ijmpc 2009

October 16, 2009 10:48 WSPC/141-IJMPC 01459

International Journal of Modern Physics CVol. 20, No. 10 (2009) 1547–1562c© World Scientific Publishing Company

STATISTICAL ANALYSIS BY STATISTICAL PHYSICS MODEL

FOR THE STOCK MARKETS

TIANSONG WANG, JUN WANG∗ and BINGLI FAN

College of Science, Beijing Jiaotong University

Beijing 100044, P. R. China∗[email protected]

Received 18 October 2008Accepted 17 May 2009

A new stochastic stock price model of stock markets based on the contact process ofthe statistical physics systems is presented in this paper, where the contact model is acontinuous time Markov process, one interpretation of this model is as a model for thespread of an infection. Through this model, the statistical properties of Shanghai StockExchange (SSE) and Shenzhen Stock Exchange (SZSE) are studied. In the present paper,the data of SSE Composite Index and the data of SZSE Component Index are analyzed,and the corresponding simulation is made by the computer computation. Further, weinvestigate the statistical properties, fat-tail phenomena, the power-law distributions,and the long memory of returns for these indices. The techniques of skewness–kurtosistest, Kolmogorov–Smirnov test, and R/S analysis are applied to study the fluctuationcharacters of the stock price returns.

Keywords: Statistical physics; stock prices; contact model; probability distribution;returns.

1. Introduction

In this paper, we study the fluctuation of stock prices and the return process in

a stock market by applying the theory of the contact model. Through computer

simulation on the financial model, we discuss the statistical behavior, the tail be-

havior, and long memory of fluctuation for the return processes, especially we study

the statistical properties of absolute value for the returns. And we also analyze the

data of Shanghai Stock Exchange (SSE) Composite Index and Shenzen Stock Ex-

change (SZSE) Component Index from 1 July 2002 to 30 June 2008, and study the

fluctuation characters of SSE Composite Index and SZSE Component Index, since

SSE Composite Index and SZSE Component Index are the most important secu-

rity indices in China. The database is from SSE and SZSE, see www.sse.com.cn

and www.sse.org.cn. The original attempt of this work is to study the financial

phenomena by statistical physics models.

Recently, some research work has been done in the field of applying the the-

ory of stochastic interacting particle dynamic systems to investigate the statistical

1547


1548 T. Wang, J. Wang & B. Fan

properties of fluctuations of stock prices in a stock market, and the corresponding

valuation and hedging of contingent claims for this price process model are also

studied, for example, see Refs. 8, 9, 16 and 17. As the stock markets are becom-

ing deregulated worldwide, the modeling of the dynamics of the forward prices is

becoming a key problem in the risk management, physical assets valuation, and

derivatives pricing, and it is also important to understand the statistical proper-

ties of fluctuations of stock price in globalized securities markets, and it is useful

for valuation and hedging of European option and American option. Stauffer and

Penna,16 Tanaka17 have studied the market fluctuation by the percolation model,

see Ref. 5. In Refs. 16 and 17, according to local interaction of percolation, the

local interaction or influence among investors in a stock market is constructed, and

a cluster of percolation is used to define the cluster of investors sharing the same

opinion about the market. In their study, they assume that the information in the

stock market leads to the stock price fluctuation and the investors in stock market

follow the effect of sheep flock, that is, the investment decision-making by other

investors’ dissemination of information, thus stock price fluctuation will eventually

depend on the investors’ investment attitude of the stock market. In this paper,

we apply a new statistical physics model (contact model) to study the fluctuation

characters of the return processes.

2. The Contact Model

The contact model is one of statistical physics models. One interpretation of this

model is a crude model of the spread of a disease or a biological population. In

one-dimensional space, one point is occupied by one individual. The virus infects

one proximate individual at a rate equal to λ, where λ is an intensity which is

called the “carcinogenic advantage” in contact model. And the individual infected

recovers at rate 1. Speaking concretely, contact model is a continuous time Markov

process ηt in the configuration 0, 1Zd

. At some random time, one individual at the

point is deemed to be infected when ηt(x) = 1 and the infected individual recovers

at rate 1; if ηt(x) = 0, the individual at the point x is healthy and will be infected

at a rate equal to λ times the number of the infected neighbors. For details, see

Ref. 12.

