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CHAOS 24, 043113 (2014)

Nonlinear scaling analysis approach of agent-based Potts financial dynamical model Weijia Hong and Jun Wanga) Institute of Financial Mathematics and Financial Engineering School of Science, Beijing Jiaotong University Beijing 100044, People’s Republic of China

(Received 12 July 2014; accepted 30 September 2014; published online 20 October 2014) A financial agent-based price model is developed and investigated by one of statistical physics dynamic systems—the Potts model. Potts model, a generalization of the Ising model to more than two components, is a model of interacting spins on a crystalline lattice which describes the interaction strength among the agents. In this work, we investigate and analyze the correlation behavior of normalized returns of the proposed financial model by the power law classification scheme analysis and the empirical mode decomposition analysis. Moreover, the daily returns of Shanghai Composite Index and Shenzhen Component Index are considered, and the comparison nonlinear analysis of statistical behaviors of returns between the actual data and the simulation data is exhibC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4898014] ited. V

As the financial markets are becoming deregulated worldwide, the modeling of dynamics of forwards prices is becoming a key problem in the risk management, physical assets valuation, and derivatives pricing. And, it is of great importance to analyze the correlations embedded in stock markets. Since a financial market can be considered like a strongly fluctuating complex system with a large number of interacting elements, many analyzing methods developed in statistical physics can be applied to characterize the time evolution of stock indexes and stock prices. The main aim of our work is to develop a new financial model and describe the correlations in Shanghai Composite Index (SSE) and Shenzhen Component Index (SZSE). In the present work, the Potts model, which can be viewed as a model for nonequilibrium statistical mechanics, is employed to establish a new financial model. Moreover, we investigate and analyze the correlation behavior of normalized returns of the proposed financial model by the power law classification scheme (PLCS) analysis and the empirical mode decomposition (EMD) analysis. The empirical results show that SSE and SZSE have much strong correlations. Meanwhile, the correlations between SSE and SZSE still exist after shuffling the time series, but the strength of correlation is decreasing. Then, we apply EMD method to decomposition the returns into intrinsic mode functions (IMFs) and investigate the correlation behaviors of the corresponding IMFs, the results show that there exist correlations between the corresponding IMFs, and the correlation behavior between IMF1s is similar to that of original returns, while the correlation strength of original returns is the highest. Furthermore, the Manhattan Distance (MD) of each pair of latter IMFs is decreasing. At last, we compare the statistical behaviors for the actual market and the simulative data, it is observable that the simulative data with proper parameters show similar properties to the actual data, the proposed financial model a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]. Tel.: (86)-10-51688453.

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developed by Potts model is reasonable for the real stock market to a certain extent.

I. INTRODUCTION

As the financial markets are becoming deregulated worldwide, the modeling of dynamics of forwards prices is becoming a key problem in the risk management, physical assets valuation, and derivatives pricing. Since a financial market can be considered like a strongly fluctuating complex system with a large number of interacting elements, many analyzing methods developed in statistical physics can be applied to characterize the time evolution of stock indexes and stock prices. So that many scientists have applied the physical theories and methods to make empirical research for economical phenomena, such as the fat-tail distribution of price changes, the power law of logarithmic returns and volumes, the volatility clustering which is described as onoff intermittency in literature of nonlinear dynamics, and the multifractality of volatility, see Refs. 1–7. Based on the perspective that the price movements are caused primarily by the arrival of new information or the idea that the price fluctuations are due to the interaction among the market investors, some research work has introduced various market models in an attempt to reproduce and analyze the fluctuation behaviors of stock markets, see Refs. 8–20. The stochastic Ising model, the most popular ferromagnetic model of statistical physics systems, has also been successfully used in modeling financial systems.21–25 These financial models describe the interaction strength among the agents, where the Ising system is supposed as a model of imitative behavior in which individuals modify their behaviors so as to conform to the behavior of other individuals in their vicinity. In the present work, the Potts model is employed to establish a new financial model. The Potts model, which can be viewed as a model for nonequilibrium statistical

