Full text of DOI:10.1111/0022-1082.00286

THE JOURNAL OF FINANCE • VOL. LV, NO. 5 • OCT. 2000
Order Flow, Transaction Clock, and
Normality of Asset Returns
THIERRY ANÉ and HÉLYETTE GEMAN*
ABSTRACT
The goal of this paper is to show that normality of asset returns can be recovered
through a stochastic time change. Clark ~1973! addressed this issue by represent-
ing the price process as a subordinated process with volume as the lognormally
distributed subordinator. We extend Clark’s results and find the following: ~i! sto-
chastic time changes are mathematically much less constraining than subordina-
tors; ~ii! the cumulative number of trades is a better stochastic clock than the
volume for generating virtually perfect normality in returns; ~iii! this clock can be
modeled nonparametrically, allowing both the time-change and price processes to
take the form of jump diffusions.
The relations among trading volume, stock prices, and price volatility, the
subject of empirical and theoretical studies over many years, have lately
received renewed attention with the increased availability of high frequency
data. A vast amount of research has focused on issues such as news arrivals,
volume, and price changes or volatility moves, usually outside any frame-
work of general or even partial equilibrium. Is the normality of returns—a
key issue, for example, in the mean-variance paradigm for portfolio choice,
or the recent study of the problems of risk management ~e.g., in Value at
Risk!—verified at any time horizon? The evidence accumulated from a num-
ber of studies that document the presence of leptokurtosis and skewness in
the distribution of returns of a wide variety of financial assets suggests that
the answer is no. Studies as early as, for example, Fama ~1965!, showed that
daily returns are more long tailed than the normal density, with the distri-
bution of returns approaching normality as the holding period is extended
to one month. In the same manner, volatility smiles and other observed
*
Ané is from the University Paris Dauphine. Geman is from the University Paris Dauphine
and ESSEC Graduate Business School. We thank the Center for International Securities and
Derivative Markets of the University of Massachusetts for providing the data. We appreciate
the comments made by participants of the Finance Seminar of the University of Chicago, the
University of Washington, the University of Maryland, Carnegie Mellon University, the Work-
shop on “Stochastics, Information and Markets” at Humboldt University, and the European
Finance Association meeting in Fontainebleau. The suggestions by René Stulz ~the editor! and
an anonymous referee were particularly helpful.
2
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The Journal of Finance
deviations from the Black and Scholes model seem to most directly contra-
dict the assumption of normality in asset returns, which has obvious impor-
tance for the pricing and hedging of derivative instruments.
Many analytical approaches have been proposed to address and analyze
the departure of returns from normality. Mandelbrot ~1963! introduced a
class of stable processes to account for the deviations of returns from Brown-
ian motion. As an alternative explanation, Clark ~1973! proposed linking the
deviations from normality to the existence of variations in volume during
different trading periods and introduced the use of subordinated processes
in finance, that is, the idea that calendar time may not be the appropriate
measure of time in financial markets. The investigation of specific subordi-
nators that might better capture asset price has been sparse in the finance
literature. Our goal is to update Clark’s results from an economic viewpoint
and to validate and generalize them from a modeling viewpoint using high
frequency data series.
Since the 1980s, the relationship between trading volume and stock prices
has been investigated in an impressive body of empirical and theoretical
literature. Virtually all empirical studies establish a positive correlation be-
tween volatility—measured as absolute or squared price changes–and vol-
ume ~see Karpoff ~1987!, Gallant, Rossi, and Tauchen ~1992!!. More recently,
Jones, Kaul, and Lipson ~1994! study daily prices of Nasdaq securities and
conclude that it is the number of trades and not their size that generates
volatility: “The average trade size has virtually no explanatory power when
volatility is conditioned on the number of transactions.”
Building on the result of Jones et al., we examine high frequency data on
two major technology stocks, Cisco Systems and Intel, to determine whether
it is the volume, as in Clark ~1973!, or the number of trades that best defines
the business time. Introducing a general stochastic time change t rather
than a subordinator and making no a priori assumption on its distribution,
we show that the clock that allows one to recover normality for asset returns
is indeed defined by the number of trades. Using classical kernel density
estimators, we construct empirical distributions of trades and stock returns,
identify the distribution of the time change through its moments, and then
reconstruct the density of the returns in the new “transaction time.” Re-
markably, this density is virtually normal. Last, because a stochastic clock
naturally leads to stochastic volatility, we show how stochastic volatility mod-
els can be related to stochastic time changes.
The remainder of the paper is organized as follows. In Section I we recall
some results on subordinating processes and explain the difference between
these special processes and stochastic time changes, from both mathemati-
cal and modeling standpoints. In Section II, we describe our database and its
statistical characteristics. In Section III we present our model and the main
result of the paper, namely, the economic identification of the stochastic clock
providing normality of asset returns. Section IV is dedicated to the relation-
ship between stochastic time changes and stochastic volatility, and Section V
contains some concluding comments.

