3.1

Duality between transition times and the magnitude of Brownian increments

Consider a Brownian motion {Bt } with constant instantaneous volatility σ and zero drift

Bt = σZ t
0dW (t), t ≥ 0, B0 ≡ 0

If the object of interest is to obtain an estimate of the instantaneous variance of the process,one can rely on the fact that the increments ht = (Bt0 +t − Bt0 ) over a given time interval t
satisfy ht ∼ N (0, σ 2 t) leading to the moment conditionhi

E h2t = σ 2 t(5)from which σ 2 can be estimated e.g. based on a sample of equispaced returns. This is ofcourse the basic logic underlying the RV estimator (4). Rather than measuring the (squared)
size of Brownian increments over a fixed time interval, one can instead measure the time ittakes the Brownian motion to travel a given distance. Denoting the fixed distance h, we canmeasure time in several alternative ways, each giving rise to a distinct estimator:
first exit time

: τ h = inf{t> 0 | |Bt | > h}max Bs − min Bs > hfirst range time

: τ h = inf t > 0

first hitting time

: τ h = inf{t > 0 | Bt = h}

0≤s≤t

0≤s≤t

(6)

In the sequel, we use the term “passage time” to denote either of these three measures ofduration. As we shall see below, estimators based on the expected first exit time are thenatural analogues to RV estimators based on (5) while the estimators based on the expected8

first range time are the natural analogues to the intraday range estimator of Christensen
and Podolskij (2007b) and the generalized range proposed by Dobrev (2007).8
The moment generating functions for the passage times τ h are well known but the probability densities (shown in Figure 1) are only known in closed form for the first hitting time(see e.g. Borodin and Salminen (2002)). From the simple form of the moment generating
functions given in Table 1, it is immediately clear (see Lemma A.2) that
D

τh =

h2
τ̃ 1
σ2

(7)

where τ̃ 1 is the passage time of a standard Brownian motion with respect to the thresholdh = 1. This implies that the case of a general volatility σ 2 and threshold h can always
be reduced to the baseline case of a standard Brownian motion and unit threshold through2
scaling of all passage times by the factor σh2 . This fact will be extremely convenient since itallows us to derive all relevant passage time properties from the baseline case.

3.2

Estimation based on passage time moments

The expected passage times can be retrieved from the moment generating functions for the
first exit time, the first range time and the first hitting time given in Table 1:






h2
σ2
1 h2
2 σ2





∞

E [τ h ] = (first exit time)(first range time)(first hitting time)(8)

A method of moments estimator of volatility can then be obtained based on an observed
sample of passage times, except in the case of the first hitting time which does not have afirst moment. Comparing (5) and (8), the duality between fixed-threshold passage times andfixed-time interval Brownian increments is clear and gives rise to two separate yet closely
related approaches to variance estimation. The moment condition for the first exit time
was used by Cho and Frees (1988) who studied the effect of price discretization on volatility
estimation and considered (8) as a more robust alternative to the existing return-based
estimation procedures. Kamstra and Milevsky (2005) used the moment for the first hitting
8

Passage times for other Brownian functionals can be defined in a similar way. For example, the maximum
drawdown of a Brownian motion can be used to define the first drawdown time, which is identical in
distribution to the first exit time and leads to estimators of exactly the same form as the first exit time.

9time, but in the case of a Brownian motion with drift so that the expected hitting time
is finite. An important, but often overlooked, issue that arises when applying the moment
conditions of the form (8) to estimate σ 2 is that they can suffer from quite severe small
sample biases induced by Jensen’s inequality. This is because the expected passage time isinversely proportional to the instantaneous variance. For a given sample of passage times ofsize N , we show in Proposition A.3, that9

"

E

#

h2
1 PN −1i=0 τ i,hN(N )= µ1 σ 2

where


2


+ O( N12 )1 + 3N(N )1µ1 =  1 + 3N
+ O( N12 )


1

N

(first exit time)(first range time)(first hitting time)(9)

In applications it is often more convenient to work with an unbiased estimator for which
no sample size dependent correction is necessary. A particularly convenient small sample
unbiased estimator can be based on the first moment of the reciprocal passage times derived
in Corollary A.4:

"

2

h
E
τh

#







2C σ 2≡ µ1 σ 2 = (4 log 2) σ 2




σ2

(first exit time)(first range time)(first hitting time)

(10)

(1)

where µ1 = µ1 and C ≈ 0.91597 is the Catalan constant. The moment conditions (10)have, to our knowledge, not previously been applied to the problem of volatility estimation.
Moreover, we see that while the expected hitting time is infinite, the expected reciprocal
hitting time is well behaved. The reciprocal passage time moment (10) leads to the local
volatility estimator, based on a single observed transition,σ̂ 2h =1 h2
µ1 τ h(11)

where the value of the scaling factor µ1 is given in equation (10) and Table 1. The estimator(11) will be the basic building block in constructing an estimator of IV in Section 4 below.
(N )9

The exact value of the Jensen correction factors µ1 can be calculated explicitly for each N. For the
(1)
(2)
(5)
5
first exit time, we have µ1 ≡ µ1 = 2C, µ1 = log(4), . . . , µ1 = 25
(18C − 11), . . . where C ≈ 0.91597 is theCatalan constant. Thus the Jensen effect is quite substantial for small N but is well approximated by (9)for N ≥ 20.

10

In Corollary A.7 we show that the estimator σ̂ 2h satisfies

E[σ̂ 2h ] = σ 2

and




0.7681 σ 4V[σ̂ 2h ] ≈ 0.4073 σ 4



4

2.0000 σ(first exit time)(first range time)(first hitting time)(12)

Note that the range time based estimator is the most efficient since it exploits more information about the joint infimum and supremum of the process than the first exit or first hitting
time. The variance of the estimator depends on the fourth moment (the so called quarticity)of the log price process but crucially does not depend on the choice of threshold h.10
When applying (8) or (10) to the estimation of volatility in markets with periodic market
closures, one faces the familiar issue of right censoring. This implies that the distribution of
the observed passage times τ h will depend on the amount of time remaining until markets
close since it is a draw from a conditional distribution where the conditioning event is thatthe transition took place prior to end-of-trading. The effect of censoring is illustrated in
Figure 2 for the baseline case (σ/h = 1). The x-axis represents the time remaining until
end-of-trading and the horizontal dashed thin lines denote the unconditional expectation of
the reciprocal stopping times given in (10). It is clear from the figure that the impact ofcensoring is quite severe for the first hitting time as we approach the censoring time but muchless pronounced for the exit time and especially the first range time. To see the effect for an
arbitrary (σ, h), the distributional identity (7) implies that the time axis should be scaledby σ 2 /h2 . Thus we see that the conditioning will tend to have a large impact whenever the(unconditional) expected passage time represents a large fraction of the remaining time until
the end of a trading day. As a practical matter, we find that the conditioning can safely beignored for estimation purposes when there is more than 3 (first exit time) or 2 (first rangetime) expected passage times until end of trading. For the first hitting time on the other
hand, the conditional expectation should always be used.
ih
The appropriate censored moment, E τ1h |τ h ≤ T , is known in closed form for the firsthitting time and can be represented as an infinite series for the first exit and first range times.
Estimation using the conditional moment is therefore feasible. However, it is important tonote the relationship between the expected reciprocal passage times and σ 2 will no longer
be linear and, moreover, the appropriate Jensen correction terms will depend on both thetime remaining as well as the true σ 2 . We therefore advocate a different approach, exploiting
10

The reciprocal passage times can be used also to define (locally) unbiased estimators of any integer power−p/2
σ based on the moment E[τ h ], as shown in Appendix A.2. In particular, local estimators of volatility,
σ, or quarticity, σ 4 , can be considered and the corresponding integrated quantities constructed as in Section
4 below. These and related developments are the subject of ongoing research.
p

11

the time-reversibility of the Brownian motion. In particular, we define the equivalent (indistribution) backward passage times:
first exit time: τ−
> 0 | |B−t | > h}h = inf{tτ−
h = inft>0

max B−s − min B−s > hfirst range time:first hitting time: τ−
h = inf{t > 0 | B−t = h}

0≤s≤t

0≤s≤t

(13)

In applications, this allows us to use the regular (forward) passage times (6) during thefirst half of a trading day and the reverse (backward) passage times (13) during the second
half thereby effectively eliminating the need for conditioning for all but the most extreme
thresholds.

3.3

Finite Activity Jumps

The occurrence of jumps is a prominent feature of many financial time series and it is clear
from (2) that RV will not be a consistent estimator of IV in this case. This fact has lead to thedevelopment of jump robust IV estimators, most notably the bi-power variation (BV) estimator of Barndorff-Nielsen, Graversen, Jacod, and Shephard (2006) and subsequent multi-power
variation estimators developed in Barndorff-Nielsen, Shephard, and Winkel (2006).11 Whilethese estimators are consistent for IV, the occurrence of large jumps still pose a significant
challenge in finite samples as seen in the Monte Carlo experiments in Section 6 below.
By contrast, the duration based volatility estimators (DV) inherently posses a significant
degree of jump robustness due to the implicit jump truncation at the chosen threshold level:
if a large jump occurs it may lead to an early threshold crossing (and hence an upward bias
in the local volatility estimate), but the amount by which the jump exceeds the threshold
will not matter. As the threshold is lowered, jumps will cease to matter in the limit.
The discreteness of actual data will in practice limit the ability to lower thresholds beyonda certain point. It is therefore of interest to study simple modifications of the baseline DV
estimator which exhibit additional jump robustness in finite samples. One such, asymptotically equivalent estimator, is the “previous tick” DV estimator. The previous tick estimator
is identical to the regular DV, except that it utilizes the threshold, h̃, corresponding to theobserved log-price at the time tick just prior to the crossing of the target threshold h. Since(large) jumps tend to lead to an immediate threshold crossing, this method eliminates the11

Numerous tests for the presence of jumps have been also developed, e.g. Barndorff-Nielsen and Shephard(2004), Ait-Sahalia and Jacod (2006), and Lee and Mykland (2007).

12

effect of jumps.12 We show in Section 6 below, that the previous tick DV estimators which welabel “robust” indeed are successful in eliminating most of the effect of jumps for empirically
relevant larger thresholds.

3.4

Microstructure Noise

It is well-known that high-frequency price observations are noisy due to various market
microstructure features and possible data errors. This leads to a natural signal-to-noise
trade-off when estimating the variance of the unobserved log-price process. An intuitive
appeal of the duration-based approach in this regard is that the threshold level can be chosenlarge enough relative to the noise level to ensure a desired degree of robustness. In orderto formally assess the level of robustness of our passage time estimators to microstructurenoise we adopt the following AR(1) noise structure:pei = pi + ui

(14)ui = ρui−1 + εi
where pei are the noisy log-price observations, pi are the unobserved log-prices, and εi ∼N (0, (1 − ρ2 )ω 2 ) are the innovations of the AR(1) noise process ui so that the noise process
has the unconditional moments E[ui ] = 0, E[u2i ] = ω 2 .
This specification allows us to explicitly study the effect of different noise levels ω 2 anddegrees of noise persistence ρ on the overall quality of our passage time estimates. The
leading special cases of interest are:(i) ρ = 0, in which case we obtain the ubiquitous Gaussian i.i.d. specification representative of transaction price data.
(ii) ρ >> 0, in which case we obtain a persistent autoregressive specification representative of quote data.13
Given a sample of noisy log-price observations pei , one can use standard projection methods to express the expectation of pi conditional on all observables, which in this case isalso equivalent to applying the Kalman filter to the state-space representation (14). For
our purposes it is of primary interest to recover more generally the expected squared ratio
12

Whenever a crossing of threshold h was due to a Brownian increment, h̃ will tend to be close to h whileh̃ may be substantially smaller than h in the case of a jump crossing. In absence of jumps and for large h,
the difference between the previous tick estimators and the original passage time estimators are negligible.
13
Hasbrouck (1999), and more recently Zhang, Russell, and Tsay (2007), report estimates of ρ as high as
0.4 for the latent cost component of 15-minute quotes, which formally corresponds to ρ > 0.99 if brought toa 3-second frequency.

