Statistical Properties of Stock Prices and Trading Volume and a Spin Model of
Stock Markets with Heterogeneous Agents
Taisei Kaizoji
Division of Social Sciences
International Christian University
Mitaka, Tokyo 181-8585, Japan
kaizoji@icu.ac.jp
http://subsite.icu.ac.jp/people/kaizoji/
1. Introduction
In the field of econophysics the findings of scaling power laws of financial time series [1] are
increasingly attracting the interest of researchers. In this paper we quantitatively investigate the
statistical properties of the stock prices and the trading volume. We analyze daily data of the price
and the trading volume for a large stock listed into the Tokyo Stock Exchange for the 26-year period
of 1975-2002. We find the power law distribution of the trading-volume change with an exponent
close to –1, the so-called Zipf distribution. We also investigate the relation between the trading
volume and the corresponding stock returns, and find the time series of the trading volume changes
show a clear on-off intermittency, while the series of the corresponding stock returns is tainted by
noise and on-off intermittency of the return series is not observed distinctly. We introduce a spin
model of stock markets originally developed by [4-6] in order to give an interpretation to our
empirical results on the relation between the trading-volume changes and the stock returns.
2. Data Analysis
2.1 On-off intermittency of the trading-volume change
We analyze the daily data on the stock price and the trading volume for the NITETUKOGYO, which
is one of the largest mining companies in Japan and is listed in the Tokyo Stock Exchange (TSE), for
the 26-year period of Jan. 1975-Jan. 2002. The time series of the stock price P(t) are shown in Fig. 1.
The figure clearly indicates that bubble and burst of the stock price occurred. From the time series,
we calculate the relative price change, that is, returns R(t), and the trading-volume changes V(t),
defined as
R( t ) l nP (t 1) l Pn t ( ) ,V t ( )S  t (  1 S) t ( ) .
(1)
Fig. 2a and Fig. 2b show the time series of the relative price change R(t), and of the trading-volume
change V(t), respectively. The series of the trading-volume change in Fig. 2b indicates clearly on-off
intermittency. Compared with the trading-volume series, the series of the relative price change in Fig.
2a seems to be polluted more by noise, and on-off intermittency is not observed distinctly.
Fig. 1a
3000
Stock Price; NITETUKOGYO (TSE);
Jan. 4, 1975-Jan. 9, 2002
2500
2000
1500
1000
500
0
750104
780705
820108
850709
890223
930315
970402
10423
Fig.2a
0.2
Log-Return; NITETUKOGYO (TSE);
July 25, 1986 - Jan. 13, 1988
0.1
0
-0.1
-0.2
860725
861204
870421
870828
880113
Fig. 2b
8000
Volume Change; NITETUKOGYO (TSE);
July 25, 1986 - Jan. 13, 1988
4000
0
-4000
-8000
-12000
860725
861204
870421
870828
880113
2.2 Correlation
The trading-volume changes as well as the relative price changes are correlated only on short time
scales (a few days). Unlike the trading-volume change, the absolute values of the trading-volume
changes show long-range autocorrelations on time scales up to a year.
The autocorrelation of the absolute values of the trading-volume changes is demonstrated to be
larger than the autocorrelation of the absolute values of the relative price changes in the first 60 lags.
Previous studies [2-4] that examine the relationship between price movements and trading
volume found several empirical regularities: (i) positive correlation between the relative price
change and the trading-volume; (ii) large price movements are followed by high trading volume.
In our case these regularities are hold, but the relationship is very weak. The correlation
coefficient between the relative price change and the trading-volume is about 0.2, and the
correlation coefficient between the relative price change and the trading-volume change is about
0.27. See Fig. 3.
Fig. 3
Correltaion
The Trading-Volume Change
12000
8000
4000
-0.3
-0.2
0
-0.1
0
-4000
0.1
0.2
0.3
-8000
-12000
The Relative Price Change
2.3 Zipf distribution of the trading-volume change
We next investigate the distribution of the trading-volume change. To test whether the scaling
behavior of the trading-volume change is observed, we compute the cumulative distribution of the
trading-volume change (Fig. 4). We find that the cumulative distribution of the trading-volume
change is consistent with a power law asymptotic behavior,
P (V  x) 
1
x
(2)
Regression fits yield estimates
1 . 1 4 f o r t h e p o s i t i v e t a i l
,
1 . 0 6 f o r t h e n e g a t i v e t a i l
 
(3)
well inside the stable Levy range. Furthermore, the exponent is close to unity. The so-called Zipf’s
law of the trading-volume change holds approximately. To test if the Zipf’s law is universal, we
repeat the same analyses to the several securities. We find that the Zipf’s law is robust.
