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Federal Reserve Bank of Chicago
Realized Volatility
Torben G. Andersen and Luca Benzoni
WP 2008-14
Realized Volatility∗
Torben G. Andersen1 and Luca Benzoni2
1
2
Kellogg School of Management, Northwestern University, Evanston, IL; NBER,
Cambridge, MA; and CREATES, Aarhus, Denmark
Federal Reserve Bank of Chicago, Chicago, IL
Summary. Realized volatility is a nonparametric ex-post estimate of the return
variation. The most obvious realized volatility measure is the sum of finely-sampled
squared return realizations over a fixed time interval. In a frictionless market the
estimate achieves consistency for the underlying quadratic return variation when
returns are sampled at increasingly higher frequency. We begin with an account
of how and why the procedure works in a simplified setting and then extend the
discussion to a more general framework. Along the way we clarify how the realized volatility and quadratic return variation relate to the more commonly applied
concept of conditional return variance. We then review a set of related and useful
notions of return variation along with practical measurement issues (e.g., discretization error and microstructure noise) before briefly touching on the existing empirical
applications.
1 Introduction
Given the importance of return volatility on a number of practical financial management decisions, there have been extensive efforts to provide good
real-time estimates and forecasts of current and future volatility. One complicating feature is that, contrary to the raw return, actual realizations of
return volatility are not directly observable. A common approach to deal with
the fundamental latency of return volatility is to conduct inference regarding
volatility through strong parametric assumptions, invoking, e.g., an ARCH or
∗
This draft: July 22, 2008. Chapter prepared for the Handbook of Financial Time
Series, Springer Verlag. We are grateful to Neil Shephard, Olena Chyruk, and
the Editors Richard Davis and Thomas Mikosch for helpful comments and suggestions. Of course, all errors remain our sole responsibility. The views expressed
herein are those of the authors and not necessarily those of the Federal Reserve
Bank of Chicago or the Federal Reserve System. The work of Andersen is supported by a grant from the NSF to the NBER and support from CREATES
funded by the Danish National Research Foundation.
2
Torben G. Andersen and Luca Benzoni
a stochastic volatility (SV) model estimated with data at daily or lower frequency. An alternative approach is to invoke option pricing models to invert
observed derivatives prices into market-based forecasts of “implied volatility”
over a fixed future horizon. Such procedures remain model-dependent and
further incorporate a potentially time-varying volatility risk premium in the
measure so they generally do not provide unbiased forecasts of the volatility
of the underlying asset. Finally, some studies rely on “historical” volatility
measures that employ a backward looking rolling sample return standard deviation, typically computed using one to six months of daily returns, as a
proxy for the current and future volatility level. Since volatility is persistent
such measures do provide information but volatility is also clearly mean reverting, implying that such unit root type forecasts of future volatility are
far from optimal and, in fact, conditionally biased given the history of the
past returns. In sum, while actual returns may be measured with minimal
(measurement) error and may be analyzed directly via standard time series
methods, volatility modeling has traditionally relied on more complex econometric procedures in order to accommodate the inherent latent character of
volatility.
The notion of realized volatility effectively reverses the above characterization. Given continuously observed price or quote data, and absent transaction
costs, the realized return variation may be measured without error along with
the (realized) return. In addition, the realized variation is conceptually related
to the cumulative expected variability of the returns over the given horizon
for a wide range of underlying arbitrage-free diffusive data generating processes. In contrast, it is impossible to relate the actual (realized) return to
the expected return over shorter sample periods in any formal manner absent
very strong auxiliary assumptions. In other words, we learn much about the
expected return volatility and almost nothing about the expected mean return
from finely-sampled asset prices. This insight has fueled a dramatic increase in
research into the measurement and application of realized volatility measures
obtained from high frequency, yet noisy, observations on returns. For liquid
financial markets with high trade and quote frequency and low transaction
costs, it is now prevailing practice to rely on intra-day return data to construct ex-post volatility measures. Given the rapidly increasing availability of
high-quality transaction data across many financial assets, it is inevitable that
this approach will continue to be developed and applied within ever broader
contexts in the future.
This chapter provides a short and largely intuitive overview of the realized
volatility concept and the associated applications. We begin with an account
of how and why the procedure works in a simplified setting and then discuss
more formally how the results apply in general settings. Next, we detail more
formally how the realized volatility and quadratic return variation relate to
the more common conditional return variance concept. We then review a set
of related and useful notions of return variation along with practical measurement issues before briefly touching on the existing empirical applications.
Realized Volatility
3
2 Measuring Mean Return versus Return Volatility
The theory of realized volatility is tied closely to the availability of asset price
observations at arbitrarily high frequencies. Hence, it is natural to consider the
volatility measurement problem in a continuous-time framework, even if we
ultimately only allow sampling at discrete intervals. We concentrate on a single
risky asset whose price may be observed at equally-spaced discrete points in
time over a given interval, [0, T ], namely t = 0, 1/n, 2/n, . . . , T − (1/n), T,
where n and T are positive integers and the unit interval corresponds to
the primary time period over which we desire to measure return volatility,
e.g., one trading day. We denote the logarithmic asset price at time t by
s(t) and the continuously compounded returns over [t − k, t] is then given by
r(t, k) = s(t) − s(t − k) where 0 ≤ t − k < t ≤ T and k = j/n for some
positive integer j. When k = 1 it is convenient to use the shorthand notation
r(t) = r(t, 1), where t is an integer 1 ≤ t ≤ T , for the unit period, or “daily,”
return.
To convey the basic rationale behind the realized volatility approach, we
initially consider a simplified setting with the continuously compounded returns driven by a simple time-invariant Brownian motion, so that
ds(t) = αdt + σdW (t), 0 ≤ t ≤ T ,
(1)
where α and σ (σ > 0) denote the constant drift and diffusion coefficients,
respectively, scaled to correspond to the unit time interval.
