arXiv subjects

Jean-Philippe Bouchaud

Publications and source records attributed to Jean-Philippe Bouchaud.

At least 145 records · Page 8Linked to original sources

arXiv2015

Sudden Trust Collapse in Networked Societies

Trust is a collective, self-fulfilling phenomenon that suggests analogies with phase transitions. We introduce a stylized model for the build-up and collapse of trust in networks, which generically displays a first order transition. The basic assumption of our model is that whereas trust begets trust, panic also begets panic, in the sense that a small decrease in trust may be amplified and ultimately lead to a sudden and catastrophic drop of trust. We show, using both numerical simulations and mean-field analytic arguments, that there are extended regions of the parameter space where two equilibrium states coexist: a well-connected network where confidence is high, and a poorly connected network where confidence is low. In these coexistence regions, spontaneous jumps from the well-connected state to the poorly connected state can occur, corresponding to a sudden collapse of trust that is not caused by any major external catastrophe. In large systems, spontaneous crises are replaced by history dependence: whether the system is found in one state or in the other essentially depends on initial conditions. Finally, we document a new phase, in which agents are connected yet distrustful.

physics.soc-ph↗

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↗

arXiv2015

The eigenvectors of Gaussian matrices with an external source

We consider a diffusive matrix process $(X_t)_{t\ge 0}$ defined as $X_t:=A+H_t$ where $A$ is a given deterministic Hermitian matrix and $(H_t)_{t\ge 0}$ is a Hermitian Brownian motion. The matrix $A$ is the "external source" that one would like to estimate from the noisy observation $X_t$ at some time $t>0$. We investigate the relationship between the non-perturbed eigenvectors of the matrix $A$ and the perturbed eigenstates at some time $t$ for the three relevant scaling relations between the time $t$ and the dimension $N$ of the matrix $X_t$. We determine the asymptotic (mean-squared) projections of any given non-perturbed eigenvector $|ψ_j^0\rangle$, associated to an eigenvalue $a_j$ of $A$ which may lie inside the bulk of the spectrum or be isolated (spike) from the other eigenvalues, on the orthonormal basis of the perturbed eigenvectors $|ψ_i^t\rangle,i\neq j$. We derive a Burgers type evolution equation for the local resolvent $(z-X_t)_{ii}^{-1}$, describing the evolution of the local density of a given initial state $|ψ_j ^0\rangle$. We are able to solve this equation explicitly in the large $N$ limit, for any initial matrix $A$. In the case of one isolated eigenvector $|ψ_j^0\rangle$, we prove a central limit Theorem for the overlap $\langle ψ_j^0|ψ_j^t\rangle$. When properly centered and rescaled by a factor $\sqrt{N}$, this overlap converges in law towards a centered Gaussian distribution with an explicit variance depending on $t$. Our method is based on analyzing the eigenvector flow under the Dyson Brownian motion.

math.PR↗

arXiv2015

Eigenvector dynamics under free addition

We investigate the evolution of a given eigenvector of a symmetric (deterministic or random) matrix under the addition of a matrix in the Gaussian orthogonal ensemble. We quantify the overlap between this single vector with the eigenvectors of the initial matrix and identify precisely a "Cauchy-flight" regime. In particular, we compute the local density of this vector in the eigenvalues space of the initial matrix. Our results are obtained in a non perturbative setting and are derived using the ideas of [O. Ledoit and S. Péché, Prob. Th. Rel. Fields, {\bf 151} 233 (2011)]. Finally, we give a robust derivation of a result obtained in [R. Allez and J.-P. Bouchaud, Phys. Rev. E {\bf 86}, 046202 (2012)] to study eigenspace dynamics in a semi-perturbative regime.

math.PR↗

arXiv2014

Turbulent fracture surfaces: A footprint of damage percolation?

We show that a length scale xi can be extracted from the spatial correlations of the "steep cliffs" that appear on fracture surface. Above xi, the slope amplitudes are uncorrelated and the fracture surface is mono-affine. Below xi, long-range spatial correlation lead to a multi-fractal behavior of the surface, reminiscent of turbulent flows. Our results support a unifying conjecture for the geometry of fracture surfaces: for scales > ξthe surface is the trace left by an elastic line propagating in a random medium, while for scales < ξthe highly correlated patterns on the surface result from the merging of interacting damage cavities.

