Can log-periodic power law structures arise from random fluctuations?

Recent research has established log-periodic power law (LPPL) patterns prior to the detonation of the German stock index (DAX) bubble in 1998. The purpose of this article is to explore whether a Langevin equation extracted from real world data can generate synthetic time series with comparable LPPL structures. To this end, we first estimate the stochastic process underlying the DAX log-returns during the period from mid-1997 until end-2003. The employed data set contains about 3.93⋅106 intraday DAX quotes at a sampling rate of 15 s. Our results indicate that the DAX log-returns can be described as a Markov process. As a consequence, a Langevin equation is derived. Based on this model equation, we run extensive simulations in order to generate 100 synthetic DAX trajectories each covering 3000 trading days. We find LPPL behavior in ten artificial time series. Moreover, we can establish a link between LPPL patterns and ensuing bubble bursts in seven synthetic 600-week windows. However, the LPPL components in most synthetic trajectories differ fundamentally from those LPPL structures that have previously been detected in real financial time series. Summarized, this paper demonstrates that LPPL structures are not necessarily the signature of imitative behavior among investors but can also stem from noise, even though the likelihood of this is extremely low. Thus, our findings confirm with high statistical confidence that the LPPL structures in the DAX development are rooted deeper than only in the random fluctuations of the German stock market.