In one-dimensional contact model, for each x and y with |x − y| = 1, let

T(x,y)n , n ≥ 1 be a Poisson process with λ, and let Ux

n , n ≥ 1 be a Poisson

process with a rate 1. At times T(x,y)n , we draw an arrow from x to y and indicate

that if x is infected then y will become infected (if it is not already). At times Uxn , we

put a δ at x. The effect of a δ is to recover the individual at x (if one is infected). For

simplicity, we give the construction of graphical representation for one-dimensional

contact model, see Fig. 1. To construct the process from this “graphical represen-

tation,” we imagine fluid entering the bottom at the points in η0 and flowing up

the structure. The δ’s are the dams and the arrows are pipes which allow the fluid

to flow in the indicated direction.


Statistical Physics Model for the Stock Markets 1549

Fig. 1. The graphical representation of one-dimensional contact model. The x-axis is the sites

and the y-axis is the time. The arrows express T(x,y)n , n ≥ 1, the Poisson process of infecting

neighbors. The δ expresses Uxn , n ≥ 1, the Poisson process of recovering himself.

Next we show the properties of contact process around the critical value λc,

which comes from Theorem 3.33 in Ref. 12. On Ω = 0, 1Z and with the cor-

responding probability P , let η0t (x) be the state of x ∈ Z at time t with the

initial point 0, and define the edge processes rt = maxy : η0t (y) = 1, lt =

miny : η0t (y) = 1, r0 = l0 = 0. Define τ = inft ≥ 0 : η

0t = ∅ and the critical

value λc as

λc = infλ : P (|η0t | > 0, for all t ≥ 0) > 0 . (1)

The estimate of the critical value λc (for one-dimension case 0, 1Z) is known as

λc ≈ 1.6494, see Ref. 12.

Then, by Theorem 3.33 in Ref. 12, we have the following. Suppose λ > λc and

η0t = 0, then

limt→∞

∑rt

y=ltη0t (y)

t= 2α(λ)ρ(λ) a.s. on τ = ∞ , (2)

where ρ(λ) ≥ 0, is a nondecreasing function of λ, and α(λ) ≥ 0. And if λ < λc, for

some positive ρ(λ), we have

P (η0t 6= ∅) ≤ e−ρt , (3)

then the process dies out exponentially fast. The estimate of the critical value λc(d)

of the d-dimensional contact process is given by

1

2d − 1≤ λc(d) ≤

2

d,

see Corollary 4.4 in Ref. 12. We can obtain similar results as (1)–(3) for d-

dimensional contact model.



3. The Financial Model Based on Contact Model

3.1. Modeling

We have introduced the contact model above. In the following, we make use of

the contact model to construct the return process of stock price on d-dimensional

integer lattice. In this paper, we also assume that the stock price fluctuates because

of the information in the stock market. The stock price fluctuation will eventually

depend on the investors’ investment attitude of the stock market. We assume that

the viruses of contact model represent the information in the stock market and

construct the stock price process based on the contact model. The information is

divided into two kinds of information (buying information and selling information).

Then the price model for the investors is interpreted as follows: At the beginning

of trading on each day, suppose that only the investor at the site 0 receives some

news. We define a random variable ξ0 for this investor, suppose that this investor

takes buying position (ξ0 = 1), selling position (ξ0 = −1) or neutral position

(ξ0 = 0) with probabilities p1, p−1 or 1 − (p1 + p−1), respectively. Then this

investor sends bullish, bearish or neutral signal to his nearest neighbors. According

to d-dimensional contact process system, investors can affect each other or the news

can be spread, which is assumed as the main factor of price fluctuations for type-2

investors. Moreover, here the investors can change their buying positions or selling

positions to neutral positions independently at a constant rate. More specifically, (I)

when ξ0 = 1 and if η0t (x) = 1, we say that the investor at x takes buying position

at time t, and this investor recovers to neutral position 0 at rate 1; if η0t (x) = 0, we

think the investor at x takes neutral position at time t, and this investor is changed

to take buying position by his nearest neighbors at rate λ∑

y:|y−x|=1 η0t (y). In

this case, the more investors with taking buying positions, the more possible the

stock price goes up. (II) When ξ0 = −1 and if η0t (x) = 1, we say that the

investor at x takes selling position at time t, also this investor recovers to neutral

position 0 at rate 1; if η0t (x) = 0, the investor is changed to take selling position

by his nearest neighbors at rate λ∑

y:|y−x|=1 η0t (y). (III) When the initial random

variable ξ0 = 0, the process η0t (x) is ignored, this means that the investors do

not affect the fluctuation of the stock price.

