Endogenous and exogenous dynamics in the fluctuations of capital fluxes

A phenomenological investigation of the endogenous and exogenous dynamics in the fluctuations of capital fluxes is carried out on the Chinese stock market using mean-variance analysis, fluctuation analysis, and their generalizations to higher orders. Non-universal dynamics have been found not only in the scaling exponent α, which is different from the universal values 1/2 and 1, but also in the distributions of the ratio η= σ^exo / σ^endo of individual stocks. Both the scaling exponent α of fluctuations and the Hurst exponent H_i increase in logarithmic form with the time scale Δt and the mean traded value per minute 〈f_i 〉, respectively. We find that the scaling exponent α^endo of the endogenous fluctuations is independent of the time scale. Multiscaling and multifractal features are observed in the data as well. However, the inhomogeneous impact model is not verified.     

Statistical regularities in the return intervals of volatility

We discuss recent results concerning statistical regularities in the return intervals of volatility in financial markets. In particular, we show how the analysis of volatility return intervals, defined as the time between two volatilities larger than a given threshold, can help to get a better understanding of the behavior of financial time series. We find scaling in the distribution of return intervals for thresholds ranging over a factor of 25, from 0.6 to 15 standard deviations, and also for various time windows from one minute up to 390 min (an entire trading day). Moreover, these results are universal for different stocks, commodities, interest rates as well as currencies. We also analyze the memory in the return intervals which relates to the memory in the volatility and find two scaling regimes, ℓ<ℓ^* with α₁=0.64±0.02 and ℓ> ℓ^* with α₂=0.92±0.04; these exponent values are similar to results of Liu et al. for the volatility. As an application, we use the scaling and memory properties of the return intervals to suggest a possibly useful method for estimating risk.     

A unified econophysics explanation for the power-law exponents of stock market activity

We survey a theory (first sketched in Nature in 2003, then fleshed out in the Quarterly Journal of Economics in 2006) of the economic underpinnings of the fat-tailed distributions of a number of financial variables, such as returns and trading volume. Our theory posits that they have a common origin in the strategic trading behavior of very large financial institutions in a relatively illiquid market. We show how the fat-tailed distribution of fund sizes can indeed generate extreme returns and volumes, even in the absence of fundamental news. Moreover, we are able to replicate the individually different empirical values of the power-law exponents for each distribution: 3 for returns, 3/2 for volumes, 1 for the assets under management of large investors. Large investors moderate their trades to reduce their price impact; coupled with a concave price impact function, this leads to volumes being more fat-tailed than returns but less fat-tailed than fund sizes. The trades of large institutions also offer a unified explanation for apparently disconnected empirical regularities that are otherwise a challenge for economic theory.     

Size matters: some stylized facts of the stock market revisited

We reanalyze high resolution data from the New York Stock Exchange and find a monotonic (but not power law) variation of the mean value per trade, the mean number of trades per minute and the mean trading activity with company capitalization. We show that the second moment of the traded value distribution is finite. Consequently, the Hurst exponents for the corresponding time series can be calculated. These are, however, non-universal: The persistence grows with larger capitalization and this results in a logarithmically increasing Hurst exponent. A similar trend is displayed by intertrade time intervals. Finally, we demonstrate that the distribution of the intertrade times is better described by a multiscaling ansatz than by simple gap scaling.     

Modeling interactions of trading volumes in financial dynamics

A dynamic herding model with interactions of trading volumes is introduced. At time t, an agent trades with a probability, which depends on the ratio of the total trading volume at time t−1 to its own trading volume at its last trade. The price return is determined by the volume imbalance and number of trades. The model can reproduce the power-law distributions of the trading volume, number of trades and price return, and the probable relation between them. The exponents are tunable by adjusting the values of the parameters, but show slight deviation from those revealed in empirical studies. Moreover, the time series generated are long-range correlated. We demonstrate that the results are rather robust, and do not depend on the particular form of the trading probability.     

