Spherical integrals and their applications to random matrix theory

2 novembre 2022

PhD Thesis

Pierre Mergny

Random matrix theory has found applications in many fields of physics (disordered systems, stability of dynamical systems, interface models, electronic transport,...) and mathematics (operator algebra, numerative combinatorics, number theory,...). A recurrent problem in many domains is to understand how the spectra of two random matrices recombine when we perform their sum or product. In this thesis, we study this problem through the prism of spherical integrals and with the help of statistical physics tools. These spherical integrals play the role of the Fourier transform in random matrix theory and their study
allows us to better understand the properties of both the limiting spectral density and the largest eigenvalue of these matrix models.

Large macroeconomic fluctuations: Self-organized criticality in firm networks, agent-based models and random matrices

1 novembre 2022

PhD Thesis

Théo Dessertaine

What is the origin of macroeconomic fluctuations? In the late XXth century, Ben Bernanke first introduced the so-called "small shocks, large business cycles" puzzle as the seeming incompatibility between small fluctuations observed at granular levels of the economy (small shocks) and large macroeconomic fluctuations (large business cycles). As an example, the Unites States’ GDP displays a steady average yearly growth rate of around 3% but with fluctuations reaching 2.7%. The conundrum is that most of this volatility cannot be linked to known exogeneous crises, such as oil shocks or the 2008 financial crisis, and must therefore be of endogeneous origin, i.e. generated by the economy itself. Numerous explanations have been proposed, the most famous of which involve the power-law distribution of firms’ sizes, rippling out at aggregate levels of the economy, or network effects responsible for amplifying micro-level shocks. However, these explanations rely on equilibrium-only economic models which picture the world as a succession of equilibria instantaneously reached without friction, and which tautologically do not account for out-of-equilibrium effects. In this thesis, the first two parts are devoted to finding mechanisms accounting for the excess volatility through those overlooked out-of-equilibrium effects. The third part is dedicated to studying more general properties of so-called conewise linear systems, which are ubiquitous in economics.

Multivariate Quadratic Hawkes Processes -- Part I: Theoretical Analysis

26 juin 2022

arXiv preprint

Cecilia Aubrun, Michael Benzaquen, J-P Bouchaud

Quadratic Hawkes (QHawkes) processes have proved effective at reproducing the statistics of price changes, capturing many of the stylised facts of financial markets. Motivated by the recently reported strong occurrence of endogenous co-jumps (simultaneous price jumps of several assets) we extend QHawkes to a multivariate framework (MQHawkes), that is considering several financial assets and their interactions. Assuming that quadratic kernels write as the sum of a time-diagonal component and a rank one (trend) contribution, we investigate endogeneity ratios and the resulting stationarity conditions. We then derive the so-called Yule-Walker equations relating covariances and feedback kernels, which are essential to calibrate the MQHawkes process on empirical data. Finally, we investigate the volatility distribution of the process and find that, as in the univariate case, it exhibits power-law behavior, with an exponent that can be exactly computed in some limiting cases.

Microfounding GARCH Models and Beyond: A Kyle-inspired Model with Adaptive Agents

22 juin 2022

SSRN Preprint

Michele Vodret, Iacopo Mastromatteo, Bence Toth, Michael Benzaquen

We relax the strong rationality assumption for the agents in the paradigmatic Kyle model of price formation, thereby reconciling the framework of asymmetrically informed traders with the Adaptive Market Hypothesis, where agents use inductive rather than deductive reasoning. Building on these ideas, we propose a stylised model able to account parsimoniously for a rich phenomenology, ranging from excess volatility to volatility clustering. While characterising the excess-volatility dynamics, we provide a microfoundation for GARCH models. Volatility clustering is shown to be related to the self-excited dynamics induced by traders' behaviour, and does not rely on clustered fundamental innovations. Finally, we propose an extension to account for the fragile dynamics exhibited by real markets during flash crashes.

Bounded Rationality and Animal Spirits: A Fluctuation-Response Approach to Slutsky Matrices

8 juin 2022

arXiv preprint

Jerome Garnier-Brun, J-P Bouchaud, Michael Benzaquen

The Slutsky equation, central in consumer choice theory, is derived from the usual hypotheses underlying most standard models in Economics, such as full rationality, homogeneity, and absence of interactions. We present a statistical physics framework that allows us to relax such assumptions. We first derive a general fluctuation-response formula that relates the Slutsky matrix to spontaneous fluctuations of consumption rather than to response to changing prices and budget. We then show that, within our hypotheses, the symmetry of the Slutsky matrix remains valid even when agents are only boundedly rational but non-interacting. We then propose a model where agents are influenced by the choice of others, leading to a phase transition beyond which consumption is dominated by herding (or `"fashion") effects. In this case, the individual Slutsky matrix is no longer symmetric, even for fully rational agents. The vicinity of the transition features a peak in asymmetry.

