Mathematical theory and stock market crashes

The mathematical characterisation of stock market movements has been a subject of intense interest. The conventional assumption has been that stock markets behave according to a random Gaussian or "normal" distribution.[23][24] Among others, mathematician Benoît Mandelbrot suggested as early as 1963 that the statistics prove this assumption incorrect.Mandelbrot observed that large movements in prices (i.e. crashes) are much more common than would be predicted in a normal distribution. Mandelbrot and others suggest that the nature of market moves is generally much better explained using non-linear analysis and concepts of chaos theory. This has been expressed in non-mathematical terms by George Soros in his discussions of what he calls reflexivity of markets and their non-linear movement.

Research at the Massachusetts Institute of Technology suggests that there is evidence the frequency of stock market crashes follows an inverse cubic power law.This and other studies such as Prof. Didier Sornette's work suggest that stock market crashes are a sign of self-organized criticality in financial markets. In 1963, Mandelbrot proposed that instead of following a strict random walk, stock price variations executed a Lévy flight. A Lévy flight is a random walk that is occasionally disrupted by large movements. In 1995, Rosario Mantegna and Gene Stanley analyzed a million records of the S&P 500 market index, calculating the returns over a five year period.Their conclusion was that stock market returns are more volatile than a Gaussian distribution but less volatile than a Lévy flight.

Researchers continue to study this theory, particularly using computer simulation of crowd behaviour, and the applicability of models to reproduce crash-like phenomena.