Order Flow, Transaction Clock, and Normality of Asset Returns 2261
I. Subordinators and Stochastic Time Changes
Subordinated processes in mathematics were first introduced in the do-
main of analysis—and not probability theory—in work on Fourier trans-
forms and Laplace transforms ~which, by today, happen to have become
classical tools in mathematical finance!. As defined by Bochner ~1955!, a
subordinator t~t! is a right-continuous increasing process that has indepen-
dent and homogeneous increments. Besides the obvious degenerate case in
which t~t! is equal to t times a positive constant, the fundamental examples
of subordinators are the Poisson process ~simple or compound!, the gamma
process ~see Madan and Seneta ~1990!!, and the stable process. For any pro-
cess X~t!, the process Y~t! ϭ X~t~t!! is called a subordinated process.
At the time of Clark’s paper, the mathematical properties of subordination
were well established and hence readily available for financial applications.
However it is clear today that the condition of independent, identical incre-
ments imposed on t~t! is far too constraining in finance and eventually un-
desirable ~e.g., inconsistent with sampling a process at irregularly spaced
dates!. General stochastic time changes, defined by an increasing process t
and a given process X with certain desirable properties, provide all the flex-
ibility necessary to represent any return process Y in the form Y~t! ϭ X~t~t!!.
t
In Clark ~1973!, as in this paper, X is chosen to be the Brownian motion.
Moreover, our choice for X is validated by the fact that any arbitrage-free
1
return process Y can be written as a time-changed Brownian motion. If the
t
time change t is continuous, the process Y~t! ϭ X~t~t!! is also continuous,
because the trajectories of the Brownian motion are everywhere continuous.
If we want to account for periods of intense market activity, during crashes,
for instance, t may include a jump component that will in turn lead to a
jump-diffusion representation for the process Yt.
The idea of “business time” can be traced back to papers published by the
NBER in the 1940s; Burns and Mitchell ~1946! transform economic data to
an alternative timescale based on stages of business cycles. This economic
clock first appeared in finance in the work of Clark ~1973! in the form of
cumulative volume. Clark’s stated goal was to “present and test an opposing
hypothesis” to Mandelbrot’s ~1963! stable processes. But we can observe that
the modeling processes of Clark and Mandelbrot are not inherently contra-
dictory, because a stable process ~of index a! is a Brownian motion subordi-
nated to another stable process ~with index a02!. They only differ by the
choice of the subordinator: log-normal for Clark, stable for Mandelbrot. Our
claims in this paper are as follows: ~a! All processes defining asset returns
can be represented as time-changed Brownian motions; hence, identifying
the asset return process is tantamount to identifying the time change; and
~
b! This time change is the key element accounting for information arrival
1
The no arbitrage assumption implies the existence of a probability measure Q under which
discounted stock prices are martingales. Hence, the stock returns processes have to be semi-
martingales under the original probability measure P. Monroe ~1978! established that any
semimartingale is a time-changed Brownian motion.

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The Journal of Finance
and market activity and must have the right modeling properties to fit the
available data on asset returns. Because the properties of the Brownian
motion are well known, we can capture in the transaction time t the specific
traits of a given market in addition to the information flow during the pe-
riod under scrutiny. As such, we adopt a nonparametric approach and do not
specify in advance any distribution for t; rather we identify the distribution
of t through its moments.
II. Empirical Analysis
We analyze tick-by-tick data of two technology stocks, namely, Cisco Sys-
tems and Intel, which are both traded on the Nasdaq. The data were ob-
tained from Reuters. We characterize an asset’s returns in a standard manner
as the change in the logarithm of the stock price over a given interval of
time. We let P denote the stock price and define the return process in cal-
t
endar time as
Pt
Yt ϭ Y~t! ϭ ln
,
~1!
PtϪ1
where the return is observed at equally spaced calendar intervals. After
introducing a time change t that transforms the calendar time into the op-
erational time through the bijective mapping s ϭ t~t!, we define the return
in operational time by
P~t~t!!
X~t~t!! ϭ ln
~2!
P~t~t! Ϫ 1!
where the length of the interval @t~t! Ϫ 1, t~t!# represents one unit of oper-
ational time. To take an elementary example, if t~t! ϭ 2t ~e.g., a determin-
istic acceleration of time!, X~t~t!!would represent the return over the calendar
interval @2t Ϫ 2, 2t#.
Our task is to identify the economic proxy of the timescale providing nor-
mality of asset returns. Numerous empirical studies examine the contempo-
raneous behavior of volume and absolute price changes and document a positive
correlation between the two; for a survey up to 1987, see Karpoff ~1987!.
More recent empirical investigations, including Gerety and Mulherin ~1989!
and Stephan and Whaley ~1990!, focus on the intraday patterns in volume
and price volatility and find similar correlations. Gallant et al. ~1992! use a
semi-nonparametric method to estimate the joint density of price change
and volume. On the other hand, Easley and O’ Hara ~1992! show that the
number of trades is a good indicator of the rate of information flow, whereas
Blume, Easley, and O’ Hara ~1994! observe that volume provides information
on the quality of market information. Hence, our purpose is to investigate
two possible representations of the time change t~t!. In the first case, as in