13

between the observed log-price difference and the underlying true log-price difference forpassage times at different threshold levels. These ratios of observed versus latent transition
magnitudes can be directly interpreted as relative bias measures for our duration-based estimators (and, conversely, the ratios of unobserved versus observed transition magnitudes canbe interpreted as the corresponding bias correction factors).
Figures 3 and 4 plot the relative bias of the first exit time and first range time estimators
respectively, obtained for three different threshold levels as a function of the noise persistence
ρ. The top graph on each figure is for the case of noise-to-signal ratio λ = 0.25 (moderate
noise level), while the bottom graph is for the case of noise-to-signal ratio λ = 1.00 (highnoise level). The chosen threshold levels 3, 5, 7 are chosen to span the range of thresholds
of greatest interest for application on real data and are expressed accordingly in units of thenotional average intraday log-spread (as in sections 5 and 6 below). It is worth noting thatthresholds much lower than 3 log-spreads are probably untenable in practice as we would
expect the noise-to-signal ratio to be prohibitively high for small thresholds.
The main conclusions to be drawn from this analysis (Figures 3 and 4) are as follows.
First, both passage time estimators are sufficiently robust to: (i) moderate noise (λ = 0.25)
with arbitrary persistence, and, (ii) persistent noise (ρ > 0.9) regardless of the noise level.
Second, the first exit time is naturally more robust to noise than the first range time. Third,
the higher the chosen threshold level, the more robust to noise are the resulting passage time
estimators. Fourth, the passage time estimators exhibit significant upward biases (above2%) only in the case of non-persistent noise (ρ ≈ 0) at high level (λ = 1.00).
These initial results provide formal evidence in support of our intuition that by setting
the threshold level large enough relative to the noise level, the duration-based estimators
do indeed possess an inherent degree of robustness to noise in many cases. If one is willing
to be parametric about the noise structure, the outlined projection approach (or Kalmanfiltering) can of course be used to develop an explicit feasible noise correction of the passage
time estimators in high-noise scenarios. We relegate this for future research, relying for
now on demonstrating the success of the duration-based approach in moderate noise and/or
persistent noise settings.

3.5

Discretely Sampled Observations

A final concern that must be addressed is time discretization, i.e. the fact that prices areobserved on a discretely spaced time grid. In order to be feasible, the proposed estimators
must be corrected for the discrepancy between the first passage times of the unobserved
process and those measured on an observed finite sample of points. In particular, the first14

passage times of a random walk overstate the Brownian first passage times. One reason
for this is that the maxima of a random walk always will understate the true unobserved
maxima of the underlying Brownian motion, so that it takes somewhat longer on average forthe random walk to break a given threshold level. The finer is the discretization step, thesmaller is the discrepancy between the random walk extrema and the Brownian extrema.
This issue is addressed by Asmussen, Glynn, and Pitman (1995). They derive exact
f of a random walk based on k
asymptotic results for the amount by which the maximum M
equidistant observations in a finite interval understates the Brownian maximum M in thesame interval (see appendix, Lemma A.8). We adapt their results to obtain an asymptotic
correction for the passage time estimators proposed above. Our main insight stemming from
Asmussen, Glynn, and Pitman (1995), is that knowledge of the moments of the discrepancy
due to discretization allows us to approximate the ratio between moments of the Brownian
maximum/range and the random walk maximum/range as a function of the number ofsample points k:
E M2

h

i

f2
≈ fM (k) × E M

h

i

e2
≈ fR (k) × E R

E R2

h

h

i

i

where the scaling factors fM (k) and fR (k) approach 1 from above as k gets large (see appendix), but crucially do not depend on the parameter of interest σ 2 .
From this it follows that for any given threshold h one needs to scale up the finite sample
volatility estimators based on the first exit time and the first range time by a factor offM (k) and fR (k) respectively, where k is the (varying) number of observations between thestart and end point of each h-excursion14 . By aggregating the resulting estimates at eachsample point on a given grid we obtain a novel and feasible realized duration-based volatility
estimator (DV) dual to RV in the sense that it is based on measuring passage times rather
than price increments. The next section lays out a unified theoretical framework for DV
based on first exit times and DV based on first range times.
14

Though easy to implement, this approach cannot yield an exact small sample correction since it is notdesigned to account for the possibility of early crossings (i.e. Brownian excursions crossing the threshold in
between random walk observations) in a passage time context. Without further small sample adjustment, thiswill lead to a downwards bias for the smallest thresholds while the bias quickly diminishes as threshold levels
increase. An exact, albeit more computationally intensive, correction is possible by viewing the unobserved
Brownian path conditional on the observed random walk as a sequence of connected Brownian bridges forwhich crossing probabilities and crossing time distributions can readily be calculated in the case of firsthitting times and first exit times. For the first range time the situation is slightly more complex and is the
subject of ongoing research.

15

4Stochastic Volatility Estimators Based on Passage
Times

When volatility is randomly varying throughout the trading day, we can no longer assume
that observed passage times are identically distributed. This leads us to consider instead local
volatility estimation as proposed by Foster and Nelson (1996) and more recently Mykland(2006). Based on consistent estimators of local volatility, an IV estimator can then be
constructed as the corresponding Riemann sum.
Consider then a fixed time grid 0 ≡ t0 < t1 < . . . < tN ≡ 1 consisting of N , notnecessarily equispaced, intervals with ∆i = ti+1 − ti denoting the mesh size and where thetime unit is chosen to ensure that the length of the trading day is unity. We will assume thatthe volatility process is “well behaved” in the sense that it can be locally approximated by apiecewise constant function as the mesh size tends to zero, e.g. due to Lipschitz continuity.
We also assume independence of the volatility process and the Brownian motion driving
returns (i.e. “no leverage”) for our theoretical results although our Monte Carlo studies
suggest no adverse effect on the DV estimators of introducing such dependence.
At a given grid point, ti , we wish to construct an estimator of the local volatility. Givena threshold size h, we can, starting from ti , look for the next passage time τ +h (ti ) and useh2
1
the estimator µ τ + (t ) . However, we can in principle do better: Since the Brownian motion
1 h i
is time reversible, we can also, starting from ti , look back in time for the next passage timeh2
1τ−h (ti ) and construct the independent estimator µ τ − (t ) .
1

τh− (ti )τh+ (ti−1 )



|

}

h

τh− (ti+1 )τh+ (ti )

~

|

ti−1

i

|

/

ti+1

ti

We thus have two independent estimators of local volatility at ti which suggests using thebi-directional local volatility estimator:
2
σbb h (ti )=σ̂ 2h (ti ) =





2
h2
+ τ −h(t )
τ+
(t
)
i
h i
h
2
1
h2
1
or µ τ −h(t )µ1 τ +
(t
)
1
h i
h i

1
2µ1

(bi-directional)(uni-directional)

16

(15)

By independence, we have the local variance estimates:2

V[σbb h (ti )]V[σ̂ 2h (ti )](first exit time)
0.3841 σ 4ti
0.2037 σ 4ti(first range time)(first hitting time) 1.0000 σ 4ti

0.7681 σ 4ti
0.4073 σ 4ti
2.0000 σ 4ti(16)

In practice, however, we do not use the bi-directional estimators due to the aforementioned censoring issue.15 Instead we use the uni-directional estimator based on τ +h (ti ) for
grid points ti that fall in the first half of the trading day and τ −h (ti ) for grid points fallingin the second half of the trading day. We thus have a sequence of local variance estimates{σ̂ 2h (ti )}i=1,...,N based on passage time durations from which we can construct the estimate
of integrated variance on [0; 1] as

d
DV

N,h =

N
−1
X

σ̂ 2h (ti ) ∆i

(17)

i=0

where ∆i = N1 in the case of an equispaced time grid. Note that the distribution of (17) will
depend on h although the distribution of each individual estimate σ̂ 2h (ti ) does not. This isbecause the covariance between neighboring volatility estimates, i.e. Cov(σ̂ 2h (ti ), σ̂ 2h (ti−1 )),will likely involve overlapping transitions when h is large relative to the mesh size. However,
the threshold h can be chosen independently of N and we can consider letting h tend to zeroat an appropriate rate so that terms in (17) have arbitrarily low correlation. In Proposition
A.16, we show that the appropriate rate is h = o(N −1/2 ) and the resulting estimator satisfies

√
N







d
DV
N,h − IV ∼ Mixed Normal 0, ν

Z 1
0



σ 4u du(18)

where ν is the constant in (16). Note that the asymptotic distribution depends on theunknown integrated quarticity. However, it is easy to show that the same localization argument can be used to consistently estimate other integrated moments of interest such as theintegrated quarticity itself, thus enabling the calculation of confidence bands for IV.
In practice, the observation record is discrete and we only observe the value of the process
15

In markets such as FX, where 24 hour trading prevails, the bidirectional approach could yield substantial
efficiency gains.

17

dat the N grid points, thus rendering the convergence rate N −1/2 of the estimator DVN,hinfeasible. Instead we can consider the construction of a set of local volatility estimates on acoarser sub-grid, consisting of K grid points {ti1 , . . . , tiK }. Here K = o(N ), K → ∞ so thatthe number of observations between adjacent grid points increases with the sample size andthe discretization error tends to zero for suitably chosen threshold levels:

√
K



d
DV

K,h − IV

∼ Mixed Normal 0, ν

Z 1
0



σ 4u du(19)

Note that the convergence rate is now the slower K −1/2 . Part of this efficiency loss canof course be mitigated by averaging the estimator over all possible sub-grids of mesh size
∆ = K −1 . This situation is analogous to the sub-sampling strategy for the RV estimator
advocated by Zhang, Mykland, and Ait Sahalia (2005).
The value of the variance factor ν is about 0.8 for the unidirectional first exit time DV
and about 0.4 for the unidirectional first range time DV (see table 1) although these values
will be reduced further by the subsampling scheme. This compares to 1.3 for subsampled
RV and about 1.5 for subsampled BV (the values for RV and BV without subsampling are
2.0 and 2.6 respectively).16 Thus, for suitably chosen threshold levels both the first exit
time DV and the first range time DV should in practice be considerably more efficient thansubsampled RV (in the absence of jumps) or subsampled BV (when jumps are present).
In what follows, we conduct extensive empirical analysis on real and simulated datawith the main goal of comparing the finite sample efficiency of the DV estimators vis-a-vis
subsampled BV as the benchmark jump-robust (and sufficiently noise-robust) estimator of
the integrated variance.17 In order to design realistic Monte Carlo experiments, we first
analyze the full set of Dow Jones 30 stocks and document that both the first exit time DV
and the first range time DV significantly outperform (optimally) subsampled BV: the DV
estimators achieve the same mean estimate of the integrated variance with a lower standard
deviation while remaining very highly correlated with the subsampled BV estimates.
16
Subsampling of RV with important extensions to optimal two-scale and multi-scale RV estimators in thepresence of microstructure noise has been advocated by Zhang, Mykland, and Ait Sahalia (2005) and Zhang(2006). Subsampling of BV has been empirically documented to have similar merits by Dobrev (2007),
although the asymptotic distributiuonal properties have yet to be studied.
17
We leave for future research the comparison to range-based estimators of IV developed by Christensen
and Podolskij (2007b), Christensen and Podolskij (2007a) and Dobrev (2007).

