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Poisson-process generalization for the trading waiting-time
distribution in a double-auction mechanism
Silvano Cincottia, Linda Pontaa, Marco Rabertoaand Enrico Scalasb
aDIBE, Universit`a di Genova, Via Opera Pia 11a, 16145 Genova, Italy
bDISTA, Universit`a del Piemonte Orientale, via Bellini 25 G, 15100 Alessandria, Italy
ABSTRACT
In the last years, intensive use of computer-based trading system produced large amount of financial data. This
resulted in a growing interest in analysis and modelling of high frequency financial time-series, in particular order
and trading waiting times. Generally speaking, for constant market activity and in the absence of memory, the
waiting time distribution is assumed to be exponential. Indeed, only an exponential distribution can results in a
memoryless process. However, recent empirical studies have shown that the trading waiting times point process
is not a Poisson process with exponential waiting-time survival function.
In this paper, empirical analyses and computational experiments are presented on high-frequency data for a
double-auction (book) market. Main objective of the paper is to generalize the order waiting time process in
order to properly model such empirical evidences.
The empirical study is performed on the best bid and best ask data of 7 U.S. financial markets, for 30-stock
time series. In particular, statistical properties of trading waiting times have been analyzed and quality of fits is
evaluated by suitable statistical tests, i.e., comparing empirical distributions with theoretical models.
Starting from the statistical studies on real data, attention has been focused on the reproducibility of such results
in an artificial market. The computational experiments have been performed within the Genoa Artificial Stock
Market. In the market model, heterogeneous agents trade one risky asset in exchange for cash. Agents have zero
intelligence and issue random limit or market orders depending on their budget constraints. The price is cleared
by means of a limit order book. The order generation is modelled with a renewal process.
Based on empirical trading estimation, the distribution of waiting times between two consecutive orders is
modelled by a mixture of exponential processes. Results show that the empirical waiting-time distribution can
be considered as a generalization of a Poisson process. Moreover, the renewal process can approximate real data
and implementation on the artificial stocks market can reproduce the trading activity in a realistic way.
Keywords: Agent-based simulation, artificial financial market, limit order book, order and trading waiting
times
1. INTRODUCTION
The dynamics of a stock market depends on the interaction between trading mechanism and trading behaviors.
The behavior of the participants is the outcome of their trading strategies, which include how they form expec-
tations or interpret signals. Conversely, the trading mechanism defines the rules of the market, which specify
how orders are placed and how the price changes.
A model of the stock market makes structural assumptions, related to the trading mechanism. In order-driven
systems, competing market makers supply liquidity by quoting bid and ask prices and volumes at which they
are willing to trade. Investors demand liquidity through the submission of market orders. A limit order book
is a snapshot at a given instant of the queues of all buy limit orders and sell limit orders, with their respective
price and volume.
All buy limit orders are below the best buy limit order, i.e., the buy limit order with the highest limit price
(the bid price). The best buy limit order is situated below the best sell limit order, i.e., the sell limit order with
Further author information: (Send correspondence to Silvano Cincotti)
Silvano Cincotti: E-mail: cincotti@dibe.unige.it, Telephone: +39 010 353 2080.
1
the lowest limit price (the ask price). All other sell limit orders are above the best sell limit order. A transaction
occurs when a trader hits the quote (the bid or the ask price) on the opposite side of the market. If a trader
issues a limit order, say a buy limit order, the order either adds to the book if its limit price is below the ask
price, or generates a trade at the ask if it is larger or equal to the ask price. In the latter case, the limit order
becomes a marketable limit order, or more simply, a market order. Conversely, if the order is a sell limit order
it becomes a market order and is executed if its limit price is below the bid price, otherwise it is stored in the
book. Limit orders with the same limit price are prioritized by time of submission, with the oldest order given
highest priority. The execution of orders often involves partial fills before it is completed, but partial fills do not
change the time priority.
In recent years, some studies on the statistical properties of the limit order book have proposed by the scientific
community.1–5 An important empirical variable is the waiting time between two consecutive transactions.
Indeed, trading via the order book is asynchronous, i.e., a transaction occurs only if a trader issues a market
order. For liquid stocks, waiting times may vary in a range between some seconds to a few minutes. Generally
speaking, the survival of order waiting times is modelled by an exponential.6However, analysis of the intra-
day trades of General Electric stock prices pointed out that trading waiting times exhibit a 1-day periodicity,
corresponding to the daily stock market activity, and that the survival probability distribution is properly fitted
by a stretched exponential.7–9 Moreover, a significative cross-correlation between waiting times and the absolute
value of log-returns was also found which suggested other memory effects.
