# Voter interacting systems applied to Chinese stock markets

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Journal Identification = MATCOM Article Identification = 3656 Date: June 16, 2011 Time: 11:26 am

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 81 (2011) 24922506

Original article

Voter interacting systems applied to Chinese stock marketsTiansong Wang, Jun Wang , Junhuan Zhang, Wen Fang

Department of Mathematics, College of Science, Beijing Jiaotong University, Beijing 100044, PR ChinaReceived 23 November 2008; received in revised form 14 March 2011; accepted 30 March 2011

Available online 17 April 2011

Abstract

Applying the theory of statistical physics systems the voter model, a random stock price model is modeled and studied inthis paper, where the voter model is a continuous time Markov process. In this price model, for the different parameters values ofthe intensity , the lattice dimension d, the initial density , and the multivariate set (, ), we discuss and analyze the statisticalbehaviors of the price model. Moreover, we investigate the power-law distributions, the long-term memory of returns and thevolatility clustering phenomena for the Chinese stock indices. The database is from the indices of Shanghai and Shenzhen in the6-year period from July 2002 to June 2008. Further, the comparisons of the empirical research and the simulation data are given. 2011 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Stock price model; Voter model; Probability distribution; Return; Computer simulation

1. Introduction

As the governments deregulate the stock markets, it is becoming an important problem to model the dynamics ofthe forwards prices in the risk management, derivatives pricing and physical assets valuation. And the research workon the statistical properties of fluctuations of stock prices in globalized securities markets is also significant. Recently,some research work has been done, by applying the interacting particle systems, to investigate the statistical behaviorsof fluctuations of stock prices, and to study the corresponding valuation and hedging of contingent claims, for instancesee [11,12,17,21,22]. Stauffer and Penna [21] and Tanaka [22] have been apply the percolation model to study themarket fluctuation, see [8]. In [21,22], they construct the local interaction or influence among investors in a stockmarket and define the cluster of investors with the same opinion about the market as a cluster of percolation. In theirfinancial models, they suppose that the stock price fluctuation is influenced by the information in the stock market, andthe investors follow the effect of sheep flock. That means that the investors decide the investment opinions by otherinvestors attitudes, so the investors investment attitudes of the stock market lead to the stock price fluctuation. In thepresent paper, we apply a statistical physics model the voter model (see [3,15]) to study the fluctuation behaviors ofthe return processes. Through the computer simulation on this financial model, we discuss the statistical behavior, thetail behavior, long term memory of fluctuation and the volatility clustering phenomena for the return processes.

In recent years, the empirical research in financial market fluctuations has been made. Some statistical properties formarket fluctuations uncovered by the high frequency financial time series, such as fat tails distribution of price changes,the power-law of logarithmic returns and volume, volatility clustering which is described as on-off intermittency in

Corresponding author. Fax: +86 10 51682867.E-mail address: wangjun@bjtu.edu.cn (J. Wang).

0378-4754/$36.00 2011 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2011.03.013

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T. Wang et al. / Mathematics and Computers in Simulation 81 (2011) 24922506 2493

Fig. 1. The chart of the voter model.

literature of nonlinear dynamics, and multifractality of volatility, etc., see [1,57,9,10,12,14,1619]. China has twostock markets, Shanghai Stock Exchange and Shenzhen Stock Exchange, and the indices studied in this paper areShanghai Stock Exchange Composite Index (SSECI) and Shenzhen Stock Exchange Component Index (SZSECI).These two indices play an important role in Chinese stock markets and Chinese economic systems. In this paper, weanalyze the daily data of SSECI and SZSECI from July 1st 2002 to June 30th 2008, and study the fluctuation propertiesof SSECI and SZSECI. The database of the indices is from the websites www.sse.com.cn and www.szse.cn.