For a given measurement period, say [0, K], where K is a positive integer,
we have n · K intraday return observations r(t, 1/n) = s(t) − s(t − 1/n) for
t = 1/n, . . . , (n − 1) · K/n, K, that are i.i.d. normally distributed with mean
α/n and variance σ 2 /n. It follows that the maximum likelihood estimator for
the drift coefficient is given by
α̂n =
n·K
1 X
r(K, K)
s(K) − s(0)
r(j/n, 1/n) =
=
.
K j=1
K
K
(2)
Hence, for a fixed interval the in-fill asymptotics, obtained by continually
increasing the number of intraday observations, are irrelevant for estimating
the expected return. The estimator of the drift is independent of the sampling
frequency, given by n, and depends only on the span of the data, K. For
example, one may readily deduce that
Var(α̂n ) =
σ2
.
K
(3)
In other words, although the estimator is unbiased, the mean drift cannot be
estimated consistently over any fixed interval. Even for the simplest case of
a constant mean, long samples (large K) are necessary for precise inference.
Thus, in a setting where the expected returns are stipulated to vary conditionally on features of the underlying economic environment, auxiliary identifying
4
Torben G. Andersen and Luca Benzoni
assumptions are required for sensible inference about α. This is the reason
why critical empirical questions such as the size of the equity premium and
the pattern of the expected returns in the cross-section of individual stocks
remain contentious and unsettled issues within financial economics.
The situation is radically different for estimation of return volatility. Even
if the expected return cannot be inferred with precision, nonparametric measurement of volatility may be based on un-adjusted or un-centered squared
returns. This is feasible as the second return moment dominates the first moment in terms of influencing the high-frequency squared returns. Specifically,
we have,
¤ α2
£
σ2
E r(j/n, 1/n)2 = 2 +
,
(4)
n
n
and
£
¤ α4
α2 σ 2
σ4
E r(j/n, 1/n)4 = 4 + 6 3 + 3 2 .
(5)
n
n
n
It is evident that the terms involving the drift coefficient are an order of
magnitude smaller, for n large, than those that pertain only to the diffusion
coefficient. This feature allows us to estimate the return variation with a high
degree of precision even without specifying the underlying mean drift, e.g.,3
σ̂ 2n =
n·K
1 X 2
r (j/n, 1/n).
K j=1
(6)
It is straightforward to establish that
£ ¤ α2
E σ̂ 2n =
+ σ2 ,
n
(7)
while some additional calculations yield
£ ¤
σ4
α2 σ 2
Var σ̂ 2n = 4 2 + 2
.
n K
nK
(8)
It follows by a standard L2 argument that, in probability, σ̂ 2n → σ 2 for n → ∞.
Hence, the realized variation measure is a biased but consistent estimator of
the underlying (squared) volatility coefficient. Moreover, it is evident that,
for n large, the bias is close to negligible. In fact, as n → ∞ we have the
distributional convergence,
√
n · K (σ̂ 2n − σ 2 ) → N (0, 2σ 4 ) .
(9)
These insights are not new. For example, within a similar context, they
were stressed by Merton [97]. However, the lack of quality intraday price data
and the highly restrictive setting have long led scholars to view them as bereft
3
The quantity (K · σ̂ 2n ) is a “realized volatility” estimator of the return variation
over [0, K] and it moves to the forefront of our discussion in the following section.
Realized Volatility
5
of practical import. This situation has changed fundamentally over the last
decade, as it has been shown that the basic results apply very generally, highfrequency data have become commonplace, and the measurement procedures,
through suitable strategies, can be adapted to deal with intraday observations
for which the relative impact of microstructure noise may be substantial.
3 Quadratic Return Variation and Realized Volatility
This section outlines the main steps in generalizing the above findings to an
empirically relevant setting with stochastic volatility. We still operate within
the continuous-time diffusive setting, for simplicity ruling out price jumps,
and assume a frictionless market. In this setting the asset’s logarithmic price
process s must be a semimartingale to rule out arbitrage opportunities (e.g.,
Back [29]). We then have,
ds(t) = µ(t)dt + σ(t) dW (t) ,
0≤t≤T,
(10)
where W is a standard Brownian motion process, µ(t) and σ(t) are predictable
processes, µ(t) is ³
of finite variation,
while σ(t) is strictly positive and square
Rt 2 ´
integrable, i.e., E 0 σ s ds < ∞. Hence, the processes µ(t) and σ(t) signify
the instantaneous conditional mean and volatility of the return. The continuously compounded return over the time interval from t − k to t, 0 < k ≤ t,
is therefore
Z t
Z t
r(t, k) = s(t) − s(t − k) =
µ(τ )dτ +
σ(τ )dW (τ ) ,
(11)
t−k
t−k
and its quadratic variation QV (t, k) is
Z
t
QV (t, k) =
σ 2 (τ )dτ .
(12)
t−k
Equation (12) shows that innovations to the mean component µ(t) do not
affect the sample path variation of the return. Intuitively, this is because the
mean term, µ(t)dt, is of lower order in terms of second order properties than
the diffusive innovations, σ(t)dW (t). Thus, when cumulated across many highfrequency returns over a short time interval of length k they can effectively
be neglected. The diffusive sample path variation over [t − k, t] is also known
as the integrated variance IV (t, k),
Z
t
IV (t, k) =
σ 2 (τ )dτ .
(13)
t−k
Equations (12) and (13) show that, in this setting, the quadratic and integrated variation coincide. This is however no longer true for more general
6
Torben G. Andersen and Luca Benzoni
return process like, e.g., the stochastic volatility jump-diffusion model discussed in Section 5 below.