18

5

A Realized Volatility Analysis of the Dow Jones 30

In recent years the Dow Jones 30 stocks have an increasingly dense (near second-by-second)
record of intraday transaction prices and quotes. This provides an excellent laboratory fortesting the empirical performance of our passage time volatility estimators on real data. For
our analysis we use NYSE TAQ data for the period January 1, 2005 to May 31, 2007. Table 2
summarizes basic descriptive statistics for the Dow Jones components, including companies
that have been included in or excluded from the index during this period. Ignoring short
trading days around major holidays we obtain a sample of 601 regular trading days of nearlysecond-by-second intraday data for each stock from 9:30 am to 4:00 pm.
We apply our first exit time and first range time estimators on the series of mid-quotes
after filtering out spread outliers as usual in order to avoid distortions due to bid-ask bounceand possible data errors. We also filter out consecutive duplicate mid-quotes since the measurement of first passage times is driven exclusively by the first time a given thresholdlevel is reached. The daily sample sizes without duplicates vary from about two to eight
thousand quotes, roughly corresponding to an effective highest sampling frequency of threeseconds. Microstructure noise still distorts the measured returns and, hence, the measured
transition times and corresponding duration-based volatility estimates. However, when considering large enough thresholds (e.g. 4-6 log-spreads) the transition time distortions will
be relatively small and the quality of our first exit time and first range time DV estimatesshould be comparable to the quality of subsampled RV and BV estimates at modest sampling frequencies such as 2 to 5 min. As we have found in section 3.4, the impact of noise on
our duration-based measurements is close to negligible for moderate noise levels and higher
degree of noise persistence typical for mid-quotes. Thus, being agnostic about the exact
structure of the noise process, the robustness argument for DV based on a large enoughthreshold is analogous to the one that justifies the use of subsampled RV or BV at a sparseenough frequency.18
We compare the finite sample efficiency of DV vis-a-vis subsampled BV as the benchmark
jump-robust estimator of the integrated variance. The results of our analysis in the form ofsignature plots are presented in Appendix C (for the average across all stocks) and Appendix
D (for individual stocks). We first determine a near-optimal sampling frequency for the
subsampled BV estimator for the stocks in our sample. Figure 8 in Appendix C represents
the cross-sectional average signature plot of BV versus RV as a function of the sampling
frequency in seconds. A key feature to notice is that BV is severely downward sloping for
18

The main advantage of DV in this regard is that the choice of threshold level directly limits the local
noise import at each grid point, while the choice of sampling frequency for RV and BV controls the averagenoise import across all grid points.

19

frequencies from 3 to 120 seconds, while RV flattens much faster above the 15 or 30 second
sampling frequency. It is known that the increasing gap between RV and BV at frequencies
higher than 2 minutes is due to the increased number of zero returns at those frequencies.
This is explicitly confirmed by the last column of Table 2 containing descriptive statistics.
The median number of zero returns lasting longer than 30 seconds on each day is 161, thoselasting longer than 1 minute are about 48 on each day, while the zero returns lasting longer
than 2 minutes are only about 11 per day. Importantly, this pattern is observed without
much variation across different stocks (reported in the table) or different trading days (notreported in the table). Therefore, the 2-minute frequency for the subsampled BV estimatorappears to be a nearly optimal choice in the cross-section.
Next, treating the 2-minute subsampled BV as the benchmark estimator of the integrated
variance, we analyze the cross-sectional average of our DV estimators for threshold levels
expressed in units of the average log-spread. As evident from Figure 9 in Appendix C, forthreshold level set to 4 log-spreads and above both our first exit time DV estimates andfirst range time DV estimates on average agree with the 2-min BV estimates. What is evenmore striking, though, is that essentially the same pattern of the signature plots emerges
also for each individual stock (see appendix D). At the same time, for thresholds larger than
4 log-spreads the standard deviation of our DV estimates is markedly lower than the one of
2-min subsampled BV (see Figure 10 in Appendix C), while the average correlation between
our DV estimates and the 2-min subsampled BV estimates exceeds 0.9 (see Figure 11 inAppendix C).
Overall, robustly across all stocks for threshold level 4 log-spreads and above we obtaina similar mean level of our DV estimates as 2-min subsampled BV, very high correlation
with BV but markedly lower variance. This provides encouraging initial evidence that our
DV estimators can significantly outperform subsampled BV as a jump-robust benchmark
estimator of the integrated variance in real-world settings of interest. The next section
further reinforces this conclusion by means of Monte Carlo experiments for a rich set ofcarefully calibrated models.

6

Monte Carlo Experiments

In this section we document the performance of the proposed duration-based DV estimators
on finite samples of simulated data from a wide selection of jump-diffusion models. For eachmodel specification, we mimic the discreteness of stock quotes observed in real data samples
from frequently traded stocks. In particular, we generate random quote arrivals at an average
20

rate of one every three seconds over the trading hours from 9:30 am to 4:00 pm during eachof 2500 trading days (roughly 10 years of data). We then analyze the performance of thefirst exit time DV and first range time DV estimators across a range of realistic thresholdlevels vis-a-vis 2-min subsampled BV (in scenarios with jumps) and 2-min subsampled RV(in scenarios without jumps).
In the Monte Carlo experiments we seek to calibrate the threshold levels for calculating
the DV estimators to ensure that the environment is reminiscent of the setting in the section
dealing with the Dow Jones 30 stock data in Section 5. To this end, we exploit the empiricallystable relationship between the average log-spread and daily volatility (the ratio typicallyfalls in the range [0.02, 0.04] with a mean of 0.03, as shown in the second to last column oftable 2). We therefore express the thresholds in units of the mean log-spread and further
assume that the log-spread equals 0.03 times the daily volatility in our simulations. In thisway, we maintain consistency between our plots on real and simulated data as a function of
the threshold expressed in units of the mean log-spread. Furthermore, we focus on thresholds
in the range from 1 to 10 times the mean log-spread, corresponding to the range from 0.03
to 0.3 times the daily volatility. This choice is reasonable as it implies a mean duration of
the first exit times ranging from about 20 seconds for the smallest threshold to about 30
minutes for the largest threshold and mean duration of the first range times ranging fromabout 10 seconds to about 15 minutes.
The full set of data generating processes is described in detail in Appendix A.1. Inorder to gain more thorough understanding of the performance of our DV estimators, wefirst consider in detail the case of ideal sampling without microstructure noise and thenrepeat the analysis for two distinct AR(1) noise structures, namely non-persistent (or i.i.d.)noise representative for transaction prices and highly-persistent noise representative for midquotes. For each model specification, we produce signature plots of the relative bias of theintegrated variance estimates as a function of the threshold level and report in table format
the corresponding relative efficiency factor vis-a-vis 2-min subsampled BV (in scenarios withjumps) or 2-min subsampled RV (in scenarios without jumps).19 The objective is to include
relevant scenarios which mimic important qualitative features of the equity market results
within an empirically relevant setting. As such, our extensive set of relative performance
results for the simulated data serve to corroborate our interpretation of the findings for the
Dow Jones 30 stock data in Section 5.
In particular, Figures 5, 6, 7 plot the relative bias of each IV estimator, while Tables 3, 4,
19

c /IV , while the relative efficiency factor at the
The relative bias is calculated as the sample mean of IVc − IV )2 /IQ, where IV and IQ are
2-min sample frequency is calculated as the sample mean of 195(IVc is the obtained IV
the true simulated integrated variance and integrated quarticity on each day, while IVestimate on each day.

21

and 5 show the relative mean squared error of each IV estimator (in the form of an efficiency
factor at the benchmark 2-min frequency) across a broad range of considered models. The
first two rows of each table correspond to the 2-min sub-sampled RV and BV estimators.
The subsequent rows represent the first exit time and first range time based estimators forthresholds ranging from 3 to 8 log-spreads. The passage time estimators are the previous
tick versions of the estimators that are more robust to jumps in finite samples (see Section
3.3). Below we discuss these results in greater detail first for our Monte Carlo experiments
without microstructure noise and then with microstructure noise.

6.1

Simulations Without Microstructure Noise

6.1.1

Deterministic Volatility

In the constant volatility model SV0-C the annualized volatility is set to 20%, so that thedaily variance is 1.59 × 10−4 (assuming 252 trading days per year). Figure 5-A shows thatthe first exit time DV and first range time DV estimators behave very similarly in terms of
their finite sample biases as a function of the threshold level and there is virtually no bias forthreshold level larger than 3 log-spreads. The first column in Table 3 displays the relative
efficiency factor of the two DV estimators for thresholds in the no-bias range from 3 to 8
log-spreads vis-a-vis the 2-min subsampled BV and RV benchmark. Both DV estimators are
considerably more efficient than the 2-min BV and RV benchmarks over a very wide rangeof thresholds from 3 to 5 log-spreads for the first exit time and from 3 to 7 log-spreads forthe first range time. The efficiency of each DV estimator decreases as the threshold level
goes up due to the inflated variance stemming from the increasing overlap between observed
passage times starting from neighboring grid points. Importantly, for any given thresholdthe first range time estimator is consistently more efficient than the first exit time estimatorand beats the 2 minute BV benchmark in relative MSE sense by a sufficiently large factor
for all thresholds in the range 3-7 log-spreads.
The overall conclusion from Figure 5 (panel A) and Table 3 (column 1) is that there isa wide range of threshold levels, roughly 3-5 log-spreads for the first exit time and 3-7 logspreads for the first range time, offering substantial efficiency gains over 2-min subsampled
BV as a measure of the daily integrated variance.

226.1.2

Stochastic Volatility

Although the asymptotics of our DV estimators is not affected by (non-explosive) intraday
variation in volatility, it is important to investigate the extent to which there still might bea detrimental impact from time-varying or stochastic volatility on the finite sample performance documented under constant volatility. To this end, we repeat the same analysis fora number of alternative data generating processes involving stochastic volatility: SV1A-C(one-factor affine), SV1L-C (one-factor log), and SV2A-C (two-factor affine). The models
are described in detail in Appendix A.1 and they are all calibrated to obtain the same average annualized volatility of 20%. We present the results for models without the "leverage"
effect (i.e. correlation between the process driving stochastic volatility and the Brownian
motion driving returns) consistent with the theoretical results derived in the Appendix. We
have, however, confirmed that simulating models with the leverage effect does not change
our results or conclusions in any significant manner.
Our main finding is that there is very little difference between the observed relative
performance of DV versus BV under each stochastic volatility model in comparison to theconstant volatility case. For example, representative for all three models, Figure 5-B formodel SV2A-C shows that both DV estimators are virtually unbiased for threshold level
larger than 3 log-spreads exactly as in the constant volatility case (c.f. figure 5-A ). Likewise,
the first four columns of Table 3 show that the relative MSE gains of DV versus 2-min
subsampled DV for each stochastic volatility model are practically about the same as those
documented under constant volatility.
The finding that our results are robust to time-varying volatility should not be surprising
given that (for small enough thresholds) volatility is locally constant to a good first order ofapproximation, which is the main requirement for our DV estimators to work well in practice.
Thus, the range of thresholds for which the DV estimators achieve superior performance
relative to the 2-min subsampled BV remains 3-5 log-spreads (first exit time) and 3-7 logspreads (first range time).

