Our research team focuses on studying fundamental problems posed by financial markets and we encourage interaction with academia and making some of our research findings public through various publications.

Trend Following: a Persistent Market Anomaly

Date: May 2015

In this note we present a study of trend following using two centuries of data. We find that trend following is a persistent market anomaly with highly significant performance over this long back-test. We also present a combined portfolio of a trend following investment and a standard basket of equities and bonds.

Hedging Out Market Factors

Date: October 2014

Large shocks in an equity portfolio are typically driven by correlated (and hence collective) moves of its constituents. This accords correlation matrices a historically central place in numerous studies on portfolio construction and risk management [1].
In this note, we illustrate how certain statistical methods enable us to identify the main market factors (or “modes”) that an equity market neutral portfolio should hedge, in order to extract value from signals, while avoiding exposure to large, collective market moves. These methods rely on the processing of stock returns correlation matrices.
However, because time series are finite, measured correlations are subject to the effects of noise: a fact that one must take into account when employing empirical correlation matrices in portfolio construction. Comparing the properties of empirical correlation matrices to those obtained in random cases, and using results from the theory of random matrices, enables us to distinguish genuine characteristics of the dependence structure of a set of stocks from noisy and unreliable features.

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

Date: October 2014

Authors Thomas Gueudré, Alexander Dobrinevski, Jean-Philippe Bouchaud

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.

Risk Premia: Asymmetric Tail Risks and Excess Returns

Date: September 2014

Authors Yves Lempérière, Cyril Deremble, Trung-Tu Nguyen, Philip Seager, Marc Potters, Jean-Philippe Bouchaud
E print http://xxx.lanl.gov/abs/1409.7720

The aim of our work is to propose a natural framework to account for all the empirically known properties of the multivariate distribution of stock returns. We define and study a "nested factor model", where the linear factors part is standard, but where the log-volatility of the linear factors and of the residuals are themselves endowed with a factor structure and residuals. We propose a calibration procedure to estimate these log-vol factors and the residuals. We find that whereas the number of relevant linear factors is relatively large (10 or more), only two or three log-vol factors emerge in our analysis of the data. In fact, a minimal model where only one log-vol factor is considered is already very satisfactory, as it accurately reproduces the properties of bivariate copulas, in particular the dependence of the medial-point on the linear correlation coefficient, as reported in Chicheportiche and Bouchaud (2012). We have tested the ability of the model to predict Out-of-Sample the risk of non-linear portfolios, and found that it performs significantly better than other schemes.

Endogenous crisis waves: a stochastic model with synchronized collective behavior

Date: September 2014

Authors Stanislao Gualdi, Jean-Philippe Bouchaud, Giulia Cencetti, Marco Tarzia, Francesco Zamponi
E print http://arxiv.org/abs/1409.3296

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.

Slow decay of impact in equity markets

Date: July 2014

Authors X. Brokmann, E. Serie, J. Kockelkoren, J.-P. Bouchaud
E print http://arxiv.org/abs/1407.3390

Using a proprietary dataset of meta-orders and prediction signals, and assuming a quasi-linear impact model, we deconvolve market impact from past correlated trades and a predictable return component to elicit the temporal dependence of the market impact of a single daily meta-order, over a ten day horizon in various equity markets. We find that the impact of single meta-orders is to a first approximation universal and slowly decays to zero (or to a small value), possibly as a power-law. We show that auto-correlated order-flows and trade information contents fully accounts for the apparent plateau observed in the raw data. We discuss the possible bias introduced by the quasi-linear assumption.

Instabilities in large economies: aggregate volatility without idiosyncratic shocks

Date: September 2014

Authors Julius Bonart, Jean-Philippe Bouchaud, Augustin Landier, David Thesmar
E print http://xxx.lanl.gov/abs/1406.5022

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.

Signal-wise performance attribution for constrained portfolio optimisation

Date: April 2014

Authors Bruno Durin
E print  http://arxiv.org/abs/1404.4798

Performance analysis, from the external point of view of a client who would only have access to returns and holdings of a fund, evolved towards exact attribution made in the context of portfolio optimisation, which is the internal point of view of a manager controling all the parameters of this optimisation. Attribution is exact, that-is-to-say no residual "interaction" term remains, and various contributions to the optimal portfolio can be identified: predictive signals, constraints, benchmark. However constraints are identified as a separate portfolio and attribution for each signal that are used to predict future returns thus corresponds to unconstrained signal portfolios. We propose a novel attribution method that put predictive signals at the core of attribution and allows to include the effect of constraints in portfolios attributed to every signal. We show how this can be applied to various trading models and portfolio optimisation frameworks and explain what kind of insights such an attribution provides.

Two Centuries of Trend Following

Date: April 2014

Authors Y. Lempérière, C. Deremble, P. Seager, M. Potters, J. P. Bouchaud
E print http://arxiv.org/abs/1404.3274

We establish the existence of anomalous excess returns based on trend following strategies across four asset classes (commodities, currencies, stock indices, bonds) and over very long time scales. We use for our studies both futures time series, that exist since 1960, and spot time series that allow us to go back to 1800 on commodities and indices. The overall t-stat of the excess returns is ~5 since 1960 and ~10 since 1800, after accounting for the overall upward drift of these markets. The effect is very stable, both across time and asset classes. It makes the existence of trends one of the most statistically significant anomalies in financial markets. When analyzing the trend following signal further, we find a clear saturation effect for large signals, suggesting that fundamentalist traders do not attempt to resist "weak trends", but step in when their own signal becomes strong enough. Finally, we study the performance of trend following in the recent period. We find no sign of a statistical degradation of long trends, whereas shorter trends have significantly withered.

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

Date: November 2014

Authors Jonathan Donier, Julius Bonart, Iacopo Mastromatteo, Jean-Philippe Bouchaud
E print http://xxx.lanl.gov/abs/1412.0141

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.

An instanton approach to large N Harish-Chandra-Itzykson-Zuber integrals

Date: March 2014

Authors Joel Bun, Jean-Philippe Bouchaud, Satya N. Majumdar, Marc Potters
E print http://arxiv.org/abs/1403.7763

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 generalised 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.

Branching ratio approximation for the self-exciting Hawkes process

Date: March 2014

Authors Stephen J. Hardiman, Jean-Philippe Bouchaud
E print http://xxx.lanl.gov/abs/1403.5227

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 process of Hawkes branching ratio estimation, proposed as a proxy for market endogeneity in recent publications and formerly estimated using numerical maximisation of likelihood. We employ this 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.

Anomalous impact in reaction-diffusion models

Date: March 2014

Authors Iacopo Mastromatteo, Bence Toth, Jean-Philippe Bouchaud
E print http://arxiv.org/abs/1403.3571

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.

Momentum Strategies with L1 Filter

Date: March 2014

Authors Tung-Lam Dao
E print http://arxiv.org/abs/1403.4069

In this article, we discuss various implementation of L1 filtering in order to detect some properties of noisy signals. This filter consists of using a L1 penalty condition in order to obtain the filtered signal composed by a set of straight trends or steps. This penalty condition, which determines the number of breaks, is implemented in a constrained least square problem and is represented by a regularization parameter which is estimated by a cross-validation procedure. Financial time series are usually characterized by a long-term trend (called the global trend) and some short-term trends (which are named local trends). A combination of these two time scales can form a simple model describing the process of a global trend process with some mean-reverting properties. Explicit applications to momentum strategies are also discussed in detail with appropriate uses of the trend configurations.

Agent-based models for latent liquidity and concave price impact

Date: November 2013

Authors Iacopo Mastromatteo, Bence Toth, Jean-Philippe Bouchaud
E print http://xxx.lanl.gov/abs/1311.6262

We revisit the "e-intelligence" model of Toth et al. (2011) [1], 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 [1] 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.

Skew and implied leverage effect: smile dynamics revisited

Date: November 2013

Authors Vincent Vargas, Tung-Lam Dao, Jean-Philippe Bouchaud
E print http://xxx.lanl.gov/abs/1311.4078

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.

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

Date: September 2013

Authors Pierre Blanc, Rémy Chicheportiche, Jean-Philippe Bouchaud
E print http://xxx.lanl.gov/abs/1309.5806

We decompose, within an ARCH framework, the daily volatility of stocks into overnight and intraday contributions. We find, as perhaps expected, that the overnight and intraday returns behave completely differently. For example, while past intraday returns affect equally the future intraday and overnight volatilities, past overnight returns have a weak effect on future intraday 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 intraday returns but infinite for overnight returns. We provide a plausible interpretation for these findings, and show that our IntraDay/Overnight model significantly outperforms the standard ARCH framework based on daily returns for Out-of-Sample predictions.

A nested factor model for non-linear dependences in stock returns

Date: September 2013

Authors Rémy Chicheportiche, Jean-Philippe Bouchaud
E print http://xxx.lanl.gov/abs/1309.3102

The aim of our work is to propose a natural framework to account for all the empirically known properties of the multivariate distribution of stock returns. We define and study a "nested factor model", where the linear factors part is standard, but where the log-volatility of the linear factors and of the residuals are themselves endowed with a factor structure and residuals. We propose a calibration procedure to estimate these log-vol factors and the residuals. We find that whereas the number of relevant linear factors is relatively large (10 or more), only two or three log-vol factors emerge in our analysis of the data. In fact, a minimal model where only one log-vol factor is considered is already very satisfactory, as it accurately reproduces the properties of bivariate copulas, in particular the dependence of the medial-point on the linear correlation coefficient, as reported in Chicheportiche and Bouchaud (2012). We have tested the ability of the model to predict Out-of-Sample the risk of non-linear portfolios, and found that it performs significantly better than other schemes.

Tipping Points in Macroeconomic Agent-Based Models

Date: July 2013

Authors Stanislao Gualdi, Marco Tarzia, Francesco Zamponi, and Jean-Philippe Bouchaud
E print http://arxiv.org/abs/1307.5319

The aim of this work is to explore the possible types of phenomena that simple macroeconomic Agent-Based models (ABM) can reproduce. Our motivation is to understand the large macro-economic fluctuations observed in the "Mark I" ABM devised by D. Delli Gatti and collaborators. Our major finding is the existence of a first order (discontinuous) phase transition between a "good economy" where unemployment is low, and a "bad economy" where unemployment is high. We show that this transition is robust against many modifications of the model, and is induced by an asymmetry between the rate of hiring and the rate of firing of the firms. This asymmetry is induced, in Mark I, by the interest rate. As the interest rate increases, the firms become more and more reluctant to take further loans. 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 also explored several natural extensions. One is to allow this hiring/firing propensity to depend on the financial fragility of firms. Quite interestingly, we find that in this case, the above transition survives but becomes second order. We also studied simple wage policies and confidence feedback effects, whereby higher unemployment increases the saving propensity of households. We observe several interesting effects, such as the appearance of acute endogenous crises, during which the unemployment rate shoots up before the economy recovers. 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.

Some applications of first-passage ideas to finance

Date: June 2013

Authors Rémy Chicheportiche, Jean-Philippe Bouchaud
E print http://arxiv.org/abs/1306.3110

Many problems in finance are related to first passage times. Among all of them, we chose three on which we contributed personally. Our first example relates Kolmogorov-Smirnov like goodness-of-fit tests, modified in such a way that tail events and core events contribute equally to the test (in the standard Kolmogorov-Smirnov, the tails contribute very little to the measure of goodness-of-fit). We show that this problem can be mapped onto that of a random walk inside moving walls. The second example is the optimal time to sell an asset (modelled as a random walk with drift) such that the sell time is as close as possible to the time at which the asset reaches its maximum value. The last example concerns optimal trading in the presence of transaction costs. In this case, the optimal strategy is to wait until the predictor reaches (plus or minus) a threshold value before buying or selling. The value of this threshold is found by mapping the problem onto that of a random walk between two walls.

A Fokker-Planck description for the queue dynamics of large tick stocks

Date: April 2013

Authors A. Garèche, G. Disdier, J. Kockelkoren, J.-P. Bouchaud
E print http://arxiv.org/abs/1304.6819

Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation, that explicitly includes state dependence, i.e. the fact that the drift and diffusion depends on the volume present on both sides of the spread. "Jump" events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on best quotes. One of our central finding is the the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.

Critical reflexivity in financial markets: a Hawkes process analysis

Date: February 2013

Authors Stephen J. Hardiman, Nicolas Bercot, and Jean-Philippe Bouchaud
E print http://arxiv.org/abs/1302.1405

We model the arrival of mid-price changes in the E-Mini S&P futures contract as a self-exciting Hawkes process. Using several estimation methods, we find that the Hawkes kernel is power-law with a decay exponent close to -1.15 at short times, less than ~ 10<sup>3</sup> seconds, and crosses over to a second power-law regime with a larger decay exponent ~ -1.45 for longer times scales in the range [10<sup>3</sup>, 10<sup>6</sup>] seconds. More importantly, we find that the Hawkes kernel integrates to unity independently of the analysed period, from 1998 to 2011. This suggests that markets are and have always been close to criticality, challenging a recent study which indicates self-reflexivity (endogeneity) has increased in recent years as a result of increased automation of trading. However, we note that the scale over which market events are correlated has decreased steadily over time with the emergence of higher frequency trading.

Eigenvectors dynamic and local density of states under free addition

Date: January 2013

Authors Romain Allez and Jean-Philippe Bouchaud
E print http://arxiv.org/abs/1301.4939

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 idea of Ledoit and Péché in [11]. Finally, we revisit our former results on the eigenspace dynamics giving a robust derivation of a result obtained in [1] in a semi-perturbative regime.

Weighted Kolmogorov-Smirnov test: Accounting for the tails

Date: October 2012

Authors Rémy Chicheportiche and Jean-Philippe Bouchaud
E print http://arxiv.org/abs/1207.7308

Accurate goodness-of-fit tests for the extreme tails of empirical distributions is a very important issue, relevant in many contexts, including geophysics, insurance, and finance. We have derived exact asymptotic results for a generalization of the large-sample Kolmogorov-Smirnov test, well suited to testing these extreme tails. In passing, we have rederived and made more precise the approximate limit solutions found originally in unrelated fields, first in [L. Turban, J. Phys. A 25, 127 (1992)] and later in [P. L. Krapivsky and S. Redner, Am. J. Phys. 64, 546 (1996)].