Starting from these results, the effects of a more general distribution of order waiting times have been investigated.
In particular, attention has been focused on the Weibull distribution that admits the exponential distribution
as a limit case.10 Results showed that in the case of exponentially distributed order waiting times, also trade
waiting times are exponentially distributed. Conversely, if order waiting times follow a Weibull, the same does
not hold for trading waiting times. Thus, such studies concluded that a single Weibull distribution of the order
waiting time could not reproduce the empirical evidences.
In this paper, order and trading waiting times are studied in the general framework of the Genoa Artificial
Stock Market.6, 11–14 A trading mechanism characterized by a double auction clearing mechanism, i.e., the limit
order book, is considered.15, 16 In particular, we use mixture of Poisson process to describe the order generation
process. The characteristics of the Poisson process in the mixture are estimated by high frequency real data that
points out changes of the average waiting time during the trading day.17 Results pointed out that a mixture of
Poisson process can reproduce the behavior of real stock market.
2. THE ARTIFICIAL STOCK MARKET
In this section, a model of artificial trading by means of a limit order book is presented. The model makes
reference to the Genoa Artificial Stock Market - GASM.10–14, 16 In the GASM, agents trade a single stock in
exchange for cash. They are modelled as liquidity traders, and the decision making process is nearly random
constrained by the finite amount of financial resources (cash + stocks) they own. At the beginning of the
simulation, cash and stocks are uniformly distributed among agents.
2.1. Trading decision making process
The GASM is populated by random trader, i.e., a trader issues a buy or a sell order with probability 50%. Let
us denote with a(th−1) and with d(th−1) the values of ask and of bid prices stored in the book at time step th−1.
Let us now assume that the order issued at time step this a sell order. Thus, the quantity of stocks offered for
sale is a random fraction of the quantity of stocks owned by the trader whereas the limit price si(th) associated
to the sell order is given by:
si(th) = ni(th)·a(th−1),(1)
where ni(th) is a random draw by trader iat time step thfrom a Gaussian distribution with constant mean
µ= 1 and standard deviation σ. If si(th)> di(th−1), the limit order is stored in the book, otherwise the order
becomes a market order and a transaction occurs at the price p(th) = d(th−1). In the latter case, the sell order
is partially or totally fulfilled and the bid price is updated.
If the order is a buy order, the associated limit price bi(th) is given by:
bi(th) = ni(th)·d(th−1) ; (2)
2
where ni(th) is determined as in the previous case. If bi(th)< a(th−1), the limit order is stored in the book,
otherwise the order becomes a market order and a transaction occurs at the price p(th) = a(th−1). The quantity
of stocks ordered to buy depends on cash endowment of the trader and on the value of bi(th).
It is worth noting that, in our framework, agents compete for the provision of liquidity. If an agent issues a
buy order, its benchmark is the best limit buy order given by the bid price. Being µ= 1, he offers in average
a more competitive buy order (if bi(th)> d(th−1)), which may result in a trade if bi(th)≥a(th−1). The same
applies for sell limit orders.
2.2. Order generation process
Trading is organized in Mdaily sections and each trading day is subdivided in Telementary time steps, say
seconds. During the trading day, at given time steps th, a trader iis randomly chosen for issuing an order. The
trading day can be divided into Lsubintervals. Within each subinterval, waiting times τh=th−th−1follow an
exponential distribution with different average waiting times τ1
0, ..., τ L
0. Recalling that the rate µiis the inverse
of the average waiting time, i.e. µi= 1/τi
0, the survival function of the i-th subinterval is given by
pi(τ) = e−µiτ,(3)
where i= 1, ..., L.
In recent years, several author have shown that, in average, trading waiting time are not uniformly distributed
during a trading day. In particular, Bertram pointed out that the number of trading in time interval of 600
seconds (i.e, 10 minutes) are variable in time with a typical smile pattern17 (see Figure 1). According to his
result, we used the average number of trading to estimate the rate parameter µiof the exponential distribution
that represents the order waiting time distribution. It is worth noting that such estimation is based on trades
instead of orders. However, one can assume that a linear relationship generally exist between the number of
issue orders and the number of transactions.
