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Cluster behavior of a simple model in financial markets
Physica A 387 (2008) 528–536 www.elsevier.com/locate/physa
Cluster behavior of a simple model in financial markets J. Jiang a,∗ , W. Li a,b , X. Cai a a Complexity Science Center, Institute of Particle Physics, Hua-Zhong (Central China) Normal University, Wuhan 430079, China b Max-Planck-Institute for Mathematics in the Science, Inselstr. 22-26, 04103 Leizig, Germany
Received 14 September 2007 Available online 29 September 2007
Abstract We investigate the cluster behavior of financial markets within the framework of a model based on a scale-free network. In this model, a cluster is formed by connected agents that are in the same state. The cumulative distribution of clusters is found to be a power-law. We find that the probability distribution of the liquidity parameter, which measures the financial markets’ energy, is rather robust. Furthermore, the time series of the liquidity parameter have the characteristics of 1/ f noise, which may indicate the fractal geometry of financial markets. c 2007 Elsevier B.V. All rights reserved.
PACS: 89.65.Gh; 87.23.Ge Keywords: Cluster behavior; Liquidity parameter; Scale-free networks; Power-law scaling; Financial markets
1. Introduction Co-interaction and evolution between different agents are known to be one of the ingredients of complex systems, such as—social, biological, economical and technological systems. Following the trend of research on complex systems, to find the universal rules and principles of these systems become more and more attractive [1–6]. In particular, the studies of financial markets prices have been found to suggest several generalized properties similar to those observed in physical systems with a large number of interacting ingredients. More and more models have been introduced to attempt to capture the universalities behind the financial markets which are the so-called stylized facts [7–9], such as sharp peaks and fat-tail distributions for the financial prices, absence of autocorrelation in return, and long-time correlations in absolute return, etc. These models include the herding multi-agent model [10–14], the related percolation model [15,16] and the dynamic games model [17–22], etc. Among the more sophisticated approaches are the multi-agent models, based on the interactions of two different agent groups (“noise” and “fundamental” traders), which reproduce some of the stylized facts of real markets but do not account for the origin of the universal characteristics. An alternative approach, the herd behavior [23] may be capable to induce the power-law asymptotic behavior in the tail of return distribution as found in the real data. But an
∗ Corresponding author.
E-mail address: [email protected] (J. Jiang). c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.09.030
J. Jiang et al. / Physica A 387 (2008) 528–536
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assumption made in its model that should not be ignored is that the probability of each cluster to sell or buy is set to be the same and remains constant throughout the whole process, which may be a good strategy for simplifying a physical model but may not be a good regulation for establishing a model which we expect to reflect the various phenomena found in the real financial markets as genuine as possible. Here we introduce a model where the probability of each agent to sell or buy varies along with the difference between the demand and supply at each time step, which may be more helpful for us to learn the nature of the markets. It is interesting to find that the various networks of the real-world, from social networks to biological networks, display scale-free degree distributions and small-world characteristics. So recently more and more models of financial markets have been proposed based on different types of networks [24–26]. It could characterize quantitatively the interaction between agents by means of a series of topological quantities, which could better capture the complex properties of the real-world. In Refs. [23,24,27], the models are in the view of the regular lattices and the famous Cont and Bouchaud model’s network structure is that of the random graph. However, in our model, we consider a different topology on the scale-free network. More importantly, we obtain some interesting results about cluster behavior which is a very common phenomena in the real-world financial market. This paper is organized as follows. In Section 2, we introduce our model. In Section 3, there are numerical simulations and some results. In Section 4, a discussion and main conclusions are given. 