Editing Benoit Mandelbrot (section)

===Randomness and fractals in financial markets===

Mandelbrot saw [[financial market]]s as an example of "wild randomness", characterized by concentration and long-range dependence. He developed several original approaches for modelling financial fluctuations.<ref>{{cite encyclopedia |last1=Cont |first1=Rama |title=Mandelbrot, Benoit |encyclopedia=Encyclopedia of Quantitative Finance |date=15 May 2010 |pages=eqf01006 |doi=10.1002/9780470061602.eqf01006|isbn = 9780470057568}}</ref> In his early work, he found that the price changes in [[financial market]]s did not follow a [[Gaussian distribution]], but rather [[Paul Lévy (mathematician)|Lévy]] [[stable distributions]] having infinite [[variance]]. He found, for example, that cotton prices followed a Lévy stable distribution with parameter ''α'' equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger [[scale parameter]].<ref>{{cite web |url=https://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |title=  New Scientist'', 19 April 1997 |publisher=Newscientist.com |date=19 April 1997 |access-date=17 October 2010 |archive-date=21 April 2010 |archive-url= https://web.archive.org/web/20100421101729/http://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |url-status=live }}</ref> The latter work from the early 60s was done with daily data of cotton prices from 1900, long before he introduced the word 'fractal'. In later years, after the concept of fractals had matured, the study of financial markets in the context of fractals became possible only after the availability of high frequency data in finance.

In the late 1980s, Mandelbrot used intra-daily tick data supplied by Olsen & Associates in Zurich<ref>{{Cite journal |last= Davidson |first= Clive |date= 15 December 1997 |title= Wildly Random Market Moves |url= https://www.joc.com/wildly-random-market-moves_19971215.html |journal= Journal of Commerce |via= JOC.com |access-date=  |archive-date= 11 July 2021 |archive-url= https://web.archive.org/web/20210711170709/https://www.joc.com/wildly-random-market-moves_19971215.html |url-status= dead }}</ref><ref>{{Cite web| last= Muldoon| first= Oliver |date= 14 October 2019|title=The Wandering Scientist Turned Father of Fractals| url=https://medium.com/swlh/the-wandering-scientist-turned-father-of-fractals-4dcdc867d4dd|access-date=19 March 2021| website= Medium.com |language=en}}</ref> to apply fractal theory to market microstructure. This cooperation led to the publication of the first comprehensive papers on scaling law in finance.<ref>{{Cite journal| last1= Müller| first1= Ulrich A.|last2=Dacorogna|first2=Michel M.|last3=Olsen|first3=Richard B.|last4=Pictet|first4=Oliver V.|last5=Schwarz|first5=Matthias|last6=Morgenegg|first6=Claude|date=Dec 1990|title=Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis|url=https://doi.org/10.1016/0378-4266(90)90009-Q|journal=Journal of Banking and Finance|volume=14|issue=6|pages=1189–1208| doi=10.1016/0378-4266(90)90009-Q|via=Elsevier Science Direct}}</ref><ref>{{Cite journal|last1=Müller|first1=U. A.|last2=Dacorogna|first2=M. M.|last3=Davé|first3=R. D.|last4=Pictet|first4=O. V.| last5= Olsen| first5= R. B.|last6=Ward|first6=J. R.|date=28 June 1995|title=FRACTALS AND INTRINSIC TIME – A CHALLENGE TO ECONOMETRICIANS| journal= Opening Lecture of the XXXIXth International Conference of the Applied Econometrics Association |citeseerx=10.1.1.197.2969}}</ref> This law shows similar properties at different time scales, confirming Mandelbrot's insight of the fractal nature of market microstructure. Mandelbrot's own research in this area is presented in his books ''Fractals and Scaling in Finance''<ref>{{Cite book| last= Mandelbrot| first= Benoit|title=Fractals and Scaling in Finance|publisher=Springer|year=1997|isbn=978-1-4757-2763-0}}</ref> and ''The (Mis)behavior of Markets''.<ref>{{Cite book| last= Mandelbrot |first= Benoit| title= The (Mis)behavior of Markets|publisher=Profile Books|year=2004|isbn=9781861977656}}</ref>