Rank-based model for weighted network with hierarchical organization and disassortative mixing

In this paper, we study a rank-based model for weighted network. The evolution rule of the network is based on the ranking of node strength, which couples the topological growth and the weight dynamics. Analytically and by simulations, we demonstrate that the generated networks recover the scale-free distributions of degree and strength in the whole region of the growth dynamics parameter (α>0). Moreover, this network evolution mechanism can also produce scale-free property of weight, which adds deeper comprehension of the networks growth in the presence of incomplete information. We also characterize the clustering and correlation properties of this class of networks. It is showed that at α=1 a structural phase transition occurs, and for α>1 the generated network simultaneously exhibits hierarchical organization and disassortative degree correlation, which is consistent with a wide range of biological networks.     

On the probability distribution of stock returns in the Mike-Farmer model

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not only the well-known power-law tails, together with several other important stylized facts. There are three key ingredients in the Mike-Farmer (MF) model: the long memory of order signs characterized by the Hurst index H_s, the distribution of relative order prices x in reference to the same best price described by a Student distribution (or Tsallis’ q-Gaussian), and the dynamics of order cancelation. They showed that different values of the Hurst index H_s and the freedom degree α_x of the Student distribution can always produce power-law tails in the return distribution f_r(r) with different tail exponent α_r. In this paper, we study the origin of the power-law tails of the return distribution f_r(r) in the MF model, based on extensive simulations with different combinations of the left part L(x) for x < 0 and the right part R(x) for x > 0 of f_x(x). We find that power-law tails appear only when L(x) has a power-law tail, no matter R(x) has a power-law tail or not. In addition, we find that the distributions of returns in the MF model at different timescales can be well modeled by the Student distributions, whose tail exponents are close to the well-known cubic law and increase with the timescale.     

Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model

We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature T_c and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Δ > 1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1 ≤ Δ < 25.6, a double re-entrant phase for 25.6 < Δ < 43.9 and no re-entrant phase for Δ > 43.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of dynamical stability for spatially inhomogeneous systems.     

Random quantum magnets with broad disorder distribution

We study the critical behavior of Ising quantum magnets with broadly distributed random couplings (J), such that P(ln J) ∼ | ln J|^{-1 - α}, α > 1, for large | ln J| (Lévy flight statistics). For sufficiently broad distributions, α < , the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, α. In one dimension, with = 2, we obtained several exact results through a mapping to surviving Riemann walks. In two dimensions the varying critical exponents have been calculated by a numerical implementation of the Ma-Dasgupta-Hu renormalization group method leading to ≈ 4.5. Thus in the region 2 < α < , where the central limit theorem holds for | ln J| the broadness of the distribution is relevant for the 2d quantum Ising model. Received 6 December 2000 and Received in final form 22 January 2001     

Thouless energy of a superconductor from non local conductance fluctuations

The three-dimensional bimodal random-field Ising model is investigated using the N-fold version of the Wang-Landau algorithm. The essential energy subspaces are determined by the recently developed critical minimum energy subspace technique, and two implementations of this scheme are utilized. The random fields are obtained from a bimodal discrete (±Δ) distribution, and we study the model for various values of the disorder strength Δ, Δ=0.5,1,1.5 and 2, on cubic lattices with linear sizes L=4–24. We extract information for the probability distributions of the specific heat peaks over samples of random fields. This permits us to obtain the phase diagram and present the finite-size behavior of the specific heat. The question of saturation of the specific heat is re-examined and it is shown that the open problem of universality for the random-field Ising model is strongly influenced by the lack of self-averaging of the model. This property appears to be substantially depended on the disorder strength.     