In other words, there are two contact processes in the financial model, the buying

process and the selling process (with the probability p1 and p−1, respectively),

which are identified by the initial investor’s attitude at the site 0 when t = 0.

The good information is spread in the buying process, where the investors take the

buying positions. And in the selling process where the bad information is spread,

the investors take the selling positions. In Sec. 3.2, the initial position of the model

is extended to the initial distribution νθ, that is at time t = 0, there is a certain

fraction of investors (with the initial density θ) knowing the information at the

beginning of the trading day which can be in buying, neutral, and selling positions

(with probabilities probability p1, 1 − (p1 + p−1) or p−1, respectively), then the

financial model evolves according to the contact system which is defined above.



According to the theory of contact process (see Ref. 12) and above definitions,

if λ > λc, the news will spread widely, so this will affect the investors’ positions,

and finally will affect the fluctuation of the stock price. If λ ≤ λc, the influence on

the stock price by the investors is limited. We compute the number of the investors

with the buying positions in buying process Σ1(t), and the number of investors with

the selling positions in selling process Σ2(t). Finally, the effect of the investment

can be calculated as

Σ(t) = Σ1(t) − Σ2(t) .

As we know, see Refs. 16 and 17, stock price S(t) is decided by the differential

equation dS/dt = αχ(t)S(t), here we let χ(t) = Σ(t)/N , where N is a large integer

and α is the constant of proportionality. According to the theory of mathematical

finance,1,10,15,18 we obtain the formula of a stock price as following

S(t + 1)

S(t)= exp

αΣ(t)

N

.

But this definition produces one problem that the extent of price change going up

is more than that of price change going down in the same absolute value of Σ(t).

It will weaken the effect of the data of price change going down, leading to getting

imprecise statistic result. So we improve the formula of stock price as

S(t + 1)

S(t)=

exp

αΣ(t)

N

, change up

2 − exp

αΣ(t)

N

, change down

.

This formula ensures that the extent of stock price fluctuation is almost the same

when the absolute values of Σ(t) (for positive value and negative value) are the

same. So the research results given by the financial model of the present paper are

more credible. According to the definition of returns (see Refs. 10, 15 and 18), we

have the formula of stock logarithmic return

r(t) = ln S(t + 1) − ln S(t) .

In this paper, we analyze the logarithmic returns for the daily price changes.

We plot the figures of the stock price series and the returns by simulating the

one-dimensional contact model, see Figs. 2 and 3.

3.2. The simulation and statistical analysis of data

In order to investigate the distribution of price returns, the supercritical contact

model is applied to study the market fluctuations (i.e. λ > λc(d)), and four param-

eters of the contact model are discussed in this paper, the intensity λ, the lattice

dimension d, the number of individual n, and the initial density θ (or the initial

distribution νθ of the model). In the following, the plots for the absolute normalized

price changes are used to show the computer simulations of the empirical data. For



0 500 1000 1500 2000 2500 3000 3500 4000 4500 50006

8

10

12

14

16

18

days

Price

s

Fig. 2. Price series of simulation.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

days

retu

rns

Fig. 3. Return series of simulation.

the different values of some parameter (for example, the intensity λ), we compare

the fluctuation characters of the normalized price changes with the corresponding

Gaussian distribution, and study the statistical properties of the financial model.

Next we consider the price changes for the different intensity values λ.

λ is the intensity, and represents the rate of information spread in the model.