2. Our model We present a model for the cluster formation and information dispersal based on a scale-free network. As a first approach to model the complicated social behaviors we consider: (1) the probability of each agent to sell or buy changes with the difference between the demand and supply of the markets, (2) agents having the same state and being linked form a cluster that makes consensual decisions, and (3) whenever a cluster forms, the information disperses instantly and each agent within the cluster randomly changes its state according to the difference between the demand and supply of the market. We then apply the model to studying the price dynamics in a financial market. In the present work, we consider a system of N agents, represented by N nodes in a scale-free network. The optional state of the agent i is represented by ϕi = {1, 0, −1} corresponding to an inactive, waiting state (ϕi = 0), and two active states of either buying (ϕi = 1) or selling (ϕi = −1). In order to mimic the scale-free network topology, we make use of the Barab´asi–Albert model [28] which is based on two main assumptions: (1) linear growth and (2) preferential attachment. The network is initialized with m 0 nodes. At each step a new node with m edges is added to the pre-existing network. The of preferential attachment Q probabilityP that an edge of the new node is linked with the ith node is expressed by (ki ) = ki / j k j (ki is the degree of node i). A cluster is formed by all connected agents that have the same states. The size si of cluster i is defined as the total number of agents in the cluster. Initially, the states of all agents are randomly determined. Each cluster buys or sells a unit of financial product such as stock, future or currency at each time step with the same probability p, or waits with probability 1 − 2 p. Here, we call p the liquidity parameter. A value of p < 0.5 allows for a finite fraction of agents not to trade during a given time window. Our model is defined in the following way. At each time step t: (1) the difference between the supply and demand of the markets is defined by: d=
a X
ϕi ,
(1)
i=1
where a is the total number of clusters in the financial markets. (2) the evolution of the probability of activity follows the rule: p(t) = p(t − 1)ed/v ,
(2)
where v is a parameter that controls the update of the cluster size and provides a measure of the liquidity of the financial markets. (3) the evolution of the financial index price follows the rule: P(t) = P(t − 1)ed/z ,
(3)
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Fig. 1. Time series of the evolution of the financial market price.
Fig. 2. The returns of simulated price fluctuation for δt = 1.
where z is the market depth [29] which is the excess demand needed to move the price by one unit: it measures the sensitivity of price to the difference between the demand and supply of the markets. (4) repeat step (1)–(3) indefinitely. In this model, v is a constant. We have to impose p(t) = p(0) if the recurrence relationship equation (2) gives values for p(t) > 0.5 or p(t) < 0. According to Eq. (3), the price would fall if d < 0 and rise if d > 0. Fig. 1 displays the time series of the price. 3. Numerical simulations and results We have performed numerical simulations for a population of N = 104 agents where the typical parameter space adopted is as follows: p(0) = 0.1, v = 65, z = 150, m 0 = 10, m = 3. The initial financial index price is 1.0. The definition of return is defined as: R(t) = ln P(t + δt) − ln P(t). Fig. 2 shows the returns corresponding to Fig. 1.
(4)
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Fig. 3. The cumulative distribution of return for different liquidity parameters. The solid (red) line shows power-laws with the exponent α ≈ 3.03 ± 0.08, well-outside the Levy stable regime 0 < α < 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. The cumulative distribution of return for different time interval. The solid (red) line shows a power-law with the exponent α ≈ 3.05 ± 0.04 for the time scale δt = 5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
We plot the cumulative distribution of the returns—the probability of a return larger than or equal to a return for the time scale δt = 1 in Fig. 3, in which the distributions for three different liquidity parameters all satisfy power-laws, characterized by an exponent α ≈ 3.03 ± 0.08, well-outside the Levy stable regime 0 < α < 2. This rather matches the empirical results of Ref. [30]. Note that in all cases one observes the power-law decaying in a range of returns, which indicates that the cumulative distribution of the return of financial product price is constant beyond the liquidity of the markets. Furthermore, we compute the distribution of returns for longer different time scale δt. Let us use C(R) to denote the cumulative distribution of the return, and R to denote the return. Fig. 4 shows the cumulative distribution of the return for time scales from five to forty. We observe good indication of the distribution converging to Gaussian denoted by black dots in the graph. We also find that the larger the time scale is, the larger volatility the price will have, which indicates that the distribution of larger returns is more close to a Gaussian distribution. This could be demonstrated in Refs. [31,32]. Note also that the distribution of returns is robust in a short range of the time scale,
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Fig. 5. The probability distribution of different liquidity parameters in double-log plot. The solid (black) line shows a power-law p −β with exponent β = 2.0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. The evolution of liquidity parameter ( p(0) = 0.01) over time. There are several large volatilities among the time series.