Order Flow, Transaction Clock, and Normality of Asset Returns 2263
Clark ~1973!, t~t! is an affine function of the cumulative traded volume up
to time t, which we denote V . In the second case, as in Jones et al. ~1994!,
t
t~t!is an affine function of the number of trades cumulated up to time t,
denoted T . Hence, the natural first step is to empirically test these two
t
choices.
We examine the one-minute and 10-minute Cisco Systems returns and
also the five-minute and 15-minute Intel returns over the period January
2
, 1997, through December 31, 1997. For each stock, our data provide the
prices, together with the time of the transaction and the trading volume;
transactions that do not give rise to a change from the last recorded price
are also included. These data are then used to create equally spaced
series at one-minute and 10-minute intervals of stock returns, trading vol-
umes, and number of trades for the Cisco Systems security and at five-
minute and 15-minute intervals for the Intel security. The trading volume
and the number of trades during a particular time interval represent the
increments of the quantities defined as V and T . The descriptive statis-
t
t
tics of the three series for each stock and time resolution are displayed in
Table I.
We first observe that each return series exhibits a slight skewness and a
kurtosis significantly greater than three—traditional evidence of non-
normality. To compare the trading volume and the number of transactions in
terms of their power to explain variance changes ~i.e., of stochastic volatil-
ity!, we follow a procedure introduced in Schwert ~1990! and compute unbi-
ased estimates of return standard deviations. We proceed in the following
order:
1
. We perform a regression of the return Y over 12 lagged returns ~any
number between 10 and 15 gives approximately the same results!,
t
1
2
Yt ϭ
( dj YtϪj ϩ «t .
~3!
jϭ1
The lagged returns are used as regressors to estimate short term move-
ments in conditional expected returns, and the residuals in regres-
sion ~3! represent unexpected returns.
2
. We define as an estimate of the volatility change at time t the quantity2
p
s[ t ϭ
6«[ 6.
t
~4!
Ί2
2
2
An elementary result on the Gaussian distribution is that if X Ϸ N~0, s !, then E~6X6! ϭ
!
20ps.

Table I
Descriptive Statistics of the Data
Descriptive statistics of the two technology stock returns, trading volume, and number of trades for the different time intervals over the period
January 2, 1997, to December 31, 1997, are presented in this table. The data have been collected from Reuters. The number of observations for
the period of analysis is n ϭ 101,707 and 10,171 for the Cisco Systems series on one-minute and 10-minute intervals and n ϭ 20,352 and 6,784
for the Intel series on 5-minute and 15-minute intervals. Whereas population skewness and kurtosis for a normal distribution are, respectively,
0
and 3, the variance coefficients of skewness and kurtosis for a size n sample extracted from a normal population are 60n and 240n, respectively.
The number of observations we are analyzing is sufficiently large to invoke the central limit theorem and to assume that sample coefficients are
normally distributed. Thus, the 95% confidence intervals for a test of stock returns normality are given by 61.96 ϫ 60n for the sample
skewness and 3 6 1.96 ϫ 240n for the sample kurtosis. For statistics computed on the returns Y , sample skewness and kurtosis values outside
these confidence intervals indicate significant departures from normality, which is clearly the case here.
!
!
t
Mean
Variance
m3~Z!
Skewness
Panel A: Cisco Systems
One-Minute Returns
m4~Z!
Kurtosis
m5~Z!
m6~Z!
Returns
Traded volume
Number
Z ϭ Yt
Z ϭ Vt
4.229270EϪ06
8.764378
4.997559EϪ07
2.6148632
8.004071EϪ11
Ϫ3.2579457
2.265552EϪ01
Ϫ7.704960EϪ01
1.020305EϪ11
42.864665
4.085210Eϩ01
6.26904645
1.286759EϪ15
493.5682
7.187722EϪ17
5899.51385
of trades
Z ϭ Tt
2.176064578
1.201186697
Ϫ1.931194EϪ02
Ϫ1.466933EϪ02
17.96898447
12.45381774
231.9889886
2457.402228
1
0-Minute Returns
Returns
Traded volume
Number
Z ϭ Yt
Z ϭ Vt
4.228270EϪ05
85.2078034
3.475961EϪ06
15.458209
2.070135EϪ09
Ϫ4.59984065
3.194379EϪ01
Ϫ0.075683991
2.059287EϪ10
528.256079
17.04383066
2.210681383
1.281678EϪ14
2077.213637
8.031443EϪ15
11038.57328
of trades
Z ϭ Tt
21.67736593
21.57279083
Ϫ27.87452855
Ϫ0.278194175
7655.118284
16.44899015
Ϫ13578.8581
122695.116
Panel B: Intel
Five-Minute Returns
Returns
Traded volume
Number
Z ϭ Yt
Z ϭ Vt
Ϫ5.2863EϪ06
17.21842406
1.024171EϪ06
4.2508945
Ϫ9.871156EϪ11
1.03457629
Ϫ9.523783EϪ02
0.118043487
2.226697EϪ11
88.4210365
21.22835857
4.893222322
Ϫ1.062492EϪ15
632.1749035
2.061844EϪ16
8013.035466
of trades
Z ϭ Tt
7.74524156
2.4295205
Ϫ1.63127814
Ϫ0.4307717
43.575287
7.382426305
457.428425
5227.11304
1
5-Minute Returns
Returns
Traded volume
Number
Z ϭ Yt
Z ϭ Vt
Ϫ1.5839EϪ05
51.85321567
7.096932EϪ07
11.0563972
Ϫ1.043905EϪ10
2.6083671
Ϫ1.746043EϪ01
0.070949325
6.179025EϪ12
876.606294
12.26814006
7.17096033
Ϫ1.360838EϪ16
3150.304752
3.038698EϪ17
14027.7482
of trades
Z ϭ Tt
23.687323
3.3259344
Ϫ5.459321
Ϫ0.900053059
331.7532738
29.99078678
1066.311298
16378.64235