6.1.3

Stochastic Volatility with U-Shaped Intraday Pattern and Jumps

It is well known that intraday stock returns exhibit a pronounced U-shaped pattern of
intraday volatility along with other activity-related variables such as trading volume, numberof transactions, and quoted spread. On average, the variance in the first hour of trading
is often three times larger than the variance in the middle of the day and peaks again toa somewhat lower level in the last hour of trading. Therefore, we augment our baseline
stochastic volatility specifications with a deterministic time-of-the-day volatility multiplier
23

having a U-shaped pattern. In particular, our time-of-the-day multiplier takes an additive
form of exponentials popular in the market microstructure literature (see Appendix A.1 fordetails of the implementation).
Likewise, to accommodate naturally occurring infrequent instantaneous price revisions
that can lead to path discontinuities (e.g. upon the arrival of big news), we add also a finite
activity jump component to our baseline stochastic volatility specifications. This allows us
to study the performance of our DV estimators on "jump" days subject also to a realistic
strong U-shaped intraday volatility pattern that interferes with the jumps by generating large
enough pure diffusive increments at the start and end of the day. In particular, we simulate
one jump on each day with a 25% average contribution to the daily diffusive integrated
variance and carry out the same relative performance analysis as above.
An important feature of the first exit time and first range time DV estimators establishedin the theory section is their asymptotic robustness to jumps due to the inherent truncation
via the chosen threshold of arbitrary large log-price increments (the smaller the threshold,the higher the degree of jump-robustness). In particular, in section 3.3 we have motivatedthe "previous tick" method for calculating the passage times involved in our DV estimatorsthat achieves nearly perfect immunity to the presence of a single large jump on each day both
asymptotically and in finite samples. Therefore we focus our analysis on these asymptoticallyequivalent "robust" versions of our first exit time DV and first range time DV estimators.
Figure 5-C for model SV2A-UJ (representative for all four models with jumps) showsthat, indeed, the combined presence of jumps and a strong U-shape volatility pattern hasvery little impact on the relative bias of our DV estimators. While the 2-min subsampled BV
exhibits a noticeable upward bias due to finite-sample jump distortion20 , both DV estimators
remain unbiased for threshold levels 3 to 6 log-spreads and less biased than BV for thresholds
up to 8 log-spreads. Importantly, as revealed by the last four columns of Table 3, this leadsalso to further improvement of the relative MSE factors of the DV estimators vis-a-vis 2min subsampled BV. Thus, the overall conclusion from Figure 5 and Table 3 is that as a
jump-robust integrated variance estimator DV for thresholds 3-6 log-spreads outperforms the
2-min subsampled BV benchmark by an even wider margin on days with jumps compared
to days without jumps.
20

It is well known that the BV estimator does not fully eliminate the impact of jumps and in fact has an
upward bias of roughly 5.4% in our simulations.

24

6.2

Simulations With Microstructure Noise

Next we further extend our simulation setup by producing noisy observations based on theAR(1) microstructure noise specification of equation 14 introduced in Section 3.4. Our goal
is to study the effect of different noise levels (noise-to-signal ratio λ) and different noise
persistence (first order autocorrelation ρ) on the relative efficiency of our DV estimators.
We focus our analysis on four distinct cases:(A) Non-persistent noise (ρ = 0) at moderate level (λ = 0.25)(B) Persistent noise (ρ = 0.99) at moderate level (λ = 0.25)(C) Non-persistent noise (ρ = 0) at high level (λ = 1.00)(D) Persistent noise (ρ = 0.99) at high level (λ = 1.00)
Based on our findings in Section 3.4 with regard to the degree of distortion of individual passage time measurements, we should expect a dramatic deterioration of the relative
efficiency of the first exit time DV and, even more so, the first range time DV only in case(C) of non-persistent noise at high level. This is indeed our main finding when we repeat
our previous analysis under the considered noise specifications first for the four SV-C models
without jumps and U-shaped volatility pattern and then for the four SV-UJ models with
U-shaped pattern and jumps.

6.2.1

Noise Impact on DV Under Stochastic Volatility

Figure 6 shows that moderate noise at any persistence (first row panels A and B) or persistentnoise at any level (second column panels B and D) have a negligible impact on the relative
bias of our DV estimators compared to the case without noise (Figure 5, panel B). By
contrast, non-persistent noise at high level (Panel C) appears to be quite detrimental. As
predicted by our analysis in Section 3.4, now there is an upward bias of our DV estimators
across the full range of empirically relevant thresholds. Likewise, comparing the MSE factors
with noise reported in Table 4 to their corresponding values without noise (Table 3) showsthat the relative efficiency of DV versus 2-min subsampled BV and RV is largely unaffectedby moderate noise at any persistence (columns with λ = 0.25) or persistent noise at any level(columns with ρ = 0). However, non-persistent noise at high level (columns with λ = 1.00)
leads to severe deterioration of the relative efficiency of DV, making it somewhat worse than
2-min subsampled BV, which unlike DV remains nearly unbiased also in this case.
The overall conclusion to be drawn from this analysis is that the presence of microstructure noise should not generally be a major concern for our duration-based approach, as longas the noise-to-signal ratio is moderate or the noise is sufficiently persistent. The special
case of non-persistent noise at a detrimentally high level is easy to detect by inspecting
25

whether the DV signature plots as a function of the threshold level are upward sloping as
the thresholds go down. Clearly, this is not the case for our sample of mid-quotes for the
Dow Jones 30 stocks (see signature plots in Appendix C and D), further corroborating theinterpretation of our empirical findings in Section 5.

6.2.2

Noise Impact on DV Under Stochastic Volatility with U-Shaped Intraday
Pattern and Jumps

As a final check of the relative performance of our DV estimators, we repeat the above
analysis with microstructure noise for our SV-UJ models featuring a pronounced U-shape
intraday pattern and jumps. We arrive at exactly the same conclusion for such "jump" days
after comparing Figure 7 with the corresponding signature plots without noise (Figure 5,
panel B), as well as Table 5 with the corresponding efficiency factors without noise (Table 3).
Leaving aside the special case of non-persistent noise at a detrimentally high level, both thefirst exit time DV and the first range time DV significantly outperform 2-min subsampled
BV as jump-robust estimators of the integrated variance over a wide range of thresholds
from 3 to 6 log-spreads.

7

Conclusion

In this paper we propose a new methodology for integrated variance estimation based ona localization argument and the theory of Brownian passage times. We exploit suitably
scaled reciprocal passage times for a given threshold as the dual analog to squared/absolute
returns and ranges for a given time unit and introduce novel duration-based estimators
DV as natural counterparts to commonly used RV estimators. Our DV estimators are
asymptotically superior to subsampled RV and BV estimators in the infeasible case with
continuous observations and no jumps. Moreover, the proposed duration-based estimators
perform very well in finite samples in both Monte Carlo experiments and an empirical analysisof the stocks in the Dow Jones 30 index. The rationale behind this performance is quite
intuitive: as long as the threshold is chosen large enough, the market microstructure noise
distortion of the observed passage times will be small. On the other hand, as long as thethreshold is small enough, individual large jumps will automatically be truncated and thus
will only affect the local volatility estimate at a single grid point. As we have demonstrated
in the Monte Carlo experiments, this jump robustness can be further strengthened by using
the previous tick versions of the DV estimators. This leaves us with an intermediate range
26

of thresholds ranging from roughly 3 to 6 log-spreads for which our estimators have goodrobustness properties with respect to both jumps and microstructure noise.21 Importantly,
the estimates are relatively stable with respect to the choice of threshold within this range andpractical implementation will therefore not need to involve any delicate first step estimationof an ‘optimal’ threshold.
Overall, the very encouraging results for the Dow Jones 30 stocks and a broad range ofMonte Carlo experiments present supportive evidence that the asymptotic efficiency of theintroduced realized duration-based volatility estimators can be exploited in practice. At thesame time, certain aspects of the behavior of our DV estimators on real data call for further
investigation. In particular, future research is needed to shed more light on what type of
assets and markets offer the most suitable high-frequency price-time data for duration-based
volatility estimation.
The duration based approach outlined in this paper for robust integrated variance estimation applies more generally to the robust estimation of other power variations, of which
the integrated quarticity is of particular interest in finance applications. The estimation ofthese quantities follows immediately from the results in Appendix A.2, which outlines thebasic duality between reciprocal passage times and certain magnitude-based functionals of
the Brownian motion. Another intriguing avenue of future research is multivariate extensions of duration based techniques to the estimation of covariances. We leave these and othernatural extensions for future work.

21

Although the asymptotic properties of the DV estimators have been derived assuming “no leverage”,
extensive Monte Carlo experiments have confirmed that leverage does not affect the performance of the DV
estimators.

27

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30

A

Technical Appendix

A.1

Monte Carlo Experiment

The models considered in our Monte Carlo experiments are SV0 (constant deterministic volatility),
SV1A (one-factor affine stochastic volatility), SV1L (one-factor log stochastic volatility), SV2A
(two-factor affine stochastic volatility). In each model, the price process follows a driftless Brownian
motion with instantaneous volatility σ(t):dp(t) = σ(t) dW1 (t)We analyze separately core model specifications SV-C and extended specifications SV-UJ incorporating a U-shaped intraday volatility pattern and jumps.

A.1.1

Core Stochastic Volatility Models SV-C

The deterministic and one-factor volatility models are described by
SV0-C
SV1A-C
SV1L-C:::σ(t)dσ 2 (t)d log σ 2 (t)

=
=
=

θ 

η θ − σ 2 (t) dt+ νσ(t) dW2 (t)η θ − log σ 2 (t) dt + ν dW2 (t)with the leverage effect in the one-factor models captured by the instantaneous correlationρ = corr(dW1 (t), dW2 (t))
The parameters for the one factor models are calibrated as in Andersen, Benzoni, and Lund
(2002).
The two-factor stochastic volatility model is described by
SV2A-C:σ 2 (t)dσ 21 (t)dσ 22 (t)===σ 21 (t)+ σ 22 (t) η 1 θ1 − σ 21 (t) dt + ν 1 σ 1 (t) dW21 (t)η 2 θ2 − σ 22 (t) dt + ν 2 σ 2 (t) dW22 (t)with W21 , W22 independent and the leverage effect captured by the instantaneous correlations
ρ1 = corr(dW1 (t), dW21 (t))
ρ2 = corr(dW1 (t), dW22 (t))
The two factor model parameters are calibrated as in Andersen, Bollerslev, and Meddahi (2005).
Across all model specifications, the unconditional IV of each day is calibrated to θ = θ1 + θ2 =0.000159 corresponding to an annualized volatility of 20% (assuming 252 trading days per year).
This roughly matches the average level of volatility observed in our DJ 30 sample between January2005 and May 2007.

A.1.2

Models SV-UJ with Intraday U-Shaped Volatility Pattern and Jumps

We extend each of the above four core models to incorporate both a deterministic intraday U-shaped
volatility pattern and price jumps, resulting in the models SV0-UJ, SV1A-UJ, SV1L-UJ, andSV2A-UJ.

31

Following Hasbrouck (1999) we model the diurnal volatility U-shape as the sum of two exponentials:ihE [σ(t)] = C + A e−at + e−b(1−t) , t ∈ [0; 1]
where the constants are calibrated to produce a strong asymmetric U-shape with variance at the
open (t = 0) more than 3 times the midday variance (t = 1/2) and variance at the close about 1.5
times the midday variance.
Price jumps are introduced by extending the log-price process as
dp(t) = σ(t) dW1 (t) + dJt
where the Poisson jump process (Jt ) is assumed independent of (W1 , W2 ). In order to stress test
our estimator, we calibrate the process to match “jump days” on which a single Gaussian jumpaccounts for an average increase of 25% in realized volatility (i.e. the jump contribution JV is 25%
of IV or, equivalently, 20% of QV = IV + JV).

A.2

Duality Results

Proposition A.1 Define the following standard Brownian functionals
Ht = sup Bθ ,Mt = sup |Bθ | ,θ∈[0;t]θ∈[0;t]Rt = sup Bθ − inf Bθθ∈[0;t]θ∈[0;t]and let (for h > 0)RT
τ HT
= inf{t|Ht = h}, τ EThh = inf{t|Mt = h}, τ h = inf{t|Rt = h}be the first range time, first exit time and first hitting time respectively, then we have the following
identities in distribution
1

D

H1 =

D

 , M1 =HT 1/2

τ1

1

1

D

 , R1 =ET 1/2

τ RT
1

τ1

1/2 ,

Proof. The scaling property of Brownian motionD

Bt/h2 =

1Bthimplies that :(1
Pr {Ht < h} = Pr sup Bt < 10≤t≤1 h
n

o

)

(

)

= Pr

= Pr τ HT> h−2 = Pr1

sup0≤t≤h−21 τ HT 1/2
1

Bt < 1<h

The corresponding results for Mt and Rt are obtained analogously.
−p/2 , [τ ET ]−p/2 , [τ RT ]−p/2
It follows that the passage time estimators of σ p based on [τ HT1 ]11are the duals to the corresponding magnitude-based estimators using H1p , M1p , R1p . The powers ofparticular interest for finance applications are p = 1, 2, 4.