which shows that α ≈ 3.05 ± 0.04 through the fitting for the time scale δt = 5 is approximately equal to the value of α for the time scale δt = 1. The liquidity parameter p appears to be the only adjustable parameter of the model, and controls the size of the cluster and the difference between the demand and supply of the markets. In Fig. 5 we show the distribution of four different liquidity parameters p(0) = 0.01, 0.1, 0.2 and 0.3. The solid line shows a power-law p −β with exponent β = 2.0. We see that in all cases one observes power-law decaying in a range of liquidity parameter. It indicates that the distribution has a approximately stable functional form for different p. Next we show the evolution of liquidity parameter ( p(0) = 0.01) in Fig. 6. From the plots, we find that there exist several large volatilities among the time series of liquidity parameter. In order to exploit the correlation property between these large volatilities, the method of detrended fluctuation analysis (DFA) [33] is employed to quantify the correlation between the volatilities. In Fig. 7, it is displayed as a power-law s γ with exponent γ = 1.01 ± 0.01, which suggests that the volatilities are long-range correlated. This is the common complex behavior in the financial markets. Moreover, according to the relationship between γ and the power spectral S( f )’s scaling
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Fig. 7. The DFA results of the time series of liquidity parameter. The solid (black) line shows a power-law s γ with exponent γ = 1.01 ± 0.01. According to the relationship between γ and the power spectral’s scaling exponent η: η = 2γ −1 [33], we could find that the time series of liquidity parameter show the behavior of 1/ f noise.
Fig. 8. The difference d between the demand and supply of the market evolves over time t.
exponent η (S( f ) ∼ f −η ) [34], we use the relation, η = 2γ − 1.
(5)
So the exponent η is equivalent to 1 approximately, which indicates that the volatility correlations among time series of the liquidity parameter are 1/ f noise-like, often observed in many systems, such as river discharge, DNA base sequence structure, cellular automata, traffic flow, financial markets and other self-organized systems. This shows the fractal or self-similar characteristics in the financial markets. With regard to the explanation of this phenomenon, we think the long-range correlations between the volatilities may account for this. In Fig. 8, we show the difference d between the demand and supply of the markets evolving over time t on the condition of p(0) = 0.01. With the time increasing, the fluctuation of the difference is decreasing, and is equivalent to zero approximately at the end which means that the market equilibrium is reached. As the difference is related to the liquidity parameter, we could see the functional relationship between them in Fig. 9. Fig. 9 shows the probability distribution of the difference for three different liquidity parameters p(0) = 0.01, 0.1, 0.3. In the plots, the distribution has the sharp peak and heavy tail. Based on the assumption that the probability of buying or selling for agents is the
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Fig. 9. The probability distribution of the difference for different liquidity parameters.
Fig. 10. The local amplification of the central part of the distribution of difference in Fig. 9. The green line is the Gaussian fit. We could find that the distribution of the difference is not fitted to the Gaussian distribution well. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
same, the distribution is symmetric. Furthermore, the smaller the liquidity parameter is, the larger the probability of difference closing to zero will be. This is because when the value of the liquidity parameter is small, few trades happen in the markets and each trade has the same volume imposed, so it is easier to reach an market equilibrium. From the plots, note also that the smaller the liquidity parameter is, the smaller the fluctuation of the difference will be, which could be seen in the tail of the picture. When p(0) = 0.01, the absolute value of the difference is 60 approximately. However, when p(0) = 0.3, the absolute value of the difference is close to 80 approximately. We also consider the above reason that caused this phenomenon. About the form of the distribution of the difference, we plot a local amplified picture of the central part of the distribution in Fig. 10. From the graph, the green line is the Gaussian fit. We could find that the distribution of the difference is not fitted to the Gaussian distribution well. The existence of clusters in financial markets is a very common phenomenon. It is also a rather important character. In Fig. 11, it shows the evolution of cumulative distribution of cluster size from time t = 1 to 5000 in double-log plot when N = 5000, p(0) = 0.25. We can divide the whole process into five parts. In the graph, we could see the five curves fit well, indicating that the cumulative distribution is independent of the time and has the characteristics of
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Fig. 11. Evolution of cumulative distribution of cluster size from time t = 1 to 5000 in double-log plot. From the plots, the five curves fit well. In the range of the cluster size s < 10, the plots can be fitted by a straight line showing the power-law behavior; in the range of the cluster size s > 2000, the number of the cluster is very small.
