Scaling in stock market data: stable laws and beyond

TL;DR

This paper reviews scale invariance in financial markets, discusses stable and truncated Levy models for price fluctuations, and explores the persistence of fluctuation scales, drawing analogies with turbulence.

Contribution

It introduces the truncated Levy flight as an alternative to stable Levy models and analyzes the dependence structure of price increments in financial data.

Findings

Stable Levy distributions fit market data but have shortcomings.

Truncated Levy flights better model price movements.

Autocorrelation of squared increments shows slow decay, indicating persistence.

Abstract

The concepts of scale invariance, self-similarity and scaling have been fruitfully applied to the study of price fluctuations in financial markets. After a brief review of the properties of stable Levy distributions and their applications to market data we indicate the shortcomings of such models and describe the truncated Levy flight as an alternative model for price movements. Furthermore, studying the dependence structure of the price increments shows that while their autocorrelation function decreases rapidly to zero, the correlation of their squares and absolute values shows a slow power law decay, indicating persistence in the scale of fluctuations, a property which can be related to the anomalous scaling of the kurtosis. In the last section we review, in the light of these empirical facts, recent attempts to draw analogies between scaling in financial markets and in turbulent flows.