In financial markets, not only prices and returns can be considered as random variables, but also the waiting time between two transactions varies randomly. In the following, we analyse the statistical properties of General Electric stock prices, traded at NYSE, in October 1999. These properties are critically revised in the framework of theoretical predictions based on a continuous-time random walk model.Duration; Continuous-time random walk; Fractional calculus; Statistical finance.

Enrico Scalas

Francesco Mainardi

Marco Raberto

English

Research Papers in Economics

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Waiting-times and returns in high-frequency
ﬁnancial data: an empirical study
Marco Raberto a, Enrico Scalas b, Francesco Mainardi c,1
aDipartimento di Ingegneria Bioﬁsica ed Elettronica, Universit` a di Genova,
via dell’Opera Pia 11a, I–16145 Genova, Italy
bDipartimento di Scienze e Tecnologie Avanzate, Universit` a del
Piemonte Orientale, via Cavour 84, I–15100 Alessandria, Italy
cDipartimento di Fisica, Universit` a di Bologna and INFN Sezione di Bologna,
via Irnerio 46, I–40126 Bologna, Italy
Abstract
In ﬁnancial markets, not only prices and returns can be considered as random vari-
ables, but also the waiting time between two transactions varies randomly. In the fol-
lowing, we analyse the statistical properties of General Electric stock prices, traded
at NYSE, in October 1999. These properties are critically revised in the framework
of theoretical predictions based on a continuous-time random walk model.
Key words: PACS: 02.50.-r, 02.50.Ey, 02.50.Wp, 89.90.+n
Stochastic processes; Continuous-time random walk; Statistical ﬁnance;
Econophysics; Autocorrelation function
1 Introduction
In ﬁnancial markets, waiting times between two consecutive transactions vary
in a stochastic fashion. In 1973, [1] Peter Clark wrote: “Instead of indexing
[time] by the integers 0,1,2,..., the [price] process could be indexed by a set
of numbers t1, t2, t3, ..., where these numbers are themselves a realization of
a stochastic process (with positive increments, so that t1 < t2 < t3 < ...).”
From this point of view the continuous time random walk (CTRW) model of
Montroll and Weiss [2] (see also Refs. [3–5]) can provide a phenomenological
description of tick-by-tick dynamics in ﬁnancial markets [6–8].
1 Corresponding author. E-mail: mainardi@bo.infn.it URL: www.fracalmo.org
Preprint submitted to Elsevier Preprint 16 June 2002Actually in CTRWs, two random variables are used: jumps ξn = x(tn+1)−x(tn)
and waiting times τn = tn+1 − tn. In the ﬁnancial interpretation of CTRWs,
x represents a log-price and ξ a log-return [6–8] (see also [9]). The physicist
can think of x as the position of a random walker performing discrete jumps
in one dimension at randomly distributed instants. Based on [7] the evolution
equation for p(x,t), the probability of occurrence of the log-price x at time t,
or of ﬁnding the random walker at position x at time instant t, can be written,
assuming the initial condition p(x,0) = δ(x) (i.e. the walker is initially at the
origin x = 0),
p(x,t) = δ(x)Ψ(t) +
t Z
0
+∞ Z
−∞
ϕ(x − x
0,t − t
0)p(x
0,t
0)dx
0dt
0 , (1.1)
where Ψ(t) is the survival probability and ϕ(ξ,τ), is the joint probability den-
sity of jumps ξn = x(tn+1)−x(tn) and of waiting times τn = tn+1−tn. Relevant
quantities are the two (marginal) probability density functions (pdf’s) deﬁned
as λ(ξ) :=
R ∞
0 ϕ(ξ,τ)dτ , ψ(τ) :=
R +∞
−∞ ϕ(ξ,τ)dξ , and called jump pdf and
waiting-time pdf, respectively. If one assumes that the jump pdf λ(ξ) is in-
dependent of the waiting-time pdf ψ(τ), we have the so-called ”decoupling”
which leads to the factorisation ϕ(ξ,τ) = λ(ξ)ψ(τ).
The Eq. (1.1) is the most general master equation of the CTRW, usually de-
rived in the Fourier-Laplace domain. The simpliﬁed form under the hypothesis
of ”decoupling” is reported in [7].
The probability that a given inter-step interval is greater or equal to τ is Ψ(τ),
which is deﬁned in terms of ψ(τ) by
Ψ(τ) =
∞ Z
τ
ψ(t
0)dt
0 = 1 −
τ Z
0
ψ(t
0)dt
0, ψ(τ) = −
d
dτ
Ψ(τ). (1.2)
We note that
R τ
0 ψ(t0)dt0 represents the probability that at least one step is
taken at some instant in the interval [0,τ), hence Ψ(τ) is the probability that
the diﬀusing quantity x does not change value during the time interval of
duration τ after a jump. We also note, recalling that t0 = 0, that Ψ(t) is the
survival probability until time instant t at the initial position x0 = 0.
A relevant choice for the survival probability is given by the Mittag-Leﬄer
function of order β (0 < β < 1), which leads to a time-fractional master
equation as shown in [7] (see also [10,11]). For reader’s convenience hereafter
we recall the main properties of this transcendental function useful for our
purposes. From its deﬁnition valid for any β > 0:
Ψ(τ) = Eβ
h
−(τ/τ0)
β
i
:=
∞ X
n=0
(−1)
n (τ/τ0)βn
Γ(β n + 1)
, β > 0, (1.3)
2one recognises that the Mittag-Leﬄer function generalises the simple expo-
nential function (recovered for β = 1) and, if 0 < β < 1, it interpolates on
the positive real axis a stretched exponential and a power law according to