Out-of-equilibrium dynamics and excess volatility in firm networks

30 avril 2022

Journal of Economic Dynamics and Control

Théo Dessertaine, Jose Moran, Michael Benzaquen, J-P Bouchaud

We study the conditions under which input-output networks can dynamically attain competitive equilibrium, where markets clear and profits are zero. We endow a classical firm network model with simple dynamical rules that reduce supply/demand imbalances and excess profits. We show that the time needed to reach equilibrium diverges as the system approaches an instability point beyond which the Hawkins-Simons condition is violated and competitive equilibrium is no longer realisable. We argue that such slow dynamics is a source of excess volatility, through accumulation and amplification of exogenous shocks. Factoring in essential physical constraints, such as causality or inventory management, we propose a dynamically consistent model that displays a rich variety of phenomena. Competitive equilibrium can only be reached after some time and within some region of parameter space, outside of which one observes periodic and chaotic phases, reminiscent of real business cycles. This suggests an alternative explanation of the excess volatility that is of purely endogenous nature. Other regimes include deflationary equilibria and intermittent crises characterised by bursts of inflation. Our model can be calibrated using highly disaggregated data on individual firms and prices, and may provide a powerful tool to describe out-of-equilibrium economies.

A New Spin on Color Quantization

25 avril 2022

arXiv preprint

Samy Lakhal, Alexandre Darmon, Michael Benzaquen

We address the problem of image color quantization using a Maximum Entropy based approach. We argue that adding thermal noise to the system yields better visual impressions than that obtained from a simple energy minimization. To quantify this observation, we introduce the coarse-grained quantization error, and seek the optimal temperature which minimizes this new observable. By comparing images with different structural properties, we show that the optimal temperature is a good proxy for complexity at different scales. Finally, having shown that the convoluted error is a key observable, we directly minimize it using a Monte Carlo algorithm to generate a new series of quantized images. Adopting an original approach based on the informativity of finite size samples, we are able to determine the optimal convolution parameter leading to the best visuals.

Scale Dependencies and Self-Similarity Through Wavelet Scattering Covariance

18 avril 2022

arXiv preprint

Rudy Morel, Gaspar Rochette, Roberto Leonarduzzi, J-P Bouchaud, Stephane Mallat

We introduce a scattering covariance matrix which provides non-Gaussian models of time-series having stationary increments. A complex wavelet transform computes signal variations at each scale. Dependencies across scales are captured by the joint covariance across time and scales of complex wavelet coefficients and their modulus. This covariance is nearly diagonalized by a second wavelet transform, which defines the scattering covariance. We show that this set of moments characterizes a wide range of non-Gaussian properties of multi-scale processes. This is analyzed for a variety of processes, including fractional Brownian motions, Poisson, multifractal random walks and Hawkes processes. We prove that self-similar processes have a scattering covariance matrix which is scale invariant. This property can be estimated numerically and defines a class of wide-sense self-similar processes. We build maximum entropy models conditioned by scattering covariance coefficients, and generate new time-series with a microcanonical sampling algorithm. Applications are shown for highly non-Gaussian financial and turbulence time-series.

The multivariate price formation process and cross-impact

29 mars 2022

PhD Thesis

Mehdi Tomas

This thesis comprises six parts. The first relates anonymous order flow and price changes using static, linear cross-impact models. We list desirable properties of such models, characterise those which satisfy them and test their predictions on different markets. The second part extends this approach to derivatives to obtain a tractable estimation method for cross-impact which is applied to SP500 options and VIX futures. In the third part, we generalise the previous setup to derive and estimate cross-impact models which account for the influence of past trades on current prices. The fourth part uses meta-order databases on stocks and futures to propose a formula for cross-impact which generalises the square-root law of market impact. In the fifth part, we propose a tick-by-tick model for price dynamics using Hawkes processes. We investigate scaling limits of prices in the high endogeneity regime to derive multivariate macroscopic price dynamics of rough Heston type. Finally, the last part solves the calibration problem of volatility models using neural networks.

The Inelastic Market Hypothesis: A Microstructural Interpretation

10 janvier 2022

arXiv preprint

J-P Bouchaud

We attempt to reconcile Gabaix and Koijen's (GK) recent Inelastic Market Hypothesis (IMH) with the order-driven view of markets that emerged within the microstructure literature in the past 20 years. We review the most salient empirical facts and arguments that give credence to the idea that market price fluctuations are mostly due to order flow, whether informed or non-informed. We show that the Latent Liquidity Theory of price impact makes a precise prediction for GK's multiplier M, which measures by how many dollars, on average, the market value of a company goes up if one buys one dollar worth of its stocks. Our central result is that M is of order unity, as found by GK, and increases with the volatility of the stock and decreases with the fraction of the market cap. traded daily. We discuss several empirical results suggesting that the lion's share of volatility is due to trading activity. We argue that the IMH holds for all asset classes, beyond the case of stock markets considered by GK.

Will Random Cone-wise Linear Systems Be Stable?

3 janvier 2022

arXiv preprint

Théo Dessertaine, J-P Bouchaud

We consider a simple model for multidimensional cone-wise linear dynamics around cusp-like equilibria. We assume that the local linear evolution is either v′=Av or Bv (with A, B independently drawn a rotationally invariant ensemble of N×N matrices) depending on the sign of the first component of v. We establish strong connections with the random diffusion persistence problem. When N→∞, we find that the Lyapounov exponent is non self-averaging, i.e. one can observe apparent stability and apparent instability for the same system, depending on time and initial conditions. Finite N effects are also discussed, and lead to cone trapping phenomena.

On Hawkes Processes with Infinite Mean Intensity

27 décembre 2021

Physical Review E

Cecilia Aubrun, Michael Benzaquen, J-P Bouchaud

The stability condition for Hawkes processes and their non-linear extensions usually relies on the condition that the mean intensity is a finite constant. It follows that the total endogeneity ratio needs to be strictly smaller than unity. In the present note we argue that it is possible to have a total endogeneity ratio greater than unity without rendering the process unstable. In particular, we show that, provided the endogeneity ratio of the linear Hawkes component is smaller than unity, Quadratic Hawkes processes are always stationary, although with infinite mean intensity when the total endogenity ratio exceeds one. This results from a subtle compensation between the inhibiting realisations (mean-reversion) and their exciting counterparts (trends).

