Bending, Stretching, and Smiling: Entropic Homeomorphisms of the Faces of Finance

Abstract

Concepts from information theory are utilized to study the effects of entropy on the behavior of finance systems and variables of interest. From this analysis, a common entropic measure was derived that determines the structure and evolution of a wide variety of financial topologies. This information theoretic variable is defined as the ratio of the arrival and processing rates of information at financial markets (R/C). Changes in R/C have intuitive effects on the distribution of returns, the volatility surface, and the yield curve. As the level and variance of (R/C) evolves so does the amount of uncertainty in each of the corresponding markets. By utilizing R/C the simplistic but often unrealistic shapes from financial theory: the normal distribution of returns, constant volatility across option strikes and terms to expiration, and the upward sloping yield curve are transformed into the complex and evolving structures of financial reality. The central role of this ratio elucidates a previously unknown and important connection between the disparate branches of finance.

Keywords: Shannon entropy; yield curve; volatility; Cauchy distribution; phase transition; implied volatility; volatility skew; volatility smile

JEL Classification: G1, G10, G12, G14, G170, D80, D84, E43