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C 2014 AIP Publishing LLC V

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mechanics, is a famous statistical physics model which is the extension of the Ising model. Like the Ising model, it has a second order phase transition separating a low-temperature ordered phase from a high-temperature disordered phase. In this paper, the two-dimensional (2D) 3-state Potts model is applied to develop a stock price model. In this financial model, the subunits in a 2D Potts model are called spins (with the interactions between the nearest neighbors), the clusters of parallel spins in the square-lattice Potts model can be defined as groups of traders acting together (or sharing the same opinion) on the market. Then we study the statistical property and the correlation relationship of the simulation data derived from the proposed model. Through comparison between the real time series and the simulated ones, the statistical properties of returns of Shanghai Stock Exchange (SSE) Composite Index and Shenzhen Stock Exchange (SZSE) Component Index are studied. The power law classification scheme (PLCS)26 is applied to characterize the correlation of returns. Furthermore, the relationships between original time series and shuffled time series after empirical mode decomposition (EMD)27,28 are investigated.

II. BRIEF DESCRIPTION OF POTTS MODEL

The Potts model is one of statistical physics systems, see Refs. 29–34. it is a generalization of the Ising model to more-than-two components (the model of general q components bears its current name), and has been a subject of increasingly intense research interest in recent years. In recent years, considerable progress has been achieved on the study of the physical phenomena of three or more phases coexistence on the Potts model. It is known that the Potts model is related to a number of outstanding problems in lattice statistics; the critical behavior has also been shown to be richer and more general than that of the Ising model. In the ensuing efforts to explore its properties, the Potts model has become an important testing ground for the different methods and approaches in the study of the critical point theory. We regard the Ising model as a system of interacting spins that can be either parallel or antiparallel. Then an appropriate generalization would be to consider a system of spins confined in a plane, with each spin pointing to one of the q equally spaced directions. We consider the two-dimensional integer lattice Z2 and denote by B the set of bonds of the lattice (pairs of nearest neighbors). In the q-state Potts model, 2 let XZ2 ¼ f1; 2; …; qgZ denote the space of spin configurations on Z2 , an element of XZ2 usually denote by r ¼ fri : i 2 Z2 g. The spin ri take one of the integer values from 1 to q, the q is a parameter of the model. We consider a q-state Potts model with the following system of Hamiltonian, for every r 2 X X X dri ;rj b dri;1 ; (1) HZ2 ;b ðrÞ ¼ J hi;ji

i

where J > 0 for the ferromagnetic system and d is the Kroeneker symbol. The first sum is over all nearest neighbors (denote by hi; ji) on the lattice and the second sum over

all lattice points. Here the applied magnetic field b acts on the (arbitrarily chosen) state 1. Then the partition function is 0 1 X X X ZZ2 ;h ðrÞ ¼ exp @K dri ;rj þ h dri;1 A; (2) r

hi;ji

i

where K ¼ bJ and h ¼ bb; b ¼ 1=ðkB TÞ; kB and T being Boltzmann’s constant and temperature respectively. The model is typically defined on a regular lattice in d-dimensions, but can, in general, be defined on any graph. In the followings, we consider q-state Potts model with zero external magnetic field (b ¼ 0 and h ¼ 0). For d 2, the model sustains an order-disorder transition, and the critical value is pﬃﬃﬃ bc ¼ lnð1 þ qÞ in d ¼ 2. For b > bc , the q-fold permutation symmetry of Eq. (1) is broken, and one of the q different ground states has been singled out. For q ¼ 2, the model is the familiar Ising model, which has a second order transition, but with increasing q, the excited states have relatively more entropy, and for q > qc , the transition is first order. For d ¼ 2, the phase transition changes order at qc ¼ 4. In an another aspect, we use the notation K ¼ bJ and v ¼ eK 1, so that the physical ranges for this temperature-dependent Boltzmann variable v are (i) v 0, corresponding to 1 T 0 for the Potts ferromagnet (J > 0), and (ii) 1 v 0, corresponding to 0 T 1 for Potts antiferromagnet (J < 0). Thus, the zero-field Potts model partition function can be a function of variables q and v, and also depending on the lattice. III. FINANCIAL PRICE MODEL DERIVED FROM POTTS MODEL