Eβ
h
−(τ/τ0)
β
i
∼
(
exp
h
−(τ/τ0)β/Γ(1 + β)
i
, τ/τ0 → 0+,
(τ/τ0)−β/Γ(1 − β), τ/τ0 → ∞,
0 < β < 1. (1.4)
For more information on the Mittag-Leﬄer function see e.g. [12–14].
The purpose of this paper is to investigate some statistical properties of the
random variables ξ and τ in ﬁnancial markets. This study is limited to a
speciﬁc equity of a given market in a deﬁnite period. Therefore caution is
necessary and our results cannot be arbitrarily generalized. In particular, the
reader will learn about General Electric stock prices, traded at NYSE, in
October 1999. This preliminary presentation is part of a broader project aimed
at studying the behaviour of all Dow-Jones-Industrial-Average stocks during
that month.
2 Empirical analysis
In Fig. 1, a scatter plot is presented for waiting times τn as a function of the
corresponding log-return ξn. By means of a contingency table analysis [15],
we have studied the independence of the two stochastic variables. A direct
inspection of Fig. 1 shows that for large values of log-returns waiting times
tend to be shorter. This indicates a possible correlation. Actually, a hypothesis
test has been performed on the empirical joint pdf ϕ(ξ,τ). According to the
contingency table presented in Tab. 1, the two random variables cannot be
considered independent. The null hypothesis of independence can be rejected
with a signiﬁcance level of 1%.
In Fig. 2, an estimate of the autocorrelation function for the absolute value of
log-returns is plotted. We have used the following estimator [16]
C(m) =
1
N − m
N−m−1 X
n=0
(|ξn+m| − |ξ|)(|ξn| − |ξ|), (2.1)
where N is the total number of points (N = 55559) and |ξ| = 1
N
PN−1
n=0 |ξn|.
The inset shows the time series of the absolute values as a function of the tick
n.
Due to scale persistence, the autocorrelation function follows a power-law
decay with a slope of −0.76. The autocorrelation is over the noise level (con-
ventionally 3/
√
N) for a lag between 20 and 30 ticks, corresponding to an
3τn
0 ÷ 10 10 ÷ 20 > 20
< −0.002 25 (38.9) 21 (10.1) 9 (6.0)
−0.002 ÷ −0.001 516 (613.6) 230 (159.5) 122 (94.9)
ξn −0.001 ÷ 0 6641 (7114.3) 2085 (1849.1) 1338 (1100.6)
0 ÷ 0.001 31661 (31008.0) 7683 (8059.2) 4520 (4797.0)
0.001 ÷ 0.002 398 (464.4) 179 (120.7) 80 (71.9)
> 0.002 34 (36.1) 10 (9.4) 7 (5.6)
Table 1
Contingency table between log-returns ξn and waiting times τn. Every cell contains
the frequency observed within the values considered and (in brackets) the theoret-
ical frequency which can be computed under the null hypothesis of independence
between ξn and τn.
average time of 250s. Therefore, within that time scale, it is not safe to as-
sume that the log-returns themselves, ξn, are independent variables. These are
well-known stylised fact for tick-by-tick ﬁnancial time series, see e.g. [17–19].
In Fig. 3, the autocorrelation function is shown for waiting-times τn. As above,
the inset shows the time series itself. Waiting times between trades are inher-
ently positive random variables. For the GE stock in October 1999, there is a
marked seasonality of waiting times with a 1-day period (nearly 3,000 trades).
Inspection of the series shows that the trading activity is slower in the mid-
dle of the day. The seasonality is outlined by the periodic behaviour of the
autocorrelation estimate, with periodicity above the conventional noise band.
In recent times, several eﬀorts have been devoted to ﬁnd a suitable measure
of time, in order to discard similar seasonalities, see e.g. [20,21].
However, as shown in Fig. 4, the survival probability Ψ(τ) can be ﬁtted by
a stretched exponential function: exp
h
−(τ/τ0)β/Γ(1 + β)
i
, with τ0 = 6.6s and
β = 0.7. The reduced chi-square of the ﬁt is 0.71.
In a previous work on bond futures [7], according to theoretical considera-
tions on the properties of continuous-time random walks, we suggested the
Mittag-Leﬄer function with a power-law decay as a suitable ﬁt for the em-
pirical survival probability. The above result does not contradict our previous
ﬁndings. In fact, whereas for bonds futures we found waiting times greater
than 10,000s, here we have only waiting times smaller than 200s, and the
Mittag-Leﬄer function is well approximated by the stretched exponential as τ
is small enough, see Eq. (1.4).
43 Summary
A preliminary study of General Electric high-frequency stock prices has been
performed. Some statistical properties of the log-return and waiting-time ran-
dom variables have been presented. This study was inspired by previous the-
oretical and empirical work, based on the phenomenological CTRW model of
ﬁnancial markets.
The main results are as follows: the two random variables cannot be considered
independent from each other; the autocorrelation of log-returns absolute values
exhibits a power-law decay and reaches the noise level after about 250 s; the
autocorrelation of waiting times shows a 1-day periodicity, corresponding to
the daily stock market activity.
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6−0.015 −0.01 −0.005 0 0.005 0.01 0.015
0
20
40
60
80
100
120
140
160
180
200
ξ
i
τ
i
 