Crises propagation and behavioural effects in multi agents self-reflexive real business cycle models

6 décembre 2021

PhD Thesis

Federico Morelli

The Global Financial Crisis of 2008 left was also a crisis for macroeconomic models. On the one hand, orthodox economists persist in using the skeleton of classical models, while on the other hand, various groups of heterodoxes have proposed different ways to change the foundations of Economics. My research aims to bridge the gap between Neo-classical Economics and complexity Economics using methods and techniques from statistical physics. Beginning with standard economic models, I study the addition of a self reflexive feedback impacting the confidence of individual economic agents. This induces large output swings despite only minor variations in economic conditions. Within this framework, economic crises propagate endogenously and are amplified by interactions. Later on, I enrich the previous framework by taking into account heterogeneities, studying how economic recessions propagate through different strata of society. In the last part of this work, I present a behavioural economic model where the stability of the economy is jeopardised by the lack of investments in risky markets.

Do fundamentals shape the price response? A critical assessment of linear impact models

5 décembre 2021

arXiv preprint

Michele Vodret, Iacopo Mastromatteo, Bence Toth, Michael Benzaquen

We compare the predictions of the stationary Kyle model, a microfounded multi-step linear price impact model in which market prices forecast fundamentals through information encoded in the order flow, with those of the propagator model, a purely data-driven model in which trades mechanically impact prices with a time-decaying kernel. We find that, remarkably, both models predict the exact same price dynamics at high frequency, due to the emergence of universality at small time scales. On the other hand, we find those models to disagree on the overall strength of the impact function by a quantity that we are able to relate to the amount of excess-volatility in the market. We reveal a crossover between a high-frequency regime in which the market reacts sub-linearly to the signed order flow, to a low-frequency regime in which prices respond linearly to order flow imbalances. Overall, we reconcile results from the literature on market microstructure (sub-linearity in the price response to traded volumes) with those relating to macroeconomically relevant timescales (in which a linear relation is typically assumed).

Universal amplitudes ratios for critical aging via functional renormalization group

5 décembre 2021

arXiv preprint

Michele Vodret, Alessio Chiocchetta, Andrea Gambassi

We discuss how to calculate non-equilibrium universal amplitude ratios in the functional renormalization group approach, extending its applicability. In particular, we focus on the critical relaxation of the Ising model with non-conserved dynamics (model A) and calculate the universal amplitude ratio associated with the fluctuation-dissipation ratio of the order parameter, considering a critical quench from a high-temperature initial condition. Our predictions turn out to be in good agreement with previous perturbative renormalization-group calculations and Monte Carlo simulations.

Exploration of the Parameter Space in Macroeconomic Agent-Based Models

15 novembre 2021

SSRN Preprint

Karl Naumann-Woleske, Max Sina Knicker, Michael Benzaquen, J-P Bouchaud

Agent-Based Models (ABM) are computational scenario-generators, which can be used to predict the possible future outcomes of the complex system they represent. To better understand the robustness of these predictions, it is necessary to understand the full scope of the possible phenomena the model can generate. Most often, due to high-dimensional parameter spaces, this is a computationally expensive task. Inspired by ideas coming from systems biology, we show that for multiple macroeconomic models, including an agent-based model and several Dynamic Stochastic General Equilibrium (DSGE) models, there are only a few stiff parameter combinations that have strong effects, while the other sloppy directions are irrelevant.
This suggest an algorithm that efficiently explores the space of parameters by primarily moving along the stiff directions. We apply our algorithm to a medium-sized agent-based model, and show that it recovers all possible dynamics of the unemployment rate. The application of this method to Agent-based Models may lead to a more thorough and robust understanding of their features, and provide enhanced parameter sensitivity analyses. Several promising paths for future research are discussed.

Capital Demand Driven Business Cycles: Mechanism and Effects

29 septembre 2021

SSRN Preprint

Karl Naumann-Woleske, Michael Benzaquen, Maxim Gusev, Dimitri Kroujiline

We develop a tractable macroeconomic model that captures dynamic behaviors across multiple timescales, including business cycles. The model is anchored in a dynamic capital demand framework reflecting an interactions-based process whereby firms determine capital needs and make investment decisions on a micro level. We derive equations for aggregate demand from this micro setting and embed them in the Solow growth economy. As a result, we obtain a closed-form dynamical system with which we study economic fluctuations and their impact on long-term growth. For realistic parameters, the model has two attracting equilibria: one at which the economy contracts and one at which it expands. This bi-stable configuration gives rise to quasiperiodic fluctuations, characterized by the economy’s prolonged entrapment in either a contraction or expansion mode punctuated by rapid alternations between them. We identify the underlying endogenous mechanism as a coherence resonance phenomenon. In addition, the model admits a stochastic limit cycle likewise capable of generating quasiperiodic fluctuations; however, we show that these fluctuations cannot be realized as they induce unrealistic growth dynamics. We further find that while the fluctuations powered by coherence resonance can cause substantial excursions from the equilibrium growth path, such deviations vanish in the long run as supply and demand converge.