fractal in time scale. It is interesting that when the cluster size s < 10, the plots can be fitted by a straight line showing the power-law behavior; when s > 2000, the number of the cluster is small. This is consistent with the real fact that the small cluster is playing the dominative role in the markets but the large cluster is extremely few. To the best of our knowledge, the problems of cluster size have few quantitative, analytical results except several results based on the Cont and Bouchaud model and there are even no empirical verifications as the statistics of the data is rare. 4. Discussion and conclusions We have presented a self-organized model for the formation of clusters based on the scale-free network and applied it to the description of cluster behavior in financial markets. We make a simple assumption for the participants in financial markets that the probability of buying financial products is the same as the probability of selling them and the expression of the evolution for liquidity parameter is in the same pattern as of financial index price’s evolution. Such a model has the only one adjustable parameter defining the liquidity parameter. Through the simulations, we have obtained some critical stylized facts which are rather similar to real financial markets. We also draw a conclusion that the time series of liquidity parameter evolves steadily with several large volatilities during the whole process, which shows the behavior of 1/ f noise and means the whole financial market has the fractal properties. According to the distribution of the clusters’ sizes, We find that it follows the power-law scaling in the range of small cluster size independent of time. It suggests that in financial markets, the number of clusters of large size is small and most of clusters have small sizes. These results were not found in Refs. [23,24,27] which are on the basis of the regular lattices or random graph. The behaviors in this regime need more study in a quantitative manner in the further investigation. Acknowledgements This work is supported in part by the National Natural Science Foundations of China under Grant Nos. 70571027, 70401020, 10647125, 10635020 and by the Ministry of Education in China under Grant No. 306022.
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Cluster behavior of a simple model in financial markets J. Jiang a,∗ , W. Li a,b , X. Cai a a Complexity Science Center, Institute of Particle Physics, Hua-Zhong (Central China) Normal University, Wuhan 430079, China b Max-Planck-Institute for Mathematics in the Science, Inselstr. 22-26, 04103 Leizig, Germany
Received 14 September 2007 Available online 29 September 2007
Abstract We investigate the cluster behavior of financial markets within the framework of a model based on a scale-free network. In this model, a cluster is formed by connected agents that are in the same state. The cumulative distribution of clusters is found to be a power-law. We find that the probability distribution of the liquidity parameter, which measures the financial markets’ energy, is rather robust. Furthermore, the time series of the liquidity parameter have the characteristics of 1/ f noise, which may indicate the fractal geometry of financial markets. c 2007 Elsevier B.V. All rights reserved.
PACS: 89.65.Gh; 87.23.Ge Keywords: Cluster behavior; Liquidity parameter; Scale-free networks; Power-law scaling; Financial markets
1. Introduction Co-interaction and evolution between different agents are known to be one of the ingredients of complex systems, such as—social, biological, economical and technological systems. Following the trend of research on complex systems, to find the universal rules and principles of these systems become more and more attractive [1–6]. In particular, the studies of financial markets prices have been found to suggest several generalized properties similar to those observed in physical systems with a large number of interacting ingredients. More and more models have been introduced to attempt to capture the universalities behind the financial markets which are the so-called stylized facts [7–9], such as sharp peaks and fat-tail distributions for the financial prices, absence of autocorrelation in return, and long-time correlations in absolute return, etc. These models include the herding multi-agent model [10–14], the related percolation model [15,16] and the dynamic games model [17–22], etc. Among the more sophisticated approaches are the multi-agent models, based on the interactions of two different agent groups (“noise” and “fundamental” traders), which reproduce some of the stylized facts of real markets but do not account for the origin of the universal characteristics. An alternative approach, the herd behavior [23] may be capable to induce the power-law asymptotic behavior in the tail of return distribution as found in the real data. But an
∗ Corresponding author.