Influence of Thomson effect on the resultant local Seebeck coefficient in thermoelectric composite materials

The resultant local Seebeck coefficient α _R (=α _S−α _T) at the interface of a thermoelement has not yet been measured, although it is an important factor governing the thermoelectric efficiency, where α _S is the local Seebeck coefficient and α _T is the one caused by the Thomson effect. It is shown in this paper that α _S, α _T, and α _R of the p- and n-type Cu/Bi–Te/Cu composites are obtained analytically and experimentally on the assumption that the local temperature of the composite on which the temperature difference ΔT is imposed varies linearly with changes in position along the composite. They were indeed estimated as a function of position from the local experimental data of R,ΔI,ΔT, and V generated by applying an additional current of ±I to the composite, where R is the electrical resistance and ΔI is a current generated by the composite. As a result, it was found that the absolute values of α _S at the hot interface of the p- and n-type composites are approximately 1.5 and 1.4 times higher than their lowest values in the middle region of the composite, respectively, while those of α _T are less than 8% of α _S all over the composite and are so small that the relation α _R≈α _S can be held. We thus succeeded in measuring α _R at the interfaces of the composite.     

Non-extensive trends in the size distribution of coding and non-coding DNA sequences in the human genome

We study the primary DNA structure of four of the most completely sequenced human chromosomes (including chromosome 19 which is the most dense in coding), using non-extensive statistics. We show that the exponents governing the spatial decay of the coding size distributions vary between 5.2 ≤r ≤5.7 for the short scales and 1.45 ≤q ≤1.50 for the large scales. On the contrary, the exponents governing the spatial decay of the non-coding size distributions in these four chromosomes, take the values 2.4 ≤r ≤3.2 for the short scales and 1.50 ≤q ≤1.72 for the large scales. These results, in particular the values of the tail exponent q, indicate the existence of correlations in the coding and non-coding size distributions with tendency for higher correlations in the non-coding DNA.     

Scaling and allometry in the building geometries of Greater London

Many aggregate distributions of urban activities such as city sizes reveal scaling but hardly any work exists on the properties of spatial distributions within individual cities, notwithstanding considerable knowledge about their fractal structure. We redress this here by examining scaling relationships in a world city using data on the geometric properties of individual buildings. We first summarise how power laws can be used to approximate the size distributions of buildings, in analogy to city-size distributions which have been widely studied as rank-size and lognormal distributions following Zipf [Human Behavior and the Principle of Least Effort (Addison-Wesley, Cambridge, 1949)] and Gibrat [Les Inégalités économiques (Librarie du Recueil Sirey, Paris, 1931)]. We then extend this analysis to allometric relationships between buildings in terms of their different geometric size properties. We present some preliminary analysis of building heights from the Emporis database which suggests very strong scaling in world cities. The data base for Greater London is then introduced from which we extract 3.6 million buildings whose scaling properties we explore. We examine key allometric relationships between these different properties illustrating how building shape changes according to size, and we extend this analysis to the classification of buildings according to land use types. We conclude with an analysis of two-point correlation functions of building geometries which supports our non-spatial analysis of scaling.     

Empirical study of recent Chinese stock market

We investigate the statistical properties of the empirical data taken from the Chinese stock market during the time period from January, 2006 to July, 2007. By using the methods of detrended fluctuation analysis (DFA) and calculating correlation coefficients, we acquire the evidence of strong correlations among different stock types, stock index, stock volume turnover, A share (B share) seat number, and GDP per capita. In addition, we study the behavior of “volatility”, which is now defined as the difference between the new account numbers for two consecutive days. It is shown that the empirical power-law of the number of aftershock events exceeding the selected threshold is analogous to the Omori law originally observed in geophysics. Furthermore, we find that the cumulative distributions of stock return, trade volume and trade number are all exponential-like, which does not belong to the universality class of such distributions found by Xavier Gabaix et al. [Xavier Gabaix, Parameswaran Gopikrishnan, Vasiliki Plerou, H. Eugene Stanley, Nature, 423 (2003)] for major western markets. Through the comparison, we draw a conclusion that regardless of developed stock markets or emerging ones, “cubic law of returns” is valid only in the long-term absolute return, and in the short-term one, the distributions are exponential-like. Specifically, the distributions of both trade volume and trade number display distinct decaying behaviors in two separate regimes. Lastly, the scaling behavior of the relation is analyzed between dispersion and the mean monthly trade value for each administrative area in China.     