The faster the information spread, the larger the value of λ is. From the simulations

in Fig. 4 and Table 1, we can see that, for fixed three parameters d = 1, n = 400,

and θ = 0.1, the behavior of the price changes depends on the intensity value λ. The

kurtosis and µ are increasing for the parameter λ. The peak distribution of returns



10−4

10−3

10−2

10−1

10−2

10−1

100

log|r|

log(

P(|r

|>x)

)

λ=10λ=25λ=30Gaussian

log–log plot

0.01 0.02 0.03 0.04 0.05

10−2

10−1

100

|r|

log(

P(|r

|>x)

)

λ=10λ=25λ=30Gaussian

semilog y plot

Fig. 4. The plots of the cumulative distributions of normalized price returns with different valuesof λ when d = 1, n = 400, and θ = 0.1.

Table 1. The statistics of values λ.

λ Min Max Mean Variance Kurtosis Skewness µ

10 −0.0833 0.0815 7.8696e − 005 5.2823e − 006 3.0333 −0.0221 5.9319

20 −0.0859 0.0817 1.0739e − 004 7.2190e − 006 3.2852 −0.0423 4.7333

25 −0.0861 0.0799 1.4972e − 004 2.2893e − 005 3.3809 0.0166 4.1475

30 −0.0846 0.0797 9.5067e − 005 4.2592e − 005 5.3933 −0.0182 3.8280

40 −0.0823 0.0771 −2.8753e − 005 5.1284e − 005 8.4874 −0.0909 3.2955

Table 2. The statistics of values n.

n Min Max Mean Variance Kurtosis Skewness µ

400 −0.0631 0.0628 −2.7328e − 005 6.4949e − 005 3.6398 −0.0283 4.3780

729 −0.0768 0.0712 −9.2875e − 005 3.3225e − 005 3.3431 −0.0260 6.0026

2197 −0.0738 0.0794 −4.6224e − 004 1.5301e − 005 3.1891 0.0485 6.7873

4096 −0.0685 0.0678 1.6685e − 005 1.1185e − 005 3.0232 −0.0344 7.9253

is obvious and the fat tail is also visible. This implies that the rate of infected speed

can affect the behavior of fat tails.

n is the number of individual investors, and also represents the size of investment

in a stock market. In Fig. 5 and Table 2, for d = 3, λ = 3, and θ = 0.01, we discuss

the statistical behavior of the price changes on the parameter n. For n = 729, the

tail of probability distribution of the simulative data departs sharply from that of

Gaussian distribution. For n = 2197 and 4096, the departure is less sharp. This

shows, for the number of individuals n, the probability distribution of the price

changes deviates from the Gaussian distribution. We can learn that the large size

of investment of stock market can weaken the fluctuation of the stock market.



10−4

10−3

10−2

10−1

10−2

10−1

100

log|r|

log(

P(|r

|>x)

)

n=400n=729n=2097Gaussian

log–log plot

0.01 0.02 0.03 0.04 0.05

10−2

10−1

100

|r|

log(

P(|r

|>x)

)

n=400n=729n=2097Gaussian

semilog y plot

Fig. 5. The plots of the cumulative distributions of normalized price returns with different valuesof n when d = 3, λ = 3, and θ = 0.01.

10−4

10−3

10−2

10−1

10−2

10−1

100

log|r|

log(

P(|r

|>x)

)

d=1d=2d=3Gaussian

log–log plot

0.01 0.02 0.03 0.04 0.05

10−2

10−1

100

|r|

log(

P(|r

|>x)

)

d=1d=2d=3Gaussian

semilog y plot

Fig. 6. The plots of the cumulative distributions of normalized price returns with different valuesof d when n = 729, λ = 3, and θ = 0.1.

Table 3. The statistics of values d.

d Min Max Mean Variance Kurtosis Skewness µ

1 −0.0521 0.0521 −4.7743e − 006 3.6827e − 007 2.8238 0.0326 7.8798

2 −0.0743 0.0730 1.0580e − 005 4.8003e − 005 3.1936 0.0142 6.0413

3 −0.0906 0.0917 −3.5756e − 005 3.1689e − 004 8.4079 −0.0984 3.3914

Next we continue to study the statistical properties of price changes for the

different dimensions d.

d is the lattice dimension. In Fig. 6 and Table 3, we can see (when increasing

dimension d) a more obvious fat-tail phenomena appear in the plots. One reason

is that interaction among individuals becomes more active in the financial contact



10−4

10−3

10−2

10−1

10−2

10−1

100

log|r|

log(

P(|r

|>x)

)

θ=0.05θ=0.1θ=0.15Gaussian

log–log plot

0.01 0.02 0.03 0.04 0.05

10−2

10−1

100

|r|

log(

P(|r

|>x)

)

θ=0.05θ=0.1θ=0.15Gaussian

semilog y plot

Fig. 7. The plots of the cumulative distributions of normalized price returns with different valuesof θ when d = 1, n = 400, and λ = 30.