E-mail address: [email protected] (J. Jiang). c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.09.030
J. Jiang et al. / Physica A 387 (2008) 528–536
529
assumption made in its model that should not be ignored is that the probability of each cluster to sell or buy is set to be the same and remains constant throughout the whole process, which may be a good strategy for simplifying a physical model but may not be a good regulation for establishing a model which we expect to reflect the various phenomena found in the real financial markets as genuine as possible. Here we introduce a model where the probability of each agent to sell or buy varies along with the difference between the demand and supply at each time step, which may be more helpful for us to learn the nature of the markets. It is interesting to find that the various networks of the real-world, from social networks to biological networks, display scale-free degree distributions and small-world characteristics. So recently more and more models of financial markets have been proposed based on different types of networks [24–26]. It could characterize quantitatively the interaction between agents by means of a series of topological quantities, which could better capture the complex properties of the real-world. In Refs. [23,24,27], the models are in the view of the regular lattices and the famous Cont and Bouchaud model’s network structure is that of the random graph. However, in our model, we consider a different topology on the scale-free network. More importantly, we obtain some interesting results about cluster behavior which is a very common phenomena in the real-world financial market. This paper is organized as follows. In Section 2, we introduce our model. In Section 3, there are numerical simulations and some results. In Section 4, a discussion and main conclusions are given. 2. Our model We present a model for the cluster formation and information dispersal based on a scale-free network. As a first approach to model the complicated social behaviors we consider: (1) the probability of each agent to sell or buy changes with the difference between the demand and supply of the markets, (2) agents having the same state and being linked form a cluster that makes consensual decisions, and (3) whenever a cluster forms, the information disperses instantly and each agent within the cluster randomly changes its state according to the difference between the demand and supply of the market. We then apply the model to studying the price dynamics in a financial market. In the present work, we consider a system of N agents, represented by N nodes in a scale-free network. The optional state of the agent i is represented by ϕi = {1, 0, −1} corresponding to an inactive, waiting state (ϕi = 0), and two active states of either buying (ϕi = 1) or selling (ϕi = −1). In order to mimic the scale-free network topology, we make use of the Barab´asi–Albert model [28] which is based on two main assumptions: (1) linear growth and (2) preferential attachment. The network is initialized with m 0 nodes. At each step a new node with m edges is added to the pre-existing network. The of preferential attachment Q probabilityP that an edge of the new node is linked with the ith node is expressed by (ki ) = ki / j k j (ki is the degree of node i). A cluster is formed by all connected agents that have the same states. The size si of cluster i is defined as the total number of agents in the cluster. Initially, the states of all agents are randomly determined. Each cluster buys or sells a unit of financial product such as stock, future or currency at each time step with the same probability p, or waits with probability 1 − 2 p. Here, we call p the liquidity parameter. A value of p < 0.5 allows for a finite fraction of agents not to trade during a given time window. Our model is defined in the following way. At each time step t: (1) the difference between the supply and demand of the markets is defined by: d=
a X
ϕi ,
(1)
i=1
where a is the total number of clusters in the financial markets. (2) the evolution of the probability of activity follows the rule: p(t) = p(t − 1)ed/v ,
(2)
where v is a parameter that controls the update of the cluster size and provides a measure of the liquidity of the financial markets. (3) the evolution of the financial index price follows the rule: P(t) = P(t − 1)ed/z ,
(3)
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Fig. 1. Time series of the evolution of the financial market price.
Fig. 2. The returns of simulated price fluctuation for δt = 1.
where z is the market depth [29] which is the excess demand needed to move the price by one unit: it measures the sensitivity of price to the difference between the demand and supply of the markets. (4) repeat step (1)–(3) indefinitely. In this model, v is a constant. We have to impose p(t) = p(0) if the recurrence relationship equation (2) gives values for p(t) > 0.5 or p(t) < 0. According to Eq. (3), the price would fall if d < 0 and rise if d > 0. Fig. 1 displays the time series of the price. 3. Numerical simulations and results We have performed numerical simulations for a population of N = 104 agents where the typical parameter space adopted is as follows: p(0) = 0.1, v = 65, z = 150, m 0 = 10, m = 3. The initial financial index price is 1.0. The definition of return is defined as: R(t) = ln P(t + δt) − ln P(t). Fig. 2 shows the returns corresponding to Fig. 1.