Order Flow, Transaction Clock, and Normality of Asset Returns 2265
To compare the performance of the volume and the number of trades in
explaining volatility changes, we run the following three regressions:
1
2
1
6
6
6
s[ 6 ϭ a ϩ b⌬V ϩ
rj 6s[ tϪj 6 ϩ h ,
~5a!
~5b!
~5c!
t
t
(
t
jϭ1
1
2
2
s[ 6 ϭ a ϩ g⌬T ϩ
rj 6s[ tϪj 6 ϩ h ,
t
t
(
t
jϭ1
1
2
3
t
s[ 6 ϭ a ϩ b⌬V ϩ g⌬T ϩ
rj 6s[ tϪj 6 ϩ h ,
t
t
t
(
jϭ1
where ⌬V~k! ϭ V~k! Ϫ V~k Ϫ 1! and ⌬T~k! ϭ T~k! Ϫ T~k Ϫ 1!. The lags of the
estimated standard deviation series are included to accommodate any per-
sistence in the futures price volatility.
For each security and under the various timescales, the results summa-
rized in Table II indicate the superiority of the number of trades in explain-
ing volatility changes, which supports the results of Jones et al. ~1994!, who
also analyze Nasdaq stock returns. For instance, the value 0.214708 ob-
2
tained for the adjusted-R of regression ~5b! in the case of one-minute Cisco
Systems returns using the number of trades as the sole explanatory variable
is strictly higher than the value 0.167614 obtained in regression ~5a!. More-
2
over, the small difference ~0.215899 Ϫ 0.214708! between the adjusted-R of
regressions ~5b! and ~5c! supports the hypothesis that the trading volume
adds virtually no explanatory power when returns are already conditioned
on the number of transactions. This observation holds for every return series
presented in this paper.
Last, we construct the empirical distributions of the stock returns, trading
volume, and number of transactions. The density estimation is traditionally
performed using the kernel method, that is, the estimator
n
1
x Ϫ x
i
fZ ~x! ϭ
K
,
~6!
(
ͩ ͪ
nh
h
iϭ1
where n ϭ number of observations, x ϭ observation i, h ϭ window width
i
~
also called the smoothing parameter!, and K ϭ the kernel estimator.
2
Ϫx 02
We choose as a kernel K~x! ϭ ~10
!
2p!e
, which ensures that fZ is a
smooth curve having derivatives of all orders and, hence, ensures the exis-
tence of the first six moments of Y , which play a key role below. Following
t
the standard approach ~see, e.g., Silverman ~1986!!, we use for the window
1
05 105
width the quantity h ϭ s~403!
n
, where s represents the standard de-
viation of the whole series. The estimated density distribution functions of
the different return series are given in Figure 1. They are more peaked
than the Gaussian distribution and exhibit thinner flanks and fatter tails.

Table II
Regression Estimates
Cisco Systems and Intel returns series are regressed, respectively, on trading volume and numbers of trades over the period January 2, 1997, to
December 31, 1997. The data were collected from Reuters. Standard errors for the coefficients of the regressions are provided in parentheses. The
2
higher adjusted-R of regressions ~5b! when compared to regressions ~5a! indicate that the number of trades T has a higher explanatory power
t
of volatility changes. The values of the coefficient g in regressions ~5c! are not statistically different from the values of this coefficient in
2
regressions ~5b!. On the contrary, the values of the coefficient b are significantly lower than in regressions ~5a!. Moreover the R in regressions
~
5b! and ~5c! are not statistically different. This indicates that not only does the number of trades have a higher explanatory power but also that
the informational content of the traded volume is included in the informational content of the number of trades.
2
2
b
g
Panel A: Cisco Systems
One-Minute Returns
—
R
b
g
Panel B: Intel
10-Minute Returns
—
R
Regression ~5a!
Regression ~5b!
Regression ~5c!
7.47216EϪ05
0.00000253!
—
0.167614
0.214708
0.215899
2.90345EϪ04
~0.000132!
—
0.140566
0.199388
0.201327
~
~
3.82534EϪ04
2.28340EϪ03
~0.000507!
2.10976EϪ03
~0.000773!
~
0.000108!
1.09674EϪ05
0.0000378!
3.54379EϪ04
1.97693EϪ04
~0.000065!
~0.000098!
Five-Minute Returns
—
15-Minute Returns
—
Regression ~5a!
Regression ~5b!
Regression ~5c!
4.39265EϪ05
0.0000208!
—
0.203811
0.286968
0.287031
9.85257EϪ05
~0.0000465!
—
0.182301
0.227614
0.229031
~
~
3.07456EϪ04
7.30843EϪ04
~0.000111!
7.08942EϪ04
~0.0000100!
~
0.000102!
2.97855EϪ06
0.0000102!
3.32056EϪ04
~0.000099!
3.20867EϪ06
~0.00000203!