Table 4. The statistics of values θ.

θ Min Max Mean Variance Kurtosis Skewness µ

0.01 −0.0817 0.0728 −6.1356e − 005 5.7115e − 006 3.2484 −0.0157 6.7635

0.05 −0.0968 0.0997 1.9895e − 005 1.9136e − 005 3.0012 0.0231 5.5280

0.10 −0.0846 0.0797 9.5067e − 005 4.2592e − 005 5.3933 −0.0182 3.8280

0.15 −0.0725 0.0669 8.0384e − 006 8.5742e − 005 7.7987 −0.1772 3.6728

0.20 −0.0603 0.0601 −6.5290e − 006 1.0407e − 004 9.3220 −0.1660 3.3129

model when we increase the number of dimension d. We learn that when n = 729,

λ = 3, θ = 0.1, and d = 1, the simulative data follows Gaussian distribution, it

means that if the interaction is not active enough, the fat-tail phenomena disappear.

But for the higher-dimension, the graph of the empirical data deviates from the

corresponding Gaussian distribution graph.

θ is the initial density, representing the proportion of individuals knowing the

information at the beginning of each trading day. In Fig. 7 and Table 4, when

increasing θ, the value of kurtosis increases, and the more obvious the fat-tail phe-

nomena can be seen in the plots. The reason is that the more the proportion of

individuals learning the information, the more the possibility of the information to

spread broadly.

4. The Statistical Comparison of SSE Composite Index, SZSE

Component Index, and the Financial Model

In this section, we compare the returns of SSE Composite Index and SZSE Com-

ponent Index with the empirical data from the financial model constructed by

the contact model. In recent years, the probability distribution in financial market



fluctuation has been studied, the empirical research results show that the distri-

bution of large returns follow a power-law distribution with exponent 3, that is

P (rt > x) ∼ x−µr , where rt is the returns of the stock prices in a given time inter-

val 4t, µr ≈ 3 , for example, see Refs. 2, 6, 7, 11, and 14. According to the statistical

methods and data analyzing methods (see Refs. 2, 3, 6, and 14), we will study the

cumulative probability distributions of daily returns and the power-law character

of daily returns for Shanghai stock market and Shenzhen stock market, and we also

simulate the corresponding cumulative probability distributions of returns by the

financial model, which is constructed based on the contact model.

4.1. The probability distribution of returns

We analyze the data from SSE Composite Index and SZSE Component Index from

1 July 2002 to 30 June 2008, and also analyze the simulative data from the model

with λ = 30, n = 729, θ = 0.15, and d = 3.

The fluctuation of returns are believed to follow a Gaussian distribution for

long time intervals but to deviate from it for short steps, so we try to study the

probability distribution of returns for SSE Index, SZSE Index, and the simulative

data. According to the data, we plot the probability density of the daily returns

figures as follows (Fig. 8).

Comparing with the Gaussian distribution, the probability densities of SSE

Composite Index, SZSE Component Index, and simulative data obviously show the

phenomena of the fat-tail and peak distribution in Fig. 8.

4.2. The skewness kurtosis

test and Kolmogorov Smirnov test of returns

In this section, we study the properties of skewness and kurtosis on the data of

returns for SSE Composite Index, SZSE Component Index, and the simulative

data. First, we give the definitions of skewness and kurtosis as following

skewness =

∑ni=1(ri − ur)

3

(n − 1)σ3, kurtosis =

∑ni=1(ri − ur)

4

(n − 1)σ4

where rt denotes the return of ith trading day, ur is the mean of r, n is the total

number of the data, and σ is the corresponding standard variance. The kurtosis

shows the centrality of data and the skewness shows the symmetry of the data. It

is known that the skewness of standard normal distribution is 0 and the kurtosis

is 3. From the data of SSE Index, SZSE Index, and the simulation, we have the

following Table 5.

