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Physica A 363 (2006) 377–382

Microscopic spin model for the dynamics of the return

distribution of the Korean stock market index

Jae-Suk Yang?, Seungbyung Chae, Woo-Sung Jung, Hie-Tae Moon

Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea

Received 16 November 2005; received in revised form 15 December 2005

Available online 23 January 2006

Abstract

In this paper, we studied the dynamics of the log-return distribution of the Korean Composition Stock Price Index

(KOSPI) from 1992 to 2004. Based on the microscopic spin model, we found that while the index during the late 1990s

showed a power-law distribution, the distribution in the early 2000s was exponential. This change in distribution shape was

caused by the duration and velocity, among other parameters, of the information that flowed into the market.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Econophysics; Emerging market; Log return; Power-law distribution; Exponential distribution

1. Introduction

Interdisciplinary research is now routinely carried out, with econophysics being one of the most active

interdisciplinary fields [1–5]. Many research papers on mature markets have already been published. However,

since emerging markets show different characteristics to those of mature markets, they represent an active field

for econophysicists. The Korean market, one of the foremost emerging markets, has already been studied by

physicists [4,5]. We concentrate on the particular properties of the Korean market through the return

distribution.

It is broadly assumed that the distribution of price changes takes the form of a Gaussian distribution, and

that all information is applied to the market immediately by the efficient market hypothesis (EMH) [6]. Using

the EMH, the trading profit with arbitrage cannot be obtained from the superiority of information. The price

changes in an efficient market cannot be predicted and change randomly. This is suited to classical economics

theory. However, experimental proofs reveal that Gaussian distributions of price changes do not exist in real

markets [7,8].

Mandelbrot determined empirically that the tail part of the distribution is wider and the center of the

distribution is sharper and higher than a Gaussian distribution by examining price changes of cotton; this

distribution of price changes is termed the Le ´ vy stable distribution [7]. Fama also found a Le ´ vy stable

distribution for the New York Stock Exchange (NYSE) [8]. After Mandelbrot’s study, the distribution of price

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?Corresponding author. Fax: +82428692510.

E-mail address: yang@kaist.ac.kr (J.-S. Yang).

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changes was identified as non-Gaussian by many researchers [9–12]. It was reported that distributions in

mature markets have a power-law tail, while those in emerging markets have an exponential tail [13,14].

Silva et al. [11] and Vicente et al. [12] reported that the distributions of price changes vary with time lag.

Price changes have a power-law distribution for a short time lag, and an exponential distribution when the

time lag is long. Moreover, for a very long time lag, the distribution becomes Gaussian. Transition problem of

the tail behavior was solved analytically by Dragulescu and Yakovenko [15] using a stochastic model [16].

Financial markets are adaptive evolving systems, so the distributions of price changes are also different for

various periods and countries. Especially, the Korean stock market has different properties compared with

other countries, and the distribution of price changes is not stable and changes with time.

In this paper, we study the characteristics of the Korean stock market using probability distribution

functions (PDFs) of the Korean Composition Stock Price Index (KOSPI) and investigate why phenomena

different to other countries occur. We also carry out a simulation using the microscopic spin model to explain

these differences.

2. Empirical data and analysis

We use the KOSPI data for the period from 1992 to 2004, and observe the PDFs of the KOSPI price

changes for a time window of 1 year. We use only intra-day returns to exclude discontinuous jumps between

the previous day’s close and the next day’s open price due to overnight effects. The price change log return is

defined by

SðtÞ ? lnYðt þ DtÞ ? lnYðtÞ,

where YðtÞ is the price at time t and Dt is the time lag.

Fig. 1a shows the log return distribution for the KOSPI. The distribution for an 1-min time lag represents a

power-law distribution, that for 10min is exponential, and that for 30min is also exponential but close to

Gaussian. These results are in accordance with previous reports [11,12].

