A mean-field model of financial markets: Reproducing long tailed distributions and volatility correlations

A model for financial market activity should reproduce the several stylized facts that have been observed to be invariant across different markets and periods. Here we present a mean-field model of agents trading a particular asset, where their decisions (to buy or to sell or to hold) is based exclusively on the price history. As there are no direct interactions between agents, the price (computed as a function of the net demand, i.e., the difference between the numbers of buyers and sellers at a given time) is the sole mediating signal driving market activity. We observe that this simple model reproduces the long-tailed distribution of price fluctuations (measured by logarithmic returns) and trading volume (measured in terms of the number of agents trading at a given instant), that has been seen in most markets across the world. By using a quenched random distribution of a model parameter that governs the probability of an agent to trade, we obtain quantitatively accurate exponents for the two distributions. In addition, the model exhibits volatility clustering, i.e., correlation between periods with large fluctuations, remarkably similar to that seen in reality. To the best of our knowledge, this is the simplest model that gives a quantitatively accurate description of financial market behavior.