Economic Fluctuations and Statistical Physics: Quantifying Extremely Rare and Much Less Rare Events

Recent analysis of truly huge quantities of empirical data suggests that classic economic theories not only fail for a few outliers, but that there occur similar outliers of every possible size. In fact, if one analyzes only a small data set (say $10^4$ data points), then outliers appear to occur as ``rare events.'' However, when we analyze orders of magnitude more data ($10^8$ data points!), we find orders of magnitude more outliers---so ignoring them is not a responsible option, and studying their properties becomes a realistic goal. We find that the statistical properties of these ``outliers'' are identical to the statistical properties of everyday fluctuations. For example, a histogram giving the number of fluctuations of a given magnitude $ x$ for fluctuations ranging in magnitude from everyday fluctuations to extremely rare fluctuations that occur with a probability of only $10^{-8}$ is a perfect straight line in a double-log plot. Quantitative analogies between financial fluctuations and earthquakes will be discussed. Two unifying principles that underlie much of the finance analysis we will present are scale invariance and universality [R. N. Mantegna and H. E. Stanley, {\it Introduction to Econophysics: Correlations \& Complexity in Finance\/} (Cambridge U. Press, 2000)]. Scale invariance is a property not about algebraic equations but rather about functional equations, which have as their solutions not numbers but rather functional forms. The key idea of universality is that the identical set of laws hold across diverse markets, and over diverse time periods. This work was carried out in collaboration with a number of students and colleagues, chief among whom are X. Gabaix (MIT and Princeton) and V. Plerou (Boston University).