32

A.3

Passage time results

Lemma A.2 Let τ h be the first exit time, the first hitting time or the first range time as given in(6) for a driftless Brownian motion with constant volatility σ 2 , then
2
D h
τ̃ 1
σ2

τh =

where τ̃ 1 is the corresponding passage time of a standard Brownian motion with threshold h = 1.
Proof. This is clear from the moment generating functions given in Table 1:


"

E e

2

−α τ̃ 1 h2

#

σ



"

=E e

2

− α h2
σ



τ̃ 1

#

= Lτ h (α)

Proposition A.3 If {τ i,h } is an i.i.d sample of N first passage times from the interval [−h; h] ofa driftless Brownian motion with B0 = 0 and constant volatility σ, then the expected value of thevolatility estimator based on the first exit time moment is given by
"

E

#

h2

(N )

1 PN
i=1 τ i,h
N

= µ1

σ2

where
R∞

1
√
N dλ
0
cosh( 2λ)][R(N )N 0∞ 
√1 2N dλ
µ1 =

cosh(
λ/2)



1
N







N

(first exit time)(first range time)(first hitting time)

Moreover, asymptotically as N → ∞
(
(N )
µ1 =

2
1 + 3N+ O( N12 )1
1 + 3N+ O( N12 )(first exit time)(first range time)

Proof. By independence the moment generating function of the convolution τ 1,h + · · · + τ N,h isgiven by [Lτ h (α)]N . We therefore have
"

E

h2

#

= h2 N1 PN

N

i=1 τ iZ ∞0[Lτ h (α)]N dα
2

The result then follows from the change of variables λ = αh
. The asymptotic approximation can
σ2
be derived for the first exit time and the first range time via a mean value expansion:
"

E

h2
1 PN
i=1 τ i
N

#

"

=E

#

h2
P
Eτ + N1 N
i=1 (τ i − Eτ )33

= σ2 × E 



1



1+

1
N

PN

(τ −Eτ )
i=1 i
Eτ




A second order Taylor expansion of 1/(1 + x) around 1 yields:≈ σ 2 × E 1 −"2

≈ σ × 1+1 PN
i=1 (τ i − Eτ )
N

+

1 PN
i=1 (τ i − Eτ )
N

Eτ

E N12

Eτ

PN

i=1 (τ i − Eτ )
2

2

#
2

+ O(1/N ) = σ

(Eτ )



!2

− · · ·

"
2

#

1 Vτ
1+
+ O(1/N 2 )N (Eτ )2

from which the result follows by calculation of the first two moments of the relevant passage timeusing the moment generating function.
Corollary A.4 The expectation of the reciprocal passage time satisfies
"

E

h2





2C σ 2(4 log 2) σ 2≡ µ1 σ 2 =


σ2

#

τh

(first exit time)(first range time)(first hitting time)

Proof. Apply Proposition A.3 with N = 1.
Proposition A.5 If {τ i,h } is an i.i.d sample of N first exit times from the interval [−h; h] of adriftless Brownian motion with B0 = 0 and constant volatility σ, then the second moment of thevolatility estimator based on the first exit time moment is given by




h4



E 

1 PN
i=1 τ i,hN(N ) 42  = µ2 σ

where

(N )

µ2







=

N 2 0∞ Λ∞1√
N dλ dΛcosh( 2λ)][RR
N 2 0∞ Λ∞ √1 2N dλ dΛRRcosh( λ/2)
3
N2






(first exit time)(first range time)(first hitting time)

Proof. By independence the moment generating function of the convolution τ 1,h + · · · + τ N,h isgiven by [Lτ h (α)]N . We therefore haveE  P1N
i=1 τ i,hZ ∞Z ∞
2  =

0A[Lτ h (α)]N dα dA

Corollary A.6 The second moment of the reciprocal passage time satisfies
"

E

#


4

 6 β(4) σh4
9 ζ(3) σ 4≡ µ2 σ 4 =

τ 2h

3 σ4

(first exit time)(first range time)(first hitting time)

Proof. Apply Proposition A.5 with N = 1.

34

2Corollary A.7 Let σ̂ 2i = µ1 τ hh(ti ) be the volatility estimate based on a single passage time obser1
vation τ h (ti ), then E[σ̂ 2i ] = σ 2 andV[σ̂ 2i ] =



µ2
−1
µ21



σ4 =

 

6 β(4)
−1σ4  4β(2)2 4






9 ζ(3) σ− 1 σ4(4log 2)2 
3− 1 σ422≈ 0.7681 σ 4(first exit time)≈ 0.4073 σ 4(first range time)

2.000 σ 4

(first hitting time)≈

Proof. Apply Corollary A.4 and A.6.

A.4

Discretization bias correction

We state the multiplicative discretization bias corrections based on the following asymptotic results
due to Asmussen, Glynn, and Pitman (1995).
Lemma A.8 Let Bt be a driftless Brownian motion on [0; 1] with volatility σ 2 and let Xt = B [tN ]Nbe the embedded (discretely observed) random walk observed at N points. Define M = sup |Bt |,t∈[0;1]

R = sup Bt − inf Bt , MN = sup |Xt |, RN = sup Xt − inf Xt . Thent∈[0;1]t∈[0;1]t∈[0;1]t∈[0;1]t∈[0;1]ζ(1/2) σ1√ + o( √ )E[MN ] = E[M ] + √2πNNζ(1/2) σ1√ + o( √ )E[RN ] = E[R] + 2 √
2π
N
N
r
2
π ζ(1/2) σ
1
2√√ + o( √ )E[MN] = E[M 2 ] + 2
2
2π
N
N
r
2
8
ζ(1/2)σ1
2√√ + o( √ )E[RN] = E[R2 ] + 4
π
2π
N
N
√
where ζ(1/2)/ 2π ≈ −0.5826.
Proof. Follows by specializing Proposition 3 in Asmussen, Glynn, and Pitman (1995) for MN andRN .
Corollary A.9 The following relationships hold approximately for the absolute maximum M andthe range R of a Brownian motion and the absolute maximum MN and the range RN of a random
walk in the same interval, as defined above:
E M2

h

i

2
≈ fM (N ) × E MN

hi2
≈ fR (N ) × E RN

E R2

h

h

i

i

where
1
1
ζ(1/2) √2C
N
2
1fR (N ) = 1 −ζ(1/2) √π ln 2NfM (N ) = 1 −35and C is Catalan’s constant.
Proof.
Obtained by rearranging
the last two expressions and substituting the known values
 2
 2
2
E R = 4 ln 2 σ and E M = 2C σ 2 .

A.5

Constant Volatility

For simplicity we here limit attention to the uni-directional IV estimation. The derivation for thebi-directional case follows analogously. Let 0 ≡ t0 , t1 , . . . , tN ≡ 1 be a set of grid points partitioning
the interval [0; 1] and let ∆i = ti+1 − ti be the mesh size. The passage time based IV estimators
are of the formd N,h =
DV

N
−1
X

µ−1
1

i=0

h2
∆i
τ h (ti )(20)

where µ1 is the constant given in Table 1 for the first hitting time, first exit time and first range
d N,h ] = σ 2 = IV so that the estimator
time based estimators respectively. It is then clear that E[DVis unbiased.
d N,h is given by
The variance of the estimator DV
d N,h ] =V ar[DVN−1X

V ar[µ−1
1

i=0

X
h2
4
∆i ∆j Cov
] ∆2i + µ−2
1 hτ h (ti )
i6=j

1
1
,
τ h (ti ) τ h (tj )

!

(21)

d N,h ] = O (1/N )We would like to show that, if h shrinks to zero at an appropriate rate, then V ar[DVand, moreover, that the local estimates of volatility at distinct grid points are asymptoticallyindependent (i.e the second term in (21) is o(1/N )).
The first term isN−1
X

V ar[µ−11i=0h2] ∆2i =
τ h (ti )



N
−1
X
µ2
4
∆2i
−
1
σ
µ21
i=0



A sufficient condition for this term to be O(1/N ) is therefore that the mesh size satisfies max ∆i =0≤i≤N

O(1/N ).
Next we consider the second term involving the covariance between reciprocal passage times
measured from distinct grid points ti < tj with ∆i,j = tj − ti (to simplify notation we writeτ h (ti ) = τ i and τ h (tj ) = τ j ) :
Cov

1 1
,
τi τj

!

1 1
=E
τi τj

!

1
1
−E
E
τi
τj




!

For the cross product of reciprocal waiting times we have22 :
1 1
E
τi τj
22

!

!

=E

1 1
τ i < ∆i,j Pr {τ i < ∆i,j } + E
τi τjNote that τ j is not independent of τ i . We exploit here that E!

1 1
τ i ≥ ∆i,j Pr {τ i ≥ ∆i,j }
τi τj


1
τj


 
τ i ≥ ∆i,j = E τ1j since the event{τ i ≥ ∆i,j }, as of time tj , only reveals information about the initial value at tj of the process but nothing
about the subsequent passage time τ j .

36



≤E

1
1
τ i < ∆i,j E
τi
τj

!

!

!



1
1
≤E
E
τi
τj




1
+E
τj

Pr {τ i < ∆i,j } + E

1
τ i ≥ ∆i,jτj!

1
Pr {τ i ≥ ∆i,j }∆i,j1
Pr {τ i ≥ ∆i,j }∆i,jThis leads to the estimate
1 1
,
τi τj

Cov

!

1
≤E
τj

!

1
σ2 1
Pr {τ i ≥ ∆i,j } = µ1 2
Pr {τ i ≥ ∆i,j }∆i,jh ∆i,jThe probability Pr {τ i ≥ ∆i,j } can be bounded by the hitting time probability (Table 1) for which:
Z ∞

2
h
2
− h 2
2tσ dt = − √
√
e
3/2
π
∆i,j σ 2πtPr {τ i ≥ ∆i,j } =

2
= −√
π

Z 0
σ

√

h
2∆i,j

"

h
= Erf pσ 2∆i,j22e−z dz = √
π

#

r

=

Z

σ

Z ∞

s

2

e

− h 2
2tσ

d

∆i,j

√h

h2
2tσ 2

2

2∆i,j

e−z dz





0

2
1
h3
p
h + O  3/2  ,π σ ∆i,j∆i,jwhere the last equality follows from the Taylor series expansion of the Erf function,
∞
2 X
(−1)n
Erf[x] = √x2n+1 .
π n=0 (2n + 1)n!

Thus, for small h,
1 1
,
τi τj

Cov

!

r

≤





2 σ 1
h
µ1
+ O  5/2 
3/2
π h∆
∆
i,j

i,j

and
4
µ−2
1 h

X

∆i ∆j Cov

i6=j

1 1
,
τi τj

!

r

≤

2 −1 3 X ∆i ∆j X ∆i ∆j  5 
µ σh
+
O h
3/2
5/2
π 1
i6=j ∆
i6=j ∆
i,j

(22)

i,j

Under the (slightly) strengthened assumption that ∀i : ∆i = O(1/N ), we have
X ∆i ∆j
3/2

i6=j

∆i,j

X ∆i ∆j
5/2

i6=j

∆i,j

√ 
1
1 X
=
O
N
≈√
N i6=j |j − i|3/2

≈

√
N



1
3/2
=
O
N
|j − i|5/2
i6=jX

so that (22) will be of order o(1/N ) if h = o(N −1/2 ). Thus the local volatility estimates corre-37

sponding to distinct grid points are asymptotically uncorrelated if the squared thresholds shrink
faster than the mesh size:
2

d N,h = N −1 µ−1 h ∆i as the IV estimator based on passage time
Proposition A.10 Define DV
1 τ h (ti )i=0durations (either the first hitting time, the first exit time or the first range time), where N denotes
the number of grid points and h denotes the threshold level. If the mesh size is ∆i = O(1/N ) andthe threshold is chosen to be h = o(N −1/2 ), then, as N → ∞P√Dd N,h − IV ) −→ N (0, (µ /µ2 − 1) σ 4 )
N (DV
2
1

where the constants µ1 , µ2 are given in Table 1.