Order Flow, Transaction Clock, and Normality of Asset Returns 2267
Figures 2 and 3 give, respectively, the empirical density distributions of the
trading volume and the number of trades for different time resolutions, also
reconstructed through the kernel method. Stephan and Whaley ~1990! show
there is an interesting U-shaped pattern when the trading volume is plotted
as a function of the time of day. We cannot directly compare our Figures 2
and 3 to the results of these authors, because our analysis does not provide
intraday patterns as a function of calendar time but rather shows return
changes as functions of market activity.
III. The Model
We know at this point the empirical distribution—hence, the empirical
moments—of the asset returns and of the cumulative number of trades. As
explained in Section I, we now search for a stochastic clock t~t! and the two
parameters m and s of an arithmetic Brownian motion such that
Y~t! ϭ X~t~t!!
~7!
and
law
2
X~s! ϭ N~ms, s s!.
~8!
The unconditional centered moments of the process Y~t! can be expressed in
terms of the parameters m, s and of the centered moments of the process
t~t!. The computation of these theoretical moments is presented in Appendix
A. However, a direct estimation procedure is not possible because any sys-
tems formed with these ~nonlinear! equations will have more unknowns than
equations—for instance, with the theoretical moments computed up to order
six, we obtain a nonlinear system of six equations with eight unknowns.
To recover sufficient information in the estimation procedure, we use the
moment generating function of the return process Y and define the following
t
minimization program:
k
theoretical
empirical
#2
Min U ϭ
@E @exp~b Y !#
Ϫ E @exp~b Y !#
~9!
(
j
t
j
t
jϭ1
for an appropriate choice of the numbers b , b ,..., b and under the equal-
1
2
k
theoretical
empirical
ity constraints m ~Y !
ϭ m ~Y !
for i ϭ 1,2,.... Appendix B
i
t
i
t
explains how the moment generating function of the return process in cal-
endar time can be expressed in terms of the unknown parameters of our
estimation problem. An empirical estimator of the moment generating func-
tion is also introduced to solve the minimization problem.

2
268
The Journal of Finance
Figure 1. Estimated densities of Cisco Systems and Intel returns. This figure represents,
for four different time resolutions, the density distributions of Cisco Systems and Intel returns.
These are reconstructed through the kernel method, the horizontal axis representing the real
values of asset returns. All distributions are more peaked than the Gaussian one and exhibit
leptokurtosicity. ~Figure continues on facing page.!
We perform our optimization procedure with several sets of values and
obtain nearly identical solutions irrespective of the values of b used. The
results of some of these minimizations are summarized in Table III.

Order Flow, Transaction Clock, and Normality of Asset Returns 2269
Figure 1. Continued.
A comparison of our findings with the empirical data presented in Table II
establishes that the moments of the time change t~t! greater than one are
perfectly matched by the moments of the cumulative number of transactions
Tt; only the mean differs significantly. If we make the simplifying ~but stan-
dard in finance! assumption that a probability distribution is defined by
the knowledge of its first several ~usually four! moments, then it appears
that the cumulated number of transactions ~up to a constant! is a good

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270
The Journal of Finance
Figure 2. Estimated densities of Cisco Systems and Intel trading volumes. This figure
displays the distributions of Cisco Systems and Intel trading volumes for the different time
resolutions examined in Figure 1. Densities are reconstructed through the kernel method. The
horizontal axis represents the trading volume in real values. ~Figure continues on facing page.!
representation of the economic time. To ascertain whether the moments of
the time change match the moments of the cumulative number of transac-
tions, we recenter the time change so that its mean equals the mean of the
number of transactions and then reconstruct the empirical density of the
time change.

Order Flow, Transaction Clock, and Normality of Asset Returns 2271
Figure 2. Continued.
We then observe that because Y ϭ X~t~t!!,
t
P~Y ʦ dy0t~t! ϭ u! ϭ P~X~u! ʦ dy0t~t! ϭ u!.
~10!
t
Assuming for simplicity, as does Clark ~1973!, the independence of X and t,
the previous relationship reduces to
P~Y ʦ dy0t~t! ϭ u! ϭ P~X~u! ʦ dy!,
~11!
t

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The Journal of Finance
Figure 3. Estimated densities of Cisco Systems and Intel number of trades. This figure
displays the distributions of Cisco Systems and Intel numbers of trades for the different time
resolutions. The estimated density is reconstructed through the kernel method. The horizontal
axis represents the real values of the number of trades. ~Figure continues on facing page.!
where X is Brownian motion. Hence, if the time change has been properly
chosen, the distribution of the return process Y, conditional on the time
change t, should be normal.
To test for this property in our data, we compute the series of the returns
conditional on the ~recentered! number of trades and use the kernel estima-
tor to reconstruct the empirical conditional density. This density is plotted in

Order Flow, Transaction Clock, and Normality of Asset Returns 2273
Figure 3. Continued.
Figure 4, together with a Gaussian distribution with the same mean and
variance, for each time resolution and stock. The near-perfect normality
strongly supports our conjecture—namely, that the time change generating
conditional normality for the return process is properly represented by the
number of trades, independent of any direct parametric representation of Y.
To confirm this result, we perform a chi-square test of goodness-of-fit on
the conditional recentered series. The critical value at the one percent con-
2
2
fidence level is x ϭ 11.6. The empirical statistics, x[ ϭ 5.21, 7.09, 6.34, and