Crises and collective socio-economic phenomena: simple models and challenges

Date: September 2012

Authors Jean-Philippe Bouchaud
E print http://arxiv.org/pdf/1209.0453.pdf

Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the so-called Random Field Ising model (rfim) provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilising self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and that account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of rfim-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith's invisible hand can fail badly at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria from being reached. As a theoretical challenge, the study of so-called "detailed-balance" violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic.

A proposal for impact-adjusted valuation: Critical leverage and execution risk

Date: August 2012

Authors F. Caccioli, J.-P. Bouchaud, J.-D. Farmer
E print http://arxiv.org/abs/1204.0922

The practice of valuation by marking-to-market with current trading prices is seriously flawed. Under leverage the problem is particularly dramatic: due to the concave form of market impact, selling always initially causes the expected leverage to increase. There is a critical leverage above which it is impossible to exit a portfolio without leverage going to infinity and bankruptcy becoming likely. Standard risk-management methods give no warning of this problem, which easily occurs for aggressively leveraged positions in illiquid markets. We propose an alternative accounting procedure based on the estimated market impact of liquidation that removes the illusion of profit. This should curb the leverage cycle and contribute to an enhanced stability of financial markets.

We've walked a million miles for one of these smiles

Date: April 2012

Authors L. De Leo, V. Vargas, S. Ciliberti, J.-P. Bouchaud

e derive a new, exact and transparent expansion for option smiles, which lends itself both to analytical approximation and to congenial numerical treatments. We show that the skew and the curvature of the smile can be computed as exotic options, for which the Hedged Monte Carlo method is particularly well suited. When applied to options on the S&P index, we find that the skew and the curvature of the smile are very poorly reproduced by the standard Edgeworth (cumulant) expansion. Most notably, the relation between the skew and the skewness is inverted at small and large vols, a feature that none of the models studied so far is able to reproduce. Furthermore, the around-the-money curvature of the smile is found to be very small, in stark contrast with the highly kurtic nature of the returns.

Empirical

Date: March 2012

Authors Joachim de Lataillade, Cyril Deremble, Marc Potters & Jean-Philippe Bouchaud

We consider the problem of the optimal trading strategy in the presence of linear costs, and with a strict cap on the allowed position in the market. Using Bellman's backward recursion method, we show that the optimal strategy is to switch between the maximum allowed long position and the maximum allowed short position, whenever the predictor exceeds a threshold value, for which we establish an exact equation. This equation can be solved explicitely in the case of a discrete Ornstein-Uhlenbeck predictor. We discuss in detail the dependence of this threshold value on the transaction costs. Finally, we establish a strong connection between our problem and the case of a quadratic risk penalty, where our threshold becomes the size of the optimal non-trading band.

The joint distribution of stock returns is not elliptical

Date: November 2011

Authors Remy Chicheportiche, Jean-Philippe Bouchaud
E print http://arxiv.org/pdf/1009.1100.pdf

Using a large set of daily US and Japanese stock returns, we test in detail the relevance of Student models, and of more general elliptical models, for describing the joint distribution of returns. We find that while Student copulas provide a good approximation for strongly correlated pairs of stocks, systematic discrepancies appear as the linear correlation between stocks decreases, that rule out all elliptical models. Intuitively, the failure of elliptical models can be traced to the inadequacy of the assumption of a single volatility mode for all stocks. We suggest several ideas of methodological interest to efficiently visualise and compare different copulas. We identify the rescaled difference with the Gaussian copula and the central value of the copula as strongly discriminating observables. We insist on the need to shun away from formal choices of copulas with no financial interpretation.

Anomalous price impact and the critical nature of liquidity in financial markets

Date: November 2011

Authors B. Toth, Y. Lemperière, C. Deremble, J. de Lataillade, J. Kockelkoren, J.-P. Bouchaud
E print http://arxiv.org/abs/1105.1694

We propose a dynamical theory of market liquidity that predicts that the average supply/demand profile is V-shaped and {\it vanishes} around the current price. This result is generic, and only relies on mild assumptions about the order flow and on the fact that prices are (to a first approximation) diffusive. This naturally accounts for two striking stylized facts: first, large metaorders have to be fragmented in order to be digested by the liquidity funnel, leading to long-memory in the sign of the order flow. Second, the anomalously small local liquidity induces a breakdown of linear response and a diverging impact of small orders, explaining the "square-root" impact law, for which we provide additional empirical support. Finally, we test our arguments quantitatively using a numerical model of order flow based on the same minimal ingredients.

The price impact of order book events: market orders, limit orders and cancellations

Date: September 2010

Authors Zoltan Eisler, Jean-Philippe Bouchaud, Julien Kockelkoren
E print http://arxiv.org/abs/0904.0900

While the long-ranged correlation of market orders and their impact on prices has been relatively well studied in the literature, the corresponding studies of limit orders and cancellations are scarce. We provide here an empirical study of the cross-correlation between all these different events, and their respective impact on future price changes. We define and extract from the data the "bare" impact these events would have, if they were to happen in isolation. For large tick stocks, we show that a model where the bare impact of all events is permanent and non-fluctuating is in good agreement with the data. For small tick stocks, however, bare impacts must contain a history dependent part, reflecting the internal fluctuations of the order book. We show that this effect can be accurately described by an autoregressive model on the past order flow. This framework allows us to decompose the impact of an event into three parts: an instantaneous jump component, the modification of the future rates of the different events, and the modification of the future gaps behind the best quotes. We compare in detail the present formalism with the temporary impact model that was proposed earlier to describe the impact of market orders when other types of events are not observed. Finally, we extend the model to describe the dynamics of the bid-ask spread.

Why is order flow so persistent?

Date: August 2011

Authors Bence Toth, Imon Palit, Fabrizio Lillo, J. Doyne Farmer
E print http://arxiv.org/abs/1108.1632

Equity order flow is persistent in the sense that buy orders tend to be followed by buy orders and sell orders tend to be followed by sell orders. For equity order flow this persistence is extremely long-ranged, with positive correlations spanning thousands of orders, over time intervals of up to several days. Such persistence in supply and demand is economically important because it influences the market impact as a function of both time and size and because it indicates that the market is in a sense out of equilibrium. Persistence can be caused by two types of behavior: (1) Order splitting, in which a single investor repeatedly places an order of the same sign, or (2) herding, in which different investors place orders of the same sign. We develop a method to decompose the autocorrelation function into splitting and herding components and apply this to order flow data from the London Stock Exchange containing exchange membership identifiers. Members typically act as brokers for other investors, so that it is not clear whether patterns we observe in brokerage data also reflect patterns in the behavior of single investors. To address this problem we develop models for the distortion caused by brokerage and demonstrate that persistence in order flow is overwhelmingly due to order splitting by single investors. At longer time scales we observe that different investors' behavior is anti-correlated. We show that this is due to differences in the response to price-changing vs. non-price-changing market orders.

Eigenvector dynamics: theory and some applications

Date: August 2011

Authors Romain Allez, Jean-Philippe Bouchaud
E print arxiv.org/abs/1203.6228

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and risk control (in which case ${\bf H}_0$ is the assets return correlation matrix). We specialize our results for the case of a Gaussian Orthogonal ${\bf H}_0$, or when ${\bf H}_0$ is a correlation matrix. We illustrate the usefulness of our framework using financial data.

Goodness-of-Fit tests with Dependent Observations

Date: August 2011

Authors Remy Chicheportiche, Jean-Philippe Bouchaud
E print http://arxiv.org/abs/1106.3016

We revisit the Kolmogorov-Smirnov and Cramer-von Mises goodness-of-fit (GoF) tests and propose a generalisation to identically distributed, but dependent univariate random variables. We show that the dependence leads to a reduction of the "effective" number of independent observations. The generalised GoF tests are not distribution-free but rather depend on all the lagged bivariate copulas. These objects, that we call "self-copulas", encode all the non-linear temporal dependences. We introduce a specific, log-normal model for these self-copulas, for which a number of analytical results are derived. An application to financial time series is provided. As is well known, the dependence is to be long-ranged in this case, a finding that we confirm using self-copulas. As a consequence, the acceptance rates for GoF tests are substantially higher than if the returns were iid random variables.

Models for the impact of all order book events

Date: July 2011

Authors Zoltan Eisler, Jean-Philippe Bouchaud, Julien Kockelkoren
E print http://arxiv.org/abs/1107.3364

We propose a general framework to describe the impact of different events in the order book, that generalizes previous work on the impact of market orders. Two different modeling routes can be considered, which are equivalent when only market orders are taken into account. One model posits that each event type has a temporary impact (TIM). The "history dependent impact" model (HDIM), on the other hand, assumes that only price-changing events have a direct impact, itself modulated by the past history of all events through an "influence matrix" that measures how much, on average, an event of a given type affects the immediate impact of a price-changing event of the same sign in the future. We find in particular that aggressive market orders tend to reduce the impact of further aggressive market orders of the same sign (and increase the impact of aggressive market orders of opposite sign). We discuss the relative merits of TIM and HDIM, in particular concerning their ability to reproduce accurately the price diffusion pattern. We find that in spite of theoretical inconsistencies, TIM appears to fare better than HDIM when compared to empirical data. We ascribe this paradox to an uncontrolled approximation used to calibrate HDIMs, calling for further work on this issue.

Smile dynamics -- a theory of the implied leverage effect:ERRATUM

Date: May 2011

Authors Stefano Ciliberti, Jean-Philippe Bouchaud, Marc Potters
E print http://arxiv.org/abs/1105.5082

We correct a mistake in the published version of our paper. Our new conclusion is that the "implied leverage effect" for single stocks is underestimated by option markets for short maturities and overestimated for long maturities, while it is always overestimated for OEX options, except for the shortest maturities where the revised theory and data match perfectly.

How does the market react to your order flow?

Date: April 2011

Authors Bence Toth, Zoltan Eisler, Fabrizio Lillo, Jean-Philippe Bouchaud, Julien Kockelkoren, J. Doyne Farmer
E print http://arxiv.org/abs/1104.0587

We present an empirical study of the intertwined behaviour of members in a financial market. Exploiting a database where the broker that initiates an order book event can be identified, we decompose the correlation and response functions into contributions coming from different market participants and study how their behaviour is interconnected. We find evidence that (1) brokers are very heterogeneous in liquidity provision -- some are consistently liquidity providers while others are consistently liquidity takers. (2) The behaviour of brokers is strongly conditioned on the actions of {\it other} brokers. In contrast brokers are only weakly influenced by the impact of their own previous orders. (3) The total impact of market orders is the result of a subtle compensation between the same broker pushing the price in one direction and the liquidity provision of other brokers pushing it in the opposite direction. These results enforce the picture of market dynamics being the result of the competition between heterogeneous participants interacting to form a complicated market ecology.

Principal Regression Analysis and the index leverage effect

Date: February 2011

Authors Pierre-Alain Reigneron, Romain Allez, Jean-Philippe Bouchaud
E print http://arxiv.org/abs/1011.5810 ;

We revisit the index leverage effect, that can be decomposed into a volatility effect and a correlation effect. We investigate the latter using a matrix regression analysis, that we call 'Principal Regression Analysis' (PRA) and for which we provide some analytical (using Random Matrix Theory) and numerical benchmarks. We find that downward index trends increase the average correlation between stocks (as measured by the most negative eigenvalue of the conditional correlation matrix), and makes the market mode more uniform. Upward trends, on the other hand, also increase the average correlation between stocks but rotates the corresponding market mode {\it away} from uniformity. There are two time scales associated to these effects, a short one on the order of a month (20 trading days), and a longer time scale on the order of a year. We also find indications of a leverage effect for sectorial correlations as well, which reveals itself in the second and third mode of the PRA.

Individual and collective stock dynamics: intra-day seasonalities

Date: September 2010

Authors Romain Allez, Jean-Philippe Bouchaud
E print http://arxiv.org/PS_cache/arxiv/pdf/1009/1009.4785v1.pdf

We establish several new stylised facts concerning the intra-day seasonalities of stock dynamics. Beyond the well known U-shaped pattern of the volatility, we find that the average correlation between stocks increases throughout the day, leading to a smaller relative dispersion between stocks. Somewhat paradoxically, the kurtosis (a measure of volatility surprises) reaches a minimum at the open of the market, when the volatility is at its peak. We confirm that the dispersion kurtosis is a markedly decreasing function of the index return. This means that during large market swings, the idiosyncratic component of the stock dynamics becomes sub-dominant. In a nutshell, early hours of trading are dominated by idiosyncratic or sector specific effects with little surprises, whereas the influence of the market factor increases throughout the day, and surprises become more frequent.

The endogenous dynamics of markets: price impact and feedback loops

Date: September 2010

Authors Jean-Philippe Bouchaud
E print http://arxiv.org/PS_cache/arxiv/pdf/1009/1009.2928v1.pdf

We review the evidence that the erratic dynamics of markets is to a large extent of endogenous origin, i.e. determined by the trading activity itself and not due to the rational processing of exogenous news. In order to understand why and how prices move, the joint fluctuations of order flow and liquidity - and the way these impact prices - become the key ingredients. Impact is necessary for private information to be reflected in prices, but by the same token, random fluctuations in order flow necessarily contribute to the volatility of markets. Our thesis is that the latter contribution is in fact dominant, resulting in a decoupling between prices and fundamental values, at least on short to medium time scales. We argue that markets operate in a regime of vanishing revealed liquidity, but large latent liquidity, which would explain their hyper-sensitivity to fluctuations. More precisely, we identify a dangerous feedback loop between bid-ask spread and volatility that may lead to micro-liquidity crises and price jumps. We discuss several other unstable feedback loops that should be relevant to account for market crises: imitation, unwarranted quantitative models, pro-cyclical regulation, etc.

Population dynamics in a random environment

Date: May 2010

Authors: Irene Giardina, Jean-Philippe Bouchaud, Marc Mezard

We investigate the competition between barrier slowing down and proliferation induced superdiffusion in a model of population dynamics in a random force field. A one-loop RG analysis close to the critical dimension d_c=2 predicts a second order phase transition between a subdiffusive regime and a superdiffusive regime, at variance with our numerical results in d=1 which suggest that a new stable mixed fixed point appears. We introduce the idea of proliferation assisted barrier crossing and give a Flory like argument to understand qualitatively the observed diffusive behaviour at this mixed fixed point.