A.6

Stochastic volatility

Assumption (SV) Assume that the volatility process is independent of the Brownian motion andthat∃κ > 0 : sup |σ 2s − σ 2t | < κ|s − t|
s,t

 

(SV1 )

1
N(SV2 )

inf σ 2t ≥ ε > 0(SV3 )

h = o(N −1/2 )(SV4 )∀i : ∆i = O
t

The IV estimation error can be decomposed as
d N,h =IV − DVZ 1
0

σ 2u du −N−1Xσ̂ 2ti ∆i =

Z 1
0

i=0

|

σ 2u du −

N
−1
X

N
−1 
X

σ 2ti ∆i +i=0



σ 2ti − σ̂ 2ti ∆i

(23)

i=0

{z

Riemann approximation

}

|

{z

}

sum of local estimation errors
2hwhere the local estimator of volatility at the grid point ti is given by σ̂ 2ti = µ−1
1 τ h (ti ) and define thepassage time with stochastic volatility as in (6) where Bt =The first term in (23) is easily dealt with:Rt

0 σ u dWu

Lemma A.11 The Riemann approximation error is O(N −1 ) as the mesh size, max ∆i → 0.
i

Proof. This follows readily from the Lipschitz assumption (SV1 ) and mesh size assumption (SV2 ),Z 10

σ 2u du −

N
−1
X
i=0

σ 2ti ∆i =N
−1 Z ∆iX
i=00(σ 2ti +u − σ 2ti ) du ≤N
−1 Z ∆i
X
i=0

0

−1
1
1 NX
∆2 = Oκu du = κ
2 i=0 i
N





(24)

This implies that, under Assumption SV, the Riemann approximation error will not matter
asymptotically.

38

Next we turn to the sum of local estimation errors which we shall first consider term by term.
We will need the following lemmas and definitions.
Lemma A.12 If Assumption
SV is satisfied, then, conditional on the value of IV = 01 σ 2u du, thePN −1  2
terms in the sum i=0 σ ti − σ̂ 2ti ∆i are asymptotically uncorrelated in the sense that
R

X





∆i ∆j Cov σ̂ 2ti , σ̂ 2tj = o(N −1 )i6=jProof. From the Lipschitz assumption, it follows that sup σ 2t ≤ σ̄ 2max ≡ IV + 12 κ must hold.
t∈[0;1]

Borrowing from the arguments in the constant volatility case we then have
Cov

h2
h2
,
τ h (ti ) τ h (tj )

!

!

1
≤h E
τ h (tj )
4

σ̄ 2
1
1
Pr {τ h (ti ) ≥ ∆i,j } ≤ h4 µ1 max
Pr {τ h (ti ) ≥ ∆i,j }2∆i,jh ∆i,jand the asymptotic uncorrelatedness follows from the conditions of Assumption SV (in particular(SV2 ) − (SV4 )) which imply that the right hand side is o(1/N ).
We are therefore justified in focusing on the asymptotic behavior of the local estimation error
at each grid point. We proceed by introducing the time changed process:Definition
A.13 For each ti , let B ∗ be a Brownian motion with B0∗ = 0 and define the time change
Rt 2
Tt = 0 σ ti +u du and the stopping times (depending on which estimator is considered)τ ∗h (ti ) =


n

inf
t|






o

BT∗t > h(first exit time)t | max B ∗ − min BT∗t > h

inf

Tt

0≤u≤t on∗ inf t | B > h(first range time)0≤u≤t(first hitting time)Ttτ̃ ∗h (ti ) =
inf t |







∗
Btσ
2
t



>h

t | max B ∗inf(first exit time)

i

2

ti
0≤u≤t tσ





∗

 inf t | Btσ2 > h∗− min Btσ
2 > h
0≤u≤t

(first range time)ti(first hitting time)

ti

∗
τ max
= inf { t | Btε> h}hLemma A.14 We have the following relationships (for a given passage time estimator)Dτ h (ti ) = τ ∗h (ti )(i)0 ≤ τ ∗h (ti ) ≤ τ max= Op (h)
h

(ii)

E

h

1
τ̃ ∗h (ti )

i

σ 2t
i
1 h2

=µ

(iii)

1
1
τ̃ ∗h (ti ) − τ ∗h (ti ) = Op (1)

(iv)

39

Proof. (i) follows directly from a standard time change argument. (ii) is immediate from thedefinition and the fact that τ max
can be seen as the first hitting time of a Brownian motion withhconstant volatility ε. Similarly (iii) follows from the interpretation of τ̃ ∗h (ti ) as the first passage
time of a constant volatility Brownian motion. To get (iv) we exploit the pathwise relationshipσ 2ti τ̃ ∗h (ti ) =R τ ∗h (ti ) 2
σti +u du and the Lipschitz assumption (SV1 ) :0

σ 2ti τ ∗h − 21 κτ ∗2h ≤R τ ∗h 2σ0

ti +u du≤ σ 2ti τ ∗h + 21 κτ ∗2h(25)

Thus we have the bounds on τ ∗h in terms of τ̃ ∗h :−σ 2ti +q

σ 4ti + 2κσ 2ti τ̃ ∗hκ≤ τ ∗h ≤σ 2ti −

q

σ 4ti − 2κσ 2ti τ̃ ∗hκfrom which it follows thatq− ∗ =τ̃ h →0 τ̃ ∗τhh

−σ 2ti + σ 4ti + 2κσ 2ti τ̃ ∗h − κτ̃ ∗hκτ ∗h − τ̃ ∗hq
 =− 2

lim
inf
= lim
∗
∗ →0
τ̃ ∗h →0 τ̃ ∗
τ̃
∗
2
∗
4
2
τ
2σ
h
ti
h h
τ̃ h −σ t + σ t + 2κσ t τ̃ h11
lim sup ∗ − ∗ =∗τhτ̃ h →0 τ̃ h

σ 2ti − σ 4ti − 2κσ 2ti τ̃ ∗h − κτ̃ ∗h
τ ∗h − τ̃ ∗hκq =+ 2
lim sup ∗ ∗ = lim∗∗
τ̃ h →0 τ̃ ∗ σ 2 −τ̃ h τ h
2σ tiτ̃ h →0
σ 4t − 2κσ 2t τ̃ ∗h
t
h

lim
inf
∗

1

1

i

i

i

q

i

i

i

Finally, since τ̃ ∗h = Op (h), we have that τ̃1∗ − τ1∗ = Op (1) as h → 0.
h

h

2

2

D

−1 h
−1 h
2
∗2
∗2
∗2
∗2
2
Lemma A.15 Let σ̂ ∗2ti = µ1 τ ∗ (ti ) and σ̃ ti = µ1 τ̃ ∗ (ti ) then σ̂ ti = σ̂ ti and σ̂ ti − σ̃ ti = Op (h ).
hhProof. Follows directly from Lemma A.14.

We are now ready to present our main result for the passage time estimator under stochastic
volatility.
d N,h denote the passage time based IV estimator (either the first hitting
Proposition A.16 Let DV
time, the first exit time or the first range time) where N denotes the number of grid points. Under
the conditions of Assumption SV we have as N → ∞√d N − IV ) IV
N (DV

D

−→ N (0, (µ2 /µ21 − 1)Z1

σ 4u du)0where ν = (µ2 /µ21 − 1) is given in (16).
Proof. First note that from Lemma A.15 and Assumption SV, we haveD(σ 2ti − σ̂ 2ti ) ∆i = (σ 2ti − σ̂ ∗2ti ) ∆i=(σ 2t − σ̃ ∗2t ) ∆i}| i {z i“constant” volatility term+∗2(σ̃ ∗2t − σ̂ t ) ∆i

| i {z i}
stochastic volatility term

40

(26)





i

h

2
2
∗2
4 2
where E (σ 2ti − σ̃ ∗2ti ) ∆i = 0 and V ar (σ ti − σ̃ ti ) ∆i = (µ2 /µ1 −1)σ ti ∆i . The “stochastic volatil∗2−4 ) and thus will not play a role asymptotically:ity” term is (σ̃ ∗2ti − σ̂ ti ) ∆i = Op (N

√
N

N
−1 
X



D √

σ 2ti − σ̂ 2ti ∆i =

N

i=0

N
−1
X

−2
(σ 2ti − σ̃ ∗2)ti ) ∆i + o(Ni=0Conditional on IV, Lemma A.12 guarantees that the terms are asymptotically uncorrelated so that(taking ∆i = 1/N ) :
"

V ar

√
N

N
−1
X

(σ 2ti − σ̃ ∗2ti ) ∆i#= (µ2 /µ21 − 1)

i=0

N
−1
X

σ 4ti ∆ii=0

which leads to
√
N

N
−1
X

D
2
(σ 2ti − σ̃ ∗2ti ) ∆i −→ N (0, (µ2 /µ1 − 1)

i=0

Z1
0

41

σ 4u du)

h2

42

1
2



1
2

Bidirectional
Estimator

Unidirectional
Variance

Bidirectional
Variance

× σ4 = 1 × σ4

× σ4 = 2 × σ4

µ2
−1
µ21



2
h2
+ µ τh− (t)µ1 τ +
(t)
1 h
h

µ2
−1
µ21



h2
µ1 τ −(t)hh2µ1 τ +(t)h

Unidirectional
Estimatorsorµ1 = 1µ2 = 3

2

Scaling
Factorsim1 = E (τ h )−1 = 1 × σh2hi4m2 = E (τ h )−2 = 3 × σh4

Reciprocal
Moments

h

−
h
2tσ 2
pτ h (t) = σ√2πt
3/2 e

√
2α

PDF

h

Lτ h (α) = e− σ

θ>0τ−h (t) = min{Wt−θ − Wt = h}Laplace
Transform

Definition

∞
P
k=−∞

2 2

i

σ 2πt3/2

e

2

1
2



1
2



or

h2
µ1 τ −(t)h

× σ 4 ≈ 0.38405 × σ 4

× σ 4 ≈ 0.76809 × σ 4

µ2
−1
µ21



2
h2
+ µ τh− (t)µ1 τ +
(t)
1 h
h

µ2
−1
µ21



h2
µ1 τ +
(t)
h

1
2



1
2



or

h2
µ1 τ −(t)h

× σ 4 ≈ 0.20367 × σ 4

× σ 4 ≈ 0.40733 × σ 4

µ2
−1
µ21



2
h2
+ µ τh− (t)µ1 τ +
(t)
1 h
h

µ2
−1
µ21



h2
µ1 τ +(t)hµ1 = 4 ln 2 ≈ 2.77259µ2 = 9ζ(3) ≈ 10.81851

h

k=1

∞
k−1 k 2 − k2 h2P4h(−1)√2tσ 2µ1 = 2β(2) ≈ 1.83193µ2 = 6β(4) ≈ 5.93367

2pτ h (t) =[t−θ,t][t,t+θ]m1 = E (τ h )−1 = 4 ln 2 × σh2hi4m2 = E (τ h )−2 = 9ζ(3) × σh4i(1+4k) h2h(1+4k)√
e− 2tσ2σ 2πt3/2[t−θ,t]Lτ h (α) = cosh2 1h √ α(σ 2 )

θ>0[t,t+θ]τ−h (t) = min{ sup Ws − inf Ws = h}

θ>0τ+h (t) = min{ sup Ws − inf Ws = h}First Range Time for Threshold h > 0

m1 = E (τ h )−1 = 2β(2) × σh2hi4m2 = E (τ h )−2 = 6β(4) × σh4hpτ h (t) =Lτ h (α) = cosh h1 √2α
(σ
)

θ>0

τ−
h (t) = min{|Wt−θ − Wt | = h}

θ>0τ+h (t) = min{|Wt+θ − Wt | = h}τ+h (t) = min{Wt+θ − Wt = h}
θ>0First Exit Time for Threshold h > 0

First Hitting Time for Threshold h 6= 0

Table 1: Summary of the properties of the first hitting time, the first exit time and the first range time estimators.