Superdiffusion and stable laws

The superdiffusion equation with a fractional Laplacian Δ ^α/2 in _N-dimensional space describes the asymptotic (t→∞) behavior of a generalized Poisson process with the range (discontinuity) distribution density ∼|x|−α−1. The solutions of this equation belong to a class of spherically symmetric stable distributions. The main properties of these solutions are given together with their representations in the form of integrals and series and the results of numerical calculations. It is shown that allowance for the finite velocity of free particle motion for α>1 merely amounts to a reduction in the diffusion coefficient with the form of the distribution remaining stable. For α<1 the situation changes radically: the expansion velocity of the diffusion packet exceeds the velocity of free particle motion and the superdiffusion equation becomes physically meaningless. Zh. éksp. Teor. Fiz. 115, 1411–1425 (April 1999)     

An evolving network model in polymer melts with community structures

In order to describe the entangled network structure in polymer melts visually, we propose an evolving network model with community structure. This network model grows according to the inner-community and inter-community preferential mechanisms of both community sizes and node degrees. Numerical simulation results indicate that the cumulative distribution of community size and node degree distribution follow power-law distributions P(S≥s)∼s^-υ and P(k)∼k^-γ respectively, with the exponents of υ≥1 and .     

Network topology of an experimental futures exchange

Many systems of different nature exhibit scale free behaviors. Economic systems with power law distribution in the wealth are one of the examples. To better understand the working behind the complexity, we undertook an experiment recording the interactions between market participants. A Web server was setup to administer the exchange of futures contracts whose liquidation prices were coupled to event outcomes. After free registration, participants started trading to compete for the money prizes upon maturity of the futures contracts at the end of the experiment. The evolving `cash' flow network was reconstructed from the transactions between players. We show that the network topology is hierarchical, disassortative and small-world with a power law exponent of 1.02±0.09 in the degree distribution after an exponential decay correction. The small-world property emerged early in the experiment while the number of participants was still small. We also show power law-like distributions of the net incomes and inter-transaction time intervals. Big winners and losers are associated with high degree, high betweenness centrality, low clustering coefficient and low degree-correlation. We identify communities in the network as groups of the like-minded. The distribution of the community sizes is shown to be power-law distributed with an exponent of 1.19±0.16.     

Transport between multiple users in complex networks

We study the transport properties of model networks such as scale-free and Erd?s-Rényi networks as well as a real network. We consider few possibilities for the trnasport problem. We start by studying the conductance G between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G, with a power-law tail distribution $\Phi_{\rm SF}(G)\sim G^{-g_G}$ , where g_G=2λ-1, and λ is the decay exponent for the scale-free network degree distribution. The power-law tail in Φ_SF(G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erd?s-Rényi networks where the tail of the conductivity distribution decays exponentially. We develop a simple physical picture of the transport to account for the results. The other model for transport is the max-flow model, where conductance is defined as the number of link-independent paths between the two nodes, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. We then extend our study to the case of multiple sources ans sinks, where the transport is defined between two groups of nodes. We find a fundamental difference between the two forms of flow when considering the quality of the transport with respect to the number of sources, and find an optimal number of sources, or users, for the max-flow case. A qualitative (and partially quantitative) explanation is also given.     

A precursor of market crashes: Empirical laws of Japan's internet bubble

In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as X(t) = S(t)/S(0), where S(0), is the stock price for an onset time of the bubble. We selected approximately 3200 stocks traded on the Japanese Stock Exchange, and formed a statistical ensemble of daily relative prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period in which internet Bubble formed and crashed in the Japanese stock market. We found that the upper tail of the complementary cumulative distribution function of the ensemble of the relative prices in the high value of the price is well described by a power-law distribution, P(S>x) ∼x^-α , with an exponent that moves over time. Furthermore we found that as the power-law exponents α approached two, the bubble burst. It is reasonable to suppose that it indicates that internet bubble is about to burst.