We make an extensive empirical study of the market impact of large orders (metaorders) executed in the U.S. equity market between 2007 and 2009. We show that the square root market impact formula, which is widely used in the industry and supported by previous published research, provides a good fit only across about two orders of magnitude in order size. A logarithmic functional form fits the data better, providing a good fit across almost five orders of magnitude. We introduce the concept of an "impact surface" to model the impact as a function of both the duration and the participation rate of the metaorder, finding again a logarithmic dependence. We show that during the execution the price trajectory deviates from the market impact, a clear indication of non-VWAP executions. Surprisingly, we find that sometimes the price starts reverting well before the end of the execution. Finally we show that, although on average the impact relaxes to approximately 2/3 of the peak impact, the precise asymptotic value of the price depends on the participation rate and on the duration of the metaorder. We present evidence that this might be due to a herding phenomenon among metaorders.

In this paper we estimate the propagation of liquidity shocks through interbank markets when the information about the underlying credit network is incomplete. We show that techniques such as maximum entropy currently used to reconstruct credit networks severely underestimate the risk of contagion by assuming a trivial (fully connected) topology, a type of network structure which can be very different from the one empirically observed. We propose an efficient message-passing algorithm to explore the space of possible network structures and show that a correct estimation of the network degree of connectedness leads to more reliable estimations for systemic risk. Such an algorithm is also able to produce maximally fragile structures, providing a practical upper bound for the risk of contagion when the actual network structure is unknown. We test our algorithm on ensembles of synthetic data encoding some features of real financial networks (sparsity and heterogeneity), finding that more accurate estimations of risk can be achieved. Finally we find that this algorithm can be used to control the amount of information that regulators need to require from banks in order to sufficiently constrain the reconstruction of financial networks.

Large deviations for fat tailed distributions, i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated on a single variable. The regime of large deviations is separated from the regime of typical fluctuations by a phase transition where the symmetry between the points in the sample is spontaneously broken. For stochastic processes with a fat tailed microscopic noise, this implies that while typical realizations are well described by a diffusion process with continuous sample paths, large deviation paths are typically discontinuous. For eigenvalues of random matrices with fat tailed distributed elements, a large deviation where the trace of the matrix is anomalously large concentrates on just a single eigenvalue, whereas in the thin tailed world the large deviation affects the whole distribution.These results find a natural application to finance. Since the price dynamics of financial stocks is characterized by fat tailed increments, large fluctuations of stock prices are expected to be realized by discrete jumps. Interestingly, we find that large excursions of prices are more likely realized by continuous drifts rather than by discontinuous jumps. Indeed, auto-correlations suppress the concentration of large deviations. Financial covariance matrices also exhibit an anomalously large eigenvalue, the market mode, as compared to the prediction of random matrix theory. We show that this is explained by a large deviation with excess covariance rather than by one with excess volatility. arXiv:1201.2817v3 [cond-mat.stat-mech]

We investigate the finite-size fluctuations due to quenched disorder of the critical temperature of the Sherrington-Kirkpatrick spin glass. In order to accomplish this task, we perform a finite-size analysis of the spectrum of the susceptibility matrix obtained via the Plefka expansion. By exploiting results from random matrix theory, we obtain that the fluctuations of the critical temperature are described by the Tracy-Widom distribution with a nontrivial scaling exponent 2/3.