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Iacopo Mastromatteo

Publications and source records attributed to Iacopo Mastromatteo.

At least 37 records · Page 2Linked to original sources
arXiv2015

Mean-field inference of Hawkes point processes

We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d-dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the Maximum Likelihood Estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters.

cs.LG↗
arXiv2015

Apparent impact: the hidden cost of one-shot trades

We study the problem of the execution of a moderate size order in an illiquid market within the framework of a solvable Markovian model. We suppose that in order to avoid impact costs, a trader decides to execute her order through a unique trade, waiting for enough liquidity to accumulate at the best quote. We find that despite the absence of a proper price impact, such trader faces an execution cost arising from a non-vanishing correlation among volume at the best quotes and price changes. We characterize analytically the statistics of the execution time and its cost by mapping the problem to the simpler one of calculating a set of first-passage probabilities on a semi-infinite strip. We finally argue that price impact cannot be completely avoided by conditioning the execution of an order to a more favorable liquidity scenario.

q-fin.TR↗
arXiv2015

Hawkes processes in finance

In this paper we propose an overview of the recent academic literature devoted to the applications of Hawkes processes in finance. Hawkes processes constitute a particular class of multivariate point processes that has become very popular in empirical high frequency finance this last decade. After a reminder of the main definitions and properties that characterize Hawkes processes, we review their main empirical applications to address many different problems in high frequency finance. Because of their great flexibility and versatility, we show that they have been successfully involved in issues as diverse as estimating the volatility at the level of transaction data, estimating the market stability, accounting for systemic risk contagion, devising optimal execution strategies or capturing the dynamics of the full order book.

q-fin.TR↗
arXiv2015

A fully consistent, minimal model for non-linear market impact

We propose a minimal theory of non-linear price impact based on a linear (latent) order book approximation, inspired by diffusion-reaction models and general arguments. Our framework allows one to compute the average price trajectory in the presence of a meta-order, that consistently generalizes previously proposed propagator models. We account for the universally observed square-root impact law, and predict non-trivial trajectories when trading is interrupted or reversed. We prove that our framework is free of price manipulation, and that prices can be made diffusive (albeit with a generic short-term mean-reverting contribution). Our model suggests that prices can be decomposed into a transient "mechanical" impact component and a permanent "informational" component.

q-fin.TR↗
arXiv2014

Agent-based models for latent liquidity and concave price impact

We revisit the "epsilon-intelligence" model of Toth et al.(2011), that was proposed as a minimal framework to understand the square-root dependence of the impact of meta-orders on volume in financial markets. The basic idea is that most of the daily liquidity is "latent" and furthermore vanishes linearly around the current price, as a consequence of the diffusion of the price itself. However, the numerical implementation of Toth et al. was criticised as being unrealistic, in particular because all the "intelligence" was conferred to market orders, while limit orders were passive and random. In this work, we study various alternative specifications of the model, for example allowing limit orders to react to the order flow, or changing the execution protocols. By and large, our study lends strong support to the idea that the square-root impact law is a very generic and robust property that requires very few ingredients to be valid. We also show that the transition from super-diffusion to sub-diffusion reported in Toth et al. is in fact a cross-over, but that the original model can be slightly altered in order to give rise to a genuine phase transition, which is of interest on its own. We finally propose a general theoretical framework to understand how a non-linear impact may appear even in the limit where the bias in the order flow is vanishingly small.

q-fin.TR↗
arXiv2014

Linear processes in high-dimension: phase space and critical properties

In this work we investigate the generic properties of a stochastic linear model in the regime of high-dimensionality. We consider in particular the Vector AutoRegressive model (VAR) and the multivariate Hawkes process. We analyze both deterministic and random versions of these models, showing the existence of a stable and an unstable phase. We find that along the transition region separating the two regimes, the correlations of the process decay slowly, and we characterize the conditions under which these slow correlations are expected to become power-laws. We check our findings with numerical simulations showing remarkable agreement with our predictions. We finally argue that real systems with a strong degree of self-interaction are naturally characterized by this type of slow relaxation of the correlations.

cond-mat.stat-mech↗
arXiv2014

Anomalous impact in reaction-diffusion models

We generalize the reaction-diffusion model A + B -> 0 in order to study the impact of an excess of A (or B) at the reaction front. We provide an exact solution of the model, which shows that linear response breaks down: the average displacement of the reaction front grows as the square-root of the imbalance. We argue that this model provides a highly simplified but generic framework to understand the square-root impact of large orders in financial markets.