Fig. 4
9
8
8
7
7
ln (Frequency)
ln (Frequency)
Power Law Distribution of Volume Changes:
The Negative Tial
Power Law Distribution of Volume Changes:
The Positive Tail
9
6
5
6
5
4
4
3
3
2
2
2
4
6
8
2
3
ln (Volume Change)
4
5
6
7
8
ln (Volume Change)
3. The spin model of stock market [7]
The trading volume has to be important to determine the movements of the price. However, as seen
above, the relation between the trading volume and the price is very weak. To consider the reason we
introduce a spin model of stock market originally developed by [5].
Let us consider a stock market. Two groups of traders with different trading strategies,
fundamentalists and interacting traders, who is a kind of noise traders, participate in the trading. The
numbers of fundamentalists m and the number of interacting traders n are assumed to be constant.
The model is designed to describe the stock price movements over short periods, such as one day. In
the following, a more precise account of the decision making of each trader type is given.
3.1 Fundamentalists
Fundamentalists are assumed to have a reasonable knowledge of the fundamental value of the stock
P*(t). If the price P(t) is below the fundamental value P*(t), a fundamentalist tends to buy the stock
(as he estimates the stock to be undervalued), and if the price is above the fundamental value, a
fundamentalist tends to sell the stock (then considered as an overvalued and risky asset). Hence we
assume that fundamentalists' buying or selling order is given by:
X F (t )  a m (ln P* (t )  ln P(t )) ,
(4)
where m is the number of fundamentalists, and a parametrizes the strength of the reaction on the
discrepancy between the fundamental price and the market price.
3.2 Interacting Traders
During each time period, an interacting trader may choose to either buy or sell the stock, and is
assumed to trade a fixed amount of the stock b in a period of trading. Interacting traders are labeled
by an integer 1  i  n . The investment attitude of interacting trader i is represented by the random
variable si and is defined as follows: If interacting trader i is a buyer of the stock during a given
period, then si  1, otherwise he sells the stock and si  1.
Now let us formulate the dynamics of the investment attitude of interacting traders in terms of the
spin model [5]. Let us consider that the investment attitude si of interacting trader i is updated with
a heat-bath dynamics according to
1

 si (t  1)  1 with p 
1  exp(2  hi (t )) ,

 s (t  1)  1 with 1  p
 i
(5)
where hi (t ) is the local field of the spin model, governing the strategic choice of the trader.
Let us consider the simplest possible scenario for local strategy changes of an interacting trader. We
assume that the decision which an interacting trader makes is influenced by two factors, local
information, as well as global information. Local information is provided by the nearby interacting
traders' behavior. To be definite, let us assume that each interacting trader may only be influenced by
its nearest neighbors in a suitably defined neighborhood. Global information, on the other hand, is
provided by the information whether the trader belongs to the majority group or to the minority
group of sellers or buyers at a given time period, and how large these groups are. The asymmetry in
size of majority versus minority groups can be measured by the absolute value of the magnetization
|M(t)| , where
M (t ) 
1 n
 si (t ).
n i 1
(6)
The goal of the interacting traders is to obtain capital gain through trading. They know that it is
necessary to be in the majority group in order to gain capital, however, this is not sufficient as, in
addition, the majority group has to expand over the next trading period. On the other hand, an
interacting trader in the majority group would expect that the larger the value of |M(t)| is, the more
difficult a further increase in size of the majority group would be. Therefore, interacting traders in
the majority group tend to switch to the minority group in order to avert the capital loss, e.g. to
escape a large crash, as the size of the majority group increases. In other words, the interacting trader
who is in the majority group tends to be a risk averter as the majority group increases. On the other
hand, an interacting agent who is in the minority group tends to switch to the majority group in order
to gain capital. An interacting agent in the minority group tends to be a risk taker as the majority
group increases.
To sum up, the larger |M(t)| is, the larger the probability with which interacting traders in the
majority group (interacting traders in the minority group, respectively) withdraw from their
coalition.
