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First price change distribution model


Inequality in price change distribution
Benoît Mandelbrot Mathematician
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So here we have a contrast that is not a contrast of a quantitative nature, but almost qualitative. On the one hand the model of Bachelier assumed continuous price change and a lack of inequality between price changes over different days. The reality forces us to envision a great degree of inequality, when a price changes drastically over a period of a year, one sensible persons begin by suspecting that probably within that year there was a small period - maybe of a day, maybe a month - where this price fluctuated extraordinarily highly. It was more or less at one level, up and down, then changed very much, and again was at one level. It didn't necessarily change in one swoop, maybe in several steps, but over a short period of time a very, very large price change occurred. That is completely contradictory with the idea of Bachelier's model in which price is very uniformly distributed. Well, the issue of inequality was studied on and off by many people between 1900 and the 1960s, I don't think very systematically. The issue of the variation of prices and whether the Bachelier model of Brownian motion and independent Gaussian implements applies, became very much an issue in the late '50s and very early '60s, when Bachelier was remembered, or in fact rediscovered by others and then remembered, and when I came up in this scene for the first time. My contribution to the problem of finance of price variation was in a certain sense to translate in terms of prices the reasoning I had been following in my study of inequality, and to allow the variance of prices to be infinite. I argued that variance is a quantity that for a physical system may be necessarily bounded because it may represent, for example, energy, which is bounded. The variance of price was not a physical, a concrete economic quantity. It was just a number, a number that was expected to characterise the variability, the volatility of prices, however well it did it. And so if one had a model in which prices varied according to Bachelier and to the Bachelier process, then the price change would be very, very equal and the invariability finite. But this need not be the case, and for reasons that I can explain but were largely accidental, a drawing on the blackboard of a colleague drew my attention to some properties of prices and made me suspect that I had something to do about it. I became very deeply immersed in that topic.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Daniel Zajdenweber Bernard Sapoval

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Duration: 3 minutes, 5 seconds

Date story recorded: May 1998

Date story went live: 29 September 2010