Next, a financial agent-based price model is developed by the 2D 3-state Potts model on a L L lattice. In the present paper, we consider q-state Potts model with zero external magnetic field. In this case, the strength of the interaction between neighboring elements, which matters a lot in the proposed financial model, varies depending on their location on the lattice, and small changes in interaction rules do not change the cooperative properties of Potts type models. So the optimal number of the states to allow in the proposed model is not restricted to 3. On the other hand, in Chinese stock markets, the majority of investors is composed of large institutional investors, institutional investors, and retail private investors. These retail investors are famous for their following behavior. Though both institutional investors and retail investors demonstrate herding behavior, the latter is much more notorious for a lack of self opinions, and is willing to make decision based on hearsay, especially the information from institutional investor. From the observation of the component of Chinese stock markets and out of the respect of making a more reasonable simulation, we are motivated to propose 3-state Potts financial model. Here, the Potts model with 3-type particles is employed to imitate the interaction of large institutional, institutional and retail investors. We classify the investors into three categories, corresponding to type 1 (for state “1”), type 2 (for state “2”) and type 3 (for state “3”) particles, respectively. We suppose that the behaviors of stock price are attributed to the number

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of traders xð1Þ ðtÞ (large institutional investors), xð2Þ ðtÞ (institutional investors), and xð3Þ ðtÞ (retail private investors). For a stock market, we consider a single stock and assume that there are L2 traders in this stock, who are located in a 2D lattice L L Z2 (similarly for d-dimensional lattice Zd ), and each trader can trade unit number of stock at each time t. Let xij be the investing position of a trader ð1 i L; 1 j LÞ at time t, and xðtÞ ¼ ðx11 ðtÞ; …; x1L ðtÞ; …; xL1 ðtÞ; …; xLL ðtÞÞ be the configuration of positions for L2 traders. A space of all configurations of positions for L2 traders from time 1 to t is given by H ¼ fx : x ¼ ðxð1Þ; …; xðtÞÞg. For a given configuration x 2 H and a trading day t, let MðkÞ ðxðtÞÞ ¼ jxðkÞ ðtÞj;

k ¼ 1; 2; 3;

(3)

which represent the numbers of xð1Þ ðtÞ; xð2Þ ðtÞ, and xð3Þ ðtÞ at time t, respectively. Take the price changes proportional to the difference between demand and supply MðkÞ ðxðtÞÞðk ¼ 1; 2; 3Þ, which is affected by the intensity parameter b, where b represents the strength of information spread. Suppose that nt is a random variable which represents the information arrived at the tth trading day, where nt ¼ þ1 for buying opinion, nt ¼ 1 for selling opinion, and nt ¼ 0 for neutral opinion with probabilities p1 , p1 , and 1 ðp1 þ p1 Þ, respectively. Then, these investors send bullish, bearish, or neutral signal to the market. According to 3state Potts model dynamic system, investors can affect each other or the information can be spread, which is supposed as the main factor of price fluctuations for the market. From the above definitions and Ref. 35, we define the stock price of the model at time tðt ¼ 1; 2; …Þ as ( ) 3 X ðkÞ hM ðtÞi Pðt 1Þ; (4) PðtÞ ¼ exp a k¼1

hMðkÞ ðtÞi ¼

ck

nkt

MðkÞ ðxðtÞÞ ; L2

k ¼ 1; 2; 3;

(5)

where að> 0Þ denotes the depth parameter of the market which measures sensitivity of price fluctuation in response to change in excess demand and ck (k ¼ 1, 2, 3) is the effect strength of each type of traders in the stock market, such that c1 þ c2 þ c3 ¼ 1. Then, we have ( ) t X 3 X ðkÞ hM ðsÞi ; (6) PðtÞ ¼ Pð0Þ exp a s¼1 k¼1