{
s
}
Fig. 1. Scatter plot of waiting times τn as a function of the corresponding log-returns
ξn.
10
0
10
1
10
2 10
−4
10
−3
10
−2
10
−1
lag
a
u
t
o
c
o
r
r
e
l
a
t
i
o
n
 
o
f
 
a
b
s
o
l
u
t
e
 
l
o
g
 
r
e
t
u
r
n
s
noise level
slope ∼ −0.76
1 10000 20000 30000 40000 50000
0
0.005
0.01
0.015
ticks
a
b
s
o
l
u
t
e
 
l
o
g
 
r
e
t
u
r
n
s
Fig. 2. Autocorrelation function for the absolute value of log-returns ξn. The inset
shows the time series.
70 2000 4000 6000 8000 10000
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
lag
a
u
t
o
c
o
r
r
e
l
a
t
i
o
n
 
o
f
 
w
a
i
t
i
n
g
 
t
i
m
e
s
noise band
1 10000 20000 30000 40000 50000
0
50
100
150
200
ticks
w
a
i
t
i
n
g
 
t
i
m
e
s
 
{
s
}
Fig. 3. Autocorrelation function for the waiting times τn. The inset shows the time
series.
10
0
10
1
10
2 10
−5
10
−4
10
−3
10
−2
10
−1
10
0
τ {s}
Ψ
(
τ
)
market data                                  
stretched exponential (τ
0=6.6, β=0.7)
standard exponential (τ
0=6.6, β=1)   
Fig. 4. Survival probability. The stretched exponential (solid line) is compared with
the standard exponential (dash-dotted line).
8

A subordinated stochastic process model with ﬁnite variance for speculative prices,

An Introduction to Econophysics: Correlations and Complexity in Finance,

An Introduction to High-Frequency Finance,

Aspects and Applications of Random Walks,

Digital Spectral Analysis with Applications, Prentice-Hall,

Fractional calculus and continuous-time ﬁnance II: the waiting-time distribution,

Fractional calculus and continuous-time ﬁnance III: the diﬀusion limit,

Fractional calculus and continuous-time ﬁnance,

Fractional calculus, integral and diﬀerential equations of fractional order,

Fractional kinetic equations: solutions and applications,

On an enriched collection of stochastic processes, in:

On Mittag-Leﬄer type functions in fractional evolution processes,

On the wonderful world of random walks,

Option Pricing: The American Put,

Random walks on lattices,

Scaling and correlation in ﬁnancial time series,

Statistical properties of ﬁnancial time series, Lectures presented at the Symposium on Mathematical Finance,

Statistics: Theory and Methods,

Time deformation: deﬁnition and comparisons,

Waiting-times and returns in high-frequency financial data: an empirical study

http://128.118.178.162/eps/fin/papers/0411/0411014.pdf

Waiting-times and returns in high-frequency financial data: an empirical study

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