Table III
Moment Estimates
This table shows the estimated values of the unknown moments of the underlying Gaussian process and of the unobserved stochastic clock t~t!
obtained by the optimization program in equation ~9!. We minimize, under equality constraints on the centered theoretical and empirical
moments, the sum of the square differences between the theoretical moment generating function of the return process ~expressed in terms of the
unknown parameters as discussed in Appendix B! and the empirical moment generating function of the return process. The minimization
program has also been run with equality constraints on the uncentered theoretical and empirical moments of the return process, and the same
values were found for the unknown parameters, exhibiting the robustness of the minimization procedure.
m
s2
E~t~t!
!
m
2
~t~t!
!
m
3
~t~t!
!
m
4
~t~t!
!
m
5
~t~t!
!
6
m ~t~t!!
Panel A: Cisco Systems
One-Minute Returns
Optimization value
Standard error
1.371329EϪ05
6.48208EϪ06
1.619712EϪ06
3.56967EϪ07
3.084067EϪ01
0.00237
1.2028836
0.57329
Ϫ1.967611EϪ02
0.00474
18.4472756
2.06474
232.0183
16.06836
2457.838915
30.68394
1
0-Minute Returns
Optimization value
Standard error
1.972051EϪ05
7.20686EϪ06
1.617121EϪ06
8.90235EϪ07
2.140840668
0.48757
21.6328604
1.07998
Ϫ27.6655439
2.08579
7654.064486
6.07687
Ϫ13577.01045
122697.2334
85.65343
23.86068
Panel B: Intel
Five-Minute Returns
Optimization value
Standard error
Ϫ8.3616EϪ06
2.46329EϪ07
1.6148EϪ06
2.07573EϪ07
6.3221EϪ01
3.30746EϪ03
2.432755
0.10750
Ϫ1.66371814
0.34524
43.8332287
1.53088
457.337425
9.04525
5229.06326
23.96368
1
5-Minute Returns
Optimization value
Standard error
Ϫ1.5290EϪ05
7.86961EϪ06
6.8437EϪ07
7.6434EϪ08
1.0358626
0.10565
3.314327
0.48460
Ϫ5.48911321
0.79994
332.614213
8.45207
1068.01078
11.96537
16380.33582
56.32110

Order Flow, Transaction Clock, and Normality of Asset Returns 2275
8
.25, respectively obtained for the one-minute and 10-minute Cisco Systems
series and for the five-minute and 15-minute Intel series, indicate that the
Gaussian hypothesis cannot be rejected. Applying the Jarque–Bera test of
normality for each sampling frequency and each security also indicates that
normality of the recentered conditional returns cannot be rejected at the one
percent confidence level ~with a critical Jarque–Bera statistic of 9.21 and
test statistics of 0.337, 3.256, 1.440, and 3.907, respectively, for the one-
minute and 10-minute Cisco Systems series and for the five-minute and
1
5-minute Intel series!. Moreover, we provide in Figure 5 the quantile–
quantile plots for each estimated density conditioned by the recentered num-
ber of trades. The perfect graphical fit is further strong evidence of the
Gaussian nature of the series.
It is worth mentioning that the normality of the return process conditional
on the number of trades was also exhibited in an empirical study we con-
ducted independently on two high frequency databases of S&P 500 Futures
prices ~see Geman and Ané ~1996!! and FTSE 100 index values. This indi-
cates that the results presented here are not an accidental property of the
data because they hold for various classes of equity instruments, each hav-
ing clocks with their own tick rates. ~The study of the multidimensional
time-change problem is left for further research.!
Last, let us illuminate a property derived from our model. An elementary
formula on conditional probabilities allows us to write
P~Y ʦ dy! ϭ
P~Y ʦ dy0t~t! ϭ u!f ~u!,
~12!
t
(
t
t
u
where f denotes the distribution of t that we assume, for the simplicity of
t
argument, to be discrete. Hence,
P~Y ʦ dy! ϭ
P~X ʦ dy!f ~u!
~13!
t
(
u
t
u
from the independence of X and t. Inspection of equation ~13! reveals that
the unconditional distribution of Y appears as a mixture of normal distribu-
tions. The mixture of distributions hypothesis ~sometimes referred to as MDH!
is a well-known representation of asset returns, and it has often been of-
fered in the financial literature to model the observed leptokurtosis in re-
turns. ~For example, Richardson and Smith ~1994! examine the MDH
hypothesis empirically and propose a direct test of it using the generalized
method of moments.! In our framework, the unconditional distribution of
returns does behave as a mixture of normals, where the number of trades
acts as the mixing variable. Obviously, the resulting process will crucially
depend on the distribution of t~t!. We already mentioned that, in our view,
in a number of markets ~e.g., the newly deregulated electricity market!, the
process ~t~t!!tՆ0 should include jumps, which would lead to a mixing process
quite different from the one classically presented in the literature.