Economic Crises in a Model with Capital Scarcity and Self-Reflexive Confidence

19 septembre 2021

SSRN Preprint

Federico Morelli, Karl Naumann-Woleske, Michael Benzaquen, Marco Tarzia, J-P Bouchaud

In the General Theory, Keynes remarked that the economy's state depends on expectations, and that these expectations can be subject to sudden swings. In this work, we develop a multiple equilibria behavioural business cycle model that can account for demand or supply collapses due to abrupt drops in consumer confidence, which affect both consumption propensity and investment. We show that, depending on the model parameters, four qualitatively different outcomes can emerge, characterised by the frequency of capital scarcity and/or demand crises. In the absence of policy measures, the duration of such crises can increase by orders of magnitude when parameters are varied, as a result of the ``paradox of thrift''. Our model suggests policy recommendations that prevent the economy from getting trapped in extended stretches of low output, low investment and high unemployment.

Exogenous and Endogenous Price Jumps Belong to Different Dynamical Classes

13 juin 2021

SSRN Preprint

Riccardo Marcaccioli, J-P Bouchaud, Michael Benzaquen

Synchronizing a database of stock specific news with 5 years worth of order book data on 300 stocks, we show that abnormal price movements following news releases (exogenous) exhibit markedly different dynamical features from those arising spontaneously (endogenous). On aver- age, large volatility fluctuations induced by exogenous events occur abruptly and are followed by a decaying power-law relaxation, while endogenous price jumps are characterized by progressively accelerating growth of volatility, also followed by a power-law relaxation, but slower than for exogenous jumps. Remarkably, our results are reminiscent of what is observed in different contexts, namely Amazon book sales and YouTube views. Finally, we show that fitting power-laws to individual volatility profiles allows one to classify large events into endogenous and exogenous dynamical classes, without relying on the news feed.

Non-parametric Estimation of Quadratic Hawkes Processes for Order Book Events

11 avril 2021

European Journal of Finance

Antoine Fosset, J-P Bouchaud, Michael Benzaquen

We propose an actionable calibration procedure for general Quadratic Hawkes models of order book events (market orders, limit orders, cancellations). One of the main features of such models is to encode not only the influence of past events on future events but also, crucially, the influence of past price changes on such events. We show that the empirically calibrated quadratic kernel is well described by a diagonal contribution (that captures past realised volatility), plus a rank-one ‘Zumbach’ contribution (that captures the effect of past trends). We find that the Zumbach kernel is a power-law of time, as are all other feedback kernels. As in many previous studies, the rate of truly exogenous events is found to be a small fraction of the total event rate. These two features suggest that the system is close to a critical point – in the sense that slightly stronger feedback kernels would lead to endogenous liquidity crises.

A new spin on optimal portfolios and ecological equilibria

31 mars 2021

Journal of Statistical Mechanics: Theory and Experiment

Jerome Garnier-Brun, Michael Benzaquen, Stefano Ciliberti, J-P Bouchaud

We consider the classical problem of optimal portfolio construction with the constraint that no short position is allowed, or equivalently the valid equilibria of multispecies Lotka-Volterra equations, in the special case where the interaction matrix is of unit rank, corresponding to a single-resource MacArthur model. We compute the average number of solutions and show that its logarithm grows as Nα, where N is the number of assets or species and α≤2/3 depends on the interaction matrix distribution. We conjecture that the most likely number of solutions is much smaller and related to the typical sparsity m(N) of the solutions, which we compute explicitly. We also find that the solution landscape is similar to that of spin-glasses, i.e. very different configurations are quasi-degenerate. Correspondingly, "disorder chaos" is also present in our problem. We discuss the consequence of such a property for portfolio construction and ecologies, and question the meaning of rational decisions when there is a very large number "satisficing" solutions.

A Stationary Kyle Setup: Microfounding propagator models

29 mars 2021

Journal of Statistical Mechanics

Michele Vodret, Iacopo Mastromatteo, Bence Toth, Michael Benzaquen

We provide an economically sound micro-foundation to linear price impact models, by deriving them as the equilibrium of a suitable agent-based system. In particular, we retrieve the so-called propagator model as the high-frequency limit of a generalized Kyle model, in which the assumption of a terminal time at which fundamental information is revealed is dropped. This allows to describe a stationary market populated by asymmetrically-informed rational agents. We investigate the stationary equilibrium of the model, and show that the setup is compatible with universal price diffusion at small times, and non-universal mean-reversion at time scales at which fluctuations in fundamentals decay. Our model suggests that at high frequency one should observe a quasi-permanent impact component, driven by slow fluctuations of fundamentals, and a faster transient one, whose timescale should be set by the persistence of the order flow.

V –, U –, L – Or W–Shaped Economic Recovery After COVID-19: Insights From an Agent Based Model

1 mars 2021

PLoS One

Dhruv Sharma, J-P Bouchaud, Stanislao Gualdi, Marco Tarzia, Francesco Zamponi

We discuss the impact of a Covid-19–like shock on a simple model economy, described by the previously developed Mark-0 Agent-Based Model. We consider a mixed supply and demand shock, and show that depending on the shock parameters (amplitude and duration), our model economy can display V-shaped, U-shaped or W-shaped recoveries, and even an L-shaped output curve with permanent output loss. This is due to the economy getting trapped in a self-sustained “bad” state. We then discuss two policies that attempt to moderate the impact of the shock: giving easy credit to firms, and the so-called helicopter money, i.e. injecting new money into the households savings. We find that both policies are effective if strong enough. We highlight the potential danger of terminating these policies too early, although inflation is substantially increased by lax access to credit. Finally, we consider the impact of a second lockdown. While we only discuss a limited number of scenarios, our model is flexible and versatile enough to accommodate a wide variety of situations, thus serving as a useful exploratory tool for a qualitative, scenario-based understanding of post-Covid recovery. The corresponding code is available on-line.

