Early warning indicator for financial crashes using the log periodic power law

Abstract

In this article, we apply the Log Periodic Power Law (LPPL), introduced by Johansen et al. (2000), for capturing the recent stock market crash in the German stock index (Deutscher Aktien Index, DAX). The contribution of this article consists not only in describing the historical crash by the LPPL, but also in demonstrating how the LPPL can be used as an early warning indicator for financial crashes.

Keywords:

• stock market crashes
• log periodic power law
• early warning

JEL Classification:

• C13
• G01

Acknowledgments

The views expressed in this article are those of the author and do not necessarily reflect the opinions of the Deutsche Bundesbank.

Notes

¹ See Ray (2011) for a survey on econophysics.

² See the survey paper by Bikhchandani and Sharma (2001) for details on herding behaviour.

³ See Gerdesmeier et al. (2010) for a typical example of econometric crash-forecasting models.

⁴ In a fully different and sophisticated setting, Sornette and Zhou (2006) also applied this method as an early warning indicator for financial crashes.

⁵ See also Brée and Joseph (2010) for the mathematical derivations of the LPPL from the crash-hazard rates.

⁶ The parameter κ ∈ (0, 1) is theoretically a rate of price drop during a crash, and it may be empirically quantified as differences between the bubble price and the fundamental value.

⁷ Johansen and Sornette (2002) reported that the values for α and ω are remarkably consistent for a large variety of speculative bubbles on different markets as α ≈ 0.67 ± 0.18 and ω ≈ 6.36 ± 1.56.

⁸ Ibid.

⁹ Ibid.

¹⁰ The use of RMSE is, to some extent, similar to the idea of Brée and Joseph (2010) who used the RMSE for determining the best fit among various parameter constellations within a sample.

¹¹ From that day on, a sequence of large drops was observed and the DAX fell from 7849.99 to 3666.41 by 6 March 2009.

¹² Lee (1998) classified market dynamics in four phases: boom, euphoria, trigger and panic phase.

¹³ Crashes could also occur at any time before the critical time as Johansen et al. (2000) emphasize. For this case, the market dynamics corresponds to the usual three-phase market dynamics of Evans (1991): boom, euphoria and panic phase (without a trigger phase, or rather, with probably a very short trigger phase, which is regarded as a (end)part of euphoria phase).

¹⁴ Abreu and Bunnermeier (2003, p. 174).