Absent microstructure noise and measurement error, the return quadratic
variation can be approximated arbitrarily well by the corresponding cumulative squared return process. Consider a partition {t − k + nj , j = 1, . . . n · k}
of the [ t − k, t ] interval. Then the realized volatility (RV) of the logarithmic
price process is
¶2
n·k µ
X
j 1
RV (t, k; n) =
r t−k+ ,
.
(14)
n n
j=1
Semimartingale theory ensures that the realized volatility measure converges
in probability to the return quadratic variation QV, previously defined in
equation (12), when the sampling frequency n increases:
RV (t, k; n) −→ QV (t, k)
as n → ∞ .
(15)
This finding extends the consistency result for the (constant) volatility coefficient discussed below equation (8) to a full-fledged stochastic volatility setting.
This formal link between realized volatility measures based on high-frequency
returns and the quadratic variation of the underlying (no arbitrage) price
process follows immediately from the theory of semimartingales (e.g., Protter
[102]) and was first applied in the context of empirical return volatility measurement by Andersen and Bollerslev [9]. The distributional result in equation
(9) also generalizes directly, as we have, for n → ∞,
!
Ã
√
RV (t, k; n) − QV (t, k)
p
n·k
→ N (0, 1) ,
(16)
2 IQ(t, k)
Rt
where IQ(t, k) ≡ t−k σ 4 (τ )dτ is the integrated quarticity, with IQ(t, k) independent from the limiting Gaussian distribution on the right hand side.
This result was developed and brought into the realized volatility literature
by Barndorff-Nielsen and Shephard [37].4
Equation (16) sets the stage for formal ex-post inference regarding the
actual realized return variation over a given period. However, the result is
not directly applicable as the so-called integrated quarticity, IQ(t, k), is unobserved and is likely to display large period-to-period variation. Hence, a
consistent estimator for the integrated quarticity must be used in lieu of the
true realization to enable feasible inference. Such estimators, applicable for any
integrated power of the diffusive coefficient, have been proposed by BarndorffNielsen and Shephard [37]. The realized power variation of order p, V (p; t, k; n)
is the (scaled) cumulative sum of the absolute p-th power of the high-frequency
returns and it converges, as n → ∞, to the corresponding power variation of
order p, V (p; t, k). That is, defining the p-th realized power variation as,
4
The unpublished note by Jacod [88] implies the identical result but this note was
not known to the literature at the time.
Realized Volatility
V (p; t, k; n) ≡ np/2−1 µ−1
p
¶
n·k ¯ µ
X
j 1 ¯¯ p
¯
¯r t − k + ,
¯ ,
n n
j=1
7
(17)
where µp denotes the p-th absolute moment of a standard normal variable,
we have, in probability,
Z t
V (p; t, k; n) →
σ p (τ )dτ ≡ V (p; t, k) .
(18)
t−k
In other words, V (4; t, k; n) is a natural choice as a consistent estimator for
the integrated quarticity IQ(t, k). It should be noted that this conclusion is
heavily dependent on the absence of jumps in the price process which is an
issue we address in more detail later. Moreover, the notion of realized power
variation is a direct extension of realized volatility as RV (t, k; n) = V (2; t, k; n)
so equation (18) reduces to equation (15) for p = 2.
More details regarding the asymptotic results and multivariate generalizations of realized volatility may be found in, e.g., Andersen et al. [16, 17],
Barndorff-Nielsen and Shephard [36, 37, 38], Meddahi [95], and Mykland [98].
4 Conditional Return Variance and Realized Volatility
This section discusses the relationship between quadratic variation or integrated variance along with its associated empirical measure, realized volatility,
and the conditional return variance. In the case of constant drift and volatility coefficients, the conditional (and unconditional) return variance equals
the quadratic variation of the log price process. In contrast, when volatility is
stochastic we must distinguish clearly between the conditional variance, representing the (ex-ante) expected size of future squared return innovations over
a certain period, and the quadratic variation, reflecting the actual (ex-post)
realization of return variation, over the corresponding horizon. Hence, the distinction is one of a priori expectations versus subsequent actual realizations
of return volatility. Under ideal conditions, the realized volatility captures
the latter, but not the former. Nonetheless, realized volatility measures are
useful in gauging the conditional return variance as one may construct well
calibrated forecasts (conditional expectations) of return volatility from a time
series of past realized volatilities. In fact, within a slightly simplified setting,
we can formally strengthen these statements. If the instantaneous return is the
continuous-time process (10) and the return, mean, and volatility processes
are uncorrelated (i.e., dW (t) and innovations to µ(t) and σ(t) are mutually
independent), then r(t, k) is normally distributed conditional on the cumulaRt
tive drift µ(t, k) ≡ t−k µ(τ )dτ and the quadratic variation QV (t, k) (which
in this setting equals the integrated variance IV (t, k) as noted in equations
(12) and (13)):
¡
¢
¡
¢
r(t, k) | µ(t, k), IV (t, k) ∼ N µ(t, k), IV (t, k) .
(19)
8
Torben G. Andersen and Luca Benzoni
Consequently, the return distribution is mixed Gaussian with the mixture governed by the realizations of the integrated variance (and integrated mean) process. Extreme realizations (draws) from the integrated variance process render
return outliers likely while persistence in the integrated variance process induces volatility clustering. Moreover, for short horizons, where the conditional
mean is negligible relative to the cumulative absolute return innovations, the
integrated variance may be directly related to the conditional variance as,
Var[ r(t, k) | Ft−k ] ≈ E[ RV (t, k; n) | Ft−k ] ≈ E[ QV (t, k) | Ft−k ] .
(20)
A volatility forecast is an estimate of the conditional return variance on the
far left-hand side of equation (20), which in turn approximates the expected
quadratic variation. Since RV is approximately unbiased for the corresponding
unobserved quadratic variation, the realized volatility measure is the natural
benchmark against which to gauge the performance of volatility forecasts.