From Ants to Fishing Vessels: A Simple Model for Herding and Exploitation of Finite Resources

23 février 2021

arXiv preprint

Jose Moran, Antoine Fosset, Alan Kirman, Michael Benzaquen

We analyse the dynamics of fishing vessels with different home ports in an area where these vessels, in choosing where to fish, are influenced by their own experience in the past and by their current observation of the locations of other vessels in the fleet. Empirical data from the boats near Ancona and Pescara shows stylized statistical properties that are reminiscent of Kirman and Föllmer's ant recruitment model, although with two ant colonies represented by the two ports. From the point of view of a fisherman, the two fishing areas are not equally attractive, and he tends to prefer the one closer to where he is based. This piece of evidence led us to extend the original ants model to a situation with two asymmetric zones and finite resources. We show that, in the mean-field regime, our model exhibits the same properties as the empirical data. We obtain a phase diagram that separates high and low herding regimes, but also fish population extinction. Our analysis has interesting policy implications for the ecology of fishing areas. It also suggests that herding behaviour here, just as in financial markets, will lead to significant fluctuations in the amount of fish landed, as the boat concentration on one area at a given point in time will diminish the overall catch, such loss not being compensated by the reproduction of fish in the other area. In other terms, individually rational behaviour will not lead to collectively optimal results.

Cross impact in derivative markets

3 février 2021

arXiv preprint

Mehdi Tomas, Iacopo Mastromatteo, Michael Benzaquen

We introduce a linear cross-impact framework in a setting in which the price of some given financial instruments (derivatives) is a deterministic function of one or more, possibly tradeable, stochastic factors (underlying). We show that a particular cross-impact model, the multivariate Kyle model, prevents arbitrage and aggregates (potentially non-stationary) traded order flows on derivatives into (roughly stationary) liquidity pools aggregating order flows traded on both derivatives and underlying. Using E-Mini futures and options along with VIX futures, we provide empirical evidence that the price formation process from order flows on derivatives is driven by cross-impact and confirm that the simple Kyle cross-impact model is successful at capturing parsimoniously such empirical phenomenology. Our framework may be used in practice for estimating execution costs, in particular hedging costs.

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the S-transform

21 janvier 2021

arXiv preprint

Pierre Mergny, Marc Potters

In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for β=2 , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called R-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all β>0 in the multiplicative case, where the asymptotic is governed by the logarithm of the S-transform. As a consequence of this result one can calculate the asymptotic behavior of complete homogeneous symmetric polynomials.

Crisis Propagation in a Heterogeneous Self-Reflexive DSGE Model

13 janvier 2021

arXiv preprint

Federico Morelli, Michael Benzaquen, J-P Bouchaud, Marco Tarzia

We study a self-reflexive DSGE model with heterogeneous households, aimed at characterising the impact of economic recessions on the different strata of the society. Our framework allows to analyse the combined effect of income inequalities and confidence feedback mediated by heterogeneous social networks. By varying the parameters of the model, we find different crisis typologies: loss of confidence may propagate mostly within high income households, or mostly within low income households, with a rather sharp crossover between the two. We find that crises are more severe for segregated networks (where confidence feedback is essentially mediated between agents of the same social class), for which cascading contagion effects are stronger. For the same reason, larger income inequalities tend to reduce, in our model, the probability of global crises. Finally, we are able to reproduce a perhaps counter-intuitive empirical finding: in countries with higher Gini coefficients, the consumption of the lowest income households tends to drop less than that of the highest incomes in crisis times.

Why does individual learning endure when crowds are wiser?

27 décembre 2020

arXiv preprint

Benoït de Courson, Léo Fitouchi, J-P Bouchaud, Michael Benzaquen

The ability to learn from others (social learning) is often deemed a cause of human species success. But if social learning is indeed more efficient (whether less costly or more accurate) than individual learning, it raises the question of why would anyone engage in individual information seeking, which is a necessary condition for social learning's efficacy. We propose an evolutionary model solving this paradox, provided agents (i) aim not only at information quality but also vie for audience and prestige, and (ii) do not only value accuracy but also reward originality -- allowing them to alleviate herding effects. We find that under some conditions (large enough success rate of informed agents and intermediate taste for popularity), both social learning's higher accuracy and the taste for original opinions are evolutionary-stable, within a mutually beneficial division of labour-like equilibrium. When such conditions are not met, the system most often converges towards mutually detrimental equilibria.