Of Songs and Men: a Model for Multiple Choice with Herding

Date: November 2009

Authors: Christian Borghesi, Jean-Philippe Bouchaud

We propose a generic model for multiple choice situations in the presence of herding and compare it with recent empirical results from a Web-based music market experiment. The model predicts a phase transition between a weak imitation phase and a strong imitation, 'fashion' phase, where choices are driven by peer pressure and the ranking of individual preferences is strongly distorted at the aggregate level. The model can be calibrated to reproduce the main experimental results of Salganik et al. (Science, 311, pp. 854-856 (2006)); we show in particular that the value of the social influence parameter can be estimated from the data. In one of the experimental situation, this value is found to be close to the critical value of the model.

Smile dynamics -- a theory of the implied leverage effect

Date: December 2008

Authors: Stefano Ciliberti, Jean-Philippe Bouchaud, Marc Potters

We study in details the skew of stock option smiles, which is induced by the so-called leverage effect on the underlying -- i.e. the correlation between past returns and future square returns. This naturally explains the anomalous dependence of the skew as a function of maturity of the option. The market cap dependence of the leverage effect is analyzed using a one-factor model. We show how this leverage correlation gives rise to a non-trivial smile dynamics, which turns out to be intermediate between the "sticky strike" and the "sticky delta" rules. Finally, we compare our result with stock option data, and find that option markets overestimate the leverage effect by a large factor, in particular for long dated options.

Price Impact

Date: November 2009

Author: Jean-Philippe Bouchaud

We define what "Price Impact" means, and how it is measured and modelled in the recent literature. Although this notion seems to convey the idea of a forceful and intuitive mechanism, we discuss why things might not be that simple. Empirical studies show that while the correlation between signed order flow and price changes is strong, the impact of trades on prices is neither linear in volume nor permanent. Impact allows private information to be reflected in prices, but by the same token, random fluctuations in order flow must also contribute to the volatility of markets.

Theory of Financial Risk and Derivative Pricing

Date: December 2003

Authors: Jean-Philippe Bouchaud, Marc Potters

Risk control and derivative pricing have become of major concern to financial institutions. The need foradequate statistical tools to measure and anticipate the amplitude of the potential moves of financialmarkets is clearly expressed, in particular for derivative markets. Classical theories, however, are based on simplified assumptions and lead to a systematic (and sometimes dramatic) underestimation of realrisks. Theory of Financial Risk and Derivative Pricing summarises recent theoretical developments, someof which were inspired by statistical physics. Starting from the detailed analysis of market data, one cantake into account more faithfully the real behaviour of financial markets (in particular the ‘rare events’)for asset allocation, derivative pricing and hedging, and risk control.

2nd English Edition

Publisher Cambridge University Press
ISBN 9780521741866
Date December 2003

Japanese Edition

Publisher Asakura Book Store
ISBN 978-4-254-29536-8

Chinese Edition
Publisher Economic Science Press
Date August 2002
ISBN 978-7-505-82805-6

Financial Applications of Random Matrix Theory: a short review

Date: October 2009

Authors: Jean-Philippe Bouchaud, Marc Potters

We discuss the applications of Random Matrix Theory in the context of financial markets and econometric models, a topic about which a considerable number of papers have been devoted to in the last decade. This mini-review is intended to guide the reader through various theoretical results (the Marcenko-Pastur spectrum and its various generalisations, random SVD, free matrices, largest eigenvalue statistics, etc.) as well as some concrete applications to portfolio optimisation and out-of-sample risk estimation.

Explaining the forward interest rate term structure

Date: September 1999

Authors: Andrew Matacz, Jean-Philippe Bouchaud

We present compelling empirical evidence for a new interpretation of the Forward Rate Curve (FRC) term structure. We find that the average FRC follows a square-root law, with a prefactor related to the spot volatility, suggesting a Value-at-Risk like pricing. We find a striking correlation between the instantaneous FRC and the past spot trend over a certain time horizon. This confirms the idea of an anticipated trend mechanism proposed earlier and provides a natural explanation for the observed shape of the FRC volatility. We find that the one-factor Gaussian Heath-Jarrow-Morton model calibrated to the empirical volatility function fails to adequately describe these features.

A shift-optimized Hill-type estimator

Date: May 2009

Authors: Éva Rácz, János Kertész, Zoltán Eisler

A wide range of natural and social phenomena result in observables whose distributions can be well approximated by a power-law decay. The well-known Hill estimator of the tail exponent provides results which are in many respects superior to other estimators in case the asymptotics of the distribution is indeed a pure power-law, however,systematic errors occur if the distribution is altered by simply shifting it. We demonstrate some related problems which typically emerge when dealing with empirical data and suggest a procedure designed to extend the applicability of the Hill estimator.

Optimal Time to Sell a Stock in Black-Scholes Model: Comment on “Thou shall buy and hold”

Date: September 2008

Authors Satya N. Majumdar, Jean-Philippe Bouchaud
E print http://xxx.lanl.gov/PS_cache/arxiv/pdf/0809/0809.2878v1.pdf

We reconsider the problem of optimal time to sell a stock studied recently by Shiryaev, Xu and Zhou using path integral methods. This method allows us to confirm the results obtained by these authors and extend them to a parameter region inaccessible to the method used by Shiryaev et. al. We also obtain the full distribution of the time t_m at which the maximum of the price is reached for arbitrary values of the drift.

Quadratic Hawkes processes for financial prices

Date: September 2015

Authors: Pierre Blanc, Jonathan Donier, Jean-Philippe Bouchaud

We introduce and establish the main properties of QHawkes ("Quadratic" Hawkes) models. QHawkes models generalize the Hawkes price models introduced in E. Bacry et al. (2014), by allowing all feedback effects in the jump intensity that are linear and quadratic in past returns. A non-parametric fit on NYSE stock data shows that the off-diagonal component of the quadratic kernel indeed has a structure that standard Hawkes models fail to reproduce. Our model exhibits two main properties, that we believe are crucial in the modelling and the understanding of the volatility process: first, the model is time-reversal asymmetric, similar to financial markets whose time evolution has a preferred direction. Second, it generates a multiplicative, fat-tailed volatility process, that we characterize in detail in the case of exponentially decaying kernels, and which is linked to Pearson diffusions in the continuous limit. Several other interesting properties of QHawkes processes are discussed, in particular the fact that they can generate long memory without necessarily be at the critical point. Finally, we provide numerical simulations of our calibrated QHawkes model, which is indeed seen to reproduce, with only a small amount of quadratic non-linearity, the correct magnitude of fat-tails and time reversal asymmetry seen in empirical time series.

How markets slowly digest changes in supply and demand

Date: September 2008

Authors: Jean-Philippe Bouchaud, Fabrizio Lillo, J. Doyne Farmer

In this article we revisit the classic problem of tatonnement in price formation from a microstructure point of view, reviewing a recent body of theoretical and empirical work explaining how fluctuations in supply and demand are slowly incorporated into prices. Because revealed market liquidity is extremely low, large orders to buy or sell can only be traded incrementally, over periods of time as long as months. As a result order flow is a highly persistent long-memory process. Maintaining compatibility with market efficiency has profound consequences on price formation, on the dynamics of liquidity, and on the nature of impact. We review a body of theory that makes detailed quantitative predictions about the volume and time dependence of market impact, the bid-ask spread, order book dynamics, and volatility. Comparisons to data yield some encouraging successes. This framework suggests a novel interpretation of financial information, in which agents are at best only weakly informed and all have a similar and extremely noisy impact on prices. Most of the processed information appears to come from supply and demand itself, rather than from external news. The ideas reviewed here are relevant to market microstructure regulation, agent-based models, cost-optimal execution strategies, and understanding market ecologies.

The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy

Date: October 2007

Authors: Giulio Biroli, Marc Potters, Jean-Philippe Bouchaud

We study a new ensemble of random correlation matrices related to multivariate Student (or more generally elliptic) random variables. We establish the exact density of states of empirical correlation matrices that generalizes the Marcenko-Pastur result. The comparison between the theoretical density of states in the Student case and empirical financial data is surprisingly good, even if we are still able to detect systematic deviations. Finally, we compute explicitely the Kullback-Leibler entropies of empirical Student matrices, which are found to be independent of the true correlation matrix, as in the Gaussian case. We provide numerically exact values for these Kullback-Leibler entropies.

Theory

Date: March 2006

Authors: Matthieu Wyart, Jean-Philippe Bouchaud, Julien Kockelkoren, Marc Potters, Michele Vettorazzo

We argue that on electronic markets, limit and market orders should have equal effective costs on average. This symmetry implies a linear relation between the bid-ask spread and the average impact of market orders. Our empirical observations on different markets are consistent with this hypothesis. We then use this relation to justify a simple, and hitherto unnoticed, proportionality relation between the spread and the volatility_per trade_. We provide convincing empirical evidence for this relation. This suggests that the main determinant of the bid-ask spread is adverse selection, if one considers that the volatility per trade is a measure of the amount of 'information' included in prices at each transaction. Symmetry between market and limit orders stems from the self-organization of liquidity in electronic markets. Our results appear to hold approximately on liquid specialist markets as well, although the spread is significantly larger.

What’s What? Long Only Smart Beta & Equity Market Neutral Alternative Beta

Date: July 2015

“Smart Beta” (long only) and “Alternative Beta” (equity market neutral) strategies both provide investors with solutions that diverge from straightforward long exposure to equity indices, the so-called “equity risk premium”. Because they share a common terminology, a systematic implementation framework, and a reliance on similar well documented effects – e.g., Momentum, Value, Quality, Low Volatility, Small-Cap vs Large-Cap – the two are seen as similar. However, in spite of the obvious commonalities, the two actually differ in many respects, specifically in their construction process, the actual exposures they provide, and where they fit in a portfolio.
This note attempts to offer some clarity. While we focus on equities here, the same logic can be (and occasionally is) applied to other asset classes.

The Excess Returns of “Quality” Stocks: A Behavioral Anomaly

Date: January 2016

Authors: Jean-Philippe Bouchaud, Stefano Ciliberti, Augustin Landier, Guillaume Simon and David Thesmar

E print: http://arxiv.org/pdf/1601.04478v1

This note investigates the causes of the quality anomaly, which is one of the strongest and most scalable anomalies in equity markets. We explore two potential explanations. The "risk view", whereby investing in high quality firms is somehow riskier, so that the higher returns of a quality portfolio are a compensation for risk exposure. This view is consistent with the Efficient Market Hypothesis. The other view is the "behavioral view", which states that some investors persistently underestimate the true value of high quality firms. We find no evidence in favor of the "risk view": The returns from investing in quality firms are abnormally high on a risk-adjusted basis, and are not prone to crashes. We provide novel evidence in favor of the "behavioral view": In their forecasts of future prices, and while being overall overoptimistic, analysts systematically underestimate the future return of high quality firms, compared to low quality firms.

Risk Premium Investing – A Tale of Two Tails

Date: November 2015

In this note we introduce Risk Premia as generically encompassing a set of strategies where investors are compensated for assuming risk. This compensation comes in the form of a regularly received premium and results in a strategy with a positive expected return. This positive performance continues until the moment the hidden risk, which is being assumed, is realized, resulting in a sharp negative move. The argument generally invoked is that this premium is proportional to the risk or volatility of the investment or instrument being held, a measure which uses both sides of the return distribution. In this paper we expand on the idea of a risk premium, with the help of a few example strategies, and show that the premium is in fact compensation for downside deviation or negative skewness risk, an idea which is justified with empirical evidence and also appeals to common sense.

White papers

Date: August 2015

In this note we explain certain subtle features of calculating correlations between time-series. Correlation is a measure of linear co-movement, to be contrasted with the quadratic nature of risk. This can lead to misleading impressions arising from correlating two time-series. We show that the correlation of a manager with a benchmark leads to an estimate of the square root of how much exposure the manager has to the benchmark. We also show that an estimate of correlation with monthly data over 5 years has an associated error of 0.13, and therefore only a correlation of greater than 0.26 should be considered significantly greater than zero.

Alternative Beta Matters - 2015 Q4 Quarterly Report

Date: February 2016

The square-root impact law also holds for option markets

Date: February 2016

Authors: Bence Toth, Zoltan Eisler, Jean-Philippe Bouchaud

E-print: http://arxiv.org/abs/1602.03043

Many independent studies on stocks and futures contracts have established that market impact is proportional to the square-root of the executed volume. Is market impact quantitatively similar for option markets as well? In order to answer this question, we have analyzed the impact of a large proprietary data set of option trades. We find that the square-root law indeed holds in that case. This finding supports the argument for a universal underlying mechanism.

Linear models for the impact of order flow on prices I. Propagators: Transient vs. History Dependent Impact

Date: February 2016

Authors: Damian Eduardo Taranto, Giacomo Bormetti, Jean-Philippe Bouchaud, Fabrizio Lillo, Bence Toth

E-print: http://arxiv.org/abs/1602.02735

Market impact is a key concept in the study of financial markets and several models have been proposed in the literature so far. The Transient Impact Model (TIM) posits that the price at high frequency time scales is a linear combination of the signs of the past executed market orders, weighted by a so-called propagator function. An alternative description -- the History Dependent Impact Model (HDIM) -- assumes that the deviation between the realised order sign and its expected level impacts the price linearly and permanently. The two models, however, should be extended since prices are a priori influenced not only by the past order flow, but also by the past realisation of returns themselves. In this paper, we propose a two-event framework, where price-changing and non price-changing events are considered separately. Two-event propagator models provide a remarkable improvement of the description of the market impact, especially for large tick stocks, where the events of price changes are very rare and very informative. Specifically the extended approach captures the excess anti-correlation between past returns and subsequent order flow which is missing in one-event models. Our results document the superior performances of the HDIMs even though only in minor relative terms compared to TIMs. This is somewhat surprising, because HDIMs are well grounded theoretically, while TIMs are, strictly speaking, inconsistent.

On the overlaps between eigenvectors of correlated random matrices

Date: March 2016

Authors: Joël Bun, Jean-Philippe Bouchaud, Marc Potters

E print: arXiv:1603.04364v1

We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in many different contexts, from quantum thermalisation to high dimensional statistics. We apply our results to the case of empirical correlation matrices, that allow us to estimate reliably the width of the spectrum of the 'true' underlying correlation matrix, even when the latter is very close to the identity matrix. We illustrate our results on the example of stock returns correlations, that clearly reveal a non trivial structure for the bulk eigenvalues.