Goodness-of-fit tests may be conducted on the residuals given by the difference between the ex-post realized volatility measure and the ex-ante forecast.
We review some of the evidence obtained via applications inspired by these
relations in Section 7. In summary, the quadratic variation is directly related
to the actual return variance as demonstrated by equation (19) and to the
expected return variance, as follows from equation (20).
Finally, note that the realized volatility concept is associated with the return variation measured over a discrete time interval rather than with the
so-called spot or instantaneous volatility. This distinction separates the realized volatility approach from a voluminous literature in statistics seeking to
estimate spot volatility from discrete observations, predominantly in a setting
with a constant diffusion coefficient. It also renders it distinct from the early
contributions in financial econometrics allowing explicitly for time-varying
volatilities, e.g., Foster and Nelson [73]. In principle, the realized volatility
measurement can be adapted to spot volatility estimation: as k goes to zero,
QV (t, k) converges to the instantaneous volatility σ 2 (t), i.e., in principle RV
converges to instantaneous volatility when both k and k/n shrink. For this to
happen, however, k/n must converge at a rate higher than k, so as the interval
shrinks we must sample returns at an ever increasing frequency. In practice,
this is infeasible, because intensive sampling over tiny intervals magnifies the
effects of microstructure noise. We return to this point in Section 6 where we
discuss the bias in RV measures when returns are sampled with error.
5 Jumps and Bipower Variation
The return process in equation (10) is continuous under the stated regularity
conditions, even if σ may display jumps. This is quite restrictive as asset
prices often appear to exhibit sudden discrete movements when unexpected
news hits the market. A broad class of SV models that allow for the presence
of jumps in returns is defined by
Realized Volatility
ds(t) = µ(t)dt + σ(t)dW (t) + ξ(t) dqt ,
9
(21)
where q is a Poisson process uncorrelated with W and governed by the jump
intensity λt , i.e., Prob(dqt = 1) = λt dt, with λt positive and finite. This assumption implies that there can only be a finite number of jumps in the price
path per time period. This is a common restriction in the finance literature,
though it rules out infinite activity Lévy processes. The scaling factor ξ(t)
denotes the magnitude of the jump in the return process if a jump occurs at
time t. While explicit distributional assumptions often are invoked for parametric estimation, such restrictions are not required as the realized volatility
approach is fully nonparametric in this dimension as well.
In this case, the quadratic return variation process over the interval from
t − k to t, 0 ≤ k ≤ t ≤ T , is the sum of the diffusive integrated variance and
the cumulative squared jumps:
Z t
X
X
QV (t, k) =
σ 2 (s)ds +
J 2 (s) ≡ IV (t, k) +
J 2 (s) , (22)
t−k
t−k≤s≤t
t−k≤s≤t
where J(t) ≡ ξ(t)dq(t) is non-zero only if there is a jump at time t.
The RV estimator (14) remains a consistent measure of the total QV in
the presence of jumps, i.e., result (15) still holds; see, e.g., Protter [102] and
the discussion in Andersen, Bollerslev, and Diebold [11]. However, since the
diffusive and jump volatility components appear to have distinctly different
persistence properties it is useful both for analytic and predictive purposes
to obtain separate estimates of these two factors in the decomposition of the
quadratic variation implied by equation (22).
To this end, the h-skip bipower variation, BV, introduced by BarndorffNielsen and Shephard [39] provides a consistent estimate of the IV component,
BV (t, k; h, n) =
¶
¶
µ
µ
n·k
π X ¯¯
ik 1 ¯¯ ¯¯
(i − h)k 1 ¯¯
,
,
r
t
−
k
+
r
t
−
k
+
¯¯
¯.
¯
2
n n
n
n
i=h+1
(23)
Setting h = 1 in definition (23) yields the ‘realized bipower variation’
BV (t, k; n) ≡ BV (t, k; 1, n). The bipower variation is robust to the presence of
jumps and therefore, in combination with RV, it yields a consistent estimate
of the cumulative squared jump component:
X
RV (t, k; n) − BV (t, k; n) −→ QV (t, k) − IV (t, k) =
J 2 (s) . (24)
n→∞
t−k≤s≤t
The results in equations (22)-(24) along with the associated asymptotic
distributions have been exploited to improve the accuracy of volatility forecasts and to design tests for the presence of jumps in volatility. We discuss
these applications in Section 7 below.
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Torben G. Andersen and Luca Benzoni
6 Efficient Sampling versus Microstructure Noise
The convergence relation in equation (15) states that RV approximates QV
arbitrarily well as the sampling frequency n increases. Two issues, however,
complicate the application of this result. First, even for the most liquid assets a
continuous price record is unavailable. This limitation introduces an inevitable
discretization error in the RV measures which forces us to recognize the presence of a measurement error. Although we may gauge the magnitude of such
errors via the continuous record asymptotic theory outlined in equations (16)(18), such inference is always subject to some finite sample distortions and
it is only strictly valid in the absence of price jumps. Second, a wide array
of microstructure effects induces spurious autocorrelations in the ultra-high
frequency return series. The list includes price discreteness and rounding, bidask bounces, trades taking places on different markets and networks, gradual
response of prices to a block trade, difference in information contained in
order of different size, strategic order flows, spread positioning due to dealer
inventory control, and, finally, data recording mistakes. Such “spurious” autocorrelations can inflate the RV measures and thus generate a traditional type
of bias-variance trade off. The highest possible sampling frequency should be
used for efficiency. However, sampling at ultra-high frequency tends to bias
the RV estimate.