First, we give the brief definitions and properties of the voter model, and also we introduce the graphical representa-tion of the model, since the graphical representation is necessary for our computer simulations. One interpretation forthe voter model is, for a collection of individuals, each of which has one of two possible positions on a political issue,at independent exponential times, an individual reassesses his view by choosing a neighbor at random with certainprobabilities and then adopting his position. Specifically, the voter model is one of the statistical physics models, wethink of the sites of the d-dimensional integer lattice as being occupied by persons who either in favor of or opposedto some issue. To write this as a set-valued process, we let s is the set of voters in favor, we can also think of the sitesin s as being occupied by cancer cells, and the other sites as being occupied by healthy cells. We can formulate thedynamics as follows: (i) an occupied site becomes vacant at a rate equal to the number of the vacant neighbors; (ii) anvacant site becomes occupied at a rate equal to times the number of the occupied neighbors, where is a intensitywhich is called the carcinogenic advantage in voter model. When = 1, the model is called the voter model, andwhen > 1, the model is called the biased voter model.

For simplicity, we give the construction of graphical representation for 1-dimensional voter model ( = 1), for moregeneral cases, see [3,15]. Thinking of 1-dimensional integer points as being laid out on a horizontal axis, with the timelines being placed vertically, above that axis. Define independent Poisson processes with rate 1 for each time lines, ateach event time (u, s), we choose one of its two neighbors with probability 1/2, and draw an arrow from that neighborpoint to (u, s), and write a at (u, s), see Fig. 1. To construct the process from this graphical representation, weimagine fluid entering the bottom at the points in 0 and flowing up the structure. The s are the dams and the arrowsare pipes which allow the fluid to flow in the indicated direction. For example in above Fig. 1, fluid can flow from thesite ( 2, 0) to the site (0, t). Let As denote the state at time s with the initial state A0 = A, then from [1,14], the votermodel As approaches total consensus in d = 1 and d = 2. But in higher dimensions d 3, the differences of opinion maypersist. For more generally, we consider the initial distribution as , the product measure with density , that is, eachsite is independently occupied with probability and let s to denote the voter model with initial distribution . Let{0}s = {0}, and {0}s (u) be the state of u Zd at time s with the initial point {0}.

More formally, the stochastic dynamics of the voter model s is a Markov process on = {0, 1}Zd whose generatorhas the form

Af () =u

c(u, )[f (u) f ()]

Journal Identification = MATCOM Article Identification = 3656 Date: June 16, 2011 Time: 11:26 am

2494 T. Wang et al. / Mathematics and Computers in Simulation 81 (2011) 24922506

where the functions f (on ) depend on finitely many coordinates, and u(v) = (v) if v /= u, u(v) = (v) if v = u,for u, v Zd . c(u, ) is the transition rate function for the process which is given by follows (see [1,14]), for any = {0, 1}Zd

c(u, ) =

v

p(u, v)(v) if (u) = 0v

p(u, v)[1 (v)] if (u) = 1

where p(u, v) 0 for u, v Zd , andvp(u, v) = 1 for all u Zd . And we assume that p(u, v) is such that the Markovchain with those transition probabilities is irreducible.

For the biased voter model ( > 1), there is a critical value for the process, that is, on = {0, 1}Zd and with thecorresponding probability P, the critical value c is defined as

c = inf{ : P(|{0}s| > 0, for all s 0) > 0}

where |{0}s | is the cardinality of {0}s . Suppose > c, then there is a convex set C so that on = {{0}s /= , for all s},we have for any > 0 and for all s sufficiently large (see [3,15])

(1 )sC Zd {0}s (1 + )sC Zd

If < c, for some positive (), we haveP({0}s /= ) es

The above results imply that, on d-dimensional lattice, if < c, the process dies out (becomes vacant) exponentiallyfast, if > c, the process survives with the positive probability.

