Comment on "tests of scaling and universality of the distributions of trade size and share volume: evidence from three distinct markets".

In a recent publication, Plerou and Stanley [Phys. Rev. E 76, 046109 (2007)] use the Meerschaert-Scheffler estimator to verify the "inverse half-cubic law" of trade size distributions. We show that this procedure systematically underestimates these tail exponents.

Volatility: a hidden Markov process in financial time series.

Volatility characterizes the amplitude of price return fluctuations. It is a central magnitude in finance closely related to the risk of holding a certain asset. Despite its popularity on trading floors, volatility is unobservable and only the price is known. Diffusion theory has many common points with the research on volatility, the key of the analogy being that volatility is a time-dependent diffusion coefficient of the random walk for the price return. We present a formal procedure to extract volatility from price data by assuming that it is described by a hidden Markov process which together with the price forms a two-dimensional diffusion process. We derive a maximum-likelihood estimate of the volatility path valid for a wide class of two-dimensional diffusion processes. The choice of the exponential Ornstein-Uhlenbeck (expOU) stochastic volatility model performs remarkably well in inferring the hidden state of volatility. The formalism is applied to the Dow Jones index. The main results are that (i) the distribution of estimated volatility is lognormal, which is consistent with the expOU model, (ii) the estimated volatility is related to trading volume by a power law of the form sigma proportional, variant V0.55, and (iii) future returns are proportional to the current volatility, which suggests some degree of predictability for the size of future returns.

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.

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.