Technical

Date: February 2016

Authors: Michael Benzaquen, Jonathan Donier, Jean-Philippe Bouchaud

E print: arXiv:1602.03011v1

We confirm and substantially extend the recent empirical result of Andersen et al. (2015), where it is shown that the amount of risk W exchanged in the E-mini S&P futures market (i.e. price times volume times volatility) scales like the 3/2 power of the number of trades N. We show that this 3/2-law holds very precisely across 12 futures contracts and 300 single US stocks, and across a wide range of times scales. However, we find that the "trading invariant" I=W/N3/2 proposed by Kyle and Obfizhaeva (2010) is in fact quite different for different contracts, in particular between futures and single stocks. Our analysis suggests I/S as a more natural candidate, where S is the bid-ask spread. We also establish two more complex scaling laws for the volatility σ and the traded volume V as a function of N, that reveal the existence of a characteristic number of trades N0 above which the expected behaviour σ∼N−−√ and V∼N hold, but below which strong deviations appear, induced by the size of the tick.

Do investors trade too much? A laboratory experiment

Date: December 2015

Authors: Joao da Gama Batista, Domenico Massaro, Jean-Philippe Bouchaud, Damien Challet, Cars Hommes

E print: arXiv:1512.03743v1

We run experimental asset markets to investigate the emergence of excess trading and the occurrence of synchronised trading activity leading to crashes in the artificial markets. The market environment favours early investment in the risky asset and no posterior trading, i.e. a buy-and-hold strategy with a most probable return of over 600%. We observe that subjects trade too much, and due to the market impact that we explicitly implement, this is detrimental to their wealth. The asset market experiment was followed by risk aversion measurement. We find that preference for risk systematically leads to higher activity rates (and lower final wealth). We also measure subjects' expectations of future prices and find that their actions are fully consistent with their expectations. In particular, trading subjects try to beat the market and make profits by playing a buy low, sell high strategy. Finally, we have not detected any major market crash driven by collective panic modes, but rather a weaker but significant tendency of traders to synchronise their entry and exit points in the market.

Optimal Trading with Linear and (small) Non-Linear Costs

Date: November 2015

Authors: Adam Rej, Raphael Benichou, Joachim de Lataillade, Gilles Zérah, Jean-Philippe Bouchaud

E print: arXiv:1511.07359v2

We reconsider the problem of optimal trading in the presence of linear and quadratic costs, for arbitrary linear costs but in the limit where quadratic costs are small. Using matched asymptotic expansion techniques, we find that the trading speed vanishes inside a band that is narrower than in the absence of quadratic costs, by an amount that scales as the one-third power of quadratic costs. Outside of the band, we find three regimes: a small boundary layer where the velocity vanishes linearly with the distance to the band, an intermediate region where the velocity behaves as a square-root of that distance, and a far region where it becomes linear. Our solution is consistent with available numerical results. We determine the conditions in which our expansion is useful in practical applications, and generalize our solution to other forms of non-linear costs.

On growth-optimal tax rates and the issue of wealth inequalities

Date: August 2015

Authors:Jean-Philippe Bouchaud (Capital Fund Management and Ecole Polytechnique)

E print: arXiv:1508.00275v2

We introduce a highly stylized, yet non trivial model of the economy, with a public and private sector coupled through a wealth tax and a redistribution policy. The model can be fully solved analytically, and allows one to address the question of optimal taxation and of wealth inequalities. We find that according to the assumption made on the relative performance of public and private sectors, three situations are possible. Not surprisingly, the optimal wealth tax rate is either 0% for a deeply dysfunctional government and/or highly productive private sector, or 100 % for a highly efficient public sector and/or debilitated/risk averse private investors. If the gap between the public/private performance is moderate, there is an optimal positive wealth tax rate maximizing economic growth, even -- counter-intuitively -- when the private sector generates more growth. The compromise between profitable private investments and taxation however leads to a residual level of inequalities. The mechanism leading to an optimal growth rate is related the well-known explore/exploit trade-off.

From Walras' auctioneer to continuous time double auctions: A general dynamic theory of supply and demand

Date: June 2015

Authors: Jonathan Donier, Jean-Philippe Bouchaud

E print: arXiv:1506.03758

In standard Walrasian auctions, the price of a good is defined as the point where the supply and demand curves intersect. Since both curves are generically regular, the response to small perturbations is linearly small. However, a crucial ingredient is absent of the theory, namely transactions themselves. What happens after they occur? To answer the question, we develop a dynamic theory for supply and demand based on agents with heterogeneous beliefs. When the inter-auction time is infinitely long, the Walrasian mechanism is recovered. When transactions are allowed to happen in continuous time, a peculiar property emerges: close to the price, supply and demand vanish quadratically, which we empirically confirm on the Bitcoin. This explains why price impact in financial markets is universally observed to behave as the square root of the excess volume. The consequences are important, as they imply that the very fact of clearing the market makes prices hypersensitive to small fluctuations.

Why Do Markets Crash? Bitcoin Data Offers Unprecedented Insights

Date: March 2015

Authors:  Jonathan Donier, Jean-Philippe Bouchaud

E print: arXiv:1503.06704

Crashes have fascinated and baffled many canny observers of financial markets. In the strict orthodoxy of the efficient market theory, crashes must be due to sudden changes of the fundamental valuation of assets. However, detailed empirical studies suggest that large price jumps cannot be explained by news and are the result of endogenous feedback loops. Although plausible, a clear-cut empirical evidence for such a scenario is still lacking. Here we show how crashes are conditioned by the market liquidity, for which we propose a new measure inspired by recent theories of market impact and based on readily available, public information. Our results open the possibility of a dynamical evaluation of liquidity risk and early warning signs of market instabilities, and could lead to a quantitative description of the mechanisms leading to market crashes.

Sudden Trust Collapse in Networked Societies

Date: September 2014

Authors: João da Gama Batista, Jean-Philippe Bouchaud, Damien Challet

E print: arXiv:1409.8321

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.

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

Date: November 2013

Authors: Jean-Philippe Bouchaud, Christian Borghesi, Pablo Jensen

E print: arXiv:1311.0810

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.

Invariant $β$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law

Date: September 2012

Authors: Romain Allez, Jean-Philippe Bouchaud, Satya N. Majumdar, Pierpaolo Vivo

E print: arXiv:1209.6171

We construct a diffusive matrix model for the β-Wishart (or Laguerre) ensemble for general continuous β∈[0,2], which preserves invariance under the orthogonal/unitary group transformation. Scaling the Dyson index β with the largest size M of the data matrix as β=2c/M (with c a fixed positive constant), we obtain a family of spectral densities parametrized by c. As c is varied, this density interpolates continuously between the Mar\vcenko-Pastur (c→∞ limit) and the Gamma law (c→0 limit). Analyzing the full Stieltjes transform (resolvent) equation, we obtain as a byproduct the correction to the Mar\vcenko-Pastur density in the bulk up to order 1/M for all β and up to order 1/M2 for the particular cases β=1,2.

The fine structure of volatility feedback I: multi-scale self-reflexivity

Date: June 2012

Authors: Rémy Chicheportiche, Jean-Philippe Bouchaud

E print: arXiv:1206.2153

We attempt to unveil the fine structure of volatility feedback effects in the context of general quadratic autoregressive (QARCH) models, which assume that today's volatility can be expressed as a general quadratic form of the past daily returns. The standard ARCH or GARCH framework is recovered when the quadratic kernel is diagonal. The calibration of these models on US stock returns reveals several unexpected features. The off-diagonal (non ARCH) coefficients of the quadratic kernel are found to be highly significant both In-Sample and Out-of-Sample, but all these coefficients turn out to be one order of magnitude smaller than the diagonal elements. This confirms that daily returns play a special role in the volatility feedback mechanism, as postulated by ARCH models. The feedback kernel exhibits a surprisingly complex structure, incompatible with models proposed so far in the literature. Its spectral properties suggest the existence of volatility-neutral patterns of past returns. The diagonal part of the quadratic kernel is found to decay as a power-law of the lag, in line with the long-memory of volatility. Finally, QARCH models suggest some violations of Time Reversal Symmetry in financial time series, which are indeed observed empirically, although of much smaller amplitude than predicted. We speculate that a faithful volatility model should include both ARCH feedback effects and a stochastic component.

Election turnout statistics in many countries: similarities, differences, and a diffusive field model for decision-making

Date: May 2012

Authors: Christian Borghesi, Jean-Claude Raynal, Jean-Philippe Bouchaud

E print: arXiv:1201.0524

We study in details the turnout rate statistics for 77 elections in 11 different countries. We show that the empirical results established in a previous paper for French elections appear to hold much more generally. We find in particular that the spatial correlation of turnout rates decay logarithmically with distance in all cases. This result is quantitatively reproduced by a decision model that assumes that each voter makes his mind as a result of three influence terms: one totally idiosyncratic component, one city-specific term with short-ranged fluctuations in space, and one long-ranged correlated field which propagates diffusively in space. A detailed analysis reveals several interesting features: for example, different countries have different degrees of local heterogeneities and seem to be characterized by a different propensity for individuals to conform to the cultural norm. We furthermore find clear signs of herding (i.e. strongly correlated decisions at the individual level) in some countries, but not in others.

Invariant β-ensembles and the Gauss-Wigner crossover

Date: May 2012

Authors: Romain Allez, Jean-Philippe Bouchaud, Alice Guionnet

E print: arXiv:1205.3598

We define a new diffusive matrix model converging towards the β -Dyson Brownian motion for all β∈[0,2] that provides an explicit construction of β-ensembles of random matrices that is invariant under the orthogonal/unitary group. For small values of β, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semi-circle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semi-circle.

Comment on Turbulent cascades in foreign exchange markets

Date: July 1996

Authors: Alain Arneodo (1), Jean-Philippe Bouchaud (2,3), Rama Cont (3,5), Jean-Francois Muzy (1), Marc Potters (3), Didier Sornette (3,4,5). ((1) Centre de Recherche Paul Pascal (2) CEA Saclay (3) Science & Finance (4) UCLA (5)

E print: arXiv:cond-mat/9607120

Recently, Ghashghaie et al. have shown that some statistical aspects of fully developed turbulence and exchange rate fluctuations exhibit striking similarities (Nature 381, 767 (1996)). The authors then suggested that the two problems might be deeply connected, and speculated on the existence of an 'information cascade' which would play the role in finance of the well known Kolmogorov energy cascade in turbulence. Here we want to convince the reader that the two problems differ on a fundamental aspect, namely, correlations.

Real-world options: smile and residual risk

Date: January 1994

Authors: Jean-Philippe Bouchaud, Giulia Iori, Didier Sornette

E print: arXiv:cond-mat/9509095

We present a theory of option pricing and hedging, designed to address non-perfect arbitrage, market friction and the presence of 'fat' tails. An implied volatility 'smile' is predicted. We give precise estimates of the residual risk associated with optimal (but imperfect) hedging.

The Black-Scholes option pricing problem in mathematical finance: generalization and extensions for a large class of stochastic processes

Date: January 1994

Authors:Jean-Philippe Bouchaud, Didier Sornette

E print:  HAL Id: jpa-00246951

Cleaning correlation matrices

Date: April 2016

Authors: Joël Bun, Jean-Philippe Bouchaud and Marc Potters

From Risk Magazine's April 2016 issue "The determination of correlation matrices is typically affected by in-sample noise. Joël Bun, Jean-Philippe Bouchaud and Marc Potters propose a simple, yet optimal, estimator of the true underlying correlation matrix and show that this new
cleaning recipe outperforms all existing estimators in terms of the out-of-sample risk of synthetic portfolios."

Spatial correlations in vote statistics: a diffusive field model for decision-making

Date: March 2010

Authors: Christian Borghesi, Jean-Philippe Bouchaud

E print: arXiv:1003.2807

We study the statistics of turnout rates and results of the French elections since 1992. We find that the distribution of turnout rates across towns is surprisingly stable over time. The spatial correlation of the turnout rates, or of the fraction of winning votes, is found to decay logarithmically with the distance between towns. Based on these empirical observations and on the analogy with a two-dimensional random diffusion equation, we propose that individual decisions can be rationalised in terms of an underlying "cultural" field, that locally biases the decision of the population of a given region, on top of an idiosyncratic, town-dependent field, with short range correlations. Using symmetry considerations and a set of plausible assumptions, we suggest that this cultural field obeys a random diffusion equation.

The (unfortunate) complexity of the economy

Date: April 2009

Author: Jean-Philippe Bouchaud

E print: arXiv:0904.0805

This article is a follow-up of a short essay that appeared in Nature 455, 1181 (2008) [arXiv:0810.5306]. It has become increasingly clear that the erratic dynamics of markets is mostly endogenous and not due to the rational processing of exogenous news. I elaborate on the idea that spin-glass type of problems, where the combination of competition and heterogeneities generically leads to long epochs of statis interrupted by crises and hyper-sensitivity to small changes of the environment, could be metaphors for the complexity of economic systems. I argue that the most valuable contribution of physics to economics might end up being of methodological nature, and that simple models from physics and agent based numerical simulations, although highly stylized, are more realistic than the traditional models of economics that assume rational agents with infinite foresight and infinite computing abilities.

Economics need a scientific revolution

Date: October 2008

Author: JP Bouchaud

E print: arXiv:0810.5306

I argue that the current financial crisis highlights the crucial need of a change of mindset in economics and financial engineering, that should move away from dogmatic axioms and focus more on data, orders of magnitudes, and plausible, albeit non rigorous, arguments.
An edited version of this essay appeared in Nature.

Stock price jumps: news and volume play a minor role

Date: March 2008

Authors: Armand Joulin, Augustin Lefevre, Daniel Grunberg, Jean-Philippe Bouchaud (CFM)

E print: arXiv:0803.1769

In order to understand the origin of stock price jumps, we cross-correlate high-frequency time series of stock returns with different news feeds. We find that neither idiosyncratic news nor market wide news can explain the frequency and amplitude of price jumps. We find that the volatility patterns around jumps and around news are quite different: jumps are followed by increased volatility, whereas news tend on average to be followed by lower volatility levels. The shape of the volatility relaxation is also markedly different in the two cases. Finally, we provide direct evidence that large transaction volumes are_not_ responsible for large price jumps. We conjecture that most price jumps are induced by order flow fluctuations close to the point of vanishing liquidity.