cond-mat.mtrl-sci↗

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

Revisiting Directed Polymers with heavy-tailed disorder

In this mostly numerical study, we revisit the statistical properties of the ground state of a directed polymer in a $d=1+1$ "hilly" disorder landscape, i.e. when the quenched disorder has power-law tails. When disorder is Gaussian, the polymer minimizes its total energy through a collective optimization, where the energy of each visited site only weakly contributes to the total. Conversely, a hilly landscape forces the polymer to distort and explore a larger portion of space to reach some particularly deep energy sites. As soon as the fifth moment of the disorder diverges, this mechanism radically changes the standard "KPZ" scaling behaviour of the directed polymer, and new exponents prevail. After confirming again that the Flory argument accurately predicts these exponent in the tail-dominated phase, we investigate several other statistical features of the ground state that shed light on this unusual transition and on the accuracy of the Flory argument. We underline the theoretical challenge posed by this situation, which paradoxically becomes even more acute above the upper critical dimension.

cond-mat.dis-nn↗

arXiv2014

Branching ratio approximation for the self-exciting Hawkes process

We introduce a model-independent approximation for the branching ratio of Hawkes self-exciting point processes. Our estimator requires knowing only the mean and variance of the event count in a sufficiently large time window, statistics that are readily obtained from empirical data. The method we propose greatly simplifies the estimation of the Hawkes branching ratio, recently proposed as a proxy for market endogeneity and formerly estimated using numerical likelihood maximisation. We employ our new method to support recent theoretical and experimental results indicating that the best fitting Hawkes model to describe S&P futures price changes is in fact critical (now and in the recent past) in light of the long memory of financial market activity.

q-fin.ST↗

arXiv2014

Endogenous crisis waves: a stochastic model with synchronized collective behavior

We propose a simple framework to understand commonly observed crisis waves in macroeconomic Agent Based models, that is also relevant to a variety of other physical or biological situations where synchronization occurs. We compute exactly the phase diagram of the model and the location of the synchronization transition in parameter space. Many modifications and extensions can be studied, confirming that the synchronization transition is extremely robust against various sources of noise or imperfections.

cond-mat.stat-mech↗

arXiv2014

Instanton Approach to Large $N$ Harish-Chandra-Itzykson-Zuber Integrals

We reconsider the large $N$ asymptotics of Harish-Chandra-Itzykson-Zuber integrals. We provide, using Dyson's Brownian motion and the method of instantons, an alternative, transparent derivation of the Matytsin formalism for the unitary case. Our method is easily generalized to the orthogonal and symplectic ensembles. We obtain an explicit solution of Matytsin's equations in the case of Wigner matrices, as well as a general expansion method in the dilute limit, when the spectrum of eigenvalues spreads over very wide regions.

cond-mat.stat-mech↗

arXiv2014

Tipping points in macroeconomic Agent-Based models

The aim of this work is to explore the possible types of phenomena that simple macroeconomic Agent-Based models (ABM) can reproduce. We propose a methodology, inspired by statistical physics, that characterizes a model through its 'phase diagram' in the space of parameters. Our first motivation is to understand the large macro-economic fluctuations observed in the 'Mark I' ABM. Our major finding is the generic existence of a phase transition between a 'good economy' where unemployment is low, and a 'bad economy' where unemployment is high. We introduce a simpler framework that allows us to show that this transition is robust against many modifications of the model, and is generically induced by an asymmetry between the rate of hiring and the rate of firing of the firms. The unemployment level remains small until a tipping point, beyond which the economy suddenly collapses. If the parameters are such that the system is close to this transition, any small fluctuation is amplified as the system jumps between the two equilibria. We have explored several natural extensions of the model. One is to introduce a bankruptcy threshold, limiting the leverage of firms. This leads to a rich phase diagram with, in particular, a region where acute endogenous crises occur, during which the unemployment rate shoots up before the economy can recover. We also introduce simple wage policies. This leads to inflation (in the 'good' phase) or deflation (in the 'bad' phase), but leaves the overall phase diagram of the model essentially unchanged. We have also started exploring the effect of simple monetary policies that attempt to contain rising unemployment and defang crises. We end the paper with general comments on the usefulness of ABMs to model macroeconomic phenomena, in particular in view of the time needed to reach a steady state that raises the issue of ergodicity in these models.

q-fin.GN↗

arXiv2014

Instabilities in large economies: aggregate volatility without idiosyncratic shocks