B
Tables and Figures

Table 2: Descriptive statistics for the 33 stocks that were part of the Dow Jones 30 index
between January 1, 2005 and May 31, 2007.

Symbol
AA
AIG
AXP
BA
BAC
C
CAT
CVX
DD
DIS
EK
GE
GM
HD
HPQ
IBM
INTC
IP
JNJ
JPM
KO
MCD
MMM
MO
MRK
MSFT
PFE
PG
T
UTX
VZ
WMT
XOM
Mean ALL:Median ALL:
Max ALL:
Min ALL:

Average
Price($)

Average2-min RV
(x10-4)

Average2-min BV

Average
Log Spread(x10-4)(x10-4)

30
64
55
74
48
49
71
63
45
29
26
35
30
39
32
86
22
34
64
42
44
36
78
74
37
27
26
58
27
68
35
48
64

2.22
1.00
0.81
1.26
0.68
0.77
1.62
1.64
1.19
1.08
2.19
0.63
3.70
1.35
1.61
0.84
1.67
1.36
0.52
0.88
0.59
1.22
0.91
1.09
1.58
0.94
1.12
0.72
1.08
1.07
0.98
0.91
1.39

2.10
0.95
0.76
1.19
0.64
0.73
1.54
1.58
1.12
1.01
1.93
0.59
3.49
1.28
1.50
0.80
1.51
1.25
0.49
0.83
0.55
1.13
0.87
0.93
1.38
0.84
1.06
0.69
1.00
1.01
0.91
0.88
1.34

47
48
86
22

1.23090
1.08398
3.69647
0.521071.14720
1.00552
3.48752
0.49326

Average Ratio
Log Spread/Sigma

Average # zero returns
per day lasting above
30s / 60s / 120s

4.80
2.73
2.84
2.97
2.87
2.70
3.01
2.88
3.48
4.46
5.48
3.35
5.09
3.69
4.32
2.43
4.63
4.28
2.47
3.27
3.15
3.90
2.74
2.38
3.84
3.80
4.63
2.69
4.94
3.16
3.81
3.01
2.27

0.03
0.03
0.03
0.03
0.04
0.03
0.02
0.02
0.03
0.04
0.04
0.04
0.03
0.03
0.04
0.03
0.04
0.04
0.04
0.04
0.04
0.04
0.03
0.02
0.03
0.04
0.05
0.03
0.05
0.03
0.04
0.03
0.02

161 / 48 / 11
127 / 30 / 5
161 / 48 / 11
113 / 21 / 3
161 / 49 / 11
144 / 42 / 9
131 / 27 / 4
82 / 15 / 2
154 / 45 / 10
177 / 67 / 19
189 / 81 / 28
182 / 70 / 20
162 / 51 / 13
149 / 39 / 8
163 / 55 / 14
89 / 14 / 2
206 / 99 / 35
191 / 71 / 20
130 / 32 / 6
163 / 52 / 12
166 / 57 / 15
164 / 55 / 14
141 / 31 / 5
142 / 33 / 5
165 / 54 / 13
199 / 106/ 43
184 / 71 / 21
144 / 38 / 7
177 / 77 / 27
127 / 27 / 4
165 / 61 / 18
130 / 32 / 6
90 / 17 / 2

0.000350.00033
0.00055
0.00023

0.03
0.03
0.05
0.02

152 / 49 / 13
161 / 48 / 11
206 / 106/ 43
82 / 14 / 2

43

Table 3: Relative mean squared error (MSE) factors for models without microstructure noise. We report the MSE factors for each estimator and each model as thed − IV )2 /IQ, where IV and IQ are the true simulated integrated variancesample mean of 195(IVand integrated quarticity on each day. The MSE factor for 2-min subsampled RV is thus ≈ 1.33
which is the theoretical value shown by Zhang, Mykland, and Aït Sahalia (2005). Each row corresponds to a given estimator and each column to one of the models described in detail in Appendix
A.1. Thus model SV0 is a model without stochastic volatility while SV1A and SV2A are one andtwo factor stochastic volatility models with affine structure. Model SV1L is a one-factor model
written in log volatility. Entries in bold denote that the MSE factor is smaller than 2-min subsampled RV (for models without jumps) or smaller than 2-min subsampled BV (for models withjumps). In the case of the passage time estimators, the previous tick method is used for models
including jumps.
Models SV-UJ with U-Shaped Volatility and
25% Mean Jump Contribution to IV
Avg Sample Freq 3sec without NoiseFirst Range Time
DVN,hFirst Exit Time
DVN,hSubsampled
BVN RVN

Core Models SV-C
Avg Sample Freq 3sec without Noise
SV0-C

SV1A-C

SV1L-C

SV2A-C

2-min

1.318

1.315

1.314

1.347

2-min

1.495

1.511

1.510

1.505

1.945

2.027

2.028

2.053

h=3

0.398

0.413

0.412

0.407

0.386

0.393

0.394

0.389

h=4

0.582

0.591

0.590

0.578

0.513

0.517

0.517

0.523

h=5

0.911

0.889

0.888

0.898

0.815

0.817

0.817

0.816

h=6

1.356

1.289

1.289

1.343

1.243

1.239

1.239

1.260

h=7

1.908

1.835

1.834

1.906

1.820

1.801

1.802

1.806

h=8

2.572

2.475

2.476

2.556

2.601

2.542

2.548

2.481

h=3

0.274

0.296

0.294

0.293

0.242

0.254

0.256

0.253

h=4

0.436

0.447

0.449

0.418

0.376

0.387

0.389

0.376

h=5

0.596

0.647

0.642

0.597

0.552

0.579

0.579

0.554

h=6

0.825

0.816

0.817

0.819

0.788

0.774

0.776

0.782

h=7

1.096

1.059

1.056

1.122

1.085

1.066

1.068

1.079

h=8

1.437

1.406

1.402

1.449

1.500

1.473

1.469

1.463

44

SV0-UJ-

SV1A-UJ

SV1L-UJ

SV2A-UJ---

Table 4: Relative mean squared error (MSE) factors for models with microstructure noise without intra-day U-shape and jumps. We report the MSE factors ford − IV )2 /IQ, where IV and IQ areeach estimator and each model as the sample mean of 195(IVthe true simulated integrated variance and integrated quarticity on each day. The MSE factor for
2-min subsampled RV is thus ≈ 1.33 which is the theoretical value shown by Zhang, Mykland,and Aït Sahalia (2005). Each row corresponds to a given estimator and each column to a given
AR(1) noise structure and one of the models described in detail in Appendix A.1. Thus model SV0
is a model without stochastic volatility while SV1A and SV2A are one and two factor stochastic
volatility models with affine structure. Model SV1L is a one-factor model written in log volatility.
We consider two levels of noise to signal ratio: moderate (λ = 0.25) and high (λ = 1.0) and twolevels of noise persistence: low (ρ = 0.0) and high (ρ = 0.99). Entries in bold denote that the MSE
factor is smaller than 2-min subsampled RV.
Core Models SV-C
Avg Sample Freq 3sec with AR(1) Microstructure Noise
Noise Persistence ρ=0.0

First Range Time
DVN,hFirst Exit Time
DVN,hSubsampled
BVN RVN

Noise-to-Signal Ratio
λ = 0.25

Noise Persistence ρ=0.99

Noise-to-Signal Ratio
λ = 1.00

Noise-to-Signal Ratio
λ = 0.25

Noise-to-Signal Ratio
λ = 1.00

SV0-C

SV2A-C

SV0-C

SV2A-C

SV0-C

SV2A-C

SV0-C

SV2A-C

2-min

1.323

1.349

1.378

1.393

1.318

1.347

1.318

1.347

2-min

1.494

1.502

1.532

1.535

1.495

1.505

1.495

1.505

h=3

0.335

0.325

3.472

3.447

0.393

0.403

0.378

0.388

h=4

0.595

0.568

1.943

1.879

0.582

0.573

0.577

0.567

h=5

0.939

0.895

1.650

1.555

0.913

0.894

0.913

0.889

h=6

1.382

1.344

1.806

1.720

1.358

1.347

1.359

1.346

h=7

1.923

1.887

2.192

2.122

1.908

1.906

1.909

1.900

h=8

2.592

2.521

2.757

2.661

2.576

2.553

2.577

2.545

h=3

0.555

0.539

7.605

7.563

0.276

0.292

0.268

0.282

h=4

0.548

0.520

3.729

3.689

0.432

0.414

0.424

0.409

h=5

0.677

0.652

2.567

2.534

0.597

0.602

0.594

0.600

h=6

0.878

0.855

2.167

2.086

0.829

0.812

0.826

0.810

h=7

1.158

1.131

2.130

1.991

1.097

1.126

1.098

1.118

h=8

1.512

1.468

2.223

2.129

1.442

1.449

1.441

1.440

45

Table 5: Relative mean squared error (MSE) factors for models with microstructure noise and including intra-day U-shape and jumps. We report the MSE factors
d − IV )2 /IQ, where IV and IQ
for each estimator and each model as the sample mean of 195(IVare the true simulated integrated variance and integrated quarticity on each day. The MSE factor
for 2-min subsampled RV is thus ≈ 1.33 which is the theoretical value shown by Zhang, Mykland,and Aït Sahalia (2005). Each row corresponds to a given estimator and each column to a given
AR(1) noise structure and one of the models described in detail in Appendix A.1. Thus model SV0
is a model without stochastic volatility while SV1A and SV2A are one and two factor stochastic
volatility models with affine structure. Model SV1L is a one-factor model written in log volatility.
We consider two levels of noise to signal ratio: moderate (λ = 0.25) and high (λ = 1.0) and twolevels of noise persistence: low (ρ = 0.0) and high (ρ = 0.99). Entries in bold denote that the
MSE factor is smaller than 2-min subsampled BV. In the case of the passage time estimators, theprevious tick method is used for all models.
Models SV-UJ with U-Shaped Volatility and 25% Mean Jump Contribution to IV
Avg Sample Freq 3sec with AR(1) Microstructure Noise
Noise Persistence ρ=0.0
Noise-to-Signal Ratio
λ = 0.25

First Range Time
DVN,hFirst Exit Time
DVN,hSubsampled
BVN RVN

SV0-UJ
2-min-

SV2A-UJ-

Noise Persistence ρ=0.99

Noise-to-Signal Ratio
λ = 1.00
SV0-UJ-

SV2A-UJ-

Noise-to-Signal Ratio
λ = 0.25
SV0-UJ-

SV2A-UJ-

Noise-to-Signal Ratio
λ = 1.00
SV0-UJ-

SV2A-UJ
-

2-min

1.986

2.096

2.151

2.265

1.946

2.054

1.948

2.056

h=3

0.289

0.290

2.873

2.811

0.380

0.384

0.364

0.369

h=4

0.556

0.551

1.848

1.804

0.513

0.521

0.511

0.513

h=5

0.887

0.887

1.723

1.690

0.813

0.818

0.815

0.816

h=6

1.322

1.331

1.980

1.937

1.241

1.256

1.248

1.257

h=7

1.902

1.873

2.456

2.385

1.820

1.801

1.825

1.804

h=8

2.677

2.543

3.163

2.996

2.602

2.480

2.605

2.490

h=3

0.564

0.555

7.026

6.956

0.245

0.254

0.242

0.251

h=4

0.586

0.565

3.737

3.682

0.376

0.377

0.375

0.375

h=5

0.722

0.731

2.722

2.707

0.547

0.555

0.551

0.559

h=6

0.967

0.937

2.485

2.408

0.793

0.784

0.796

0.785

h=7

1.256

1.225

2.498

2.411

1.092

1.074

1.099

1.072

h=8

1.677

1.620

2.720

2.593

1.497

1.466

1.500

1.471

46

First Range
Time (τRT)

2First Hitting
Time (τHT)

First Exit
Time (τET)

1

0
0

1

2

Duration (τ)

3

4

5

Figure 1: The passage time densities for the baseline case ( σh = 1). The first
range time (thin line), the first exit time (dashed line) and the first hitting time (thick line).