cond-mat.stat-mech↗
arXiv2013

On the typical properties of inverse problems in statistical mechanics

In this work we consider the problem of extracting a set of interaction parameters from an high-dimensional dataset describing T independent configurations of a complex system composed of N binary units. This problem is formulated in the language of statistical mechanics as the problem of finding a family of couplings compatible with a corresponding set of empirical observables in the limit of large N. We focus on the typical properties of its solutions and highlight the possible spurious features which are associated with this regime (model condensation, degenerate representations of data, criticality of the inferred model). We present a class of models (complete models) for which the analytical solution of this inverse problem can be obtained, allowing us to characterize in this context the notion of stability and locality. We clarify the geometric interpretation of some of those aspects by using results of differential geometry, which provides means to quantify consistency, stability and criticality in the inverse problem. In order to provide simple illustrative examples of these concepts we finally apply these ideas to datasets describing two stochastic processes (simulated realizations of a Hawkes point-process and a set of time-series describing financial transactions in a real market).

cond-mat.stat-mech↗
arXiv2013

On sampling and modeling complex systems

The study of complex systems is limited by the fact that only few variables are accessible for modeling and sampling, which are not necessarily the most relevant ones to explain the systems behavior. In addition, empirical data typically under sample the space of possible states. We study a generic framework where a complex system is seen as a system of many interacting degrees of freedom, which are known only in part, that optimize a given function. We show that the underlying distribution with respect to the known variables has the Boltzmann form, with a temperature that depends on the number of unknown variables. In particular, when the unknown part of the objective function decays faster than exponential, the temperature decreases as the number of variables increases. We show in the representative case of the Gaussian distribution, that models are predictable only when the number of relevant variables is less than a critical threshold. As a further consequence, we show that the information that a sample contains on the behavior of the system is quantified by the entropy of the frequency with which different states occur. This allows us to characterize the properties of maximally informative samples: in the under-sampling regime, the most informative frequency size distributions have power law behavior and Zipf's law emerges at the crossover between the under sampled regime and the regime where the sample contains enough statistics to make inference on the behavior of the system. These ideas are illustrated in some applications, showing that they can be used to identify relevant variables or to select most informative representations of data, e.g. in data clustering.

physics.data-an↗
arXiv2012

Impact of meta-order in the Minority Game

We study the market impact of a meta-order in the framework of the Minority Game. This amounts to studying the response of the market when introducing a trader who buys or sells a fixed amount h for a finite time T. This perturbation introduces statistical arbitrages that traders exploit by adapting their trading strategies. The market impact depends on the nature of the stationary state: We find that the permanent impact is zero in the unpredictable (information efficient) phase, while in the predictable phase it is non-zero and grows linearly with the size of the meta-order. This establishes a quantitative link between information efficiency and trading efficiency (i.e. market impact). By using statistical mechanics methods for disordered systems, we are able to fully characterize the response in the predictable phase, to relate execution cost to response functions and obtain exact results for the permanent impact.

q-fin.TR↗
arXiv2012

Beyond inverse Ising model: structure of the analytical solution for a class of inverse problems

I consider the problem of deriving couplings of a statistical model from measured correlations, a task which generalizes the well-known inverse Ising problem. After reminding that such problem can be mapped on the one of expressing the entropy of a system as a function of its corresponding observables, I show the conditions under which this can be done without resorting to iterative algorithms. I find that inverse problems are local (the inverse Fisher information is sparse) whenever the corresponding models have a factorized form, and the entropy can be split in a sum of small cluster contributions. I illustrate these ideas through two examples (the Ising model on a tree and the one-dimensional periodic chain with arbitrary order interaction) and support the results with numerical simulations. The extension of these methods to more general scenarios is finally discussed.

cond-mat.stat-mech↗
arXiv2012

Reconstruction of financial network for robust estimation of systemic risk

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 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 regulators need to require from banks in order to sufficiently constrain the reconstruction of financial networks.