A useful tool to assess this trade-off is the volatility signature plot, which
depicts the sample average of the RV estimator over a long time span as a
function of the sampling frequency. The long time span mitigates the impact
of sampling variability so, absent microstructure noise, the plot should be
close to a horizontal line. In practice, however, for transaction data obtained
from liquid stocks the plot spikes at high sampling frequencies and decays
rather smoothly to stabilize at frequencies in the 5- to 40-minute range. In
contrast, the opposite often occurs for returns constructed from bid-ask quote
midpoints as asymmetric adjustment of the spread induces positive serial correlation and biases the signature plot downward at the very highest sampling
frequencies. Likewise, for illiquid stocks the inactive trading induces positive
return serial autocorrelation which renders the signature plot increasing at
lower sampling frequencies, see, e.g., Andersen, Bollerslev, Diebold, and Labys
[14]. Aı̈t-Sahalia, Mykland, and Zhang [4] and Bandi and Russell [33] extend
this approach by explicitly trading off efficient sampling versus bias-inducing
noise to derive optimal sampling schemes.
Other researchers have suggested dealing with the problem by using alternative QV estimators that are less sensitive to microstructure noise. For
instance, Huang and Tauchen [87] and Andersen, Bollerslev, and Diebold [12]
note that using staggered returns and BV helps reduce the effect of noise, while
Andersen, Bollerslev, Frederiksen, and Nielsen [20] extend volatility signature
plots to include power and h-skip bipower variation. Other studies have instead relied on the high-low price range measure (e.g., Alizadeh, Brandt, and
Diebold [5], Brandt and Diebold [48], Brandt and Jones [49], Gallant et al.
Realized Volatility
11
[74], Garman and Klass [76], Parkinson [99], Schwert [104], and Yang and
Zhang [114]) to deal with situations in which the noise to signal ratio is high.
Christensen and Podolskij [55] and Dobrev [62] generalize the range estimator
to high-frequency data in distinct ways and discuss the link to RV.
A different solution to the problem is considered in the original contribution of Zhou [119] who seeks to correct the bias of RV style estimators by
explicitly accounting for the covariance in lagged squared return observations.
Hansen and Lunde [82] extend Zhou’s approach to the case of non-i.i.d. noise.
In contrast, Aı̈t-Sahalia et al. [4] explicitly determine the requisite bias correction when the noise term is i.i.d. normally distributed, while Zhang et al. [116]
propose a consistent volatility estimator that uses the entire price record by
averaging RVs computed from different sparse sub-samples and correcting for
the remaining bias. Aı̈t-Sahalia et al. [3] extend the sub-sampling approach to
account for certain types of serially correlated errors. Another prominent and
general approach is the recently proposed kernel-based technique of BarndorffNielsen et al. [40, 41].
7 Empirical Applications
Since the early 1990s transaction data have become increasingly available to
academic research. This development has opened the way for a wide array
of empirical applications exploiting the realized return variation approach.
Below we briefly review the progress in different areas of research.
7.1 Early Work
Hsieh [86] provides one of the first estimates of the daily return variation
constructed from intra-daily S&P500 returns sampled at the 15-minute frequency. The investigation is informal in the sense that there is no direct association with the concept of quadratic variation. More in-depth applications
were pursued in publications by the Olsen & Associates group and later surveyed in Dacorogna et al. [60] as they explore both intraday periodicity and
longer run persistence issues for volatility related measures. Another significant early contribution is a largely unnoticed working paper by Dybvig [65]
who explores interest rate volatility through the cumulative sum of squared
daily yield changes for the three-month Treasury bill and explicitly refers to
it as an empirical version of the quadratic variation process used in analysis
of semimartingales. More recently, Zhou [119] provides an initial study of RV
style estimators. He notes that the linkage between sampling frequency and
autocorrelation in the high-frequency data series may be induced by sampling noise and he proposes a method to correct for this bias. Andersen and
Bollerslev [8, 10] document the simultaneous impact of intraday volatility
patterns, the volatility shocks due to macroeconomic news announcements,
and the long-run dependence in realized volatility series through an analysis
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Torben G. Andersen and Luca Benzoni
of the cumulative absolute and squared five-minute returns for the Deutsche
Mark-Dollar exchange rate. The pronounced intraday features motivate the
focus on (multiples of) one trading as the basic aggregation unit for realized volatility measures since this approach largely annihilates repetitive high
frequency fluctuations and brings the systematic medium and low frequency
volatility variation into focus. Comte and Renault [58] point to the potential
association between RV measures and instantaneous volatility. Finally, early
empirical analyses of daily realized volatility measures are provided in, e.g.,
Andersen et al. [15] and Barndorff-Nielsen and Shephard [36].
7.2 Volatility Forecasting
As noted in Section 3, RV is the natural benchmark against which to gauge
volatility forecasts. Andersen and Bollerslev [9] stress this point which is further developed by Andersen et al. [17, 22, 23] and Patton [100] through different analytic means.
Several studies pursue alternative approaches in order to improve predictive performance. Ghysels et al. [77] consider Mixed Data Sampling (MIDAS)
regressions that use a combination of volatility measures estimated at different
frequencies and horizons. Related, Engle and Gallo [67] exploit the information in different volatility measures, modelled with a multivariate extension
of the multiplicative error model suggested by Engle [66], to predict multistep volatility. A rapidly growing literature studies jump detection (e.g., Aı̈tSahalia and Jacod [1], Andersen et al. [21, 19], Fleming and Paye [71], Huang
and Tauchen [87], Jiang and Oomen [89], Lee and Mykland [90], Tauchen and
Zhou [108], and Zhang [117]). Andersen et al. [12] show that separating the
jump and diffusive components in QV estimates enhances the model forecasting performance. Related, Liu and Maheu [91] and Forsberg and Ghysels [72]
show that realized power variation, which is more robust to the presence of
jumps than RV, can improve volatility forecasts.