Large dimension forecasting models and random singular value spectra

Date: February 2008

Authors: Jean-Philippe Bouchaud, Laurent Laloux, M. Augusta Miceli, Marc Potters

E print: arXiv:physics/0512090

We present a general method to detect and extract from a finite time sample statistically meaningful correlations between input and output variables of large dimensionality. Our central result is derived from the theory of free random matrices, and gives an explicit expression for the interval where singular values are expected in the absence of any true correlations between the variables under study. Our result can be seen as the natural generalization of the Marcenko-Pastur distribution for the case of rectangular correlation matrices. We illustrate the interest of our method on a set of macroeconomic time series.

Trend followers lose more often than they gain

Date: August 2005

Authors: Marc Potters, Jean-Philippe Bouchaud

E print: arXiv:physics/0508104

We solve exactly a simple model of trend following strategy, and obtain the analytical shape of the profit per trade distribution. This distribution is non trivial and has an option like, asymmetric structure. The degree of asymmetry depends continuously on the parameters of the strategy and on the volatility of the traded asset. While the average gain per trade is always exactly zero, the fraction f of winning trades decreases from f=1/2 for small volatility to f=0 for high volatility, showing that this winning probability does not give any information on the reliability of the strategy but is indicative of the trading style.

Financial Applications of Random Matrix Theory: Old Laces and New Pieces

Date: July 2005

Authors: Marc Potters, Jean-Philippe Bouchaud, Laurent Laloux

E print: arXiv:physics/0507111

This contribution to the proceedings of the Cracow meeting on 'Applications of Random Matrix Theory' summarizes a series of studies, some old and others more recent on financial applications of Random Matrix Theory (RMT). We first review some early results in that field, with particular emphasis on the applications of correlation cleaning to portfolio optimisation, and discuss the extension of the Marcenko-Pastur (MP) distribution to a non trivial 'true' underlying correlation matrix. We then present new results concerning different problems that arise in a financial context: (a) the generalisation of the MP result to the case of an empirical correlation matrix (ECM) constructed using exponential moving averages, for which we give a new elegant derivation (b) the specific dynamics of the 'market' eigenvalue and its associated eigenvector, which defines an interesting Ornstein-Uhlenbeck process on the unit sphere and (c) the problem of the dependence of ECM's on the observation frequency of the returns and its interpretation in terms of lagged cross-influences.

Theory of collective opinion shifts: from smooth trends to abrupt swings

Date: July 2005

Authors: Quentin Michard, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0504079

We unveil collective effects induced by imitation and social pressure by analyzing data from three different sources: birth rates, sales of cell phones and the drop of applause in concert halls. We interpret our results within the framework of the Random Field Ising Model, which is a threshold model for collective decisions accounting both for agent heterogeneity and social imitation. Changes of opinion can occur either abruptly or continuously, depending on the importance of herding effects. The main prediction of the model is a scaling relation between the height h of the speed of variation peak and its width w of the form h ~ w^{-kappa}, with kappa = 2/3 for well connected populations. Our three sets of data are compatible with such a prediction, with kappa ~ 0.62 for birth rates, kappa ~ 0.71 for cell phones and kappa ~ 0.64 for clapping. In this last case, we in fact observe that some clapping samples end discontinuously (w=0), as predicted by the model for strong enough imitation.

Theory of collective opinion shifts: from smooth trends to abrupt swings

Date: April 2005

Authors:  Quentin Michard, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0504079

We unveil collective effects induced by imitation and social pressure by analyzing data from three different sources: birth rates, sales of cell phones and the drop of applause in concert halls. We interpret our results within the framework of the Random Field Ising Model, which is a threshold model for collective decisions accounting both for agent heterogeneity and social imitation. Changes of opinion can occur either abruptly or continuously, depending on the importance of herding effects. The main prediction of the model is a scaling relation between the height h of the speed of variation peak and its width w of the form h ~ w^{-kappa}, with kappa = 2/3 for well connected populations. Our three sets of data are compatible with such a prediction, with kappa ~ 0.62 for birth rates, kappa ~ 0.71 for cell phones and kappa ~ 0.64 for clapping. In this last case, we in fact observe that some clapping samples end discontinuously (w=0), as predicted by the model for strong enough imitation.

The Dynamics of Financial Markets - Mandelbrot's multifractal cascades, and beyond

Date: January 2005

Authors: Lisa Borland, Jean-Philippe Bouchaud, Jean-Francois Muzy, Gilles Zumbach

E print: arXiv:cond-mat/0501292

This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in W ilmott magazine. We discuss how multiplicative cascades and related multifractal ideas might be relevant to model the main statistical features of financial time series, in particular the intermittent, long-memory nature of the volatility. We describe in details the Bacry-Muzy-Delour multifractal random walk. We point out some inadequacies of the current models, in particular concerning time reversal symmetry, and propose an alternative family of multi-timescale models, intermediate between GARCH models and multifractal models, that seem quite promising.

Experts' earning forecasts: bias, herding and gossamer information

Date: October 2004

Authors: Olivier Guedj, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0410079

We study the statistics of earning forecasts of US, EU, UK and JP stocks during the period 1987-2004. We confirm, on this large data set, that financial analysts are on average over-optimistic and show a pronounced herding behavior. These effects are time dependent, and were particularly strong in the early nineties and during the Internet bubble. We furthermore find that their forecast ability is, in relative terms, quite poor and comparable in quality, a year ahead, to the simplest no change' forecast. As a result of herding, analysts agree with each other five to ten times more than with the actual result. We have shown that significant differences exist between US stocks and EU stocks, that may partly be explained as a company size effect. Interestingly, herding effects appear to be stronger in the US than in the Eurozone. Finally, we study the correlation of errors across stocks and show that significant sectorization occurs, some sectors being easier to predict than others. These results add to the list of arguments suggesting that the tenets of Efficient Market Theory are untenable.

A Non-Gaussian Option Pricing Model with Skew

Date: March 2004

Authors: L. Borland (Evnine-Vaughan Associates), Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0403022

Closed form option pricing formulae explaining skew and smile are obtained within a parsimonious non-Gaussian framework. We extend the non-Gaussian option pricing model of L. Borland (Quantitative Finance, {\bf 2}, 415-431, 2002) to include volatility-stock correlations consistent with the leverage effect. A generalized Black-Scholes partial differential equation for this model is obtained, together with closed-form approximate solutions for the fair price of a European call option. In certain limits, the standard Black-Scholes model is recovered, as is the Constant Elasticity of Variance (CEV) model of Cox and Ross. Alternative methods of solution to that model are thereby also discussed. The model parameters are partially fit from empirical observations of the distribution of the underlying. The option pricing model then predicts European call prices which fit well to empirical market data over several maturities.

Option pricing and hedging with minimum expected shortfall

Date: August 2003

Authors: Benoît Pochart, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0308570

We propose a versatile Monte-Carlo method for pricing and hedging options when the market is incomplete, for an arbitrary risk criterion (chosen here to be the expected shortfall), for a large class of stochastic processes, and in the presence of transaction costs. We illustrate the method on plain vanilla options when the price returns follow a Student-t distribution. We show that in the presence of fat-tails, our strategy allows to significantly reduce extreme risks, and generically leads to low Gamma hedging. Similarly, the inclusion of transaction costs reduces the Gamma of the optimal strategy.

Comment on Two-phase behaviour of financial markets

Date: April 2003

Authors: Marc Potters, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0304514

In a recent article [Nature 421, 130 (2003)], Plerou, Gopikrishnan and Stanley report some evidence for an intriguing two-phase behavior of financial markets when studying the distribution of volume imbalance conditional to the local intensity of its fluctuations. We show here that this apparent phase transition is a generic consequence of the conditioning and exists even in the absence of any non trivial collective phenomenon.

Self-referential behaviour, overreaction and conventions in financial markets

Date: March 2003

Authors: Matthieu Wyart, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0303584

We study a generic model for self-referential behaviour in financial markets, where agents attempt to use some (possibly fictitious) causal correlations between a certain quantitative information and the price itself. This correlation is estimated using the past history itself, and is used by a fraction of agents to devise active trading strategies. The impact of these strategies on the price modify the observed correlations. A potentially unstable feedback loop appears and destabilizes the market from an efficient behaviour. For large enough feedbacks, we find a phase transition' beyond which non trivial correlations spontaneously set in and where the market switches between two long lived states, that we call conventions. This mechanism leads to overreaction and excess volatility, which may be considerable in the convention phase. A particularly relevant case is when the source of information is the price itself. The two conventions then correspond then to either a trend following regime or to a contrarian (mean reverting) regime. We provide some empirical evidence for the existence of these conventions in real markets, that can last for several decades.

Multiple time scales in volatility and leverage correlation: A stochastic volatility model

Date: February 2003

Authors: Josep Perello, Jaume Masoliver, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0302095

Financial time series exhibit two different type of non linear correlations: (i) volatility autocorrelations that have a very long range memory, on the order of years, and (ii) asymmetric return-volatility (or leverage') correlations that are much shorter ranged. Different stochastic volatility models have been proposed in the past to account for both these correlations. However, in these models, the decay of the correlations is exponential, with a single time scale for both the volatility and the leverage correlations, at variance with observations. We extend the linear Ornstein-Uhlenbeck stochastic volatility model by assuming that the mean reverting level is itself random. We find that the resulting three-dimensional diffusion process can account for different correlation time scales. We show that the results are in good agreement with a century of the Dow Jones index daily returns (1900-2000), with the exception of crash days.

CTAs in a Regime of Rising Rates

Date: May 2016

The advent of the Federal Reserve’s recent interest rate “lift-off” and the future unwinding of other developed market central banks’ ultra-loose monetary policies prompts the question as to whether this new regime presents a problem for interest rate trend following. In this short note we examine the performance of the trend applied to interest rate futures and also to the other asset classes employed by CTAs - namely equities, commodities and FX, in periods of rising and falling rates and also in regimes of upward and downward sloping yield curves.

Alternative Beta Matters - 2016 Q1 Quarterly Report

Date: May 2016

Within this report we recap major developments of the quarter for Equities, Fixed Income / Credit, FX and Commodities, as well as Alternatives. All discussion is agnostic to particular approaches or techniques, and where alternative benchmark strategy results are presented, the exact methodology used is given.

We have also included one white paper and an extended academic abstract from a paper published during the quarter. Our hope is that these publications, which convey our views on topics related to Alternative Beta that have arisen in our many discussions with clients, can be used as a reference for our readers, and can stimulate conversations on these topical issues.

An introduction to statistical finance

Date: October 2002

Author: Jean-Philippe Bouchaud

We summarize recent research in a rapid growing field, that of statistical finance, also called 'econophysics'. There are three main themes in this activity: (i) empirical studies and the discovery of interesting universal features in the statistical texture of financial time series, (ii) the use of these empirical results to devise better models of risk and derivative pricing, of direct interest for the financial industry, and (iii) the study of 'agent-based models' in order to unveil the basic mechanisms that are responsible for the statistical 'anomalies' observed in financial time series. We give a brief overview of some of the results in these three directions.

Bubbles, crashes and intermittency in agent based market models

Date: June 2002

Authors: Irene Giardina, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0206222

We define and study a rather complex market model, inspired from the Santa Fe artificial market and the Minority Game. Agents have different strategies among which they can choose, according to their relative profitability, with the possibility of not participating to the market. The price is updated according to the excess demand, and the wealth of the agents is properly accounted for. Only two parameters play a significant role: one describes the impact of trading on the price, and the other describes the propensity of agents to be trend following or contrarian. We observe three different regimes, depending on the value of these two parameters: an oscillating phase with bubbles and crashes, an intermittent phase and a stable 'rational' market phase. The statistics of price changes in the intermittent phase resembles that of real price changes, with small linear correlations, fat tails and long range volatility clustering. We discuss how the time dependence of these two parameters spontaneously drives the system in the intermittent region. We analyze quantitatively the temporal correlation of activity in the intermittent phase, and show that the 'random time strategy shift' mechanism that we proposed earlier allows one to understand the observed long ranged correlations. Other mechanisms leading to long ranged correlations are also reviewed. We discuss several other issues, such as the formation of bubbles and crashes, the influence of transaction costs and the distribution of agents wealth.

The skewed multifractal random walk with applications to option smiles

Date: January 2002

Authors: B. Pochart, J.-P. Bouchaud

E print: arXiv:cond-mat/0204047

We generalize the construction of the multifractal random walk (MRW) due to Bacry, Delour and Muzy to take into account the asymmetric character of the financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. We compute the leading behaviour of q-moments of the process, that behave as power-laws of the time lag with an exponent zeta_q=p-2p(p-1) lambda^2 for even q=2p, as in the symmetric MRW, and as zeta_q=p(1-2p lambda^2)+1-alpha (q=2p+1), where lambda and alpha are parameters. We show that this extended model reproduces the HARCH' effect or causal cascade' reported by some authors. We illustrate the usefulness of this skewed MRW by computing the resulting shape of the volatility smiles generated by such a process, that we compare to approximate cumulant expansions formulas for the implied volatility. A large variety of smile surfaces can be reproduced.

Theory

Date: March 2001

Authors: Marc Potters, Jean-Philippe Bouchaud, Dragan Sestovic

Published in: Risk Magazine 14 3, 133-136

Microscopic models for long ranged volatility correlations

Date: May 2001

Authors: Irene Giardina, Jean-Philippe Bouchaud, Marc Mézard

E print: arXiv:cond-mat/0105076

We propose a general interpretation for long-range correlation effects in the activity and volatility of financial markets. This interpretation is based on the fact that the choice between 'active' and 'inactive' strategies is subordinated to random-walk like processes. We numerically demonstrate our scenario in the framework of simplified market models, such as the Minority Game model with an inactive strategy, or a more sophisticated version that includes some price dynamics. We show that real market data can be surprisingly well accounted for by these simple models.

Hedged Monte-Carlo: low variance derivative pricing with objective probabilities

Date: August 2000

Authors: Marc Potters, Jean-Philippe Bouchaud, Dragan Sestovic

E print: arXiv:cond-mat/0008147

We propose a new 'hedged' Monte-Carlo (HMC) method to price financial derivatives, which allows to determine simultaneously the optimal hedge. The inclusion of the optimal hedging strategy allows one to reduce the financial risk associated with option trading, and for the very same reason reduces considerably the variance of our HMC scheme as compared to previous methods. The explicit accounting of the hedging cost naturally converts the objective probability into the 'risk-neutral' one. This allows a consistent use of purely historical time series to price derivatives and obtain their residual risk. The method can be used to price a large class of exotic options, including those with path dependent and early exercise features.