Macroeconomic Agent-based Models: A Statistical Physics perspective

25 novembre 2020

PhD Thesis

Dhruv Sharma

The object of this thesis is to advance a methodology commonly used in statistical physics and apply it to the study of two macroeconomic agent-based models. In both models studied here, we first determine the “phase-diagram” of the model to identify the relevant macroscopic regimes to develop an intuitive understanding of the macrodynamics using a small subset of parameters. The first ABM presented here builds upon the paradigm of constraint satisfaction problems (CSPs) and integrates it within the model’s behavioral rules via agents’ budgetary constraints. These constraints, similar to the well-studied perceptron CSP, reveal the existence of three regimes and underscore the importance of debt for macroeconomic stability: at low-levels of debt, the economy remains structure-less with frequent bankruptcies while high debt leads to endogenous business cycles. Between these two extremes, an intermediate regime of relative stability is found with low levels of bankruptcies for all times. Within this ABM, agents’ preferences, serving as the source of disorder in the CSP, evolve continuously in time. We thus study a simple dynamical scheme for the perceptron and discover that a rugged landscape can indeed exist with dynamic, annealed disorder. Finally, we extend the Mark-0 ABM to simulate exogenous consumption and productivity shocks due to the Covid pandemic. Whereas standard approaches design a model to understand a particular outcome, this model can generate a variety of scenarios after a Covid-like shock. Furthermore, we also investigate the efficacy of several policies, including the much-debated “helicopter money” drop, in avoiding economic collapse. We thus highlight the importance of ABMs as multi-purpose “scenario generators”, for producing outcomes that are difficult to foresee due to the intrinsic complexity of macro-economic dynamics.

Statistical physics and anomalous macroeconomic fluctuations

12 octobre 2020

PhD Thesis

Jose Moran

How do microscopic perturbations at the level of an individual grow to become macroscopic fluctuations of the whole economy? Despite decades of effort, this puzzle remains open. In this work, I tackle this problem using methods and techniques from statistical physics. Beginning with a thorough analysis of power law distributions, I argue that understanding their origin and properties helps in elucidating their socio-economic consequences. I then explore a model of an economy where firms interact through a production network in a way that causes them to be intrinsically prone to amplify fluctuations. Later on, I conduct an empirical survey of the statistical properties of firm growth rates and provide a framework to study their dynamics. I finally move onto models where non trivial collective phenomena arise from imitation and memory effects at the level of the individual, highlighting the need of accounting for complexity in economic modelling.

How to build a cross-impact model from first principles: Theoretical requirements and empirical results

12 septembre 2020

arXiv preprint

Mehdi Tomas, Iacopo Mastromatteo, Michael Benzaquen

Trading a financial instrument induces a price response on itself and on other correlated instruments, a phenomenon known as cross-impact. Unfortunately, empirical measures of cross-impact are affected by a large estimation error due to both the large number of interactions to infer and the strongly fluctuating nature of price returns. In this study we propose a principled approach that leverages simple consistency criteria (symmetries, no-arbitrage conditions, correlation and liquidity limit-case properties) in order to impose ex-ante properties that might be required for practical applications. We validate our approach on empirical data for several asset classes, thus determining which properties are desirable across multiple markets. In particular, our results show that two cross-impact models perform well in all markets studied but only one is suitable for other applications, such as optimal execution.

Endogenous Liquidity Crises in Financial Markets

8 septembre 2020

PhD Thesis

Antoine Fosset

Recent empirical analyses have revealed the existence of the Zumbach effect. This discovery has led to the development of quadratic Hawkes processes, which are suitable for reproducing this effect. Since this model is not linked with the price formation process, we extended it to order book modeling with a generalized quadratic Hawkes process (GQ-Hawkes). Using market data, we showed that there is a Zumbach-like effect that decreases future liquidity. Microfounding the Zumbach effect, it is responsible for a destabilization of financial markets. Moreover, the exact calibration of a GQ-Hawkes process tells us that the markets are on the verge of criticality. This empirical evidence therefore prompted us to analyse an order-book model constructed upon a Zumbach-like feedback. We therefore introduced the quadratic Santa Fe model and proved numerically that there is a phase transition between a stable market and an unstable market subject to liquidity crises. Thanks to a finite size scaling we were able to determine the critical exponents of this transition, which appears to belong to a new universality class. As this was not analytically tractable, it led us to introduce simpler models to describe liquidity crises. Setting aside the microstructure of the order book, we obtain a class of spread models where we computed the critical parameters of their transitions. Even if these exponents are not those of the quadratic Santa Fe transition, these models open new horizons for modelling spread dynamics. One of them has a non-linear coupling that reveals a metastable state. This elegant alternative scenario does not need critical parameters to obtain an unstable market, even if the empirical evidence is not in its favour. Finally, we looked at the order book dynamics from another point of view: the reaction-diffusion one. We have modelled a liquidity that appears in the order book with a certain frequency. The resolution of this model at equilibrium reveals that there is a condition of stability on the parameters beyond which the order book empties completely, corresponding to a liquidity crisis. By calibrating it on market data we were able to qualitatively analyse the distance to this unstable region.

Schrödinger's ants: a continuous description of Kirman's recruitment model

3 août 2020

Journal of Physics: Complexity

Jose Moran, Antoine Fosset, Michael Benzaquen, J-P Bouchaud

We show how the approach to equilibrium in Kirman's ants model can be fully characterized in terms of the spectrum of a Schrödinger equation with a Pöschl–Teller (tan2) potential. Among other interesting properties, we have found that in the bimodal phase where ants visit mostly one food site at a time, the switch time between the two sources only depends on the 'spontaneous conversion' rate and not on the recruitment rate. More complicated correlation functions can be computed exactly, and involve higher and higher eigenvalues and eigenfunctions of the Schrödinger operator, which can be expressed in terms of hypergeometric functions.

