# SFB 303 Discussion Paper No. B - 458

**Author**: Lux, Thomas, and Didier Sornette

**Title**: On Rational Bubbles and Fat Tails

**Abstract**: This paper addresses the statistical properties of time series driven by rational bubbles a la Blanchard and Watson (1982). Using insights on the behavior of multiplicative stochastic processes, we demonstrate that the tails of the unconditional distribution emerging from such bubble processes follow power-laws (exhibit hyperbolic decline). More precisely, we find that rational bubbles predict a fat power tail for both the bubble component and price differences with an exponent m smaller than 1. The distribution of returns is dominated by the same power-law over an extended range of large returns. Although power-law tails are a pervasive feature of empirical data, these numerical predictions are in disagreement with the usual empirical estimates. It, therefore, appears that exogenous rational bubbles are hardly reconcilable with some of the stylized facts of financial data at a very elementary level.

**Keywords**: rational bubbles, random difference equations, multiplicative processes,
rational bubbles, random difference equations, multiplicative processes, fat tails.

**JEL-Classification-Number**: G12, D84, C32

**Creation-Date**: October 1999

**URL**: ../1999/b/bonnsfb458.pdf
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