XðtÞ ¼

N X

Ci ðtÞ þ rðtÞ;

(7)

IV. METHODOLOGIES FOR NONLINEAR STATISTICAL ANALYSIS

The EMD method is an empirical, intuitive, direct, and self-adaptive data processing method which is proposed especially for nonlinear and nonstationary data.27,28 The main idea of EMD is to locally decompose any time series into a sum of a local trend and a local detail, correspondingly, for a low

(8)

i¼1

where r(t) is the residual, which is usually a monotonic function representing either the mean trend or a constant. Ci ðtÞ is the ith IMF (i ¼ 1; 2; …; N), which is the amplitude- and frequency-modulated function of time. Power law classification scheme (PLCS). PLCS is helpful to introduce the notion of distance in an attempt to measure the correlation between two time series. Manhattan Distance (MD), as one of the simplest definition of a distance, is quite frequently used in econophysics. Let us consider two time series, X and Y, for which the elements are denoted as xi and yi, respectively. Then, the MD between X and Y is defined as MDðX; YÞ ¼

where Pð0Þ is the stock price at time 0. From Ref. 35, the formula of stock logarithmic return is defined as follows: RðtÞ ¼ log Pðt þ 1Þ log PðtÞ:

frequency part and a high frequency part. The former is called the residual and the latter is called the intrinsic mode functions (IMFs), which represent the different scales of the original time series and form the adaptive and physical basis of the data. To be an IMF, an approximation to the so-called moncomponent signal, it must satisfy the following two conditions: (i) In the whole set of data, the number of local extrema and the number of zero crossings must be equal or differ by 1 at most; (ii) at any time point, the mean value of the “upper envelope” (defined by the local maxima) and the “lower envelope” (defined by the local minima) must be zero. In practice, the algorithm to create IMFs in EMD is rather elegant, in which the IMFs are extracted through a sifting process, and it can be briefly described in the following steps: (1) Identify the local extrema of the signal X(t); (2) Construct upper envelope Emax ðtÞ by using the local maxima through a cubic spline interpolation, construct a lower envelop Emin ðtÞ by using the local minma; (3) Define the mean value mðtÞ ¼ ðEmax ðtÞ þ Emin ðtÞÞ=2; (4) Remove the mean value from the signal, providing the local detail hðtÞ ¼ XðtÞ mðtÞ; (5) Check if the component h(t) satisfies the above conditions to be an IMF. If yes, take it as the first IMF C1 ðtÞ ¼ hðtÞ. This IMF mode is then removed from the original signal, and the first residual, rðtÞ ¼ XðtÞ C1 ðtÞ, is taken as the new series in step (1). If h(t) is not an IMF, a procedure called the sifting process is applied as many times as necessary to obtain an IMF (not detailed here). And, we finally obtain

n X

jxi yi j:

(9)

i¼1

The time series X and Y must be of the same length. In other cases, the distance between the overlapping (in time) parts can be calculated. One of the key features of MD is that the distance between the time series is a non-decreasing function of the time series length n. The definition of power law classification scheme is derived from Manhattan distance, see Ref. 26. Calculate the series of cumulative MD between given time series in the following equation: MDðX; YÞk ¼

k X

jxi yi j:

(10)

i¼1

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All the subseries begin at the same origin, and k is the number of elements in the sum. (1) Plot the series of cumulative MD as a function of time series length in a log-log plot. (2) Fit the linear function to the data and find the slope coefficient q corresponding to the k integrated correlation function. (3) Let the class of the correlation function be labeled as q 1. Additionally, calculate the quality of the fit, measured by the statistical significance probability e. It gives some information about the stability of the found correlation in the considered k interval. V. EMPIRICAL ANALYSIS OF ACTUAL AND SIMULATION DATA