2
276
The Journal of Finance
Figure 4. Estimated densities of Cisco Systems and Intel returns conditioned by the
recentered number of trades. For each security and each time interval, the return density
conditioned on the recentered number of trades is estimated using the kernel method and
plotted here with the horizontal axis representing the real values of returns. On the same chart
is represented the graph of the Gaussian distribution with the same mean and variance. The
near-perfect identity of the two densities indicates that a time change that provides normality
of returns is well proxied by an affine function of the number of trades. ~Figure continues on
facing page.!

Order Flow, Transaction Clock, and Normality of Asset Returns 2277
Figure 4. Continued.
IV. Stochastic Time Changes and Stochastic Volatility
The second line of equation ~A1! in Appendix A shows that the variance of
the unconditional return process ~Y ! evolves stochastically because the mean
t
and the standard deviation of the number of trades are not constant over
time. Hence, our representation of the return process in calendar time falls

2
278
The Journal of Finance
Figure 5. Q-Q plots of Cisco Systems and Intel conditional returns. This figure provides
the quantile-quantile plots for return densities conditioned on the recentered number of trades,
for the four time resolutions analyzed all along. The Q-Q plot is a well-known criterion of
normality when the set of points is a 458 line. The excellent fits we obtained are a strong
evidence of return normality in transaction time. ~Figure continues on facing page.!
into the group of stochastic volatility models. Moreover, the above formula
shows that the volatility is monotonically related to the average of the num-
ber of trades but also to the unexpected component of the trading activity

Order Flow, Transaction Clock, and Normality of Asset Returns 2279
Figure 5. Continued.
~
represented by Var~t~t!!!. This result is in agreement with the findings of
Bessembinder and Seguin ~1993!, who specifically examine the effects on the
volatility of the expected and unexpected transacted volumes.
More generally, our setting can accommodate any type of stochastic vola-
tility model for the return process, the changes in volatility being translated
by contraction or dilatation of time ~the mathematics associated with the

2
280
The Journal of Finance
construction of t are described in the Appendixes!. In one of his last papers,
Black ~1992! comments on the limits of the Black–Scholes formula and its
extensions. In particular, he argues that an alternative to the addition of a
jump component in the stock price ~to model the effects of the arrival of
news! is to incorporate a higher volatility in the formula ~hence, the impor-
tance of the problem of proper volatility estimation!. Our modest claim is
that the introduction of a business clock t allows one to reconcile these two
possible representations of non-normal returns.
Last, let us illustrate by an example that in option pricing and hedging,
the stochastic volatility of the underlying asset price dynamics can be trans-
formed into constant volatility through a stochastic time change. Consider
for instance a financial institution that has sold standard European options
at time zero on the basis of a Black–Scholes–Merton volatility equal to b,
where b denotes a positive constant. Let the asset price dynamics be de-
scribed by
dSt
ϭ m dt ϩ s dWt ,
~14!
St
where ~W !
is a Brownian motion on the probability space ~⍀, F, F , P!
t tՆ0
t
representing the randomness of the economy, P is the statistical probability
and F is the filtration of information available at time t. If s is itself sto-
t
chastic on the interval @0, T #, the value of the self-financing portfolio built
at time zero with the premium recovered from the sale of the option and
dynamically readjusted over time to replicate the option will not equal at
maturity T the payoff of the option. Instead of experiencing a deviation from
the target, one may be interested in finding the first time t at which perfect
b
replication is achieved, that is, when
tb
2
2
͵
@s~s!# ds ϭ b T.
~15!
0
Obviously, t is a stopping time whose probability distribution is extremely
b
valuable to option traders. As an example, assume the Hull and White ~1987!
framework, that is, asset price dynamics given by the equations
dSt
1
ϭ m dt ϩ s dW ,
~16!
~17!
t
t
St
dYt
Yt
2
t
ϭ u dt ϩ z dW ,
2
where Y is equal to s and r is the correlation between the Brownian mo-
t
1
t
2
tions W and W . Geman and Yor ~1993! provide the Laplace transform of
the density f of the stopping time t , namely, the quantity *
ϩ`
Ϫlx
fb~x!e
dx,
b
b
0

