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 α+iω = τ α 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 1 variable is again time t. Call c(t) the index as a function of time. Truncating at the first harmonic of a general log-periodic correction, we can then write for c(t) the same formula as that given in [6] for the ion concentrationAs a first test of this idea let us consider the crash which occured in New York on October 19, 1987. As the relevant index let us use the S&P 500, which dropped by more than 20% that day. In figure 1a we present a fit of the 1986-1987 weekly S&P 500 using Eq. (1) . One can see clearly two full periods of the log-periodic oscillation and some more oscillatory behavior close to the time of the crash. We only fit data up to three weeks before the crash, where the fit starts very fast oscillations. A reasonable fit is obtained this way with parameters given in Table 1 (where we omitted the parameters A, B and ϕ, which depend on the arbitrary normalization of the index or on the time scale).Error bars of ±10 were assigned to each data point for purposes of calculating χ 2 . To a certain extent this is arbitrary, but it also reflects the possibility of higher harmonics neglected in our fit and of noise. This error assignment will be used...

No abstract

Motivated by the hypothesis that financial crashes are macroscopic examples of critical phenomena associated with a discrete scaling symmetry, we reconsider the evidence of log-periodic precursors to financial crashes and test the prediction that log-periodic oscillations in a financial index are embedded in the mean function of this index. In particular, we examine the first differences of the logarithm of the S&P 500 prior to the October 87 crash and find the log-periodic component of this time series is not statistically significant if we exclude the last year of data before the crash. We also examine the claim that two separate mechanisms are needed to explain the frequency distribution of draw downs in the S&P 500 and find the evidence supporting this claim to be unconvincing.

A puzzle in consumption theory is the observation of a hump in age-consumption profiles. This paper studies a general equilibrium life-cycle economy with capital in which households include both consumption and leisure in their period utility function. A calibrated version of the model shows that a significant hump in life-cycle consumption is a feature of the equilibrium. Thus inclusion of leisure in household preferences may provide part of the explanation of observed lifecycle consumption humps.