In this section, we apply PLCS to investigate the correlation characteristics of original returns and IMFs which are obtained through the EMD performance on the returns of simulative data, SSE and SZSE, and also study the correlation of shuffled data. The empirical data comes from intraday data of SSE and SZSE from January 4, 2000 to September 10, 2013, and two pairs of simulated data are generated from the proposed model with different parameter values (S.data1: b ¼ bc & b ¼ 2, and S.data2: b ¼ 2 & b ¼ 3). We select the

Chaos 24, 043113 (2014)

proper parameter values by Monte Carlo method and ensure these two pairs of simulative financial time series close to the real markets data in some degree. It has been shown in Ref. 26 that PLCS does not only recognize linear correlations properly but also allows to point out converging time series as well as to distinguish nonlinear correlations.

A. Correlation exploration of returns and IMFs

We take return R(t) of simulative prices and actual indexes as the primary time series, and then perform EMD to decompose R(t) and shuffled R(t) into different number of IMFs. In this section, the PLCS is applied to analyze the correlations between SSE and SZSE, and two pairs of simulated data. Moreover, we shuffle the simulated data, SSE, and SZSE to discuss their correlation behaviors. The empirical results are shown in Fig. 1. From the log-log plot of Fig. 1, MD and k are linearly correlated. Besides, the value of MD from shuffled data is larger than that of the original data, and the linear correlation is steadier. This behaviors are reflected in Fig. 1, the red line is on the top of blue one, and the red

FIG. 1. (a) Cumulative MD of actual data, S.data1, and S.data2. (b) Cumulative MD of actual data and their shuffled data. (c) Cumulative MD of S.data1 and their shuffled data. (d) Cumulative MD of S.data2 and their shuffled data.

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FIG. 2. (a) Cumulative MD between first four IMFs of SSE and SZSE and their shuffled data. (b) Cumulative MD between first four IMFs of S.data1 and their shuffled data. (c) Cumulative MD between first four IMFs of S.data2 and their shuffled data.

Chaos 24, 043113 (2014)

line is smoother. In addition, the correlation behavior between S.data2 is closer to the real time series. Then, we apply EMD method to the original time series and shuffled time series, and study the correlation behaviors between the corresponding IMFs, i.e., iIMF of SSE and iIMF of SZSE, with the aim to investigate whether the IMFs obtained from EMD method also possess correlation behavior as the original return series. The results are shown in Fig. 2. For the sake of clarity, a part of the results are presented, others are similar. It is known that the first four IMFs are approximated as the trend of the original R(t), but they are heavily polluted by the noises. This can be also obviously noticed in the figure. In this analysis, it is very important to note that IMF1 is not equal to time series R(t). If we evaluate some quantities specially defined for R(t) from IMFs, the results may be quite different. Actually, it is not reasonable to copy all the fundamental statistics primarily performed on R(t) to IMFs. The linear correlation between MD and k in a log-log plot is obvious, as the increase of k, this kind of relationship is steadier, especially in the result of shuffled data. Furthermore, we study the IMFs in a vertical comparison, see Figs. 3–5. Each IMF is independent from the others, and each IMF can not be represented by other IMFs decomposed from the same primary time series. IMF1 is the first mode separated from R(t) after the sifting process, and it has the highest frequency among all IMFs. It is obviously seen from Figs. 3–5 that IMF1 characterizes the the most similar correlation behavior of R(t) for actual data, simulative data, and their shuffled data. We can see that all IMFs present linear correlation, moreover, the MD distance from later IMF is smaller, showing a regularly deceasing. This behavior is more obvious in shuffled data. At last, we fit the linear function to the data and find the slope coefficient q, and q is the strenth of correlations, which is the key feature of PLCS. The high value of the correlation strength means that the MD distance between time series is growing with time, therefore, the time series are divergent. On the other hand, a high value of the correlation strength means that small changes in one time series leads to strong changes in the second one. The results are shown in Table I. From Table I, we can obtain the following conclusions: (1)

FIG. 3. (a) Cumulative MD between all IMFs of SSE and SZSE. (b) Cumulative MD between all IMFs of shuffled time series of SSE and SZSE.