Power-laws in economics and finance: some ideas from physics

Date: August 2000

Author: Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0008103

We discuss several models in order to shed light on the origin of power-law distributions and power-law correlations in financial time series. From an empirical point of view, the exponents describing the tails of the price increments distribution and the decay of the volatility correlations are rather robust and suggest universality. However, many of the models that appear naturally (for example, to account for the distribution of wealth) contain some multiplicative noise, which generically leads to *non universal exponents*. Recent progress in the empirical study of the volatility suggests that the volatility results from some sort of multiplicative cascade. A convincing 'microscopic' (i.e. trader based) model that explains this observation is however not yet available. It would be particularly important to understand the relevance of the pseudo-geometric progression of natural human time scales on the long range nature of the volatility correlations.

Hedging large risks reduces the transaction costs

Date: May 2000

Authors: Farhat Selmi, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0005148

As soon as one accepts to abandon the zero-risk paradigm of Black-Scholes, very interesting issues concerning risk control arise because different definitions of the risk become unequivalent. Optimal hedges then depend on the quantity one wishes to minimize. We show that a definition of the risk more sensitive to the extreme events generically leads to a decrease both of the probability of extreme losses and of the sensitivity of the hedge on the price of the underlying (the 'Gamma'). Therefore, the transaction costs and the impact of hedging on the price dynamics of the underlying are reduced.

Wealth condensation in a simple model of economy

Date: February 2000

Authors: Jean-Philippe Bouchaud, Marc Mezard

E print: arXiv:cond-mat/0002374

We introduce a simple model of economy, where the time evolution is described by an equation capturing both exchange between individuals and random speculative trading, in such a way that the fundamental symmetry of the economy under an arbitrary change of monetary units is insured. We investigate a mean-field limit of this equation and show that the distribution of wealth is of the Pareto (power-law) type. The Pareto behaviour of the tails of this distribution appears to be robust for finite range models, as shown using both a mapping to the random 'directed polymer' problem, as well as numerical simulations. In this context, a transition between an economy dominated by a few individuals from a situation where the wealth is more evenly spread out, is found. An interesting outcome is that the distribution of wealth tends to be very broadly distributed when exchanges are limited, either in amplitude or topologically. Favoring exchanges (and, less surprisingly, increasing taxes) seems to be an efficient way to reduce inequalities.

Option pricing and hedging with temporal correlations

Date: November 2000

Authors: Lorenzo Cornalba, Jean-Philippe Bouchaud, Marc Potters

E print: arXiv:cond-mat/0011506

We consider the problem of option pricing and hedging when stock returns are correlated in time. Within a quadratic-risk minimisation scheme, we obtain a general formula, valid for weakly correlated non-Gaussian processes. We show that for Gaussian price increments, the correlations are irrelevant, and the Black-Scholes formula holds with the volatility of the price increments on the scale of the re-hedging. For non-Gaussian processes, further non trivial corrections to the 'smile' are brought about by the correlations, even when the hedge is the Black-Scholes Delta-hedge. We introduce a compact notation which eases the computations and could be of use to deal with more complicated models.

Back to basics: historical option pricing revisited

Date: August 1998

Authors: Jean-Philippe Bouchaud, Marc Potters

E print: arXiv:cond-mat/9808206

We reconsider the problem of option pricing using historical probability distributions. We first discuss how the risk-minimisation scheme proposed recently is an adequate starting point under the realistic assumption that price increments are uncorrelated (but not necessarily independent) and of arbitrary probability density. We discuss in particular how, in the Gaussian limit, the Black-Scholes results are recovered, including the fact that the average return of the underlying stock disappears from the price (and the hedging strategy). We compare this theory to real option prices and find these reflect in a surprisingly accurate way the subtle statistical features of the underlying asset fluctuations.

Elements for a theory of financial risks

Date: June 1998

Author: Jean-Philippe Bouchaud

E print: arXiv:cond-mat/9806101

Estimating and controlling large risks has become one of the main concern of financial institutions. This requires the development of adequate statistical models and theoretical tools (which go beyond the traditionnal theories based on Gaussian statistics), and their practical implementation. Here we describe three interrelated aspects of this program: we first give a brief survey of the peculiar statistical properties of the empirical price fluctuations. We then review how an option pricing theory consistent with these statistical features can be constructed, and compared with real market prices for options. We finally argue that a true 'microscopic' theory of price fluctuations (rather than a statistical model) would be most valuable for risk assessment. A simple Langevin-like equation is proposed, as a possible step in this direction.

A Langevin approach to stock market fluctuations and crashes

Date: January 1998

Authors: Jean-Philippe Bouchaud, Rama Cont

E print: arXiv:cond-mat/9801279

We propose a non linear Langevin equation as a model for stock market fluctuations and crashes. This equation is based on an identification of the different processes influencing the demand and supply, and their mathematical transcription. We emphasize the importance of feedback effects of price variations onto themselves. Risk aversion, in particular, leads to an up-down symmetry breaking term which is responsible for crashes, where 'panic' is self reinforcing. It is also responsible for the sudden collapse of speculative bubbles. Interestingly, these crashes appear as rare, 'activated' events, and have an exponentially small probability of occurence. We predict that the shape of the falldown of the price during a crash should be logarithmic. The normal regime, where the stock price exhibits behavior similar to that of a random walk, however reveals non trivial correlations on different time scales, in particular on the time scale over which operators perceive a change of trend.

Taming large events: portfolio selection for strongly fluctuating assets

Date: January 1998

Authors: Jean-Philippe Bouchaud, Didier Sornette, Christian Walter and Jean-Pierre Aguilar

Published: International Journal of Theoretical and Applied Finance 1, 25, (1998)

We propose a method of optimization of asset allocation in the case where the stock price variations are supposed to have "fat" tails represented by power laws. Generalizing over previous works using stable Lévy distributions, we distinguish three distinct components of risk described by three different parts of the distributions of price variations: unexpected gains (to be kept), harmless noise inherent to financial activity, and unpleasant losses, which is the only component one would like to minimize. The independent treatment of the tails of distributions for positive and negative variations and the generalization to large events of the notion of covariance of two random variables provide explicit formulae for the optimal portfolio. The use of the probability of loss (or equivalently the Value-at-Risk), as the key quantity to study and minimize, provides a simple solution to the problem of optimization of asset allocations in the general case where the characteristic exponents are different for each asset.

Herd behavior and aggregate fluctuations in financial markets

Date: December 1997

Authors: Rama Cont, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/9712318

We present a simple model of a stock market where a random communication structure between agents gives rise to a heavy tails in the distribution of stock price variations in the form of an exponentially truncated power-law, similar to distributions observed in recent empirical studies of high frequency market data. Our model provides a link between two well-known market phenomena: the heavy tails observed in the distribution of stock market returns on one hand and 'herding' behavior in financial markets on the other hand. In particular, our study suggests a relation between the excess kurtosis observed in asset returns, the market order flow and the tendency of market participants to imitate each other.

Financial modeling and option theory with the truncated Lévy process

Date: October 1997

Author: Andrew Matacz

E print: arXiv:cond-mat/9710197

In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed excess kurtosis at short timescales, along with the slow convergence to Gaussian at longer timescales. I further test the truncated Levy paradigm using high frequency data from the Australian All Ordinaries share market index. I then consider, for the early Levy dominated regime, the issue of option hedging for two different hedging strategies that are in some sense optimal. These are compared with the usual delta hedging approach and found to differ significantly. I also derive the natural generalization of the Black-Scholes option pricing formula when the underlying security is modeled by a geometric TLP. This generalization would not be possible without the truncation.

Missing information and asset allocation

Date: July 1997

Authors:  Jean-Philippe Bouchaud, Marc Potters, Jean-Pierre Aguilar

E print: arXiv:cond-mat/9707042

 Jean-Philippe Bouchaud, Marc Potters, Jean-Pierre Aguilar

When the available statistical information is imperfect, it is dangerous to follow standard optimisation procedures to construct an optimal portfolio, which usually leads to a strong concentration of the weights on very few assets. We propose a new way, based on generalised entropies, to ensure a minimal degree of diversification.

Option pricing in the presence of extreme fluctuations

Date: January 1997

Authors: Jean-Philippe Bouchaud, Didier Sornette, Marc Potters

We discuss recent evidence that B. Mandelbrot's proposal to model market fluctuations as a Lévy stable process is adequate for short enough time scales, crossing over to a Brownian walk for larger time scales. We show how the reasoning of Black and Scholes should be extended to price and hedge options in the presence of these 'extreme' fluctuations. A comparison between theoretical and experimental option prices is also given.

Exponential Weighting and Random-Matrix-Theory-Based Filtering of Financial Covariance Matrices for Portfolio Optimization

Date: February 2004

Authors: Szilard Pafka, Marc Potters, Imre Kondor

E print: arXiv:cond-mat/0402573

We introduce a covariance matrix estimator that both takes into account the heteroskedasticity of financial returns (by using an exponentially weighted moving average) and reduces the effective dimensionality of the estimation (and hence measurement noise) via techniques borrowed from random matrix theory. We calculate the spectrum of large exponentially weighted random matrices (whose upper band edge needs to be known for the implementation of the estimation) analytically, by a procedure analogous to that used for standard random matrices. Finally, we illustrate, on empirical data, the superiority of the newly introduced estimator in a portfolio optimization context over both the method of exponentially weighted moving averages and the uniformly-weighted random-matrix-theory-based filtering.

Statistical models for company growth

Date: October 2002

Authors: Matthieu Wyart, Jean-Philippe Bouchaud

Published in: Physica A, Elsevier, 2003, 326, pp.241-255

We study Sutton's microcanonical' model for the internal organisation of firms, that leads to non trivial scaling properties for the statistics of growth rates. We show that the growth rates are asymptotically Gaussian in this model, at variance with empirical results. We also obtain the conditional distribution of the number and size of sub-sectors in this model. We formulate and solve an alternative model, based on the assumption that the sector sizes follow a power-law distribution. We find in this new model both anomalous scaling of the variance of growth rates and non Gaussian asymptotic distributions. We give some testable predictions of the two models that would differentiate them further. We also discuss why the growth rate statistics at the country level and at the company level should be identical.

Path dependent option pricing: the path integral partial averaging method

Date: May 2000

Author: Andrew Matacz

E print: arXiv:cond-mat/0005319

In this paper I develop a new computational method for pricing path dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying risk-neutral diffusion process. This result greatly eases the computational burden placed on the subsequent numerical evaluation. For short-medium term options it leads to a general approximation formula that only requires the evaluation of a one dimensional integral. I illustrate the application of the method to Asian options and occupation time derivatives.

Worst fluctuation method for fast value-at-risk estimates

Date: September 1999

Authors: Jean-Philippe Bouchaud, Marc Potters

E print: arXiv:cond-mat/9909245

We show how one can actually take advantage of the strongly non-Gaussian nature of the fluctuations of financial assets to simplify the calculation of the Value-at-Risk of complex non linear portfolios. The resulting equations are not hard to solve numerically, and should allow fast VaR and ΔVaR estimates of large portfolios, where {\it by construction} the influence of rare events is taken into account reliably. Our method can be seen as a correctly probabilized 'scenario' calculation (or 'stress-testing').

Rational Decisions, Random Matrices and Spin Glasses

Date: January 1998

Authors: Stefano Galluccio, Jean-Philippe Bouchaud, Marc Potters

E Print: arXiv:cond-mat/9801209

We consider the problem of rational decision making in the presence of nonlinear constraints. By using tools borrowed from spin glass and random matrix theory, we focus on the portfolio optimisation problem. We show that the number of ''optimal'' solutions is generically exponentially large: rationality is thus de facto of limited use. In addition, this problem is related to spin glasses with L\'evy-like (long-ranged) couplings, for which we show that the ground state is not exponentially degenerate.

Universality Classes for Extreme Value Statistics

Date: July 1997

Authors: Jean-Philippe Bouchaud, Marc Mezard

E print: arXiv:cond-mat/9707047

The low temperature physics of disordered systems is governed by the statistics of extremely low energy states. It is thus rather important to discuss the possible universality classes for extreme value statistics. We compare the usual probabilistic classification to the results of the replica approach. We show in detail that one class of independent variables corresponds exactly to the so-called one step replica symmetry breaking solution in the replica language. This universality class holds if the correlations are sufficiently weak. We discuss the relation between the statistics of extremes and the problem of Burgers turbulence in decay.

Random walks, liquidity molasses and critical response in financial markets

Date: June 2004

Authors: Jean-Philippe Bouchaud, Julien Kockelkoren, Marc Potters

E print: arXiv:cond-mat/0406224

Stock prices are observed to be random walks in time despite a strong, long term memory in the signs of trades (buys or sells). Lillo and Farmer have recently suggested that these correlations are compensated by opposite long ranged fluctuations in liquidity, with an otherwise permanent market impact, challenging the scenario proposed in Quantitative Finance 4, 176 (2004), where the impact is *transient*, with a power-law decay in time. The exponent of this decay is precisely tuned to a critical value, ensuring simultaneously that prices are diffusive on long time scales and that the response function is nearly constant. We provide new analysis of empirical data that confirm and make more precise our previous claims. We show that the power-law decay of the bare impact function comes both from an excess flow of limit order opposite to the market order flow, and to a systematic anti-correlation of the bid-ask motion between trades, two effects that create a 'liquidity molasses' which dampens market volatility.

Stiff Field Theory of Interest Rates and Psychological Future Time

Date: March 2004

Authors: Belal Baaquie, Jean-Philippe Bouchaud

E printarXiv:cond-mat/0403713

The simplest field theory description of the multivariate statistics of forward rate variations over time and maturities, involves a quadratic action containing a gradient squared rigidity term. However, this choice leads to a spurious kink (infinite curvature) of the normalized correlation function for coinciding maturities. Motivated by empirical results, we consider an extended action that contains a squared Laplacian term, which describes the bending stiffness of the FRC. With the extra ingredient of a 'psychological' future time, describing how the perceived time between events depends on the time in the future, our theory accounts extremely well for the phenomenology of interest rate dynamics.