E[1/τRT|τRT<T]

E[1/τET|τET<T]E[1/τHT|τHT<T]5

4 log2
2C
1

0
0

1

2
3
Time Remaining until Censoring (T)45

Figure 2: Conditional expectation of reciprocal passage times for the baseline case ( σh = 1). The expected reciprocal first range time (thin line), the expected
reciprocal first exit time (dashed line), and the expected reciprocal hitting time (thick line).
The respective unconditional expectations are indicated by three horizontal thin dashedlines, where C ≈ 0.91597 is the Catalan constant.

47

Figure 3: This Figure shows the relative (upward) bias of the first exit time passage time estimatoras a function of the persistence of the noise. The x-axis is the first order autocorrelation of thenoise which is of the AR(1) type. Panel A shows the “moderate” noise scenario where the noise tosignal ratio is λ = 0.25. Panel B shows the “high” noise scenario where the noise to signal ratio is
λ = 1.0

Bias Faactor

First Exit Time - Noise to Signal Ratio λ = 0.25
3

1.2
1.18
1.16
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
0.98

5

7

Noise PersistencePanel A

Noise PersistencePanel B

48

0.999

0.99

0.95

0.9

0.75

0.5

0.4

0.25

0.15

0.1

0.05

7

0

-0.05

5

-0.1

-0.25

-0.4

-0.5

-0.75

-0.9

-0.95

-0.99

-0.15

3

1.2
1.18
1.16
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
0.98
-0.999Bias Fa
actor

First Exit Time - Noise to Signal Ratio λ = 1.00

Figure 4: This Figure shows the relative (upward) bias of the first range time passage time estimatoras a function of the persistence of the noise. The x-axis is the first order autocorrelation of thenoise which is of the AR(1) type. Panel A shows the “moderate” noise scenario where the noise
to signal is λ = 0.25. Panel B shows the “high” noise scenario where the noise to signal ratio is
λ = 1.0.

Bias Fa
actor

First Range Time - Noise to Signal Ratio λ = 0.25
1.2
1.18
1.16
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
0.98

3

5

7

Noise PersistencePanel A

Bias Fa
actor

First Range Time - Noise to Signal Ratio λ = 1.00
1.2
1.18
1.16
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
0.98

3

5

Noise PersistencePanel B

49

7

Figure 5: Signature plots based on Monte Carlo experiments described in Section 5. The
d /IV of the first exit time DV and first range time DV
figure shows the mean relative bias IV
as a function of the threshold.
Core Model SV0-C
Avg Sample Freq 3sec without Microstructure Noise

Relative Bias

Robust Exit Time DV

Robust Range Time DV

2min Subsampled BV

2min Subsampled RV

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
1

2

3

4
5
6
7
Threshold Factor (x Average Log Spread)

8

9

10

Panel A: Deterministic Volatility Model SV0-C
Core Model SV2A-C
Avg Sample Freq 3sec without Microstructure Noise

Relative Bias

Robust Exit Time DV

Robust Range Time DV

2min Subsampled BV

2min Subsampled RV

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
1

2

3

4
5
6
7
Threshold Factor (x Average Log Spread)

8

9

10

Panel B: Stochastic Volatility Model SV2A-C
Model SV0-UJ with U-Shaped Pattern and 25% Mean Jump Contribution to IV
Avg Sample Freq 3sec without Microstructure Noise

Relative Bias

Robust Exit Time DV

Robust Range Time DV

2min Subsampled BV

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
1

2

3

4
5
6
7
Threshold Factor (x Average Log Spread)

8

9

10

Panel C: Stochastic Volatility Model SV2A-UJ with U-Shaped Pattern and Jumps

50

51

Relative Bias

Relative Bias

45
6
7
Threshold Factor (x Average Log Spread)892min Subsampled RV

10

0.8
0.7

1
0.9

1.2
1.1

1.4
1.3

1.5

4
5
6
7
Threshold Factor (x Average Log Spread)892min Subsampled RV

10

0.8
0.7

1
0.9

1.2
1.1

1.4
1.3

1.5

Panel C: Noise-to-signal ratio λ = 1.0, persistence ρ = 0.0

0.5

32min Subsampled BV

0.6

2

Robust Range Time DV

0.5

1

Robust Exit Time DV

0.6

0.8
0.7

0.9

1.1
1

1.2

1.4
1.3

1.5

Core Model SV2A-C
Avg Sample Freq 3sec with Non-Persistent Noise at High Level

Panel A: Noise-to-signal ratio λ = 0.25, persistence ρ = 0.0

0.5

32min Subsampled BV

0.6

2

Robust Range Time DV

0.5

1

Robust Exit Time DV

0.6

0.8
0.7

0.9

1.1
1

1.2

1.4
1.3

1.5

Core Model SV2A-C
Avg Sample Freq 3sec with Non-Persistent Noise at Moderate Level

1

1

3

2min Subsampled BV

4567Threshold Factor (x Average Log Spread)

Robust Range Time DV

892min Subsampled RV

32min Subsampled BV

4567Threshold Factor (x Average Log Spread)

Robust Range Time DV

892min Subsampled RV

Panel D: Noise-to-signal ratio λ = 1.0, persistence ρ = 0.99

2

Robust Exit Time DV

Core Model SV2A-C
Avg Sample Freq 3sec with Persistent Noise at High Level

Panel B: Noise-to-signal ratio λ = 0.25, persistence ρ = 0.99

2

Robust Exit Time DV

Core Model SV2A-C
Avg Sample Freq 3sec with Persistent Noise at Moderate Level

Figure 6: The impact of microstructure noise on the first exit time DV and first range time DV estimators for the two factor
stochastic volatility model without intra-day U-shape and jumps (SV2A-C). The observation frequency is 3 seconds and thenoise structure is assumed to be AR(1) as described in Section 3.4.

Relative Bias
Relative Bias

10

10

52

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5

Relative Bias

Relative Bias

1

1

3

4
5
6
7
Threshold Factor (x Average Log Spread)

Robust Range Time DV

892min Subsampled BV

3

4
5
6
7
Threshold Factor (x Average Log Spread)

Robust Range Time DV

892min Subsampled BV

Panel C: Noise-to-signal ratio λ = 1.0, persistence ρ = 0.0

2

Robust Exit Time DV

Model SV2A-UJ with U-Shaped Pattern and 25% Mean Jump Contribution to IV
Avg Sample Freq 3sec with Non-Persistent Noise at High Level

Panel A: Noise-to-signal ratio λ = 0.25, persistence ρ = 0.0

2

Robust Exit Time DV

Model SV2A-UJ with U-Shaped Pattern and 25% Mean Jump Contribution to IV
Avg Sample Freq 3sec with Non-Persistent Noise at Moderate Level

10

10

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
1

1

3

4
5
6
7
Threshold Factor (x Average Log Spread)

Robust Range Time DV

892min Subsampled BV

3

4
5
6
7
Threshold Factor (x Average Log Spread)

Robust Range Time DV

892min Subsampled BV

Panel D: Noise-to-signal ratio λ = 1.0, persistence ρ = 0.99

2

Robust Exit Time DV

Model SV2A-UJ with U-Shaped Pattern and 25% Mean Jump Contribution to IV
Avg Sample Freq 3sec with Persistent Noise at High Level

Panel B: Noise-to-signal ratio λ = 0.25, persistence ρ = 0.99

2

Robust Exit Time DV

Model SV2A-UJ with U-Shaped Pattern and 25% Mean Jump Contribution to IV
Avg Sample Freq 3sec with Persistent Noise at Moderate Level

Figure 7: The impact of microstructure noise on the first exit time DV and first range time DV estimators for the two factor
stochastic volatility model witt intra-day U-shape pattern and jumps (SV2A-UJ). The observation frequency is 3 seconds andthe noise structure is assumed to be AR(1) as described in Section 3.4

Relative Bias
Relative Bias

10

10

C

Cross-Sectional Average DJ30 Signature PlotsThis section displays the average performance of the duration-based integrated variance estimators(first exit time DV and first range time DV) across the 33 stocks which were part of the Dow Jones 30
index at any point during the period January 2005 through May 2007. The performance is comparedto the 2-minute subsampled bi-power estimator (BV or BV1) and the 2 minute subsampled realized
volatility estimator (RV). The distinction between the two BV estimators is whether adjacentabsolute returns are multiplied (BV) or whether one return is skipped (BV1) as suggested byAndersen, Bollerslev, and Diebold (2007).

Ticker Symbol: (All)Subsampled BV

0.00025

Subsampled BV1

Subsampled RV

Mean

0.000200.000150.000100.00005
0.00000
3

15

30
60
120
Sample Frequency (seconds)

180

300

600

Figure 8: The cross-sectional average signature plot for the sub-sampled BV and RV estimators as a function of the sample frequency in seconds. The average is taken across the 33
DJ30 stocks in our sample.

Ticker symbol: (All)0.00025

Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV12min Subsampled BV

Mean

0.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Figure 9: The cross-sectional average signature plot for duration-based integrated varianceestimators (DV) as a function of the threshold level (measured in units of the average intraday
log-spread).

53

Ticker symbol: (All)0.00025

Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV12min Subsampled BV

StdDev

0.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Figure 10: The cross-sectional average standard deviation of duration-based integrated variance estimators (DV) as a function of the threshold level (measured in units of the average
intraday log-spread).

Exit Time DV

Range Time DV

Correlationnwith 2min Subsam
mpled BV

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1

2

3

4

5

6

7

8

9

10

Threshold Factor (x Average Log Spread)

Figure 11: The cross-sectional average correlation of the 2-minute subsampled BV estimatorwith duration-based integrated variance estimators (DV) as a function of the threshold level(measured in units of the average intraday log-spread).

54

D

Individual DJ30 Signature Plots

Signature plots for the duration-based integrated variance estimators for the 33 Dow Jones 30
index from January 2005 through May 2007. The benchmark estimators are the 2-min subsampled
realized volatility (RV) and bi-power (BV) estimators.

Ticker symbol: AA

Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: AIG
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: AXP
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

55

7

8

9

10

Ticker symbol: BA
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: BAC
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: C
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

56

7

8

9

10

Ticker symbol: CAT
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: CVX
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: DD
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

57

7

8

9

10

Ticker symbol: DIS
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: EK
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: GE
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

58

7

8

9

10

Note different scale

Ticker symbol: GM
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000350.000300.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: HD
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: HPQ
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

59

7

8

9

10

Ticker symbol: IBM
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: INTC
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: IP
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

60

7

8

9

10

Ticker symbol: JNJ
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: JPM
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: KO
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

61

7

8

9

10

Ticker symbol: MCD
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: MMM
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: MO
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

62

7

8

9

10

Ticker symbol: MRK
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: MSFT
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: PFE
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

63

7

8

9

10

Ticker symbol: PG
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: T
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: UTX
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

64

7

8

9

10

Ticker symbol: VZ
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: WMT
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.00025
0 000200.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

7

8

9

10

Threshold factor (x average log spread)

Ticker symbol: XOM
Range Time DV

Exit Time DV

2min Subsampled RV

2min Subsampled BV22min Subsampled BV1

0.000250.000200.000150.000100.00005
0.00000
1

2

3

4

5

6

Threshold factor (x average log spread)

65

7

8

9

10