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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 toolto detect bubbles. The LPPL model considers the faster-than-exponential (powerlaw with finite-time singularity) increase in asset prices decorated byaccelerating oscillations as the main diagnostic of bubbles. It embodies apositive feedback loop of higher return anticipations competing with negativefeedback spirals of crash expectations. The power of the LPPL model isillustrated by two recent real-life predictions performed recently by ourgroup: the peak of the Oil price bubble in early July 2008 and the burst of abubble on the Shanghai stock market in early August 2009. We then present theconcept of "negative bubbles", which are the mirror images of positive bubbles.We argue that similar positive feedbacks are at work to fuel these accelerateddownward price spirals. We adapt the LPPL model to these negative bubbles andimplement a pattern recognition method to predict the end times of the negativebubbles, which are characterized by rebounds (the mirror images of crashesassociated with the standard positive bubbles). The out-of-sample testsquantified by error diagrams demonstrate the high significance of theprediction performance.

Diagnosis and Prediction of Tipping Points in Financial Markets: Crashes and Rebounds