Zoltan Eisler

Fluctuation scaling in complex systems: Taylor's law and beyond1

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form ‘fluctuations ≈ constant × averageα’, where the exponent α is predominantly in the range [1/2, 1]. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names Taylors law or fluctuation scaling. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling. 1Dedicated to the memory of L. R. Taylor (1924–2007).

Size matters : some stylized facts of the stock market revisited

Abstract.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.

Multiscaling and non-universality in fluctuations of driven complex systems

For many externally driven complex systems neither the noisy driving force, nor the internal dynamics are a priori known. Here we focus on systems for which the time-dependent activity of a large number of components can be monitored, allowing us to separate each signal into a component attributed to the external driving force and one to the internal dynamics. We propose a formalism to capture the potential multiscaling in the fluctuations and apply it to the high-frequency trading records of the New York Stock Exchange. We find that on the time scale of minutes the dynamics is governed by internal processes, while on a daily or longer scale the external factors dominate. This transition from internal to external dynamics induces systematic changes in the scaling exponents, offering direct evidence of non-universality in the system.

Scaling theory of temporal correlations and size-dependent fluctuations in the traded value of stocks

Records of the traded value of fi stocks display fluctuation scaling, a proportionality between the standard deviation sigma(i) and the average : sigma(i) is proportional to alpha, with a strong time scale dependence alpha(Delta(t)). The nontrivial (i.e., neither 0.5 nor 1) value of alpha may have different origins and provides information about the microscopic dynamics. We present a set of stylized facts and then show their connection to such behavior. The functional form alpha(Delta(t)) originates from two aspects of the dynamics: Stocks of larger companies both tend to be traded in larger packages and also display stronger correlations of traded value. The results are integrated into a general framework that can be applied to a wide range of complex systems.

Multifractal model of asset returns with leverage effect

Multifractal processes are a relatively new tool of stock market analysis. Their power lies in the ability to take multiple orders of autocorrelations into account explicitly. In the first part of the paper we discuss the framework of the Lux model and refine the underlying phenomenological picture. We also give a procedure of fitting all parameters to empirical data. We present a new approach to account for the effective length of power-law memory in volatility. The second part of the paper deals with the consequences of asymmetry in returns. We incorporate two related stylized facts, skewness and leverage autocorrelations into the model. Then from Monte Carlo measurements we show, that this asymmetry significantly increases the mean squared error of volatility forecasts. Based on a filtering method we give evidence on similar behavior in empirical data.

Liquidity and the multiscaling properties of the volume traded on the stock market

We investigate the correlation properties of transaction data from the New York Stock Exchange. The trading activity f(t) of each stock displays a crossover from weaker to stronger correlations at time scales 60-390 minutes. In both regimes, the Hurst exponent H depends logarithmically on the liquidity of the stock, measured by the mean traded value per minute. All multiscaling exponents tau(q) display a similar liquidity dependence, which clearly indicates the lack of a universal form assumed by other studies. The origin of this behavior is both the long memory in the frequency and the size of consecutive transactions.

Random walks on complex networks with inhomogeneous impact

In many complex systems, for the activity f(i) of the constituents or nodes i a power-law relationship was discovered between the standard deviation sigma(i) and the average strength of the activity: sigma(i) proportional variant alpha: universal values alpha=1/2 or 1 were found, however, with exceptions. With the help of an impact variable we present a random walk model where the activity is the product of the number of visitors at a node and their impact. If the impact depends strongly on the node connectivity and the properties of the carrying network are broadly distributed (as in a scale-free network) we find both analytically and numerically nonuniversal alpha values. The exponent always crosses over to the universal value of 1 if the external drive dominates.

The limit order book on different time scales

Financial markets can be described on several time scales. We use data from the limit order book of the London Stock Exchange (LSE) to compare how the fluctuation dominated microstructure crosses over to a more systematic global behavior.

The dynamics of traded value revisited

We conclude from an analysis of high resolution NYSE data that the distribution of the traded value fi (or volume) has a finite variance σi for the very large majority of stocks i, and the distribution itself is non-universal across stocks. The Hurst exponent of the same time series displays a crossover from weakly to strongly correlated behavior around the time scale of 1 day. The persistence in the strongly correlated regime increases with the average trading activity 〈fi〉 as Hi=H0+γlog〈fi〉, which is another sign of non-universal behavior. The existence of such liquidity dependent correlations is consistent with the empirical observation that σi∝〈fi〉α, where α is a non-trivial, time scale dependent exponent.

Why do Hurst exponents of traded value increase as the logarithm of company size

The common assumption of universal behavior in stock market data can sometimes lead to false conclusions. In statistical physics, the Hurst exponents characterizing long-range correlations are often closely related to universal exponents. We show, that in the case of time series of the traded value, these Hurst exponents increase logarithmically with company size, and thus are non-universal. Moreover, the average transaction size shows scaling with the mean transaction frequency for large enough companies. We present a phenomenological scaling framework that properly accounts for such dependencies.