We study a dynamical model of interconnected firms which allows for certain market imperfections and frictions, restricted here to be myopic price forecasts and slow adjustment of production. Whereas the standard rational equilibrium is still formally a stationary solution of the dynamics, we show that this equilibrium becomes linearly unstable in a whole region of parameter space. When agents attempt to reach the optimal production target too quickly, coordination breaks down and the dynamics becomes chaotic. In the unstable, "turbulent" phase, the aggregate volatility of the total output remains substantial even when the amplitude of idiosyncratic shocks goes to zero or when the size of the economy becomes large. In other words, crises become endogenous. This suggests an interesting resolution of the "small shocks, large business cycles" puzzle.

econ.GN↗

arXiv2014

The fine structure of volatility feedback II: overnight and intra-day effects

We decompose, within an ARCH framework, the daily volatility of stocks into overnight and intra-day contributions. We find, as perhaps expected, that the overnight and intra-day returns behave completely differently. For example, while past intra-day returns affect equally the future intra-day and overnight volatilities, past overnight returns have a weak effect on future intra-day volatilities (except for the very next one) but impact substantially future overnight volatilities. The exogenous component of overnight volatilities is found to be close to zero, which means that the lion's share of overnight volatility comes from feedback effects. The residual kurtosis of returns is small for intra-day returns but infinite for overnight returns. We provide a plausible interpretation for these findings, and show that our Intra-Day/Overnight model significantly outperforms the standard ARCH framework based on daily returns for Out-of-Sample predictions.

q-fin.ST↗

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↗

arXiv2014

Critical dynamical heterogeneities close to continuous second-order glass transitions

We analyse, using Inhomogenous Mode-Coupling Theory, the critical scaling behaviour of the dynamical susceptibility at a distance epsilon from continuous second-order glass transitions. We find that the dynamical correlation length xi behaves generically as epsilon^{-1/3} and that the upper critical dimension is equal to six. More surprisingly, we find activated dynamic scaling, where xi grows with time as [ln(t)]^2 exactly at criticality. All these results suggest a deep analogy between the glassy behaviour of attractive colloids or randomly pinned supercooled liquids and that of the Random Field Ising Model.

cond-mat.dis-nn↗

arXiv2014

On the emergence of an "intention field" for socially cohesive agents

We argue that when a social convergence mechanism exists and is strong enough, one should expect the emergence of a well defined "field", i.e. a slowly evolving, local quantity around which individual attributes fluctuate in a finite range. This condensation phenomenon is well illustrated by the Deffuant-Weisbuch opinion model for which we provide a natural extension to allow for spatial heterogeneities. We show analytically and numerically that the resulting dynamics of the emergent field is a noisy diffusion equation that has a slow dynamics. This random diffusion equation reproduces the long-ranged, logarithmic decrease of the correlation of spatial voting patterns empirically found in [1, 2]. Interestingly enough, we find that when the social cohesion mechanism becomes too weak, cultural cohesion breaks down completely, in the sense that the distribution of intentions/opinions becomes infinitely broad. No emerging field exists in this case. All these analytical findings are confirmed by numerical simulations of an agent-based model.

physics.soc-ph↗

arXiv2013

Explore or exploit? A generic model and an exactly solvable case

Finding a good compromise between the exploitation of known resources and the exploration of unknown, but potentially more profitable choices, is a general problem, which arises in many different scientific disciplines. We propose a stylized model for these exploration-exploitation situations, including population or economic growth, portfolio optimisation, evolutionary dynamics, or the problem of optimal pinning of vortices or dislocations in disordered materials. We find the exact growth rate of this model for tree-like geometries and prove the existence of an optimal migration rate in this case. Numerical simulations in the one-dimensional case confirm the generic existence of an optimum.

cond-mat.dis-nn↗

arXiv2013

Skew and implied leverage effect: smile dynamics revisited

We revisit the ``Smile Dynamics'' problem, which consists in relating the implied leverage (i.e. the correlation of the at-the-money volatility with the returns of the underlying) and the skew of the option smile. The ratio between these two quantities, called ``Skew-Stickiness Ratio'' (SSR) by Bergomi (Smile Dynamics IV, RISK, 94-100, December 2009), saturates to the value 2 for linear models in the limit of small maturities, and converges to 1 for long maturities. We show that for more general, non-linear models (such as the asymmetric GARCH model), Bergomi's result must be modified, and can be larger than 2 for small maturities. The discrepancy comes from the fact that the volatility skew is, in general, different from the skewness of the underlying. We compare our theory with empirical results, using data both from option markets and from the underlying price series, for the S&P500 and the DAX. We find, among other things, that although both the implied leverage and the skew appear to be too strong on option markets, their ratio is well explained by the theory. We observe that the SSR indeed becomes larger than 2 for small maturities.

q-fin.ST↗