Discrete Scaling in Stock Markets Before Crashes

We propose a picture of stock market crashes as critical points in a hierachical system with discrete scaling. The critical exponent is then complex, leading to log-periodic fluctuations in stock market indexes. We present “experimental” evidence in favor of this prediction. This picture is in the spirit of the known earthquake-stock market analogy and of recent work on log-periodic fluctuations associated with earthquakes. The study of earthquakes as critical points has been of interest for some time now [1, 2, 3, 4, 5]. At a critical point one expects a scaling regime to set in. Recently it has been suggested [6, 7] that the underlying scale invariance is discrete, as expected for a hierarchical system. Then the critical exponent is complex and the scaling law near the critical point is “decorated” by logperiodic corrections (Rτ = τ cos(ω log τ)) Evidence for such log-periodic fluctuations was found [6] in measurements of the concentration of Cl− and SO−− 4 ions in mineral water collected over the 20 months immediately preceeding the 1995 Kobe earthquake at a source close to its epicenter. Similar evidence was also found [7] in the cumulative Benioff strain in connection with the 1989 Loma Prieta earthquake. It was proposed [6] that monitoring log-periodic fluctuations may ultimately prove useful in earthquake prediction. Such fluctuations seem generic in hierarchically organized rupture processes. In the spirit of an earthquake-stock market analogy, this has led us to consider the possibility that log-periodic fluctuations may appear in stock market indices over a period preceding a crash. The stock market index (S&P 500, Dow-Jones, NIKKEI, ...) is to play here the same role as the Cl− ion concentration played in the the analysis of the Kobe earthquake. Fortunately such indices are closely monitored and good data are plentiful. The scaling