We present a simple model of a stock market where a random communication structure between agents generically gives rise to 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. Keywords: heavy tails, financial markets, herd behavior, market organization, intermittency, random graphs, percolation. JEL Classification number: C0, D49, G19 1 R. Cont gratefully acknowledges an AMX fellowship from Ecole Polytechnique (France) and thanks Science & Financ...

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 0 as the formation of bubbles and crashes, the influence of transaction costs and the distribution of agents wealth. 1 1

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 e#ects 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. The model leads to a specific "shape" of the falldown of the price during a crash, which we compare with the October 1987 data. 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.

We show, by studying in detail the market prices of options on liquid markets, that the market has empirically corrected the simple, but inadequate Black-Scholes formula to account for two important statistical features of asset fluctuations: "fat tails" and correlations in the scale of fluctuations. These aspects, although not included in the pricing models, are very precisely reflected in the price fixed by the market as a whole. Financial markets thus behave as rather efficient adaptive systems.

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. We show that real market data can be surprisingly well accounted for by these simple models. A well documented ‘stylized fact ’ of financial markets is volatility clustering [1, 2, 3, 4]. Figure 1 compares the time series of the daily returns of the Dow-Jones index since 1900 and that of a Brownian random walk. Two features are immediately obvious to the eye: the volatility does indeed 1 have rather strong intermittent fluctuations, and these fluctuations tend to persist in time. A more quantitative analysis shows that the daily volatility

Introduction Since the 1980s, the deterioration of the academic job market in physics has been attracting a large number of physicists to investment banks: many of them are now working as \quants", designing sophisticated new derivative products or developing numerically intensive data analysis techniques for price and volatility forecasting. More recently, several teams of physicists have launched their own rms, oering services in the elds of nancial software design and forecasting. There exists however another set of motivations { scientic ones { which have also been prompting theoretical physicists { especially those with a background in statistical physics { to become interested in nance. Although this phenomenon may seem a bit mysterious to the outsider, we will attempt to convince the reader that it is not: nancial markets may well be considered as objects of high potential interest for researchers in statistical physics. 1.1 Motivations Statist

We discuss several models in order to shed light on the origin of powerlaw 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. 1

We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first present data sources and discuss the choice of a time scale when constructing financial time series. Various statistical properties of asset returns are then described: distributional properties, tail analysis and extreme fluctuations, linear and non-linear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. The last part deals with interest rates: we present some issues encountered in constructing yield curves from empirical data and discuss the statistical properties of the term structure fluctuations.

Over the last few years there has been a surge of activity within the physics community in the emerging field of Econophysics -- the study of economic systems from a physicist's perspective. Physicists tend to take a different view than economists and other social scientists, being interested in such topics as phase transitions and fluctuations. In this dissertation two simple models of stock exchange...

In this paper we overview the pricing of several so-called exotic options in the nowdays quite popular exponential Lévy models.