2. Modeling a stock price model by the voter dynamic systems

In this part, the return process of a stock price on d-dimensional integer lattice is constructed, based on the votermodel. We assume that the information in the stock market lead to the fluctuation of a stock price, and there are threekinds of information: buying information, selling information and neutral information. And the fluctuation of a stockprice relies on the investors investment attitudes, which accordingly classify buying stock, selling stock and holdingstock. Consider a model of auctions for the same stock defined above, we can derive the stock price process fromthe auctions. Assume that each trader can trade the stock several times at each day t {1, 2, . . ., n}, but at most oneunit number of the stock at each time. Let l be the time length of trading time in each trading day, we denote thestock price at time s in the t th trading day by St(s), where s [0, l]. Suppose that this stock consists of m2 + 1 (m2 islarge enough) investors, who are located in a line {m2/2, . . . , 1, 0, 1, . . . , m2/2} Z (similarly for d-dimensionallattice Zd). At the beginning of trading in each day, suppose that only the investor at the site {0} receives some news.We define a random variable t({0}) for this investor, suppose that this investor taking buying position (t({0}) = 1),selling position (t({0}) = 1) or neutral position (t({0}) = 0) with probability p1, p1 or 1 (p1 + p1) respectively.Then this investor sends bullish, bearish or neutral signal to his nearest neighbors. According to d-dimensional voterprocess system, investors can affect each other or the news can be spread, which is assumed as the main factor of pricefluctuations for the investors. Moreover, here the investors can change their buying positions or selling positions toneutral positions independently at a constant rate. More specifically, (I) when t({0}) = 1 and if {0}s (u) = 1, we saythat the investor at u takes buying position at time s, and this investor recovers to neutral position 0 at a rate equalto the number of the vacant neighbors; if {0}s (u) = 0, we think the investor at u takes neutral position at time s, andthis investor is changed to take buying position by his nearest neighbors at rate

v:|vu|=1

{0}s (v). In this case, the

more investors with taking buying positions, the more possible the stock price goes up. (II) When t({0}) = 1 and if{0}s (u) = 1, we say that the investor at u takes selling position at time s, also this investor recovers to neutral position 0

at a rate equal to the number of the vacant neighbors; if {0}s (u) = 0, the investor is changed to take selling position by

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0 100 200 300 400 500 600 700 800 900 10009.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

Days

Pric

es

Fig. 2. The fluctuations of the stock prices model.

his nearest neighbors at rate

v:|vu|=1{0}s (v). (III) When the initial random variable t({0}) = 0, the process {0}s (u)

is ignored, this means that the investors do not affect the fluctuation of the stock price.For a fixed s [0, l] (l large enough), let

Bt(s) = t({0}) |{0}s |

m2(n), s [0, l]

where |{0}s | =m2/2

u=m2/2{0}s (u), and m2 depends on the trading days n. From above definitions and [1,12,19,22], we

define the stock price at t th trading day t(t = 1, 2, . . ., n) asSt(s) = eBt (s)St1(s), s [0, l]

where > 0, represents the depth parameter of the market. Then we have

St(s) = S0 exp{t

k=1Bk(s)}, s [0, l]

where S0 is the stock price at time 0. According to the theory of the voter process (see [15]) and above definitions, if > c, the news will spread widely, so this will affect the investors positions, and finally will affect the fluctuation ofthe stock price. If c, the influence on the stock price by the investors is limited.

Next we define the returns of stock prices, see [13,20,23], we have the formula of the stock logarithmic return asfollows

r(t) = ln St+1(s)St(s)

, t = 1, 2, . . . , n

In this paper, we analyze the logarithmic returns for the daily price changes. In the followings, we let s = 2000,then we plot the fluctuations of the stock prices and the corresponding returns by simulating the one-dimension votermodel, see Figs. 2 and 3.

3. Data analysis by the computer simulation

In recent years, the probability distribution in the financial market fluctuation has been studied, the empiricalresearch results show that the distribution of the large returns follow a power-law distribution with the exponent 3, thatis P(r(t) > x) xr , where r(t) is the returns of the stock prices in a given time interval t, r 3, for example see[5,9,10,14,19]. In this section, Section 3.1 presents the statistical analysis of the simulation data for univariate process.

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0 100 200 300 400 500 600 700 800 900 10000.03

0.02

0.01

0

0.01

0.02

0.03

Days

Ret

urns

Fig. 3. The fluctuations of market returns.

Section 3.2 investigates the returns for multivariate process. Finally, in Section 3.3, the conclusions of the analysisbetween univariate process and multivariate process are summarized.

3.1. Analysis for univariate process

In this section, we discuss the three parameters of the financial model, the intensity , the lattice dimension d, theinitial density (or the initial distribution of the model). And we show some figures and tables to describe thepower-law distribution of the returns for the simulation data. Further, we compare the power-law distribution of thereturns with the corresponding Gaussian distribution, and study the statistical properties of the price model by thedifferent values of one parameter. First, we discuss the behaviors of the returns for the different values of .

According to the computer computatio...