Beauty and structural complexity

26 mai 2020

Physical Review Research

Samy Lakhal, Alexandre Darmon, J-P Bouchaud, Michael Benzaquen

We revisit the long-standing question of the relation between image appreciation and its statistical properties. We generate two different sets of random images well distributed along three measures of entropic complexity. We run a large-scale survey in which people are asked to sort the images by preference, which reveals maximum appreciation at intermediate entropic complexity. We show that the algorithmic complexity of the coarse-grained images, expected to capture structural complexity while abstracting from high frequency noise, is a good predictor of preferences. Our analysis suggests that there might exist some universal quantitative criteria for aesthetic judgment.

By force of habit: Self-trapping in a dynamical utility landscape

10 mai 2020


Jose Moran, Antoine Fosset, Davide Luzzati, J-P Bouchaud, Michael Benzaquen

Historically, rational choice theory has focused on the utility maximization principle to describe how individuals make choices. In reality, there is a computational cost related to exploring the universe of available choices and it is often not clear whether we are truly maximizing an underlying utility function. In particular, memory effects and habit formation may dominate over utility maximization. We propose a stylized model with a history-dependent utility function, where the utility associated to each choice is increased when that choice has been made in the past, with a certain decaying memory kernel. We show that self-reinforcing effects can cause the agent to get stuck with a choice by sheer force of habit. We discuss the special nature of the transition between free exploration of the space of choice and self-trapping. We find, in particular, that the trapping time distribution is precisely a Zipf law at the transition, and that the self-trapped phase exhibits super-aging behavior.

Confidence collapse in a multihousehold, self-reflexive DSGE model

27 avril 2020


Federico Morelli, Michael Benzaquen, Marco Tarzia, J-P Bouchaud

We investigate a multihousehold dynamic stochastic general equilibrium (DSGE) model in which past aggregate consumption impacts the confidence, and therefore consumption propensity, of individual households. We find that such a minimal setup is extremely rich and leads to a variety of realistic output dynamics: high output with no crises; high output with increased volatility and deep, short-lived recessions; and alternation of high- and low-output states where a relatively mild drop in economic conditions can lead to a temporary confidence collapse and steep decline in economic activity. The crisis probability depends exponentially on the parameters of the model, which means that markets cannot efficiently price the associated risk premium. We conclude by stressing that within our framework, narratives become an important monetary policy tool that can help steer the economy back on track.

How Much Income Inequality Is Too Much?

11 avril 2020

arXiv preprint

J-P Bouchaud

We propose a highly schematic economic model in which, in some cases, wage inequalities lead to higher overall social welfare. This is due to the fact that high earners can consume low productivity, non essential products, which allows everybody to remain employed even when the productivity of essential goods is high and producing them does not require everybody to work. We derive a relation between heterogeneities in technologies and the minimum Gini coefficient required to maximize global welfare. Stronger inequalities appear to be economically unjustified. Our model may shed light on the role of non-essential goods in the economy, a topical isue when thinking about the post-Covid-19 world

Are trading invariants really invariant? Trading costs matter

6 avril 2020

Quantitative Finance

Frédéric Bucci, Fabrizio Lillo, J-P Bouchaud, Michael Benzaquen

We revisit the trading invariance hypothesis recently proposed by Kyle and Obizhaeva by empirically investigating a large dataset of bets, or metaorders, provided by ANcerno. The hypothesis predicts that the quantity I := R/N^3/2, where R is the exchanged risk (volatility × volume × price) and N is the number of bets, is invariant. We find that the 3/2 scaling between R and N works well and is robust against changes of year, market capitalisation and economic sector. However our analysis clearly shows that I is not invariant. We find a very high correlation (> 0.8) between I and the total trading cost (spread and market impact) of the bet. We propose new invariants defined as a ratio of I and costs and find a large decrease in variance. We show that the small dispersion of the new invariants is mainly driven by (i) the scaling of the spread with the volatility per transaction, (ii) the near invariance of the distribution of metaorder size and of the volume and number fractions of bets across stocks.

Endogenous Liquidity Crises

18 février 2020

arXiv preprint

Antoine Fosset, J-P Bouchaud, Michael Benzaquen

Empirical data reveals that the liquidity flow into the order book (depositions, cancellations andmarket orders) is influenced by past price changes. In particular, we show that liquidity tends todecrease with the amplitude of past volatility and price trends. Such a feedback mechanism inturn increases the volatility, possibly leading to a liquidity crisis. Accounting for such effects withina stylized order book model, we demonstrate numerically that there exists a second order phasetransition between a stable regime for weak feedback to an unstable regime for strong feedback,in which liquidity crises arise with probability one. We characterize the critical exponents, whichappear to belong to a new universality class. We then propose a simpler model for spread dynamicsthat maps onto a linear Hawkes process which also exhibits liquidity crises. If relevant for thereal markets, such a phase transition scenario requires the system to sit below, but very close tothe instability threshold (self-organised criticality), or else that the feedback intensity is itself timedependent and occasionally visits the unstable region. An alternative scenario is provided by a classof non-linear Hawkes process that show occasional "activated" liquidity crises, without having to bepoised at the edge of instability.