Other researchers have been investigating the role of microstructure noise
on forecasting performance (e.g., Aı̈t-Sahalia and Mancini [2], Andersen et
al. [24, 25], and Ghysels and Sinko [78]) and the issue of how to use noisy
overnight return information to enhance volatility forecasts (e.g., Hansen and
Lunde [81] and Fleming et al. [70]).
A critical feature of volatility is the degree of its temporal dependence.
Correlogram plots for the (logarithmic) RV series show a distinct hyperbolic
decay that is described well by a fractionally-integrated process. Andersen and
Bollerslev [8] document this feature using the RV series for the Deutsche MarkDollar exchange rate. Subsequent studies have documented similar properties
across financial markets for the RV on equities (e.g., Andersen et al. [13], Areal
and Taylor [28], Deo et al. [61], Martens [93]), currencies (e.g., Andersen and
Bollerslev [10], Andersen et al. [16, 17], and Zumbach [120]), and bond yields
(e.g., Andersen and Benzoni [6]). This literature concurs on the value of the
fractional integration coefficient, which is estimated in the 0.30–0.48 range,
Realized Volatility
13
i.e., the stationarity condition is satisfied. Accounting for long memory in
volatility can prove useful in forecasting applications (e.g., Deo et al. [61]). A
particularly convenient approach to accommodate the persistent behavior of
the RV series is to use a component-based regression to forecast the k-stepahead quadratic variation (e.g., Andersen et al. [12], Barndorff-Nielsen and
Shephard [36], and Corsi [59]):
RV (t+k, k) = β 0 +β D RV (t, 1)+β W RV (t, 5)+β M RV (t, 21)+ε(t+k) . (25)
Simple OLS estimation yields consistent estimates for the coefficients in the
regression (25), which can be used to forecast volatility out of sample.
7.3 The Distributional Implications of the No-Arbitrage Condition
Equation (19) implies that, approximately, the daily return r(t) follows a
Gaussian mixture directed by the IV process. This is reminiscent of the
mixture-of-distributions hypothesis analyzed by, e.g., Clark [57] and Tauchen
and Pitts [106]. However, in the case of equation (19) the mixing variable is
directly measurable by the RV estimator which facilitates testing the distributional restrictions implied by the no-arbitrage condition embedded in the
return dynamics (10). Andersen et al. [15] and Thomakos and Wang [110]
find that returns standardized by RV are closer to normal than the standardized residuals from parametric SV models estimated at the daily frequency.
Any remaining deviation from normality may be due to a bias in RV stemming from microstructure noise or model misspecification. In particular, when
returns jump as in equation (21), or if volatility and return innovations correlate, condition (19) no longer holds. Peters and de Vilder [101] deal with the
volatility-return dependence by sampling returns in ‘financial time,’ i.e., they
identify calendar periods that correspond to equal increments to IV, while
Andersen et al. [19] extend their approach for the presence of jumps. Andersen et al. [21] apply these insights, in combination with alternative jumpidentification techniques, to different data sets and find evidence consistent
with the mixing condition. Along the way they document the importance of
jumps and the asymmetric return-volatility relation. Similar issues are also
studied in Fleming and Paye [71] and Maheu and McCurdy [92].
7.4 Multivariate Quadratic Variation Measures
A growing number of studies uses multivariate versions of realized volatility
estimators, i.e., realized covariance matrix measures, in portfolio choice (e.g.,
Bandi et al. [34] and Fleming et al. [70]) and risk measurement problems
(e.g., Andersen et al. [13, 18] and Bollerslev and Zhang [44]). Multivariate
applications, however, are complicated by delays in the security price reactions
to price changes in related assets as well as by non-synchronous trading effects.
Sheppard [105] discusses this problem but how to best deal with it remains
14
Torben G. Andersen and Luca Benzoni
largely an open issue. Similar to Scholes and Williams [103], some researchers
include temporal cross-correlation terms estimated with lead and lag return
data in covariance measures (e.g., Hayashi and Yoshida [83, 84] and Griffin
and Oomen [79]). Other studies explicitly trade off efficiency and noise-induced
bias in realized covariance estimates (e.g., Bandi and Russell [32] and Zhang
[115]), while Bauer and Vorkink [42] propose a latent-factor model of the
realized covariance matrix.
7.5 Realized Volatility, Model Specification and Estimation
RV gives empirical content to the latent variance variable and is therefore useful for specification testing of the restrictions imposed on volatility by parametric models previously estimated with low-frequency data. For instance,
Andersen and Benzoni [6] examine the linkage between the quadratic variation and level of bond yields embedded in some affine term structure models
and reject the condition that volatility is spanned by bond yields in the U.S.
Treasury market. Christoffersen et al. [56] reject the Heston [85] model implication that the standard deviation dynamics are conditionally Gaussian by
examining the distribution of the changes in the square-root RV measure for
S&P 500 returns.
Further, RV measures facilitate direct estimation of parametric models.
Barndorff-Nielsen and Shephard [37] decompose RV into actual volatility and
realized volatility error. They consider a state-space representation for this
decomposition and apply the Kalman filter to estimate different flavors of the
SV model. Bollerslev and Zhou [45] and Garcia et al. [75] build on the results
of Meddahi [96] to obtain efficient moment conditions which they use in the
estimation of continuous-time stochastic volatility processes. Todorov [112]
extends the analysis for the presence of jumps.
8 Possible Directions for Future Research
In recent years the market for derivative securities offering a pure play on
volatility has grown rapidly in size and complexity. Well-known examples are
the over-the-counter markets for variance swaps, which at maturity pay the
difference between realized variance and a fixed strike price, and volatility
swaps with payoffs linked to the square root of realized variance. These financial innovations have opened the way for new research on the pricing and
hedging of these contracts. For instance, while variance swaps admit a simple
replication strategy through static positions in call and put options combined
with dynamic trading in the underlying asset (e.g., Britten-Jones and Neuberger [50] and Carr and Madan [53]), it is still an open issue to determine
the appropriate replication strategy for volatility swaps and other derivatives
that are non-linear functions of realized variance (e.g., call and put options).
