Imprints of log-periodic self-similarity in the stock market

TL;DR

This paper investigates log-periodic structures as indicators of impending stock market crashes, demonstrating their presence in the German DAX index and other markets, supporting the idea of discrete scale-invariance in financial dynamics.

Contribution

It provides empirical evidence of log-periodic precursors to crashes in stock markets, highlighting a consistent scaling ratio and extending the concept to multiple market indices.

Findings

Log-periodic structures appeared before the 1998 DAX crash.

Similar oscillations observed in smaller market declines.

Consistent scaling ratio λ ≈ 2 across different systems.

Abstract

Detailed analysis of the log-periodic structures as precursors of the financial crashes is presented. The study is mainly based on the German Stock Index (DAX) variation over the 1998 period which includes both, a spectacular boom and a large decline, in magnitude only comparable to the so-called Black Monday of October 1987. The present example provides further arguments in favour of a discrete scale-invariance governing the dynamics of the stock market. A related clear log-periodic structure prior to the crash and consistent with its onset extends over the period of a few months. Furthermore, on smaller time-scales the data seems to indicate the appearance of analogous log-periodic oscillations as precursors of the smaller, intermediate decreases. Even the frequencies of such oscillations are similar on various levels of resolution. The related value $\lambda \approx 2$ of preferred scaling ratios is amazingly consistent with those found for a wide variety of other complex systems. Similar analysis of the major American indices between September 1998 and February 1999 also provides some evidence supporting this concept but, at the same time, illustrates a possible splitting of the dynamics that a large market may experience.