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FIG. 4. (a) Cumulative MD between all IMFs of S.data1. (b) Cumulative MD between all IMFs of shuffled time series of S.data1.

FIG. 5. (a) Cumulative MD between all IMFs of S.data2. (b) Cumulative MD between all IMFs of shuffled time series of S.data2.

Comparing with the shuffled data, the value correlation strength of original data is higher; (2) The correlation strength of R(t) is stronger than that of IMFs from R(t) in most cases; (3) Different from the regularly decrease of MD distance, the correlation strength of IMFs fluctuates a lot. TABLE I. Slope coefficient q of IMFs of returns for SSE, SZSE, and two simulated data (S.Real, S.S.data1 and S.S.data2 denote for shuffled real data, shuffled S.data1, and shuffled S.data2, correspondingly). Data Orig. IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8 IMF9 IMF10

Real

S.Real

S.data1

S.S.data1

S.data.2

S.S.data2

1.1769 1.1819 1.1824 1.1866 1.2614 1.1779 1.2032 1.2075 1.6353 0.8407 1.2245

1.0079 0.9961 1.0301 0.9317 1.0398 0.9295 0.9691 1.0814 0.8736 0.9344 0.6558

0.9957 0.9922 0.9447 0.9683 1.0281 1.1962 0.8974 0.8314 0.9137 0.9590 1.0023

0.9892 0.9705 0.9817 1.0869 0.9860 0.9622 1.0999 1.1179 0.6809 0.6753 …

0.9514 0.9899 0.9480 0.9141 0.9676 0.8398 1.0188 1.3680 0.9060 1.0595 …

0.9910 0.9718 1.0114 1.0117 1.0033 0.8642 1.0620 1.3409 0.9586 1.4375 …

B. Extension of PLCS method

In this part, we change Eq. (9) into the form below ( MDð X; Y Þk ¼

k X

)1p jxi yi j

p

:

(11)

i¼1

Then we plot the log-log plot of MD and k for different number p. The results are shown in Figs. 6–8. From the results, we can find that q is smaller as the increase of p, the lines are deceasing regularly, and present stair-step shape. This is a very useful result, the stair-step shape of the line meaning at some point, there is a strong fluctuation in the market. Although we have not shown the evidence for these changes, it will be studied in future research. VI. CONCLUSION

In the present paper, a financial agent-based time series model is developed by the Potts model, in an attempt to reproduce the interactions among market traders which cause price fluctuations. By applying PLCS analysis, the empirical

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FIG. 6. (a) Cumulative MD of actual data for different values of p. (b) Cumulative MD of shuffled actual data for different values of p.

FIG. 7. (a) Cumulative MD of S.data1 for different values ofvalues of p. (b) Cumulative MD of shuffled S.data1 for different values of p.

FIG. 8. (a) Cumulative MD of S.data2 for different values of p. (b) Cumulative MD of shuffled S.data2 for different values of p.

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results show that SSE and SZSE have much strong correlations. Meanwhile, we study the effect of shuffling data to the correlation relationship of stock market, the results show that the correlations between SSE and SZSE still exist after shuffling the time series, but the strength of correlation is decreasing. Moreover, the PLCS method is applied to investigate the correlation behaviors of the corresponding IMFs of original returns R(t), which are decomposed by EMD method. The empirical results indicate that there exist correlations between the corresponding IMFs, and correlation behavior between IMF1s is similar to that of R(t), while the correlation strength of original returns is the highest. Furthermore, the MD distance of each pair of latter IMFs is decreasing. Through the comparisons of the above statistical behaviors for the actual market and the simulative data, it is observable that the simulative data with proper parameters show similar properties to the actual data, the proposed financial model developed by Potts model is reasonable for the real stock market to a certain extent. ACKNOWLEDGMENTS

The authors were supported by NSFC Grant Nos. 71271026 and 10971010. 1

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