Carr and Lee [52] make an interesting contribution in this direction.
Realized Volatility
15
Realized volatility is also a useful source of information to learn more about
the volatility risk premium. Recent contributions have explored the issue by
combining RV measures with model-free option-implied volatility gauges like
the VIX (e.g., Bollerslev et al. [43], Carr and Wu [54], and Todorov [112]).
Other studies are examining the linkage between volatility risk and equity
premia (Bollerslev and Zhou [46]), bond premia (Wright and Zhou [113]),
credit spreads (Tauchen and Zhou [109] and Zhang et al. [118]), and hedgefund performance (Bondarenko [47]). In addition, new research is studying the
pricing of volatility risk in individual stock options (e.g., Bakshi and Kapadia
[30], Carr and Wu [54], Driessen et al. [63], and Duarte and Jones [64]) and
in the cross section of stock returns (e.g., Ang et al. [26, 27], Bandi et al. [31],
and Guo et al. [80]).
Finally, more work is needed to better understand the linkage between
asset return volatility and fluctuations in underlying fundamentals. Several
studies have proposed general equilibrium models that generate low-frequency
conditional heteroskedasticity (e.g., Bansal and Yaron [35], Campbell and
Cochrane [51], McQueen and Vorkink [94], and Tauchen [107]). Related, Engle and Rangel [69] and Engle et al. [68] link macroeconomic variables and
long-run volatility movements. An attempt to link medium and higher frequency realized volatility fluctuations in the bond market to both business
cycle variation and macroeconomic news releases is initiated in Andersen and
Benzoni [7], but clearly much more work on this front is warranted.
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Inflation Experiences by Demographic Group: 1983-2005
Leslie McGranahan and Anna Paulson
WP-05-20
Universal Access, Cost Recovery, and Payment Services
Sujit Chakravorti, Jeffery W. Gunther, and Robert R. Moore
WP-05-21
Supplier Switching and Outsourcing
Yukako Ono and Victor Stango
WP-05-22
Do Enclaves Matter in Immigrants’ Self-Employment Decision?
Maude Toussaint-Comeau
WP-05-23
The Changing Pattern of Wage Growth for Low Skilled Workers
Eric French, Bhashkar Mazumder and Christopher Taber
WP-05-24
U.S. Corporate and Bank Insolvency Regimes: An Economic Comparison and Evaluation
Robert R. Bliss and George G. Kaufman
WP-06-01
Redistribution, Taxes, and the Median Voter
Marco Bassetto and Jess Benhabib
WP-06-02
Identification of Search Models with Initial Condition Problems
Gadi Barlevy and H. N. Nagaraja
WP-06-03
Tax Riots
Marco Bassetto and Christopher Phelan
WP-06-04
The Tradeoff between Mortgage Prepayments and Tax-Deferred Retirement Savings
Gene Amromin, Jennifer Huang,and Clemens Sialm
WP-06-05
2
Working Paper Series (continued)
Why are safeguards needed in a trade agreement?
Meredith A. Crowley
WP-06-06
Taxation, Entrepreneurship, and Wealth
Marco Cagetti and Mariacristina De Nardi
WP-06-07
A New Social Compact: How University Engagement Can Fuel Innovation
Laura Melle, Larry Isaak, and Richard Mattoon
WP-06-08
Mergers and Risk
Craig H. Furfine and Richard J. Rosen
WP-06-09
Two Flaws in Business Cycle Accounting
Lawrence J. Christiano and Joshua M. Davis
WP-06-10
Do Consumers Choose the Right Credit Contracts?
Sumit Agarwal, Souphala Chomsisengphet, Chunlin Liu, and Nicholas S. Souleles
WP-06-11
Chronicles of a Deflation Unforetold
François R. Velde
WP-06-12
Female Offenders Use of Social Welfare Programs Before and After Jail and Prison:
Does Prison Cause Welfare Dependency?
Kristin F. Butcher and Robert J. LaLonde
Eat or Be Eaten: A Theory of Mergers and Firm Size
Gary Gorton, Matthias Kahl, and Richard Rosen
Do Bonds Span Volatility Risk in the U.S. Treasury Market?
A Specification Test for Affine Term Structure Models
Torben G. Andersen and Luca Benzoni
WP-06-13
WP-06-14
WP-06-15
Transforming Payment Choices by Doubling Fees on the Illinois Tollway
Gene Amromin, Carrie Jankowski, and Richard D. Porter
WP-06-16
How Did the 2003 Dividend Tax Cut Affect Stock Prices?
Gene Amromin, Paul Harrison, and Steven Sharpe
WP-06-17
Will Writing and Bequest Motives: Early 20th Century Irish Evidence
Leslie McGranahan
WP-06-18
How Professional Forecasters View Shocks to GDP
Spencer D. Krane
WP-06-19
Evolving Agglomeration in the U.S. auto supplier industry
Thomas Klier and Daniel P. McMillen
WP-06-20
3
Working Paper Series (continued)
Mortality, Mass-Layoffs, and Career Outcomes: An Analysis using Administrative Data
Daniel Sullivan and Till von Wachter
The Agreement on Subsidies and Countervailing Measures:
Tying One’s Hand through the WTO.
Meredith A. Crowley
WP-06-21
WP-06-22
How Did Schooling Laws Improve Long-Term Health and Lower Mortality?
Bhashkar Mazumder
WP-06-23
Manufacturing Plants’ Use of Temporary Workers: An Analysis Using Census Micro Data
Yukako Ono and Daniel Sullivan
WP-06-24
What Can We Learn about Financial Access from U.S. Immigrants?
