  Keeping a basic tenet of economic theory, rational expectations, we model the
nonlinear positive feedback between agents in the stock market as an interplay
between nonlinearity and multiplicative noise. The derived hyperbolic
stochastic finite-time singularity formula transforms a Gaussian white noise
into a rich time series possessing all the stylized facts of empirical prices,
as well as accelerated speculative bubbles preceding crashes. We use the
formula to invert the two years of price history prior to the recent crash on
the Nasdaq (april 2000) and prior to the crash in the Hong Kong market
associated with the Asian crisis in early 1994. These complex price dynamics
are captured using only one exponent controlling the explosion, the variance
and mean of the underlying random walk. This offers a new and powerful
detection tool of speculative bubbles and herding behavior.
