Phenomenological Model for Stock Portfolios. Note Quote.

The data analysis and modeling of financial markets have been hot research subjects for physicists as well as economists and mathematicians in recent years. 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. From the analysis of the high-frequency time series of market indices, a universal property was found in the probability distributions. The central part of the distribution agrees well with Levy stable distribution, while the tail deviate from it and shows another power law asymptotic behavior. In probability theory, a distribution or a random variable is said to be stable if a linear combination of two independent copies of a random sample has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. 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. 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. 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.

Portfolio is a set of stock issues. 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 fluctuation-response theorem. 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 S_i (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 J_ij, which will be phenomenologically determined later, and the dynamical part δJ_ij. The Hamiltonian including the interaction with external fields h_i (i = 1,2,…,N) is defined as

H [S, δ, J, h] = ∑_<i,j>[δJ_ij²/2Δ_ij – (J_ij + δJ_ij)S_iS_j] – ∑_ih_iS_i —– (1)

The summation is taken over all pairs of stock issues. This form of Hamiltonian is that of annealed spin glass. The fluctuations δJ_ij are assumed to distribute according to Gaussian function. The main part of statistical physics is the evaluation of partition function that is given by the following functional in this case

Z[h] = ∑_{si} ∫∏_<i,j> dδJ_ij/√(2πΔ_ij) e^{-H [S, δ, J, h]} —– (2)

The integration over the variables δJij is easily performed and gives

Z[h] = A ∑_{si} e^{-H_eff[S, h]} —– (3)

Here the effective Hamiltonian H_eff[S,h] is defined as

H_eff[S, h] = – ∑_<i,j>J_ijS_iS_j – ∑_ih_iS_i —– (4)

and A = e(1/2 ∆_ij) is just a normalization factor which is irrelevant to the following step. This form of Hamiltonian with constant J_ij is that of quenched spin glass.

The constant interaction coefficients J_ij are still undetermined. We use fluctuation-response theorem which relates the susceptibility χ_ij with the covariance C_ij between dynamical variables in order to determine those constants, which is given by the equation,

χ_ij = ∂m_i/∂h_j |_h=0 = C_ij —– (5)

Thouless-Anderson-Palmer (TAP) equation for quenched spin glass is

m_i =tanh(∑_jJ_ijm_j + h_i – ∑_jJ_ij²(1 – m_j²)m_i —– (6)

Equation (5) and the linear approximation of the equation (6) yield the equation

∑_k(δ_ik − J_ik)C_kj = δ_ij —– (7)

Interpreting C_ij as the time average of empirical data over a observation time rather than ensemble average, the constant interaction coefficients J_ij is phenomenologically determined by the equation (7).

The energy spectra of the system, simply the portfolio energy, is defined as the eigenvalues of the Hamiltonian H_eff[S,0]. The probability density of the portfolio energy can be obtained in two ways. We can calculate the probability density from data by the equation

p(E) ΔE = p(E – ΔE/2 ≤ E ≤ E + ΔE/2) —– (8)

This is a fully consistent phenomenological model for stock portfolios, which is expressed by the effective Hamiltonian (4). This model will be also applicable to other financial markets that show collective time evolutions, e.g., foreign exchange market, options markets, inter-market interactions.

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