We empirically study the market impact of trading orders. We are specifically interested in large trading orders that are executed incrementally, which we call hidden orders. These are reconstructed based on information about market member codes using data from the Spanish Stock Market and the London Stock Exchange. We find that market impact is strongly concave, approximately increasing as the square root of order size. Furthermore, as a given order is executed, the impact grows in time according to a power-law; after the order is finished, it reverts to a level of about 0.5 − 0.7 of its value at its peak. We observe that hidden orders are executed at a rate that more or less matches trading in the overall market, except for small deviations at the beginning and end of the order.

Computational methods in statistical physics and nonlinear dynamics.Abstract. -We provide numerical indications of the q-generalised central limit theorem that has been conjectured (Tsallis 2004) in nonextensive statistical mechanics. We focus on N binary random variables correlated in a scale-invariant way. The correlations are introduced by imposing the Leibnitz rule on a probability set based on the so-called q-product with q ≤ 1. We show that, in the large N limit (and after appropriate centering, rescaling, and symmetrisation), the emerging distributions are qe-Gaussians, i.e., p(x) ∝ [1 − (1 − qe) β(N )x 2 ] 1/(1−qe) , with qe = 2 − 1 q , and with coefficients β(N ) approaching finite values β(∞). The particular case q = qe = 1 recovers the celebrated de Moivre-Laplace theorem.c EDP Sciences

We demonstrate that the dynamics toward and within the Feigenbaum attractor combine to form a q -deformed statistical-mechanical construction. The rate at which ensemble trajectories converge to the attractor (and to the repellor) is described by a q entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor. The values of the q indices involved are given by the unimodal map universal constants, while the thermodynamic structure is closely related to that formerly developed for multifractals. As an essential component in our demonstration we expose, in great detail, the features of the dynamics of trajectories that either evolve toward the Feigenbaum attractor or are captured by its matching repellor. The dynamical properties of the family of periodic superstable cycles in unimodal maps are seen to be key ingredients for the comprehension of the discrete scale invariance features present at the period-doubling transition to chaos. Elements in our analysis are the following. (i) The preimages of the attractor and repellor of each of the supercycles appear entrenched into a fractal hierarchical structure of increasing complexity as period doubling develops. (ii) The limiting form of this rank structure results in an infinite number of families of well-defined phase-space gaps in the positions of the Feigenbaum attractor or of its repellor. (iii) The gaps in each of these families can be ordered with decreasing width in accordance with power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. (iv) The power law with log-periodic modulation associated with the rate of approach of trajectories toward the attractor (and to the repellor) is explained in terms of the progression of gap formation. (v) The relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated. (vi) A "mean field" evaluation of the atypical partition function, a thermodynamic interpretation of the time evolution process, and a crossover to ordinary exponential statistics are given. We make clear the dynamical origin of the anomalous thermodynamic framework existing at the Feigenbaum attractor.

We scrutinize the anomalies in diffusion observed in an extended long-range system of classical rotors, the HMF model. Under suitable preparation, the system falls into long-lived quasi-stationary states presenting super-diffusion of rotor phases. We investigate the diffusive motion of phases by monitoring the evolution of their probability density function for large system sizes. These densities are shown to be of the q-Gaussian form, P (x) ∝ (1 + (q − 1) [x/β] 2 ) 1/(1−q) , with parameter q increasing with time before reaching a steady value q ≃ 3/2. From this perspective, we also discuss the relaxation to equilibrium and show that diffusive motion in quasi-stationary trajectories strongly depends on system size.