  By combining (i) the economic theory of rational expectation bubbles, (ii)
behavioral finance on imitation and herding of investors and traders and (iii)
the mathematical and statistical physics of bifurcations and phase transitions,
the log-periodic power law (LPPL) model has been developed as a flexible tool
to detect bubbles. The LPPL model considers the faster-than-exponential (power
law with finite-time singularity) increase in asset prices decorated by
accelerating oscillations as the main diagnostic of bubbles. It embodies a
positive feedback loop of higher return anticipations competing with negative
feedback spirals of crash expectations. The power of the LPPL model is
illustrated by two recent real-life predictions performed recently by our
group: the peak of the Oil price bubble in early July 2008 and the burst of a
bubble on the Shanghai stock market in early August 2009. We then present the
concept of "negative bubbles", which are the mirror images of positive bubbles.
We argue that similar positive feedbacks are at work to fuel these accelerated
downward price spirals. We adapt the LPPL model to these negative bubbles and
implement a pattern recognition method to predict the end times of the negative
bubbles, which are characterized by rebounds (the mirror images of crashes
associated with the standard positive bubbles). The out-of-sample tests
quantified by error diagrams demonstrate the high significance of the
prediction performance.