Fluctuations and response in financial markets: the subtle nature of `random' price changes

Date: July 2003

Authors: Jean-Philippe Bouchaud, Yuval Gefen, Marc Potters, Matthieu Wyart

E print:arXiv:cond-mat/0307332

Using Trades and Quotes data from the Paris stock market, we show that the random walk nature of traded prices results from a very delicate interplay between two opposite tendencies: long-range correlated market orders that lead to super-diffusion (or persistence), and mean reverting limit orders that lead to sub-diffusion (or anti-persistence). We define and study a model where the price, at any instant, is the result of the impact of all past trades, mediated by a non constant 'propagator' in time that describes the response of the market to a single trade. Within this model, the market is shown to be, in a precise sense, at a critical point, where the price is purely diffusive and the average response function almost constant. We find empirically, and discuss theoretically, a fluctuation-response relation. We also discuss the fraction of truly informed market orders, that correctly anticipate short term moves, and find that it is quite small.

More statistical properties of order books and price impact

Date: October 2002

Authors: Marc Potters, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0210710

We investigate present some new statistical properties of order books. We analyse data from the Nasdaq and investigate (a) the statistics of incoming limit order prices, (b) the shape of the average order book, and (c) the typical life time of a limit order as a function of the distance from the best price. We also determine the 'price impact' function using French and British stocks, and find a logarithmic, rather than a power-law, dependence of the price response on the volume. The weak time dependence of the response function shows that the impact is, surprisingly, quasi-permanent, and suggests that trading itself is interpreted by the market as new information.

Statistical properties of stock order books: empirical results and models

Date: March 2002

Authors: Jean-Philippe Bouchaud, Marc Mezard, Marc Potters

E print: arXiv:cond-mat/0203511

We investigate several statistical properties of the order book of three liquid stocks of the Paris Bourse. The results are to a large degree independent of the stock studied. The most interesting features concern (i) the statistics of incoming limit order prices, which follows a power-law around the current price with a diverging mean; and (ii) the humped shape of the average order book, which can be quantitatively reproduced using a 'zero intelligence' numerical model, and qualitatively predicted using a simple approximation.

Introducing Variety in Risk Management

Date: July 2001

Authors: Fabrizio Lillo, Rosario N. Mantegna, Jean-Philippe Bouchaud, Marc Potters

E print: arXiv:cond-mat/0107208

We review the recently introduced concept of variety of a financial portfolio and we sketch its importance for risk control purposes. The empirical behaviour of variety, correlation, exceedance correlation and asymmetry of the probability density function of daily returns is discussed. The results obtained are compared with the ones of a one-factor model showing strengths and limitations of this model.

More stylized facts of financial markets: leverage effect and downside correlations

Date: January 2001

Authors: Marc Potters, Jean-Philippe Bouchaud

Published:  Physica A: Statistical Mechanics and its Applications - Volume 299, Issues 1–2, 1 October 2001, Pages 60–70

We discuss two more universal features of stock markets: the so-called leverage effect (a negative correlation between past returns and future volatility), and the increased downside correlations. For individual stocks, the leverage correlation can be rationalized in terms of a new ‘retarded’ model which interpolates between a purely additive and a purely multiplicative stochastic process. For stock indices a specific market panic phenomenon seems to be necessary to account for the observed amplitude of the effect. As for the increase of correlations in highly volatile periods, we investigate how much of this effect can be explained within a simple non-Gaussian one-factor description with time independent correlations. In particular, this one-factor model can explain the level and asymmetry of empirical exceedance correlations, which reflects the fat-tailed and negatively skewed distribution of market returns.

The leverage effect in financial markets: retarded volatility and market panic

Date: January 2001

Authors: Jean-Philippe Bouchaud, Andrew Matacz, Marc Potters

E print: arXiv:cond-mat/0101120

We investigate quantitatively the so-called leverage effect, which corresponds to a negative correlation between past returns and future volatility. For individual stocks, this correlation is moderate and decays exponentially over 50 days, while for stock indices, it is much stronger but decays faster. For individual stocks, the magnitude of this correlation has a universal value that can be rationalized in terms of a new 'retarded' model which interpolates between a purely additive and a purely multiplicative stochastic process. For stock indices a specific market panic phenomenon seems to be necessary to account for the observed amplitude of the effect.

Correlation structure of extreme stock returns

Date: June 2000

Authors: Pierre Cizeau, Marc Potters, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/0006034

It is commonly believed that the correlations between stock returns increase in high volatility periods. We investigate how much of these correlations can be explained within a simple non-Gaussian one-factor description with time independent correlations. Using surrogate data with the true market return as the dominant factor, we show that most of these correlations, measured by a variety of different indicators, can be accounted for. In particular, this one-factor model can explain the level and asymmetry of empirical exceedance correlations. However, more subtle effects require an extension of the one factor model, where the variance and skewness of the residuals also depend on the market return.

An empirical investigation of the forward interest rate term structure

Date: July 1999

Authors: Andrew Matacz, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/9907297

In this paper we study empirically the Forward Rate Curve (FRC) of 5 different currencies. We confirm and extend the findings of our previous investigation of the U.S. Forward Rate Curve. In particular, the average FRC follows a square-root law, with a prefactor related to the spot volatility, suggesting a Value-at-Risk like pricing. We find a striking correlation between the instantaneous FRC and the past spot trend over a certain time horizon, in agreement with the idea of an extrapolated trend effect. We present a model which can be adequately calibrated to account for these effects.

Apparent multifractality in financial time series

Date: June 1999

Authors: Jean-Philippe Bouchaud, Marc Potters, Martin Meyer

E print: arXiv:cond-mat/9906347

We present a exactly soluble model for financial time series that mimics the long range volatility correlations known to be present in financial data. Although our model is 'monofractal' by construction, it shows apparent multiscaling as a result of a slow crossover phenomenon on finite time scales. Our results suggest that it might be hard to distinguish apparent and true multifractal behavior in financial data. Our model also leads to a new family of stable laws for sums of correlated random variables.

Random matrix theory and financial correlations

Date: March 1999

Authors: Laurent Laloux, Pierre Cizeau, Jean-Philippe Bouchaud, Marc Potters

Published: International Journal of Theoretical and Applied Finance - Volume 03, Issue 03, July 2000

We show that results from the theory of random matrices are potentially of great interest when trying to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate financial time series. We find a remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). Finally, we give a specific example to show how this idea can be sucessfully implemented for improving risk management.

Noise Dressing of Financial Correlation Matrices

Date: October 1998

Authors: Laurent Laloux, Pierre Cizeau, Jean-Philippe Bouchaud, Marc Potters

E print: arXiv:cond-mat/9810255

We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of price fluctuations. The central result of the present study is the remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). In particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management.

Are financial crashes predictable?

Date: April 1998

Authors: Jean-Philippe Bouchaud, Nicolas Sagna, Rama Cont, Nicole El-Karoui and Marc Potters

E print: arXiv:cond-mat/9804111

We critically review recent claims that financial crashes can be predicted using the idea of log-periodic oscillations or by other methods inspired by the physics of critical phenomena. In particular, the October 1997 'correction' does not appear to be the accumulation point of a geometric series of local minima.

Are financial crashes predictable?

Date: April 1998

Authors: Jean-Philippe Bouchaud, Nicolas Sagna, Rama Cont, Nicole El-Karoui and Marc Potters

E print: arXiv:cond-mat/9804111

We critically review recent claims that financial crashes can be predicted using the idea of log-periodic oscillations or by other methods inspired by the physics of critical phenomena. In particular, the October 1997 'correction' does not appear to be the accumulation point of a geometric series of local minima.

Phenomenology of the interest rate curve

Date: December 1997

Authors: Jean-Philippe Bouchaud, Nicolas Sagna, Rama Cont, Nicole El-Karoui, Marc Potters

E print: arXiv:cond-mat/9712164

This paper contains a phenomenological description of the whole U.S. forward rate curve (FRC), based on an data in the period 1990-1996. We find that the average FRC (measured from the spot rate) grows as the square-root of the maturity, with a prefactor which is comparable to the spot rate volatility. This suggests that forward rate market prices include a risk premium, comparable to the probable changes of the spot rate between now and maturity, which can be understood as a 'Value-at-Risk' type of pricing. The instantaneous FRC however departs form a simple square-root law. The distortion is maximum around one year, and reflects the market anticipation of a local trend on the spot rate. This anticipated trend is shown to be calibrated on the past behaviour of the spot itself. We show that this is consistent with the volatility 'hump' around one year found by several authors (and which we confirm). Finally, the number of independent components needed to interpret most of the FRC fluctuations is found to be small. We rationalize this by showing that the dynamical evolution of the FRC contains a stabilizing second derivative (line tension) term, which tends to suppress short scale distortions of the FRC. This shape dependent term could lead, in principle, to arbitrage. However, this arbitrage cannot be implemented in practice because of transaction costs. We suggest that the presence of transaction costs (or other market 'imperfections') is crucial for model building, for a much wider class of models becomes eligible to represent reality.

Scaling in stock market data: stable laws and beyond

Date: May 1997

Authors: Rama Cont, Marc Potters, Jean-Philippe Bouchaud

E print: arXiv:cond-mat/9705087

The concepts of scale invariance, self-similarity and scaling have been fruitfully applied to the study of price fluctuations in financial markets. After a brief review of the properties of stable Levy distributions and their applications to market data we indicate the shortcomings of such models and describe the truncated Levy flight as an alternative model for price movements. Furthermore, studying the dependence structure of the price increments shows that while their autocorrelation function decreases rapidly to zero, the correlation of their squares and absolute values shows a slow power law decay, indicating persistence in the scale of fluctuations, a property which can be related to the anomalous scaling of the kurtosis. In the last section we review, in the light of these empirical facts, recent attempts to draw analogies between scaling in financial markets and in turbulent flows.

Linear models for the impact of order flow on prices II. The Mixture Transition Distribution model

Date: April 2016

Authors: Damian Eduardo Taranto, Giacomo Bormetti, Jean-Philippe Bouchaud, Fabrizio Lillo, Bence Toth

E print: arXiv:1604.07556

Modeling the impact of the order flow on asset prices is of primary importance to understand the behavior of financial markets. Part I of this paper reported the remarkable improvements in the description of the price dynamics which can be obtained when one incorporates the impact of past returns on the future order flow. However, impact models presented in Part I consider the order flow as an exogenous process, only characterized by its two-point correlations. This assumption seriously limits the forecasting ability of the model. Here we attempt to model directly the stream of discrete events with a so-called Mixture Transition Distribution (MTD) framework, introduced originally by Raftery (1985). We distinguish between price-changing and non price-changing events and combine them with the order sign in order to reduce the order flow dynamics to the dynamics of a four-state discrete random variable. The MTD represents a parsimonious approximation of a full high-order Markov chain. The new approach captures with adequate realism the conditional correlation functions between signed events for both small and large tick stocks and signature plots. From a methodological viewpoint, we discuss a novel and flexible way to calibrate a large class of MTD models with a very large number of parameters. In spite of this large number of parameters, an out-of-sample analysis confirms that the model does not overfit the data.

Why have asset price properties changed so little in 200 years

Date: May 2016

Authors: Jean-Philippe Bouchaud, Damien Challet

E print: arXiv:1605.00634

We first review empirical evidence that asset prices have had episodes of large fluctuations and been inefficient for at least 200 years. We briefly review recent theoretical results as well as the neurological basis of trend following and finally argue that these asset price properties can be attributed to two fundamental mechanisms that have not changed for many centuries: an innate preference for trend following and the collective tendency to exploit as much as possible detectable price arbitrage, which leads to destabilizing feedback loops.

Tail Protection for Long Investors: Convexity at Work

Date: May 2016

Authors: Jean-Philippe Bouchaud, Trung-Lam Dao, Cyril Deremble, Yves Lemperière, Trung-Tu Nguyen, Marc Potters

E print: http://ssrn.com/abstract=2777657

We relate the performance of trend following strategy to the difference between a long-term and a short-term variance. We show that this result is rather general, and holds for various definitions of the trend. We use this result to explain the positive convexity property of CTA performance and show that it is a much stronger effect than initially thought. This result also enable us to highlight interesting connections with Risk Parity portfolio.

Finally, we propose a new portfolio of options that gives us a pure exposure to the variance of the underlying, shedding some light on the link between trend and volatility, and also helping us understanding the exact role of hedging.

Alternative Beta Matters - 2016 Q2 Quarterly Report

Date: September 2016

Within this report we recap major developments of the quarter for Equities, Fixed Income / Credit, FX and Commodities, as well as Alternatives. All discussion is agnostic to particular approaches or techniques, and where alternative benchmark strategy results are presented, the exact methodology used is given.

We have also included one white paper ( In-Sample Overfitting - Avoiding the Pitfalls in Dataminig) and an extended academic abstract from a paper published during the quarter (The Excess Returns of Quality Stocks – a Behavioral Anomaly). Our hope is that these publications, which convey our views on topics related to Alternative Beta that have arisen in our many discussions with clients, can be used as a reference for our readers, and can stimulate conversations on these topical issues.

Dissecting cross-impact on stock markets: An empirical analysis

Date: September 2016

Authors: Michael Benzaquen, Iacopo Mastromatteo, Zoltan Eisler, Jean-Philippe Bouchaud

The vast majority of recent studies in market impact assess each product individually, and the interactions between their order flows are disregarded. This strong approximation may lead to an underestimation of trading costs and possible contagion effects. Transactions mediate a significant part of the interaction between different instruments. In turn, liquidity shares the sectorial structure of market correlations, which can be encoded as a set of eigenvalues and eigenvectors. We introduce a multivariate linear propagator model that successfully describes such a structure, and reproduces well the response and a significant fraction of the covariance matrix of returns. We explain in detail the various dynamical mechanisms that contribute to these quantities. We also define two simplified models with substantially less parameters to reduce overfitting, and show that they have superior out-of-sample performance.

Price impact without order book: A study of the OTC credit index market

Date: September 2016

Authors: Zoltan Eisler, Jean-Philippe Bouchaud

E print: http://arxiv.org/abs/1609.04620

We present a study of price impact in the over-the-counter credit index market, where no limit order book is used. Contracts are traded via dealers, that compete for the orders of clients. Despite this distinct microstructure, we successfully apply the propagator technique to estimate the price impact of individual transactions. Because orders are typically split less than in multilateral markets, impact is observed to be mainly permanent, in line with theoretical expectations. A simple method is presented to correct for errors in our classification of trades between buying and selling. We find a very significant, temporary increase in order flow correlations during late 2015 and early 2016, which we attribute to increased order splitting or herding among investors. We also find indications that orders advertised to less dealers may have lower price impact. Quantitative results are compatible with earlier findings in other more classical markets, further supporting the argument that price impact is a universal phenomenon, to a large degree independent of market microstructure.