(4)
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Fig. 3. The cumulative distribution of return for different liquidity parameters. The solid (red) line shows power-laws with the exponent α ≈ 3.03 ± 0.08, well-outside the Levy stable regime 0 < α < 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. The cumulative distribution of return for different time interval. The solid (red) line shows a power-law with the exponent α ≈ 3.05 ± 0.04 for the time scale δt = 5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
We plot the cumulative distribution of the returns—the probability of a return larger than or equal to a return for the time scale δt = 1 in Fig. 3, in which the distributions for three different liquidity parameters all satisfy power-laws, characterized by an exponent α ≈ 3.03 ± 0.08, well-outside the Levy stable regime 0 < α < 2. This rather matches the empirical results of Ref. [30]. Note that in all cases one observes the power-law decaying in a range of returns, which indicates that the cumulative distribution of the return of financial product price is constant beyond the liquidity of the markets. Furthermore, we compute the distribution of returns for longer different time scale δt. Let us use C(R) to denote the cumulative distribution of the return, and R to denote the return. Fig. 4 shows the cumulative distribution of the return for time scales from five to forty. We observe good indication of the distribution converging to Gaussian denoted by black dots in the graph. We also find that the larger the time scale is, the larger volatility the price will have, which indicates that the distribution of larger returns is more close to a Gaussian distribution. This could be demonstrated in Refs. [31,32]. Note also that the distribution of returns is robust in a short range of the time scale,
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Fig. 5. The probability distribution of different liquidity parameters in double-log plot. The solid (black) line shows a power-law p −β with exponent β = 2.0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. The evolution of liquidity parameter ( p(0) = 0.01) over time. There are several large volatilities among the time series.
which shows that α ≈ 3.05 ± 0.04 through the fitting for the time scale δt = 5 is approximately equal to the value of α for the time scale δt = 1. The liquidity parameter p appears to be the only adjustable parameter of the model, and controls the size of the cluster and the difference between the demand and supply of the markets. In Fig. 5 we show the distribution of four different liquidity parameters p(0) = 0.01, 0.1, 0.2 and 0.3. The solid line shows a power-law p −β with exponent β = 2.0. We see that in all cases one observes power-law decaying in a range of liquidity parameter. It indicates that the distribution has a approximately stable functional form for different p. Next we show the evolution of liquidity parameter ( p(0) = 0.01) in Fig. 6. From the plots, we find that there exist several large volatilities among the time series of liquidity parameter. In order to exploit the correlation property between these large volatilities, the method of detrended fluctuation analysis (DFA) [33] is employed to quantify the correlation between the volatilities. In Fig. 7, it is displayed as a power-law s γ with exponent γ = 1.01 ± 0.01, which suggests that the volatilities are long-range correlated. This is the common complex behavior in the financial markets. Moreover, according to the relationship between γ and the power spectral S( f )’s scaling
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Fig. 7. The DFA results of the time series of liquidity parameter. The solid (black) line shows a power-law s γ with exponent γ = 1.01 ± 0.01. According to the relationship between γ and the power spectral’s scaling exponent η: η = 2γ −1 [33], we could find that the time series of liquidity parameter show the behavior of 1/ f noise.
Fig. 8. The difference d between the demand and supply of the market evolves over time t.
exponent η (S( f ) ∼ f −η ) [34], we use the relation, η = 2γ − 1.
