Herd Behavior and Fat Tails in Financial Markets

Abstract

This paper demonstrates that a generic herd behavior model generates a fat-tailed distribution of traders' aggregate actions. We consider a simultaneous-move game of traders who infer other traders' private information on the value of assets by observing their actions and decide whether to buy the asset or not. The number of buying actions in a Bayesian Nash equilibrium is characterized by a sum of a binomial process by introducing a fictitious tatonnement. Under a broad class of distributions for the private information, we show that the aggregate actions follow a power-law distribution with an exponential truncation. The empirical distribution of the daily returns of S&P 500 stocks is fitted by the model prediction, when the aggregate actions are translated into price movements either by an empirical volume-price impact function or by a market-maker who sets the price by incorporating the available information. This model nests the benchmark herd behavior model and the recent models of critical phenomena in the network of traders. The latter showed that the aggregate actions follow a power-law tailed distribution when the connectivity of networked traders is set at a critical level. In this context, we provide an economic reason why at all the rational herding behavior exhibits criticality in a general setting. Suppose that a good private information leads a trader to buy, whereas the other traders do not buy despite their observation of the action. Then their inactions reveal their private information partially. The total impact of the action on the revealed information is thus of order 1/N, where N is the total number of traders, if the private information is equally informative across the traders. When this is the case, the mean impact of the initial action on the other actions is roughly equal to one. The tatonnement triggered by the initial action becomes a martingale, in which the distribution of the total number of buying actions during the tatonnement exhibits a power-law tail. We further show that, when the static game is repeated over time, the triggering action almost surely occurs and the mean impact of the action in the chain reaction evolves toward the critical level. This implies that the rational learning of traders self-organizes their beliefs to the critical state at which a power-law clustering of actions emerges.