q-fin.RM↗
arXiv2011

On the criticality of inferred models

Advanced inference techniques allow one to reconstruct the pattern of interaction from high dimensional data sets. We focus here on the statistical properties of inferred models and argue that inference procedures are likely to yield models which are close to a phase transition. On one side, we show that the reparameterization invariant metrics in the space of probability distributions of these models (the Fisher Information) is directly related to the model's susceptibility. As a result, distinguishable models tend to accumulate close to critical points, where the susceptibility diverges in infinite systems. On the other, this region is the one where the estimate of inferred parameters is most stable. In order to illustrate these points, we discuss inference of interacting point processes with application to financial data and show that sensible choices of observation time-scales naturally yield models which are close to criticality.

physics.data-an↗
arXiv2009

Cosmic ray knee and new physics at the TeV scale

We analyze the possibility that the cosmic ray knee appears at an energy threshold where the proton-dark matter cross section becomes large due to new TeV physics. It has been shown that such interactions could break the proton and produce a diffuse gamma ray flux consistent with MILAGRO observations. We argue that this hypothesis implies knees that scale with the atomic mass for the different nuclei, as KASKADE data seem to indicate. We find that to explain the change in the spectral index in the flux from E^{-2.7} to E^{-3.1} the cross section must grow like E^{0.4+β} above the knee, where β=0.3-0.6 parametrizes the energy dependence of the age (τ\propto E^{-β}) of the cosmic rays reaching the Earth. The hypothesis also requires mbarn cross sections (that could be modelled with TeV gravity) and large densities of dark matter (that could be clumped around the sources of cosmic rays). We argue that neutrinos would also exhibit a threshold at E=(m_χ/m_p)E_{knee}\approx 10^8 GeV where their interaction with a nucleon becomes strong. Therefore, the observation at ICECUBE or ANITA of standard neutrino events above this threshold would disprove the scenario.

hep-ph↗
arXiv2009

Cosmic-ray knee and flux of secondaries from interactions of cosmic rays with dark matter

We discuss possible implications of a large interaction cross section between cosmic rays and dark matter particles due to new physics at the TeV scale. In particular, in models with extra dimensions and a low fundamental scale of gravity the cross section grows very fast at transplanckian energies. We argue that the knee observed in the cosmic ray flux could be caused by such interactions. We show that this hypothesis implies a well defined flux of secondary gamma rays that seems consistent with MILAGRO observations.

hep-ph↗
arXiv2009

Signatures of TeV gravity from the evaporation of cosmogenic black holes

TeV gravity models provide a scenario for black hole formation at energies much smaller than G_N^(-1/2) \sim 10^19 GeV. In particular, the collision of a ultrahigh energy cosmic ray with a dark matter particle in our galactic halo or with another cosmic ray could result into a black hole of mass between 10^4 and 10^11 GeV. Once produced, such object would evaporate into elementary particles via Hawking radiation. We show that the interactions among the particles exiting the black hole are not able to produce a photosphere nor a chromosphere. We then evaluate how these particles evolve using the jet-code HERWIG, and obtain a final diffuse flux of stable 4-dimensional particles peaked at 0.2 GeV. This flux consists of an approximate 43% of neutrinos, a 28% of electrons, a 16% of photons and a 13% of protons. Emission into the bulk would range from a 1.4% of the total energy for n=2 to a 16% for n=6.

hep-ph↗
arXiv2008

Cosmic-ray knee and diffuse gamma, e+ and pbar fluxes from collisions of cosmic rays with dark matter

In models with extra dimensions the fundamental scale of gravity M_D could be of order TeV. In that case the interaction cross section between a cosmic proton of energy E and a dark matter particle χwill grow fast with E for center of mass energies \sqrt{2m_χE} above M_D, and it could reach 1 mbarn at E\approx 10^9 GeV. We show that these gravity-mediated processes would break the proton and produce a diffuse flux of particles/antiparticles, while boosting χwith a fraction of the initial proton energy. We find that the expected cross sections and dark matter densities are not enough to produce an observable asymmetry in the flux of the most energetic (extragalactic) cosmic rays. However, we propose that unsuppressed TeV interactions may be the origin of the knee observed in the spectrum of galactic cosmic rays. The knee would appear at the energy threshold for the interaction of dark matter particles with cosmic protons trapped in the galaxy by \muG magnetic fields, and it would imply a well defined flux of secondary antiparticles and TeV gamma rays.

hep-ph↗