Large financial crashes

We propose that large stock market crashes are analogous to critical points studied in statistical physics with log-periodic correction to scaling. We extend our previous renormalization group model of stock market prices prior to and after crashes (D. Sornette, A. Johansen, J.P. Bouchaud, J, Phys. I France 6 (1996) 167) by including the first non-linear correction, This predicts the existence of a log-frequency shift over time in the log-periodic oscillations prior to a crash. This is tested on the two largest historical crashes of the century, the October 1929 and October 1987 crashes, by fitting the stock market index over an interval of 8 yr prior to the crashes. The good quality of the fits, as well as the consistency of the parameter values obtained from the two crashes, promote the theory that crashes have their origin in the collective ''crowd'' behavior of many interacting agents.