Generally speaking, the average number of trading in a period of 600 seconds is a function of time (see Figure
1). In this study, we have modelled the average number of transections by means of a polynomial approximation
i.e.,
n(t) =
J
X
j=0
ajtj(4)
Moreover, least squared fit on the empirical data pattern reported by Bertram17 points out that Eq. 4 with
J= 8 is in good agreement with the empirical results (as shown in Figure 1). Thus, Eq. 4 allows one to evaluate
the rate of the exponential distribution in the sub-interval as
µi=n(ti)
600 (5)
where i= 1, ..., L. It is worth noting that n(ti) denotes the value of the polynomial calculated in the middle
of i-th interval (see circle in Figure 1). As a consequence, in any time interval, agents issue orders according to
an exponential distribution whose rate changes in time, i.e., order waiting times are distributed according to a
mixture of Poisson processes.
3. EMPIRICAL ANALYSIS
In this paper, the trading waiting time for 30 DJIA titles traded at NYSE in October 1999 have been consid-
ered. The time-series have been statistically analyzed and Table 1 points out main results. The first column
reports the Anderson-Darling statistics A2. In these cases, the critical value is 1.9 and, as clearly stated, the null
hypothesis of exponential distribution is rejected for all real time-series. This conclusion is in perfect agrement
with previous results, thus pointing out that the point process of trading is not a Poisson process.8, 10
The second column shows estimation of Weibull exponent parameter β. The Weibull process generalizes the
3
Poisson process. Indeed, the exponential distribution is a particular case of the Weibull distribution for β= 1.
In the case β < 1, the Weibull distribution assumes the form of the so-called stretched exponential and great
values of trading waiting times occur with higher probability than in the case of β= 1.8, 10 As shown, the
estimated βare lower than 1, i.e., the trading waiting times distribution are stretched exponentials.
The third column reports Ljung-Box test statistics Qat lag 15. The critical value of the test is 24.99 at the 5%
significance level. As clearly stated, the null hypothesis of no serial correlation is generally rejected, except for
the case of Citigroup Inc. (denoted by ∗), that is the only case in which test is not rejected.
Furthermore, Table 2 summarizes serial correlation results. First, second and third columns reports autocorre-
lation value of trading waiting times for different lags, measured in seconds, for IBM stock in December 1990,
January 1991, October 1999. Fourth column shows the same statistics of the trading times for Citigroup Inc.
(C), i.e., the only stock for which the null hypothesis of no serial correlation is not rejected.
The data sets used to produce Table 2 have different characteristics. First sample exhibits an average time
interval between trades of 26.2 seconds, with minimum and maximum interval of 0 and of 4,592 seconds (i.e.,
about 1 hour and 15 minutes), respectively. Moreover, standard deviation results of 53.9 seconds and skewness
is equal to 42.3. Second sample has an average of 27.07 seconds with minimum and maximum interval of 0
and of 426 seconds (i.e., about 7 minutes), respectively. The standard deviations is 37.69 seconds. The third
sample has an average time interval between trades of 9.8 seconds, minimum interval is 0 seconds, and maximum
interval is 3,600 seconds. The standard deviation is 34.3 seconds and skewness is 91.4. The fourth sample has
an average time interval between trades of 9.2 seconds, with minimum and maximum interval of 0 seconds and
3,655 seconds respectively, the standard deviation is 41.8 seconds and skewness is 82.2.
The autocorrelations of trading waiting times in Table 2 point out the presence of correlation, i.e., autocorre-
lation functions decrease slowly with exception of C asset whose autocorrelation function decrease faster. This
result is confermed by the Ljung-Box statistical test. Indeed, Ljung-Box statistics are very large, except for the
stock C. Thus, the null hypothesis of white noise is generally rejected based on the critical value of 24.99 at the
5% significance level.
4. COMPUTATIONAL EXPERIMENTS
Besides real data analysis, some computational experiments have been considered. The timing parameters of
every simulation have been set as follows: M= 50 daily sections, each characterized by a length of T= 21,000
s. Each simulation is characterized by a different number of exponential distributions used in the mixture. In
our simulation we can choose how many exponential distributions use. We have chosen 7 different value for
L: 2,3,4,5,15,25,35. The orders lifespan has been set to 600 s À hτoi.
