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
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
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
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
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.
Tâtonnement, Approach to Equilibrium and Excess Volatility in Firm Networks
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.
Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the S-transform
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
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?
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.
How to build a cross-impact model from first principles: Theoretical requirements and empirical results
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.
Schrödinger's ants: a continuous description of Kirman's recruitment model
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
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.
Non-parametric Estimation of Quadratic Hawkes Processes for Order Book Events
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 stronger feedback kernels would lead to instabilities.
By force of habit: Self-trapping in a dynamical utility landscape
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
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.
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
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.
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.
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
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
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
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
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?
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.
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.
ENDOGENOUS LIQUIDITY CRISES IN FINANCIAL MARKETS, BY ANTOINE FOSSET