Una Okonkwo Osili and Anna Paulson
WP-06-25
Bank Imputed Interest Rates: Unbiased Estimates of Offered Rates?
Evren Ors and Tara Rice
WP-06-26
Welfare Implications of the Transition to High Household Debt
Jeffrey R. Campbell and Zvi Hercowitz
WP-06-27
Last-In First-Out Oligopoly Dynamics
Jaap H. Abbring and Jeffrey R. Campbell
WP-06-28
Oligopoly Dynamics with Barriers to Entry
Jaap H. Abbring and Jeffrey R. Campbell
WP-06-29
Risk Taking and the Quality of Informal Insurance: Gambling and Remittances in Thailand
Douglas L. Miller and Anna L. Paulson
WP-07-01
Fast Micro and Slow Macro: Can Aggregation Explain the Persistence of Inflation?
Filippo Altissimo, Benoît Mojon, and Paolo Zaffaroni
WP-07-02
Assessing a Decade of Interstate Bank Branching
Christian Johnson and Tara Rice
WP-07-03
Debit Card and Cash Usage: A Cross-Country Analysis
Gene Amromin and Sujit Chakravorti
WP-07-04
The Age of Reason: Financial Decisions Over the Lifecycle
Sumit Agarwal, John C. Driscoll, Xavier Gabaix, and David Laibson
WP-07-05
Information Acquisition in Financial Markets: a Correction
Gadi Barlevy and Pietro Veronesi
WP-07-06
Monetary Policy, Output Composition and the Great Moderation
Benoît Mojon
WP-07-07
4
Working Paper Series (continued)
Estate Taxation, Entrepreneurship, and Wealth
Marco Cagetti and Mariacristina De Nardi
WP-07-08
Conflict of Interest and Certification in the U.S. IPO Market
Luca Benzoni and Carola Schenone
WP-07-09
The Reaction of Consumer Spending and Debt to Tax Rebates –
Evidence from Consumer Credit Data
Sumit Agarwal, Chunlin Liu, and Nicholas S. Souleles
WP-07-10
Portfolio Choice over the Life-Cycle when the Stock and Labor Markets are Cointegrated
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-07-11
Nonparametric Analysis of Intergenerational Income Mobility
with Application to the United States
Debopam Bhattacharya and Bhashkar Mazumder
WP-07-12
How the Credit Channel Works: Differentiating the Bank Lending Channel
and the Balance Sheet Channel
Lamont K. Black and Richard J. Rosen
WP-07-13
Labor Market Transitions and Self-Employment
Ellen R. Rissman
WP-07-14
First-Time Home Buyers and Residential Investment Volatility
Jonas D.M. Fisher and Martin Gervais
WP-07-15
Establishments Dynamics and Matching Frictions in Classical Competitive Equilibrium
Marcelo Veracierto
WP-07-16
Technology’s Edge: The Educational Benefits of Computer-Aided Instruction
Lisa Barrow, Lisa Markman, and Cecilia Elena Rouse
WP-07-17
The Widow’s Offering: Inheritance, Family Structure, and the Charitable Gifts of Women
Leslie McGranahan
WP-07-18
Demand Volatility and the Lag between the Growth of Temporary
and Permanent Employment
Sainan Jin, Yukako Ono, and Qinghua Zhang
WP-07-19
A Conversation with 590 Nascent Entrepreneurs
Jeffrey R. Campbell and Mariacristina De Nardi
WP-07-20
Cyclical Dumping and US Antidumping Protection: 1980-2001
Meredith A. Crowley
WP-07-21
The Effects of Maternal Fasting During Ramadan on Birth and Adult Outcomes
Douglas Almond and Bhashkar Mazumder
WP-07-22
5
Working Paper Series (continued)
The Consumption Response to Minimum Wage Increases
Daniel Aaronson, Sumit Agarwal, and Eric French
WP-07-23
The Impact of Mexican Immigrants on U.S. Wage Structure
Maude Toussaint-Comeau
WP-07-24
A Leverage-based Model of Speculative Bubbles
Gadi Barlevy
WP-08-01
Displacement, Asymmetric Information and Heterogeneous Human Capital
Luojia Hu and Christopher Taber
WP-08-02
BankCaR (Bank Capital-at-Risk): A credit risk model for US commercial bank charge-offs
Jon Frye and Eduard Pelz
WP-08-03
Bank Lending, Financing Constraints and SME Investment
Santiago Carbó-Valverde, Francisco Rodríguez-Fernández, and Gregory F. Udell
WP-08-04
Global Inflation
Matteo Ciccarelli and Benoît Mojon
WP-08-05
Scale and the Origins of Structural Change
Francisco J. Buera and Joseph P. Kaboski
WP-08-06
Inventories, Lumpy Trade, and Large Devaluations
George Alessandria, Joseph P. Kaboski, and Virgiliu Midrigan
WP-08-07
School Vouchers and Student Achievement: Recent Evidence, Remaining Questions
Cecilia Elena Rouse and Lisa Barrow
WP-08-08
Does It Pay to Read Your Junk Mail? Evidence of the Effect of Advertising on
Home Equity Credit Choices
Sumit Agarwal and Brent W. Ambrose
WP-08-09
The Choice between Arm’s-Length and Relationship Debt: Evidence from eLoans
Sumit Agarwal and Robert Hauswald
WP-08-10
Consumer Choice and Merchant Acceptance of Payment Media
Wilko Bolt and Sujit Chakravorti
WP-08-11
Investment Shocks and Business Cycles
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti
WP-08-12
New Vehicle Characteristics and the Cost of the
Corporate Average Fuel Economy Standard
Thomas Klier and Joshua Linn
WP-08-13
6
Working Paper Series (continued)
Realized Volatility
Torben G. Andersen and Luca Benzoni
WP-08-14
7
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