(5)
So the exponent η is equivalent to 1 approximately, which indicates that the volatility correlations among time series of the liquidity parameter are 1/ f noise-like, often observed in many systems, such as river discharge, DNA base sequence structure, cellular automata, traffic flow, financial markets and other self-organized systems. This shows the fractal or self-similar characteristics in the financial markets. With regard to the explanation of this phenomenon, we think the long-range correlations between the volatilities may account for this. In Fig. 8, we show the difference d between the demand and supply of the markets evolving over time t on the condition of p(0) = 0.01. With the time increasing, the fluctuation of the difference is decreasing, and is equivalent to zero approximately at the end which means that the market equilibrium is reached. As the difference is related to the liquidity parameter, we could see the functional relationship between them in Fig. 9. Fig. 9 shows the probability distribution of the difference for three different liquidity parameters p(0) = 0.01, 0.1, 0.3. In the plots, the distribution has the sharp peak and heavy tail. Based on the assumption that the probability of buying or selling for agents is the
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Fig. 9. The probability distribution of the difference for different liquidity parameters.
Fig. 10. The local amplification of the central part of the distribution of difference in Fig. 9. The green line is the Gaussian fit. We could find that the distribution of the difference is not fitted to the Gaussian distribution well. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
same, the distribution is symmetric. Furthermore, the smaller the liquidity parameter is, the larger the probability of difference closing to zero will be. This is because when the value of the liquidity parameter is small, few trades happen in the markets and each trade has the same volume imposed, so it is easier to reach an market equilibrium. From the plots, note also that the smaller the liquidity parameter is, the smaller the fluctuation of the difference will be, which could be seen in the tail of the picture. When p(0) = 0.01, the absolute value of the difference is 60 approximately. However, when p(0) = 0.3, the absolute value of the difference is close to 80 approximately. We also consider the above reason that caused this phenomenon. About the form of the distribution of the difference, we plot a local amplified picture of the central part of the distribution in Fig. 10. From the graph, the green line is the Gaussian fit. We could find that the distribution of the difference is not fitted to the Gaussian distribution well. The existence of clusters in financial markets is a very common phenomenon. It is also a rather important character. In Fig. 11, it shows the evolution of cumulative distribution of cluster size from time t = 1 to 5000 in double-log plot when N = 5000, p(0) = 0.25. We can divide the whole process into five parts. In the graph, we could see the five curves fit well, indicating that the cumulative distribution is independent of the time and has the characteristics of
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Fig. 11. Evolution of cumulative distribution of cluster size from time t = 1 to 5000 in double-log plot. From the plots, the five curves fit well. In the range of the cluster size s < 10, the plots can be fitted by a straight line showing the power-law behavior; in the range of the cluster size s > 2000, the number of the cluster is very small.
fractal in time scale. It is interesting that when the cluster size s < 10, the plots can be fitted by a straight line showing the power-law behavior; when s > 2000, the number of the cluster is small. This is consistent with the real fact that the small cluster is playing the dominative role in the markets but the large cluster is extremely few. To the best of our knowledge, the problems of cluster size have few quantitative, analytical results except several results based on the Cont and Bouchaud model and there are even no empirical verifications as the statistics of the data is rare. 4. Discussion and conclusions We have presented a self-organized model for the formation of clusters based on the scale-free network and applied it to the description of cluster behavior in financial markets. We make a simple assumption for the participants in financial markets that the probability of buying financial products is the same as the probability of selling them and the expression of the evolution for liquidity parameter is in the same pattern as of financial index price’s evolution. Such a model has the only one adjustable parameter defining the liquidity parameter. Through the simulations, we have obtained some critical stylized facts which are rather similar to real financial markets. We also draw a conclusion that the time series of liquidity parameter evolves steadily with several large volatilities during the whole process, which shows the behavior of 1/ f noise and means the whole financial market has the fractal properties. According to the distribution of the clusters’ sizes, We find that it follows the power-law scaling in the range of small cluster size independent of time. It suggests that in financial markets, the number of clusters of large size is small and most of clusters have small sizes. These results were not found in Refs. [23,24,27] which are on the basis of the regular lattices or random graph. The behaviors in this regime need more study in a quantitative manner in the further investigation. Acknowledgements This work is supported in part by the National Natural Science Foundations of China under Grant Nos. 70571027, 70401020, 10647125, 10635020 and by the Ministry of Education in China under Grant No. 306022.
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