In the Korean stock market, the log return distribution for an 1-min time lag shows some peculiar

phenomena. Fig. 1b shows the PDFs of the KOSPI log return from 1998 to 2002, while Fig. 1c is a graph of

the tail index, power-law exponent of tail part of the log return distribution, as a function of time from 1993 to

2004. The shape of the distribution in 1998 ðBÞ in Fig. 1b is close to a Le ´ vy distribution and the tail part

shows a power-law distribution. However, the tail index of the PDFs increases over the years and the shape of

tail part changes to exponential with increasing time. This phenomenon can be confirmed in Fig. 1c. As well

the tail index in the early 1990s is approximately 2.0, increases from the mid-1990s to the early 2000s, finally

changes to an exponential tail. Although a discontinuity in the increasing trend occurred during the 1997

Asian financial crisis, the tail index continued to increase thereafter.

In Fig. 2a, the decay time for the autocorrelation function of log return is continuously decreasing, while the

tail index abruptly varied around the time of the 1997 Asian financial crisis. This suggests that the decay time

is related to the increasing trend of the tail index, regardless of the Asian financial crisis. Thus, we investigated

the relation between the decay time for the autocorrelation function of log return and the tail index of the log

return distribution to identify why this phenomenon has happened. Fig. 2b shows the relation between the tail

index and the decay time. The decay time of the autocorrelation function is inversely proportional to the tail

index by a factor of 4.

The decay time of the autocorrelation function decreases as the log return distribution of the KOSPI

changes from a power-law to an exponential distribution (Fig. 2). This decrease in decay time means that the

duration of information in the market is diminished compared with the past, that is, past information

decreases more rapidly when the decay time is shorter. Information and communication technology such as

high-speed internet connections and electronic trading systems were not fully utilized in the past, so it took a

longer time to deliver information to the market. Moreover, the information flow was small because social

structures in the past were relatively simpler. For this reason, much more information flows into the market

for a specific time interval now compared with the past. Therefore, the time scale of the past differs from the

current scale. The amount of information and its velocity in reaching the market for 1min now may be the

same as that for 2 or 3min or more in the past.

(1)

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(c)

(a)

(b)

Fig. 1. (a) PDFs of log return for the KOSPI. Rectangle(’) represents 1-min time lag, circle(?) 10-min, and triangle(m) 30-min,

respectively. (b) PDFs of log return for the KOSPI from 1998 to 2002: grey (B) 1998, yellow (5) 1999, blue (4) 2000, green (?) 2001, and

red (&) 2002. (c) Evolution of the KOSPI tail index. The tail first increases and then changes to exponential. After 2003, tail index cannot

be calculated because of the distribution is changed to exponential. The dashed line represents the 1997 Asian financial crisis.

(a)

(b)

Fig. 2. (a) Evolution of the decay time for the autocorrelation function (ACF) and the tail index. (b) Relation between decay time and tail

index for the ACF.

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3. Model and results

Eguiluz and Zimmermann [17], Krawiecki et al. [18], Chowdhruy and Stauffer [19], and Cont and Bouchaud

[20] used microscopic models to simulate the financial market, and these models describe well the

characteristics of the financial market. Krawiecki et al. described the market as a spin model. Agents and

traders comprising the market are represented by spins, and interaction between agents and external

information is represented by fields.

We modify the Krawiecki microscopic model of many interacting agents to simulate the variation of log

return distribution for the Korean stock market. The number of agents is N, and we consider i ¼ 1;2;...;N

agents with orientations siðtÞ ¼ ?1, corresponding to the decision to sell ð?1Þ or buy ðþ1Þ stock at discrete

time-steps t. The orientation of agent i at the next step, siðt þ 1Þ, depends on the local field:

IiðtÞ ¼1

N

j

X

AijðtÞsjðtÞ þ hiðtÞ,(2)

where AijðtÞ represent the time-dependent interaction strength among agents, and hiðtÞ is an external field

reflecting the effect of the environment. The time-dependent interaction strength among agents is

