Hamiltonian in Financial Markets

A statistical physics model for the time evolutions of stock portfolios is proposed. In this model the time series of price changes are coded into the sequences of up and down spins. The Hamiltonian of the system is introduced and is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. The interaction coefficients between two stocks are determined by empirical data coded into up and down spin sequences using fluctuation-response theorem. Monte Carlo simulations are performed and the resultant probability densities of the system energy and magnetization show good agreement with empirical data. The data analysis and modeling of financial markets have been hot research subjects for physicists as well as economists and mathematicians in recent years [1]. The non-Gaussian property of the probability distributions of price changes, in stock markets and foreign exchange markets, has been one of main problems in this field [1] [2] [3]. From the analysis of the high-frequency time series of market indices, e.g., S&P500;, Nikkei225, a universal property was found in the probability distributions. The central part of the distribution agrees well with Levy stable distribution [4], while the tail deviate from it and shows another power law asymptotic behavior. The scaling property on the sampling time interval of data is also well described by the crossover of the two distributions. Several stochastic models of the fluctuation dynamics of stock prices are proposed, which reproduce power law behavior of the probability density [1] [2] [5]. The auto-correlation of financial time series is also an important problem for markets. There is no time correlation of price changes in daily scale, while from more detailed data analysis an exponential decay with a characteristic time τ = 4 minutes was found [1] [3]. The fact that there is no auto-correlation in daily scale is not equal to the independence of the time series in the scale. In fact there is auto-correlation of volatility (absolute value of price change) with a power law tail [1] [6]. Recently, the cross-correlation between pairs of stock issues was deeply investigated for the time series of price changes, and the hierarchical structure in the subdominant ultrametric space [7] was found [1] [8]. This result suggests the complex collective time evolution of financial markets as in frustrated disordered systems like spin glass [9] in which the ultrametricity has been established. Those problems listed here have been studied using the methods and the concepts developed in material sciences especially in the studies of complex systems. In this paper, a statistical physics model for the collective time evolutions of stock portfolios is proposed. Portfolio is a set of stock issues. In this model we deal with the time 1 series of price changes coded into the sequences of up and down spins. A sample of coding procedure is shown in Fig. 1. The Hamiltonian of the system is introduced and is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. The interaction coefficients between two stocks are phenomenologically determined by empirical data. They are derived from the covariance of sequences of up and down spins using fluctuationresponse theorem. We investigate the stocks listed in Dow-Jones industrial average as a portfolio for the test of this model. Monte Carlo simulations using Gibbs weight as a transition probability reproduce the probability densities of the energy and the magnetization of the portfolio, whose definitions are given later. We start with the Hamiltonian expression of our system that contain N stock issues. It is a function of the configuration S consisting of N coded price changes Si (i = 1, 2, ..., N) at equal trading time. The interaction coefficients are also dynamical variables, because the interactions between stocks are thought to change from time to time. We divide a coefficient into two parts, the constant part Jij, which will be phenomenologically determined later, and the dynamical part δJij. The Hamiltonian including the interaction with external fields hi (i = 1, 2, ..., N) is defined as H [S, δJ, h] = ∑ [ δJ ij 2∆ij − (Jij + δJij)SiSj)]− ∑