Ordered phase and non-equilibrium fluctuation in stock market

Abstract

We analyze the statistics of daily price change of stock market in the framework of a statistical physics model for the collective fluctuation of stock portfolio. In this model the time series of price changes are coded into the sequences of up and down spins, and the Hamiltonian of the system is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. Through the analysis of Dow-Jones industrial portfolio consisting of 30 stock issues by this model, we find a non-equilibrium fluctuation mode on the point slightly below the boundary between ordered and disordered phases. The remaining 29 modes are still in disordered phase and well described by Gibbs distribution. The variance of the fluctuation is outlined by the theoretical curve and peculiarly large in the non-equilibrium mode compared with those in the other modes remaining in ordinary phase.The number of days in which prices of all stock issues in Dow-Jones industrial portfolio moved in the same direction is 80 within 3636 trading days in the period from 9/Jul/86 to 22/Nov/00, while its probability is 2 −29 when we assume Bernoulli trials. How can we explain the factor 10 7 in the difference between these values? The methods and the concepts, as scaling and criticality, developed in material science have been applied to the study of financial markets [1, 2, 3]. Recently a framework based on spin glass model to study the collective price changes of stock portfolios was proposed [4, 5]. The application to 1-minute price changes in D-J portfolio made clear that the concept of energy works even in financial markets as well as the above physical concepts. Here we study the properties of daily price changes in D-J portfolio based on the same model and attempt an explanation of the factor 10 7 by Gibbs factor of canonical distribution. Through this study we clarify the applicability of the spin glass picture to the price fluctuations in a wide range of time scale and find a novel feature of stock market. That is a non-equilibrium fluctuation mode on the point close to the boundary between ordered and disordered phases. D-J portfolio has been quenched into the unstable region, but does not reach equilibrium. The variance of the fluctuation, which is physically equal to susceptibility and is called (the square of) risk in financial economy, is outlined by the theoretical curve and peculiarly large in the non-equilibrium mode compared with those in the other modes remaining in ordinary phase.In this paper, we concentrate on the statistics of the sign of price change [4, 5]. The time series of price changes are coded into the sequences of up and down spins S i = ±1 (i=1 to portfolio size N) and the Hamiltonian is expressed by long-range spin-spin interactions as Sherrington-Kirkpatrick model of spin glass [6], which is given byWe consider a portfolio as a subset of the whole stock market, and the complement of the subset works as heat reservoir. Various observable quantities are obtained as the statistical averages with Gibbs weight ...

U.S. stock market interaction network as learned by the Boltzmann machine

“…Within the small correlation approximation, IP algorithm does not perform well for all historical dates, while quality of the SM couplings is comparable to the MF cases (however being still lower in general) and decreases for the periods where higher correlations are observed. Thus, the observed behavior indeed justifies use of TAP as a reliable approximation of true couplings and external fields (making use if the diagonal trick), which has been extensively used in the previous studies [15,[21][22][23]. However, special care should be taken about overestimated positive outliers, where SM algorithm can be helpful.…”

“…(1) play essential role not only in statistical physics but also in neuroscience (models of neural networks [18,19]) and machine learning (also known as Boltzmann machines [20]). Recently, applications of the pairwise interaction models to financial markets have been also explored [15,[21][22][23]. Given topological similarities between neural and financial networks [24], these systems can be considered as examples of complex adaptive systems [25], which are characterized by the adaptation ability to changing environment, trying to stay in equilibrium with it.…”

“…In this case, diagonal elements of the coupling matrix, J ii , are zero because s 2 i = 1 terms do not contribute to the Hamiltonian. It is worth stressing that the current investigation develops ideas outlined in the previous studies [15,[21][22][23] in a way that the effect of binarization, comparison of learning algorithms, evolution and scaling properties of the parameters distributions are studied in a more systematic fashion.…”

Detection of statistical asymmetries in non-stationary sign time series: Analysis of foreign exchange data

“…Econophysics is a relatively new research field that largely explores big data to access the characteristics and construct models in order to improve our understanding of economic systems. It applies methods traditionally developed in the scope of physics together with established economics ideas, strongly relying on statistical analysis of data [ 2 – 4 ]: Analysis of scaling properties [ 5 – 7 ], analogies with physical systems as the Brownian particle [ 8 – 11 ] and the Ising model [ 12 , 13 ], studies related to equilibrium as in thermodynamics [ 14 – 16 ]. Aside the name, econophysics has become a broader discipline, also using concepts from other areas as network theory [ 17 – 19 ] and information theory [ 20 – 22 ].…”

On Some Processes and Distributions in a Collective Model of Investors' Behavior

“…recently there were results on models combining minority and majority features Kaizoji, Bornholdt, Fujivara [4], Badshah, Boyer and Theodosopoulos [9], [10]. We refer to works [11][12][13][14][15][16][17][18][19][20][21] for different aspects of agent-based models.…”