In Table 5, the value of kurtosis of returns for the actual and simulative markets

are both more than 3, and the values of skewness are close to −0.1. This implies

that the statistical distribution of the data deviates from the Gaussian distribution,

and it also implies that the fluctuation of returns for SSE Index, SZSE Index, and

the simulative data is greater than that of the corresponding Gaussian distribution.



−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

Normalize r

Pro

babi

lity

Data of SSEData of Gaussian

(a)

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

Normalize r

Pro

babi

lity

Data of SZSEData of Gaussian

(b)

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

Normalize r

Pro

babi

lity

Data of SimulationData of Gaussian

(c)

Fig. 8. The probability distributions of (a) SSE Composite Index, (b) SZSE Component Index,and (c) the simulative data.

Table 5. The statistical properties of returns of SSE Composite Index, SZSEComponent Index, and the simulative data.

SSE Composite Index SZSE Component Index Simulative

Sample size 1455 1454 5000

Min −0.0884 −0.0929 −0.0801

Max 0.0929 0.0959 0.0955

Mean 4.5992e − 004 8.3990e − 004 3.8051e − 004

Variance 2.7555e − 004 3.3090e − 004 2.1963e − 004

Skewness −0.2300 −0.1541 −0.1004

Kurtosis 6.8189 6.2569 5.9844

For deeply analyzing the character of distribution of SSE Index, SZSE Index,

and the simulative data, we make another single-sample Kolmogorov–Smirnov test

by the statistical soft, MATLAB. We use the MATLAB to calculate the value of



Table 6. The Kolmogorov–Smirnov test.

SSE Composite Index SZSE Component Index Simulative

Sample size 1455 1454 5000

H 1 1 1

CV 0.0287 0.0294 0.0387

Two-tail test P 0.000 0.000 0.000

Statistics of K–S 0.4749 0.4720 0.4889

the basic statistics of the returns of SSE Index, SZSE Index, and the simulative

data, see Table 6.

The value of two-tail test P is 0.000, thus the hypothesis is denied that the

distribution of returns follow the Gaussian distribution.

Through the above two tests, the distribution of SSE Index, SZSE Index, and

the simulative data obviously have the fat-tail character, and have little distinct

skewness. So if we properly choose the parameters of the financial model, we can fit

the actual distribution of SSE Composite Index and SZSE Component Index well.

4.3. The power-law behavior

Power-law scaling is the universal property that characterizes collective phenomena

that emerge from complex systems composed of many interacting units. Power-law

scaling has been observed not only in physical systems, but also in economic and

financial systems. In this section, we study the cumulative probability distribution

of returns for SSE Composite Index, SZSE Component Index, and the simulative

data, and try to show the power-law distribution of returns. In recent years, the

empirical research has shown the power-law tails in the return fluctuations, that

is, P (rt > x) ∼ x−µ, and some research results show that the distribution of

large returns follow a power-law distribution with exponent µ = 3. We plot the

cumulative probability distribution of returns for SSE Index, SZSE Index, and the

simulative data as followings.

In Fig. 9, the probability distribution of returns follows a power-law, which is

determined by ordinary least-squares regression in log–log coordinates. The dis-

tribution follows the power-law behavior in the large price range, but gradually

deviates from the power-law as the absolute return becomes small.

