Editing Stock market crash (section)

==Mathematical theory==
===Random walk theory===
The conventional assumption is that stock markets behave according to a random [[log-normal distribution]].<ref>{{cite book | last=Malkiel | first=Burton G. | author-link=Burton Malkiel | title=A Random Walk Down Wall Street | edition=6th | publisher=W.W. Norton & Company, Inc. | year=1973| isbn=978-0-393-06245-8}}</ref> This implies that the expected [[volatility (finance)|volatility]] is the same all the time. Among others, mathematician [[Benoit Mandelbrot]] suggested as early as 1963 that the statistics prove this assumption incorrect.<ref>{{cite web | url=http://www.bearcave.com/bookrev/misbehavior_of_markets.html | title=The (Mis)behavior of Markets | work=bearcave.com}}</ref> Mandelbrot observed that large movements in prices (i.e. crashes) are much more common than would be predicted from a log-normal distribution. Mandelbrot and others suggested that the nature of market moves is generally much better explained using [[non-linear]] analysis and concepts of [[chaos theory]].<ref>{{Cite web | url=http://news.mit.edu/2006/father-fractals-takes-stock-market | title='Father of Fractals' takes on the stock market | first=Katharine Stoel | last=Gammon | work=[[Massachusetts Institute of Technology]] | date=November 16, 2006}}</ref> 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.<ref>{{Cite book | url=https://archive.org/details/alchemyoffinance00geor_0 | url-access=registration | last=Soros | first=George | author-link=George Soros | title=Alchemy of Finance | publisher=[[Simon & Schuster]] | year=1988| isbn=9780671662387}}</ref> George Soros said in late October 1987, "Mr. [[Robert Prechter]]'s reversal proved to be the crack that started the avalanche."<ref>{{cite news | url=https://www.marketwatch.com/story/elliott-wave-adviser-now-aggressively-bearish-2009-11-25 | title=Short term vs. long-term | first=Mark | last=Hulbert | author-link=Mark Hulbert | work=[[Marketwatch]] | date=November 25, 2009}}</ref><ref>{{cite news | url=https://www.nytimes.com/2007/10/13/business/13speculate.html | work=[[The New York Times]] | title=The Man Who Won as Others Lost | first=Landon | last=Thomas Jr | date=October 13, 2007 | url-access=subscription}}</ref>

===Self-organized criticality===
Research at the [[Massachusetts Institute of Technology]] suggests that there is evidence that the frequency of stock market crashes follows an inverse cubic [[power law]].<ref>{{cite web | url=http://news.mit.edu/2003/powerlaw | title=Stock trade patterns could predict financial earthquakes | first=Xavier | last=Gavaix | work=[[Massachusetts Institute of Technology]] | date=May 14, 2003}}</ref> This and other studies such as [[Didier Sornette]]'s work suggest that stock market crashes are a sign of [[self-organized criticality]] in financial markets.<ref>{{cite journal | title=Self-Organized Percolation Model for Stock Market Fluctuations | first=Didier | last=Sornette | journal=Physica A: Statistical Mechanics and Its Applications | volume=271 | issue=3–4 | pages=496–506 | author-link=Didier Sornette | date=29 June 1999| arxiv=cond-mat/9906434 | doi=10.1016/S0378-4371(99)00290-3 | bibcode=1999PhyA..271..496S | s2cid=18641082 }}</ref>

===Lévy flight===
In 1963, Mandelbrot proposed that instead of following a strict [[random walk]], stock price variations executed a [[Lévy flight]].<ref>{{cite journal | url=https://ideas.repec.org/a/ucp/jnlbus/v36y1963p394.html | title=The Variation of Certain Speculative Prices | journal=The Journal of Business | first=Benoit | last=Mandelbrot | date=1 January 1963 | volume=36 | issue=[object Attr] | page=394 | via=RePEc – IDEAS | doi=10.1086/294632| url-access=subscription }}</ref> 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 Index]], calculating the returns over a five-year period.<ref>{{cite journal | title=Scaling behaviour in the dynamics of an economic index | first1=Rosario N. | last1=Mantegna | first2=H. Eugene | last2=Stanley | date=6 July 1995 | journal=Nature | volume=376 | issue=6535 | pages=46–49 | doi=10.1038/376046a0 | bibcode=1995Natur.376...46M| s2cid=4326594 }}</ref> Researchers continue to study this theory, particularly using [[computer simulation]] of crowd behavior, and the applicability of models to reproduce crash-like phenomena.

===Result of investor imitation===
In 2011, using statistical analysis tools of [[complex systems]], research at the [[New England Complex Systems Institute]] found that the panics that lead to crashes come from a dramatic increase in [[imitation]] among investors, which always occurred during the year before each market crash. When investors closely follow each other's cues, it is easier for panic to take hold and affect the market. This work is a mathematical demonstration of a significant advance warning sign of impending market crashes.<ref>{{cite arXiv | eprint=1102.2620 | title=Predicting economic market crises using measures of collective panic | first1=Dion | last1=Harmon | first2=Marcus A. M. | last2=de Aguiar | first3=David D. | last3=Chinellato | first4=Dan | last4=Braha | first5=Irving R. | last5=Epstein | first6=Yaneer | last6=Bar-Yam | date=13 February 2011 | class=q-fin.ST}}</ref><ref>{{cite magazine | url=https://www.wired.com/2011/03/market-panic-signs/ | title=Possible Early Warning Sign for Market Crashes | first=Brandon | last=Keim | magazine=[[Wired (magazine)|Wired]] | date=March 18, 2011}}</ref>