Sell and buy limit prices are computed following Eq. 1 and Eq. 2 respectively. The random number ni(th)
is a random draw by trader ifrom a Gaussian distribution with constant mean µ= 1 and standard deviation
σ= 0.005.
The number of agents is set to 10,000. At the beginning of the simulation, the stock price is set at 100.00
units of cash, say dollars and each trader is endowed with an equal amount of cash and of shares of the risky
stocks. These amounts are 100,000 dollars and 1,000 shares, respectively.
The effects of the number of exponential distribution on the order and trading waiting times have been
considered. In particular, mixtures of 2,3,4,5,15,25,35 exponential distributions have been studied, being 35
the maximum time resolution allowed by empirical data.17
Figure 2 shows the survival functions of order and trading waiting times. As clearly stated, the effect of large
numbers of exponential distribution on the survivals of order and trading waiting times results almost negligible,
thus suggesting that already a mixture of two Poisson processes can properly represent the empirical evidences.
It is worth noting that a mixture of two exponential process per trading day corresponds to order waiting times
identically distributed at the beginning and at the end of the trading day. In fact, we can think that the Poisson
process at the end of one day is exactly the same at the beginning of the next day.
Figure 2 points out that waiting times generated by the GASM are not distributed according to a Poisson process.
This is further confirmed by the Anderson-Darling test, shown in Table 3. According to the critical value of 1.9,
we reject the null hypothesis of exponential distribution.10, 18 Conversely, Weibull distribution has been verified
with the Kolmogorov-Smirnov test. As shown in Table 4 the null hypothesis is never rejected, thus allowing the
4
possibility of Weibull processes for order and trading waiting times.
In addition to Kolmogorov-Smirnov test, estimation of β(i.e., the Weibull parameter) for GASM data points out
values quite close to those obtained in the case of 30 DJIA stocks traded at NYSE in October 1999 (see Tables
5 and 1 for a comparison). This suggest applicability of the proposed model in order to reproduce empirical
evidences. Moreover, in the case of βestimation, results point out again an almost negligible effect of the number
of exponential distribution in the mixture.
Concerning serial correlation, we have calculated Ljung-Box test statistics Qat lag 15 for the order τOand
trading τTwaiting times. As shown in Table 6, the null hypothesis is rejected. Moreover, fifth column in Table 2
points out the autocorrelation of trading waiting times for different lags in the case of 35 exponential distribution
in the mixture.
It is worth noting that GASM data are referred to 50 trading days, instead of the one month trading data set
for columns 1-4. Moreover, the GASM data are characterized by an average time interval between trades of
130.61 seconds, with minimum and maximum intervals of 0 and 3109 seconds, respectively. Standard deviation
is equal to 181.06 seconds and skewness is 5.08. The statistical properties of GASM intertrade durations are not
in contradiction with those measured for real data. This confirm that the GASM can reproduce the beavior of
real stock market.
5. CONCLUSION
Empirical analysis and computational experiments of high-frequency data for a double-auction (book) market
have been presented. Empirical analysis confirmed that trading waiting times are not exponentially distributed,
whereas a Weibull process cannot be rejected. This result deserves attention because exponentially distributed
trade waiting times only result from a finite thinning of a Poisson order process. Consequently, the distribution
of order waiting times should be a more general distribution than an exponential distribution underlying a simple
memoryless Poisson process.
In order to better understand the relationship between order and trading waiting times, computer experiments
on the Genoa Artificial Stock Market have been considered. In particular, a double-auction mechanism (i.e.,
limit order book) with order waiting times characterized by a mixture of Poisson process has been modelled and
implemented. The characteristics of Poisson process in the mixture have been properly estimated by real data.
It has been shown that, both order and trading waiting times, reject hypothesis of exponentially distributed
point process. Conversely, the hypothesis of Weibull distribution cannot be rejected and estimation of the
corresponding Weibull parameter βpointed out values quite close to those obtained in the case of 30 DJIA
stocks traded at NYSE in October 1999. Finally, the influence of the number of exponential distribution of the
mixture appears almost negligible as far as survivals of order and trading waiting times and βdo not change
significantly. This allows one to conclude that already a mixture of two Poisson process can be sufficient in order
to reproduce the behavior of real stock market.
ACKNOWLEDGMENTS
This work is partially supported by the University of Genoa and the Italian Ministry of Education, University
and Research (MIUR) under grants FIRB 2001 and COFIN 2004.
