Bubbles, Jumps, and Scaling from Properly Anticipated Prices

Abstract

Abstract. Prices in financial markets exhibit extreme jumps far more often than can be accounted for by external news. Further, magnitudes of price changes are correlated over long times. These so called stylized facts are quantified by scaling laws similar to, for example, turbulent fluids. They are believed to reflect the complex interactions of heterogenous agents which give rise to irrational herding. Therefore, the stylized facts have been argued to provide evidence against the efficient market hypothesis which states that prices rapidly reflect available information and therefore are described by a martingale. Here we show, that in very simple bidding processes efficiency is not opposed to, but causative to scaling properties observed in real markets. Thereby, we link the stylized facts not only to price efficiency, but also to the economic theory of rational bubbles. We then demonstrate effects predicted from our normative model in the dynamics of groups of real human subjects playing a modified minority game. An extended version of the latter can be played online at seesaw.neuro.uni-bremen.de.

Quantification of market efficiency based on informational-entropy

“…Since there are no restrictions to the given information of head or tail, the conditional probabilities p(y = ′ t ′ |x = ′ t ′ ) and p(y = ′ h ′ |x = ′ h ′ ) are 16 The definition can be used for any information processing system where an informational entropy can be calculated (e.g. minority game [7] or the seesaw game [34]). 17 Alternatively, information could also be based on beliefs of the agent, which are correct with a certain probability p(x|y).…”