AijðtÞ ¼ AxðtÞ þ aZijðtÞ,

with xðtÞ and ZijðtÞ determined randomly in every step. A is an average interaction strength and a is a deviation

of the individual interaction strength. The external field reflecting the effect of the environment is

(3)

hi¼ h

X

1

k¼0

ziðt ? kÞe?k=t,(4)

where h is an information diffusion factor, and ziðtÞ is an event happening at time t and influencing the ith

agent. t is the duration time of the information, which represents how long the event at time t retains influence

on the opinion of agents on market prices. At every step, agents are assumed to receive newly generated

information. The later the event, the greater is the influence on agents and the market. The most recent event

has a strong influence on agents, while old events have a weak influence. Influence on the market is considered

to be reduced exponentially. Moreover, for larger t, an event at time t retains a relatively strong influence on

agent opinions and market price changes for a long time. On the other hand, for shorter t, the information

from an event is rapidly delivered to agents and applied to market prices immediately, so the information

quickly vanishes from the market. If we assume that the information flow is the same (¼ 1) whether t is long

or short, then h is equal to 1/t.

From the local field determined as above, agent opinions in the next step are determined by

(

siðt þ 1Þ ¼

þ1

?1

with probability p;

with probability 1 ? p;

(5)

where p ¼ 1=ð1 þ expf?2IiðtÞgÞ. In this model, price changes are:

xðtÞ ¼1

N

X

siðtÞ.(6)

We simulate a stock market with 1000 agents, and the values xðtÞ, ZðtÞ, and zðtÞ are generated randomly within

the range ½?1;1?.

Fig. 3a shows the relation between the diffusion factor, h, and the tail index of the PDFs. The tail index is

directly proportional to h. This is similar to the trend line (solid line) in Fig. 1c. Fig. 3b and c shows the PDFs

of log return for various h values. Similar to the experimental result, a power-law tail is evident for small h

(large t), and an exponential tail for large h (small t). For very large h, the distribution is close to Gaussian

(Fig. 3d).

The time lag, Dt, can be determined for 1, 10min, 1h, 1day, etc., when price changes are calculated.

However, time is not uniform, because the volume, the number of contracts, and the information flow into the

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market are not homogeneous. In other words, 1min of today may not be same as 1min of tomorrow. Thus, it

is necessary to study the time uniformity.

4. Conclusions

The distribution of the KOSPI log return has recently taken the form of an exponential, while it showed a

power-law tail in the 1990s. Moreover, the decay time of the autocorrelation function is continually

decreasing. Thus, the duration of information received by agents is also decreasing as the amount of

information increases. According to the EMH, the distribution of log return becomes Gaussian when the

velocity of information flow is very fast, and all information received immediately affects the opinion of agents

in the market. We could identify and confirm a relationship between the distribution of price changes, the

velocity of information flow, and the duration of the influence of information for a time series of the Korean

stock market. A similar phenomenon occurred in Japan around 1990, as identified from daily data [21–23].

However, in mature markets, including the NYSE, the tail index is not increasing or changing in shape. The

reason for this is the robustness and maturity of the market. Mature markets are solid enough to endure

external shocks, while emerging markets are susceptible to shocks and environmental changes. Modeling of

the robustness and maturity of the market is planned as further work in the near future.

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(a)(b)

(c) (d)

Fig. 3. (a) Relation between hð¼ 1=tÞ and the tail index of PDFs. (b) Semi-log plot for the PDFs of price changes as a function of h.

&; h ¼ 0; ?; h ¼ 1; 4; h ¼ 2; and 5; h ¼ 3. (c) Log–log plot for the PDFs. The straight line is guide line for power-law distribution and

the curved line is for exponential distribution. Symbols are as for (b). (d) PDF of price changes when h is very large (h ¼ 10).

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Acknowledgements

We wish to thank H. Jeong and O. Kwon for the helpful discussions and supports.

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