Following [5], the local field hi (t ) containing the interactions discussed above is specified by
n
hi (t )   J ij s j (t )   s j (t ) | M (t ) |
(7)
j 1
with a global coupling constant   0 . The first term is chosen as a local Ising Hamiltonian with
nearest neighbor interactions J ij  J and J ij  0 for all other pairs. We assume that the
interacting-traders' excess demand for the stock is approximated as
X I (t )  b n M (t ) .
(8)
3.3 Market price and trading volume
Let us leave the traders' decision-making processes and turn to the determination of the market price.
We assume a market clearing system. In the system a market maker mediates the trading and adjusts
the market price to the market clearing values. The market transaction is executed when the buying
orders are equal to the selling orders.
The balance of demand and supply is written as
X F (t )  X I (t )  a m (ln P* (t )  ln P(t ))  n b M (t )  0
(9)
Hence the market price and the trading volume are calculated as
ln P(t )  ln P* (t )   M (t ),  
1 | M (t ) |
bn
) . (10)
, and V (t )  b n (
2
am
Using the price equation (10) we can categorize the market situations as follows: If M(t) = 0, the
market price P(t) is equal to the fundamental price P*(t). If M(t) > 0, the market price P(t) exceeds
the fundamental price P*(t) (bull market regime). If M(t) < 0, the market price P(t) is less than the
fundamental price P*(t) (bear market regime). Using (8), the logarithmic relative change of price,
the so-called log-return, is defined as
ln P(t )  ln P(t  1)  (ln P* (t )  ln P* (t 1))   ( M (t )  M (t 1)) .
(11)
Let us consider the fundamental price P*(t) follows a stochastic process. Then it is not always that in
principle the log-return R(t) = ln P(t) - ln P(t-1) correlates with the trading-volume change V(t).
3.4 Simulation results
It is easily shown that the on-off intermittency in dynamics of the increments of the magnetization
M(t) appear form the spin model considered here. The bull (bear) market M(t) > 0 (M(t) < 0)
becomes unstable, and the transition from a metastable phase to a phase of high volatility occurs,
when the absolute magnetization |M(t)| approaches some critical value. Noting that trading volume is
defined as b n (1+|M(t)|)/2, this suggests that also some critical trading volume exists near the onset
of turbulent phases. The origin of the on-off intermittency can be seen in the local field hi (7)
representing the external influences on the decision-making of interacting-trader i. In particular, the
second term of hi tends to encourage a spin flip when magnetization gets large. Thus each
interacting-trader frequently switches his strategy to the opposite value if the trading volume gets
large. As a consequence, the bull (bear) market is unstable and the phase of the high volatility
appears.
These give an reasonable explanation to the empirical regularities we find above: the series of the
trading-volume change indicates clearly on-off intermittency, but the series of the relative price
change seems to be polluted more by noise, and on-off intermittency is not observed distinctly.
4. Concluding remarks
In this paper we analyze daily data of the stock prices and the trading volume for a large stock listed
into the Tokyo Stock Exchange for the 26-year period of 1975-2002. We find the power law
distribution of the trading-volume changes with an exponent close to –1, the so-called Zipf
distribution. We also investigate the relation between the trading volume and the corresponding
stock returns, and find the time series of the trading volume changes show a clear on-off
intermittency, while the series of the corresponding stock returns is tainted by noise and on-off
intermittency of the return series is not observed distinctly. We introduce a spin model of stock
markets, and give an interpretation to our empirical results on the relation between the
trading-volume changes and the corresponding stock returns.
The Zipf law of the distribution of the trading-volume change showed here is an interesting finding
theoretically. The theoretical study will be left for future work.
5. References
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correlation in financial time series, Physica A 287 362 (2000).
[2] J. M. Karpoff, The relation between price changes and trading volume: a survey, Journal of
Financial and Quantitative Analysis 22, 109-126 (1987).
[3] A.R. Gallant, P.E. Rossi, and G. Tauchen, Stock prices and volume, Review of Financial Studies
5 2, 199 (1992).
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traded in financial markets, Phys. Rev. E. (Rapid Comm.), 62 R4493 (2000).
[5] S. Bornholdt, Expectation bubbles in a spin model of markets: Intermittency from frustration
across scales, Int. J. Mod. Phys. C 12 667 (2001).
[6] T. Yamano, Bornholdt's spin model of a marketdynamics in high dimensions, Int. J. Mod. Phys. C
13 89 (2002).
[7] T. Kaizoji, S. Bornoldt, and Y. Fujiwara, Dynamics of price and trading volume in a spin model
of stock markets with heterogeneous agents, to be submitted into Physica A (2002).