Agnostic Risk Parity: Taming Known and Unknown-Unknowns

Date: October 2016

Authors Raphael Benichou, Yves Lempérière, Emmanuel Sérié, Julien Kockelkoren, Philip Seager, Jean-Philippe Bouchaud, Marc Potters
E print arXiv:1610.08818

Markowitz' celebrated optimal portfolio theory generally fails to deliver out-of-sample diversification. In this note, we propose a new portfolio construction strategy based on symmetry arguments only, leading to "Eigenrisk Parity" portfolios that achieve equal realized risk on all the principal components of the covariance matrix. This holds true for any other definition of uncorrelated factors. We then specialize our general formula to the most agnostic case where the indicators of future returns are assumed to be uncorrelated and of equal variance. This "Agnostic Risk Parity" (AGP) portfolio minimizes unknown-unknown risks generated by over-optimistic hedging of the different bets. AGP is shown to fare quite well when applied to standard technical strategies such as trend following.

Alternative Beta Matters - 2016 Q3 Quarterly Report

Date: September 2016

Within this report we recap major developments of the quarter for Equities, Fixed Income / Credit, FX and Commodities, as well as Alternatives. All discussion is agnostic to particular approaches or techniques, and where alternative benchmark strategy results are presented, the exact methodology used is given.

We have also included one white paper and an extended academic abstract from a paper produced during the quarter. Our hope is that these publications, which convey our views on topics related to Alternative Beta that have arisen in our many discussions with clients, can be used as a reference for our readers, and can stimulate conversations on these topical issues.

Trading Lightly: Cross-Impact and Optimal Portfolio Execution

Date: February 2017

Authors Iacopo Mastromatteo, Michael Benzaquen, Zoltan Eisler, Jean-Philippe Bouchaud
E print https://arxiv.org/abs/1702.03838

We model the impact costs of a strategy that trades a basket of correlated instruments, by extending to the multivariate case the linear propagator model previously used for single instruments. Our specification allows us to calibrate a cost model that is free of arbitrage and price manipulation. We illustrate our results using a pool of US stocks and show that neglecting cross-impact effects leads to an incorrect estimation of the liquidity and suboptimal execution strategies. We show in particular the importance of synchronizing the execution of correlated contracts.

Alternative Beta Matters - 2017 Q1 Newsletter

Date: March 2017

Within this report we recap major developments of the previous quarter for Equities, Fixed Income / Credit, FX and Commodities, as well as Alternatives. All discussion is agnostic to particular approaches or techniques, and where alternative benchmark strategy results are presented, the exact methodology used is given.
We have also included one white paper and an extended academic abstract from a paper produced during the quarter. Our hope is that these publications, which convey our views on topics related to Alternative Beta that have arisen in our many discussions with clients, can be used as a reference for our readers, and can stimulate conversations on these topical issues.

Alternative Beta Matters - 2017 Q2 Newsletter

Date: May 2017

Within this report we recap major developments of the quarter for Equities, Fixed Income / Credit, FX and Commodities, as well as Alternatives. All discussion is agnostic to particular approaches or techniques, and where alternative benchmark strategy results are presented, the exact methodology used is given.

We have also included one white paper and an extended academic abstract from a paper published during the quarter. Our hope is that these publications, which convey our views on topics related to Alternative Beta that have arisen in our many discussions with clients, can be used as a reference for our readers, and can stimulate conversations on these topical issues.

Financial crises: how CFM navigates the markets

Date: June 2017

In a world that feels particularly uncertain, CFM believes that crises are the norm - there were 431 financial crises from 1970 to 2011 (IMF) - and we design portfolios to take them into account.

Asset management firms must adapt or die

Date: June 2017

Data and technology are increasingly seen as enablers of the development, analysis and execution of investment strategies. The asset management firms of the future must invest in the right human capital to manage this trend.

The "Size Premium" in Equity Markets: Where is the Risk?

Date: August 2017

Authors Stefano Ciliberti, Emmanuel Sérié, Guillaume Simon, Yves Lempérière, Jean-Philippe Bouchaud
E print https://arxiv.org/abs/1708.00644

We find that when measured in terms of dollar-turnover, and once β-neutralised and Low-Vol neutralised, the Size Effect is alive and well. With a long term t-stat of 5.1, the "Cold-Minus-Hot" (CMH) anomaly is certainly not less significant than other well-known factors such as Value or Quality. As compared to market-cap based SMB, CMH portfolios are much less anti-correlated to the Low-Vol anomaly. In contrast with standard risk premia, size-based portfolios are found to be virtually unskewed. In fact, the extreme risk of these portfolios is dominated by the large cap leg; small caps actually have a positive (rather than negative) skewness. The only argument that favours a risk premium interpretation at the individual stock level is that the extreme drawdowns are more frequent for small cap/turnover stocks, even after accounting for volatility. This idiosyncratic risk is however clearly diversifiable.

Alternative Beta Matters - 2017 Q3 Newsletter

Date: July 2017

Within this report we recap major developments of the quarter for Equities, Fixed Income / Credit, FX and Commodities, Alternative Benchmark Performance as well as Trading Regulations. All discussion is agnostic to particular approaches or techniques, and where alternative benchmark strategy results are presented, the exact methodology used is given.

We have also included one white paper and an extended academic abstract from a paper published during the quarter. Our hope is that these publications, which convey our views on topics related to Alternative Beta that have arisen in our many discussions with clients, can be used as a reference for our readers, and can stimulate conversations on these topical issues.

You are in a drawdown. When should you start worrying?

Date: July 2017

Authors Adam Rej, Philip Seager, Jean-Philippe Bouchaud
E print https://arxiv.org/abs/1707.01457

Trading strategies that were profitable in the past often degrade with time. Since unlucky streaks can also hit "healthy" strategies, how can one detect that something truly worrying is happening? It is intuitive that a drawdown that lasts too long or one that is too deep should lead to a downward revision of the assumed Sharpe ratio of the strategy. In this note, we give a quantitative answer to this question based on the exact probability distributions for the length and depth of the last drawdown for upward drifting Brownian motions. We also point out that both managers and investors tend to underestimate the length and depth of drawdowns consistent with the Sharpe ratio of the underlying strategy.

Universal scaling and nonlinearity of aggregate price impact in financial markets

Date: June 2017

Authors Felix Patzelt, Jean-Philippe Bouchaud
E print  https://arxiv.org/abs/1706.04163

How and why stock prices move is a centuries-old question still not answered conclusively. More recently, attention shifted to higher frequencies, where trades are processed piecewise across different timescales. Here we reveal that price impact has a universal non-linear shape for trades aggregated on any intra-day scale. Its shape varies little across instruments, but drastically different master curves are obtained for order-volume and -sign impact. The scaling is largely determined by the relevant Hurst exponents. We further show that extreme order flow imbalance is not associated with large returns. To the contrary, it is observed when the price is "pinned" to a particular level. Prices move only when there is sufficient balance in the local order flow. In fact, the probability that a trade changes the mid-price falls to zero with increasing (absolute) order-sign bias along an arc-shaped curve for all intra-day scales. Our findings challenge the widespread assumption of linear aggregate impact. They imply that market dynamics on all intra-day timescales are shaped by correlations and bilateral adaptation in the flows of liquidity provision and taking.

The short-term price impact of trades is universal

Date: February 2017

Authors Bence Toth, Zoltan Eisler, Jean-Philippe Bouchaud
E print https://arxiv.org/abs/1702.08029

We analyze a proprietary dataset of trades by a single asset manager, comparing their price impact with that of the trades of the rest of the market. In the context of a linear propagator model we find no significant difference between the two, suggesting that both the magnitude and time dependence of impact are universal. This result is important as optimal execution policies often rely on propagators calibrated on anonymous data. We also find evidence that in the wake of a trade the order flow of other market participants first adds further copy-cat trades enhancing price impact on very short time scales. The induced order flow then quickly inverts, thereby contributing to impact decay.

Nonlinear price impact from linear models

Date: August 2017

Authors Felix Patzelt, Jean-Philippe Bouchaud
E print http://lanl.arxiv.org/abs/1708.02411

The impact of trades on asset prices is a crucial aspect of market dynamics for academics, regulators and practitioners alike. Recently, universal and highly nonlinear master curves were observed for price impacts aggregated on all intra-day scales [1]. Here we investigate how well these curves, their scaling, and the underlying return dynamics are captured by linear "propagator" models. We find that the classification of trades as price-changing versus non-price-changing can explain the price impact nonlinearities and short-term return dynamics to a very high degree. The explanatory power provided by the change indicator in addition to the order sign history increases with increasing tick size. To obtain these results, several long-standing technical issues for model calibration and -testing are addressed. We present new spectral estimators for two- and three-point cross-correlations, removing the need for previously used approximations. We also show when calibration is unbiased and how to accurately reveal previously overlooked biases. Therefore, our results contribute significantly to understanding both recent empirical results and the properties of a popular class of impact models.

A fractional reaction-diffusion description of supply and demand

Date: April 2017

Authors Michael Benzaquen, Jean-Philippe Bouchaud
E print https://arxiv.org/abs/1704.02638

We suggest that the broad distribution of time scales in financial markets could be a crucial ingredient to reproduce realistic price dynamics in stylised Agent-Based Models. We propose a fractional reaction-diffusion model for the dynamics of liquidity in financial markets, where agents are very heterogeneous in terms of their characteristic frequencies. Several features of our model are amenable to an exact analytical treatment. We find in particular that the impact is a concave function of the transacted volume (aka the "square-root impact law"), as in the normal diffusion limit. However, the impact kernel decays as t−β with β=1/2 in the diffusive case, which is inconsistent with market efficiency. In the sub-diffusive case the decay exponent β takes any value in [0,1/2], and can be tuned to match the empirical value β≈1/4. Numerical simulations confirm our theoretical results. Several extensions of the model are suggested.

On a multi-timescale statistical feedback model for volatility fluctuations

Date: July 2005

Authors: Lisa Borland, Jean-Philippe Bouchaud
E-print: https://arxiv.org/abs/physics/0507073

We study, both analytically and numerically, an ARCH-like, multiscale model of volatility, which assumes that the volatility is governed by the observed past price changes on different time scales. With a power-law distribution of time horizons, we obtain a model that captures most stylized facts of financial time series: Student-like distribution of returns with a power-law tail, long-memory of the volatility, slow convergence of the distribution of returns towards the Gaussian distribution, multifractality and anomalous volatility relaxation after shocks. At variance with recent multifractal models that are strictly time reversal invariant, the model also reproduces the time assymmetry of financial time series: past large scale volatility influence future small scale volatility. In order to quantitatively reproduce all empirical observations, the parameters must be chosen such that our model is close to an instability, meaning that (a) the feedback effect is important and substantially increases the volatility, and (b) that the model is intrinsically difficult to calibrate because of the very long range nature of the correlations. By imposing the consistency of the model predictions with a large set of different empirical observations, a reasonable range of the parameters value can be determined. The model can easily be generalized to account for jumps, skewness and multiasset correlations.

Demystifying smart and alternative beta

Date: August 2017

Published in Pensions&Investments on August 11, 2017 - This article seeks to clarify the two categories of Smartbeta and Alternative Beta with respect to investment objectives, implementation and fees. The comparative analysis focuses on equity-based strategies, but the arguments presented are asset class agnostic. Ultimately, this comparison intends to help investors make well-informed investment decisions.

Deconstructing the Low-Vol Anomaly

Date: October 2015

Authors: S. Ciliberti, Y. Lempérière, A. Beveratos, G. Simon, L. Laloux, M. Potters, J.-P. Bouchaud

E print: http://arxiv.org/abs/1510.01679

We study several aspects of the so-called low-vol and low-beta anomalies, some already documented (such as the universality of the effect over different geographical zones), others hitherto not clearly discussed in the literature. Our most significant message is that the low-vol anomaly is the result of two independent effects. One is the striking negative correlation between past realized volatility and dividend yield. Second is the fact that ex-dividend returns themselves are weakly dependent on the volatility level, leading to better risk-adjusted returns for low-vol stocks. This effect is further amplified by compounding. We find that the low-vol strategy is not associated to short term reversals, nor does it qualify as a Risk-Premium strategy, since its overall skewness is slightly positive. For practical purposes, the strong dividend bias and the resulting correlation with other valuation metrics (such as Earnings to Price or Book to Price) does make the low-vol strategies to some extent redundant, at least for equities.

Rotational invariant estimator for general noisy matrices

Date: February 2015

Authors Joël Bun, Romain Allez, Jean-Philippe Bouchaud, Marc Potters

We investigate the problem of estimating a given real symmetric signal matrix C from a noisy observation matrix M in the limit of large dimension. We consider the case where the noisy measurement M comes either from an arbitrary additive or multiplicative rotational invariant perturbation. We establish, using the Replica method, the asymptotic global law estimate for three general classes of noisy matrices, significantly extending previously obtained results. We give exact results concerning the asymptotic deviations (called overlaps) of the perturbed eigenvectors away from the true ones, and we explain how to use these overlaps to "clean" the noisy eigenvalues of M. We provide some numerical checks for the different estimators proposed in this paper and we also make the connection with some well known results of Bayesian statistics.

Monetary Policy and Dark Corners in a stylized Agent-Based Model

Date: January 2015

Authors Stanislao Gualdi, Marco Tarzia, Francesco Zamponi, Jean-Philippe Bouchaud

We generalise the stylised macroeconomic Agent-Based model introduced in "Tipping Points in Macroeconomic Agent Based Models" [JEDC 50, 29-61 (2015)], with the aim of investigating the role and efficacy of monetary policy of a 'Central Bank', that sets the interest rate such as to steer the economy towards a prescribed inflation and unemployment level. Our major finding is that provided its policy is not too aggressive (in a sense detailed in the paper) the Central Bank is successful in achieving its goals. However, the existence of different equilibrium states of the economy, separated by phase boundaries (or "dark corners"), can cause the monetary policy itself to trigger instabilities and be counter-productive. In other words, the Central Bank must navigate in a narrow window: too little is not enough, too much leads to instabilities and wildly oscillating economies. This conclusion strongly contrasts with the prediction of DSGE models.