4.4. The long memory testing for returns

The evaluation of the long memory of one series can be made through various

methodologies. Among the methodologies which are used today for the identification

and the quantification of long memory, the classic R/S analysis is one of the most

common methodologies. It can be used without prior knowledge of the factors that

act in the price generating process, taking into consideration only the series of



10−4

10−3

10−2

10−1

10−2

10−1

100

log|r|

log(

P(|r

|>x)

)

10−1

10−2

log|r|lo

g(P

(|r|>

x))

k=−2.8651

(a)

10−4

10−3

10−2

10−1

10−2

10−1

100

log|r|

log(

P(|r

|>x)

)

10−1

10−2

log|r|

log(

P(|r

|>x)

) k=−2.7336

(b)

10−4

10−3

10−2

10−1

10−3

10−2

10−1

100

log|r|

log(

P(|r

|>x)

)

10−1

10−3

10−2

10−1

log|r|

log(

P(|r

|>x)

)

k=3.1043

(c)

Fig. 9. The cumulative probability distributions of returns for SSE Composite Index (a), SZSEComponent Index (b) and simulative data (c), and µ = 2.8651 (Shanghai), µ = 2.7336 (Shenzhen),µ = 3.1043 (Simulation).



returns or volatilities for which it is desired to estimate long memory. In this paper,

the long memory is measured by the Hurst exponent H , calculated through the

classic R/S analysis. When 0 < H < 0.5, the analyzed series is anti-persistent,

presenting reversion to the mean; if H = 0.5, the series presents random walk and

if 0.5 < H < 1, the series is persistent, with the maintenance of tendency. In order

to calculate the Hurst exponent H , the calculations of the (R/S)n statistic are

made for diverse lengths of the n block, given that n < N (number of observations

in the series). At the end of these calculations, there is a table of n values and

(R/S)n corresponding statistics. The Hurst exponent H is obtained by the following

expression, defined by Mandelbrot and Wallis,13

ln

(

R

S

)

n

= ln C + H ln n .

The standard error of H is obtained through the regression by the method of the

ordinary least squares, this shows that the data in different points (n, (R/S)n) are

correlated, because they are constructed from aggregations of the same series of

data.4

In Fig. 10, we obtain the relation between the exponent H and n. The way

we calculate is that n = 100 is the starting point of the regression; gradually

increasing n to obtain a value of H regressing once each additional day, up to 30%

of the length of the sequence. We also find that the values of H are all bigger than

0.5 for SSE Composite Index, SZSE Component Index, and the data of simulation.

Furthermore, the maximum value of H is 0.6363 at n = 282 for SSE Composite

Index, the maximum value is 0.6367 at n = 282 for SZSE Component Index, and

the maximum value is 0.6091 at n = 89 for the data of simulation. Finally, the

exponent H of returns for SSE Composite Index is 0.6363, the length of memory

is 282 days; the exponent H of the returns for SZSE Component Index is 0.6367,

the length of memory is 321 days; the exponent H of the returns for the data of

simulation is 0.6091, the length of memory is 89 days. These series are persistent

with the maintenance of tendency.

5. Conclusion

In the present paper, we apply the statistical model — the contact model to con-

struct a financial model (a price changes model). From this model, we try to un-

derstand the statistical properties (for example, the fat-tail phenomena and the

power-law behavior) of the price changes at the supercritical case. The statistical

properties of SSE Composite Index and SZSE Component Index are also studied

in this paper. And we compare the statistical properties of actual and simulative

markets. We hope that the work can help the investors to understand the fluctua-

tion of a stock market better and to invest more reasonably. This paper also gives a

new approach to study the fluctuation of a stock market by applying the financial

model based on the contact model.



2 2.5 3 3.5 4 4.5 5 5.5 6 6.51

1.5

2

2.5

3

3.5

log(n)

log

(R/S

)

150 200 250 300 350 400 4500.58

0.59

0.6

0.61

0.62

0.63

0.64

n

The v

alu

e o

f H

(a)

2 2.5 3 3.5 4 4.5 5 5.5 6 6.51

1.5

2

2.5

3

3.5

log(n)

log

(R/S

)

150 200 250 300 350 400 4500.6

0.605

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0.645

n

Th

e v

alu

e o

f H

(b)

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 71

1.5

2

2.5

3

3.5

4

log(n)

log

(R/S

)

0 100 200 300 400 500 600 700 800 9000.51

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

0.61

n

The v

alu

e o

f H

(c)

Fig. 10. The fluctuation of exponent H of returns for (a) SSE Composite Index, (b) SZSEComponent Index, and (c) the simulative data.



Acknowledgments

The authors are supported in part by National Natural Science Foundation of China

Grant No. 70771006, BJTU Foundation No. 2006XM044.

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