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6
10:10 12:30 14:10 16:00
5
10
15
20
25
30
35
40
45
50
55
trading hours
n
Figure 1. Average number of transactions during the trading day. Dotted line represents real data,17 continuous line
the polynomial fit and dashed curve the discretization of polynomial fit in 15 intervals.
0 500 1000 1500 2000 2500 3000
10−5
10−4
10−3
10−2
10−1
100
Survival probability distribution
τ {s}
L = 2
L = 3
L = 4
L = 5
L = 15
L = 25
L = 35
Figure 2. Survival probability distributions of order and trading waiting times for different mixtures of exponential
distributions. Continuous line: order distributions; dashed curves: trading distributions.
7
Table 1. Anderson-Darling test statistics A2,βestimated and Ljung-Box test statistics Qat lag 15 for trading waiting
times τTof the 30 DJIA titles traded at NYSE in October 1999.
A2β Q
Stocks τT
AA inf 0.74 569
ALD 111 0.72 1058
AXP inf 0.58 401
BA inf 0.73 815
C inf 0.48 20∗
CAT 291 0.76 754
CHV inf 0.74 403
DD inf 0.68 63
DIS inf 0.64 238
EK 123 0.85 997
GE inf 0.55 110
GM 144 0.84 1131
GT 262 0.78 3098
HWP inf 0.57 634
IBM inf 0.51 208
IP inf 0.76 439
JNJ 107 0.64 1812
JPM inf 0.66 797
KO 134 0.61 2537
MCD 177 0.75 2007
MMM 211 0.80 1264
MO inf 0.60 551
MRK inf 0.55 601
PG 126 0.61 2252
S inf 0.75 655
T inf 0.55 347
UK 182 0.99 1529
UTX 169 0.80 777
WMT inf 0.58 58
XON 399 0.68 3469
8
Table 2. Trading-interval autocorrelations function and Ljung-Box statistics for 15 lags. The first two columns are taken
from Engle and Russell (1994).19
IBM (Dec 1990) IBM (Jan 1991) IBM (Oct 1999) C (Oct 1999) GASM (L=35)
lag 1 0.168 0.129 0.061 0.003 0.165
lag 2 0.090 0.120 0.009 0.011 0.157
lag 3 0.068 0.106 0.005 0.005 0.140
lag 4 0.074 0.119 0.005 0.005 0.133
lag 5 0.059 0.107 0.007 0.004 0.151
lag 6 0.069 0.096 0.003 0.006 0.144
lag 7 0.051 0.100 0.002 0.003 0.104
lag 8 0.046 0.099 0.003 0.004 0.102
lag 9 0.045 0.123 0.004 0.005 0.114
lag 10 0.042 0.085 0.003 0.005 0.074
lag 11 0.043 0.105 0.002 0.003 0.101
lag 12 0.045 0.087 0.001 0.004 0.060
lag 13 0.047 0.089 0.005 0.004 0.075
lag 14 0.037 0.089 0.001 0.004 0.054
lag 15 0.025 0.083 0.002 0.002 0.058
Q1272 2423 208 20 1583
Table 3. Anderson-Darling test statistics, A2for order τOand trading τTwaiting times. Critical value of the test is 1.9.
In all cases, the null hypothesis of exponential distribution is rejected.
A2
L τOτT
2 52 53
3 66 71
4 20 22
5 31 34
15 61 57
25 84 84
35 73 64
9
Table 4. Kolmogorov-Smirnov test statistics Dfor order waiting times τOand trade waiting times τT.
D
L τOτTCritical Value
2 0.033 0.029 0.052
3 0.030 0.039 0.050
4 0.025 0.018 0.051
5 0.026 0.024 0.051
15 0.032 0.026 0.052
25 0.034 0.025 0.051
35 0.033 0.026 0.51
Table 5. Value of βfor trading waiting times and orders waiting times for GASM data.
L βOβT
2 0.93 0.87
3 0.92 0.86
4 0.96 0.91
5 0.95 0.89
15 0.92 0.87
25 0.91 0.85
35 0.92 0.86
Table 6. Ljung-Box test statistics Qat lag 15 for order waiting times τOand trade waiting times τT. The critical value
of the test is 24.99. In all the cases, the null hypothesis of no serial correlation up to lag 15 is rejected.
Q
L τOτT
2 6405 1543
3 7970 1155
4 2371 562
5 3429 526
15 7442 1380
25 7662 1792
35 8717 1583
10