Order Flow, Transaction Clock, and Normality of Asset Returns 2281
in terms of the parameters u, z defining the volatility process dynamics; as
usual, the drift of the stock price plays no role. By inversion of the Laplace
transform, it is possible to obtain the complete density f ~x!; if one only
b
wishes to compute the moments of the “perfect replication time” t , one can,
b
using standard properties of Laplace transforms, repeatedly differentiate
the transform the appropriate number of times ~n times for the nth uncen-
tered moment! and evaluate at zero. In particular, one can obtain the aver-
age time between issuance at a volatility level b and optimal replication of
the option. ~Strictly speaking, the replication process need not be unique,
because the presence of stochastic volatility that does not depend solely on
S means that markets are incomplete.!
t
Obviously, the use of time changes can be extended to any stochastic vol-
atility model and provides a general and powerful technique to handle prob-
lems related to stochastic volatility. In particular, the economic identification
of the appropriate time change t and its density may facilitate the recovery
of option prices as the Black–Scholes price for a fixed time integrated against
the density of t. More specifically, the European fixed maturity option price
in the case of stochastic volatility will depend on the distribution of the
cumulative quadratic variation associated with the underlying asset, which
will depend on the nature of the time change. We leave these extensions to
future research as our goal here was to provide a precise analysis of the
methodology and empirics of the recovery of normality of stock returns.
V. Conclusion
This paper looks at the distribution of a high frequency database of tech-
nology stock returns using stochastic time changes rather than subordina-
tors. It is shown that, to recover normality in asset returns, the number of
trades is a better time change than the traditionally used trading volume.
The near-perfect normality in transaction time is exhibited through the re-
construction of the time-changed return process. No particular distribution
is assumed for the time change itself. Instead, the time change is charac-
terized through its moments. Due to this flexibility, the time change can be
represented by a lognormal or another continuous distribution; it can also
include jumps to account for periods of high market activity, leading in turn
to a jump-diffusion model for the asset return process.
Appendix A
In this Appendix, we give the theoretical form of the first six centered
moments of the return process in terms of the corresponding moments of the
2
unobservable stochastic time change t~t! and the two moments ~m, s ! of the
underlying Gaussian distribution. Denoting by m ~Y ! the centered moment
i
of order i of the variable Y, and using the assumption of independence of X
and t, we can express the theoretical moments of the return process Y as
t
follows:

2
282
E~Y ! ϭ mE~t~t!!,
The Journal of Finance
t
2
2
Var ~ Y ! ϭ m ~ Y ! ϭ s E ~t~t!! ϩ m Var ~t~t!!,
t
2
t
2
3
m ~Y ! ϭ 3ms Var ~t~t!! ϩ m m ~t~t!!,
3
t
3
4
2
2
2
2
m ~Y ! ϭ m m ~t~t!! ϩ 6s m m ~t~t!! ϩ 6s m E~t~t!!Var ~t~t!!
4
t
4
3
4
2
ϩ 3s @Var ~t~t!! ϩ @E~t~t!!# #,
~
A1!
5
2
3
2
3
m ~Y ! ϭ m m ~t~t!! ϩ 10s m m ~t~t!! ϩ ~10s m E~t~t!!
5
t
5
4
4
4
ϩ 15ms !m ~t~t!! ϩ 30ms Var ~t~t!!,
3
6
6
3
m ~Y ! ϭ m m ~t~t!! ϩ 15s ~m ~t~t!! ϩ 3m ~t~t!!E~t~t!! ϩ ~E~t~t!!! !
6
t
6
3
2
4
2
2
4
ϩ 15m s ~m ~t~t!! ϩ m ~t~t!!E~t~t!!!! ϩ 45m s ~m ~t~t!!
5
4
4
2
ϩ 2m ~t~t!!E~t~t!! ϩ m ~t~t!!~E~t~t!!! !.
3
2
Appendix B
Using the properties of conditional expectations and the assumption of
Brownian motion for the process X~s!, the moment generating function of
the return process Y can be expressed in terms of the moment generating
t
function of the time change process t~t!,
E @exp~bY !# ϭ E @E~exp~bY !0t~t!#
t
t
1
2
_
2
2
ϭ E @exp~bm ϩ b s !t~t!#
~B1!
ϭ E @exp~At~t!!#,
1
2
_
2
2
where A ϭ bm ϩ b s .
The nonparametric setting we adopt in this study gives us no direct in-
formation on the distribution of the time change process or on its moment
generating function. To obtain the moment generating function of the return
process Y in terms of m, s and the moments of t, we write a series expan-
t
sion of exp~At~t!! around the mean of the time change t~t!:

Order Flow, Transaction Clock, and Normality of Asset Returns 2283
A2
exp~At~t!! Х exp~AE~t~t!!! *
ͫ
1 ϩ A~t~t! Ϫ E~t~t!! ϩ
~t~t! Ϫ E~t~t!!2
2
A3
A4
3
~t~t! Ϫ E~t~t!!4
.
ϩ
~t~t! Ϫ E~t~t!! ϩ
ͬ
B2!
6
24
~
The expectation of this series expansion yields a theoretical expression of
the moment generating function of the directing process t~t! and hence of
the return process Yt,
theoretical
E @exp~b Y !#
j
t
~
B3!
A2
A3
A4
AE~t~t!
Х e
*
ͫ
1 ϩ
Var ~t~t!! ϩ
m ~t~t!! ϩ
3
m ~t~t!! .
ͬ
4
2
6
24
For any given value of b, say, b , the moment generating function of the
j
return process Y may also be approximated empirically by
t
n
1
empirical
E @exp~b Y !#
ϭ
exp~b y !.
~B4!
j
t
(
j
t
i
n
iϭ1
The unknown parameters can then be obtained by minimizing the expression
k
theoretical
empirical
#2
Min U ϭ
@E @exp~b Y !#
Ϫ E @exp~b Y !#
~B5!
(
j
t
j
t
jϭ1
theoretical
empirical
under the constraints on equality m ~Y !
ϭ m ~Y !
for i ϭ
i
t
i
t
1
,2,.... We minimize the sum of squared differences between the sample
data moment generating function and the theoretical moment generating
function computed for the values b , b ,...b of b. Obviously the choice of
1
2
k
the values b , b ,...b is crucial. Simulations we ran suggest that very large
1
2
k
and very small values of b should be avoided.
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