Zooming In on Equity Factor Crowding

12 janvier 2020

arXiv preprint

Valerio Volpati, Michael Benzaquen, Zoltan Eisler, Iacopo Mastromatteo, Bence Toth, J-P Bouchaud

Crowding is most likely an important factor in the deterioration of strategy performance, the increase of trading costs and the development of systemic risk. We study the imprints of crowding on both anonymous market data and a large database of metaorders from institutional investors in the U.S. equity market. We propose direct metrics of crowding that capture the presence of investors contemporaneously trading the same stock in the same direction by looking at fluctuations of the imbalances of trades executed on the market. We identify significant signs of crowding in well known equity signals, such as Fama-French factors and especially Momentum. We show that the rebalancing of a Momentum portfolio can explain between 1–2% of order flow, and that this percentage has been significantly increasing in recent years.

Conditional Correlations and Principal Regression Analysis for Futures

9 janvier 2020

arXiv preprint

Armine Karami, Raphael Benichou, Michael Benzaquen, J-P Bouchaud

We explore the effect of past market movements on the instantaneous correlations between assets within the futures market. Quantifying this effect is of interest to estimate and manage the risk associated to portfolios of futures in a non-stationary context. We apply and extend a previously reported method called the Principal Regression Analysis (PRA) to a universe of 84 futures contracts between 2009 and 2019. We show that the past up (resp. down) 10 day trends of a novel predictor -- the eigen-factor -- tend to reduce (resp. increase) instantaneous correlations. We then carry out a multifactor PRA on sectorial predictors corresponding to the four futures sectors (indexes, commodities, bonds and currencies), and show that the effect of past market movements on the future variations of the instantaneous correlations can be decomposed into two significant components. The first component is due to the market movements within the index sector, while the second component is due to the market movements within the bonds sector.

Slow Decay of Impact in Equity Markets: Insights from the ANcerno Database

6 août 2019

Market Microstructure and Liquidity

Frédéric Bucci, Michael Benzaquen, Fabrizio Lillo, J-P Bouchaud

We present an empirical study of price reversion after the executed metaorders. We use a data set with more than 8 million metaorders executed by institutionalinvestors in the US equity market. We show that relaxation takes place as soonas the metaorder ends:while at the end of the same day it is on average ≈2/3 of the peak impact, the decay continues the next days, following a power-law functionat short time scales, and converges to a non-zero asymptotic value at long timescales (∼50 days) equal to ≈1/2 of the impact at the end of the first day. Dueto a significant, multiday correlation of the sign of executed metaorders, a carefuldeconvolution of theobservedimpact must be performed to extract the estimate ofthe impact decay of isolated metaorders.

Impact is not just volatility

4 juillet 2019

Quantitative Finance

Frédéric Bucci, Iacopo Mastromatteo, Michael Benzaquen, J-P Bouchaud

The notion of market impact is subtle and sometimes misinterpreted. Here we argue that impact should not be misconstrued as volatility. In particular, the so-called “square-root impact law”, which states that impact grows as the square-root of traded volume, has nothing to do with price diffusion, i.e. that typical price changes grow as the square-root of time. We rationalise empirical findings on impact and volatility by introducing a simple scaling argument and confronting it to data.

Crossover from Linear to Square-Root Market Impact

14 mars 2019

Physical Review Letters

Frédéric Bucci, Michael Benzaquen, Fabrizio Lillo, J-P Bouchaud

Using a large database of 8 million institutional trades executed in the U.S. equity market, we establish a clear crossover between a linear market impact regime and a square-root regime as a function of the volume of the order. Our empirical results are remarkably well explained by a recently proposed dynamical theory of liquidity that makes specific predictions about the scaling function describing this crossover. Allowing at least two characteristic timescales for the liquidity (“fast” and “slow”) enables one to reach quantitative agreement with the data.

How does latent liquidity get revealed in the limit order book?

7 janvier 2019

Journal of Statistical Mechanics: Theory and Experiment

Lorenzo Dall’Amico, Antoine Fosset, J-P Bouchaud, Michael Benzaquen

Latent order book models have allowed for significant progress in our understanding of price formation in financial markets. In particular they are able to reproduce a number of stylized facts, such as the square-root impact law. An important question that is raised—if one is to bring such models closer to real market data—is that of the connection between the latent (unobservable) order book and the real (observable) order book. Here we suggest a simple, consistent mechanism for the revelation of latent liquidity that allows for quantitative estimation of the latent order book from real market data. We successfully confront our results to real order book data for over a hundred assets and discuss market stability. One of our key theoretical results is the existence of a market instability threshold, where the conversion of latent order becomes too slow, inducing liquidity crises. Finally we compute the price impact of a metaorder in different parameter regimes.

Market impact with multi-timescale liquidity

9 octobre 2017

Quantitative Finance

Michael Benzaquen, J-P Bouchaud

We present an extended version of the recently proposed "LLOB" model for the dynamics of latent liquidity in financial markets. By allowing for finite cancellation and deposition rates within a continuous reaction-diffusion setup, we account for finite memory effects on the dynamics of the latent order book. We compute in particular the finite memory corrections to the square root impact law, as well as the impact decay and the permanent impact of a meta-order. The latter is found to be linear in the traded volume and independent of the trading rate, as dictated by no-arbitrage arguments. In addition, we consider the case of a spectrum of cancellation and deposition rates, which allows us to obtain a square root impact law for moderate participation rates, as observed empirically. Our multi-scale framework also provides an alternative solution to the so-called price diffusivity puzzle in the presence of a long-range correlated order flow.



Antoine Fosset

29 Sep 2020


Jose Moran

13 Oct 2020


Dhruv Sharma

26 Nov 2020


Federico Morelli

7 December 2021


Mehdi Tomas

30 March 2022