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The importance of the eye


Inequality and finance; differences between Bachelier and Mandelbrot
Benoît Mandelbrot Mathematician
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One must understand the extraordinary depth of difference of the predictions that Bachelier, and all my models, make about certain probabilities of risk. When Bachelier models would say that the probability of risk is one millionth, or perhaps one ten millionth, or perhaps one hundred thousandth -the detail doesn't count, it is certainly completely wrong. It suggests that some portfolios are riskless, that a miracle has been accomplished. In the case of the models which I proposed, either the '63 model or the '65 model, which had only dependence, or the more recent model, the probability of risk may be one twentieth or one tenth, one fifth - enormously larger. Therefore it is not a matter of seeing whether the models can be tested separately and their qualities compared statistically. What statistics catch is not the important reality, but some quantities selected for the purpose of statistics. The important differences are the ones that precisely indicate these measurements of risk. If one wants to measure risk one has no time to wait for economic theory to be established; no time to wait for even the mathematics of my models to be fully developed: it is prudent to take estimates of risks much higher than the estimates which are given by Bachelier. Now, what about other uses of these ideas? I think that quantitative economics, quantitative finance has its own needs, but you also must continue to an understanding of the functioning of economic phenomena. I am returning, in a certain sense, to the idea of reduction and to say what I would like to do is to use a description of data that is not a matter of curve fitting and millions of unrelated relations, but a matter of assuming scaling, scaling variants, etc., etc., of using that to understand the nature of economic systems. Now I would like to link the two parts of what I've been talking about in economics: the phenomenon of inequality and the phenomenon of finance. It is clear that they all belong to one economic system, and an economic system in which some parts will be subjected to deep inequality, like incomes, like size of firms, like sizes of cities, etc., etc., etc., when almost any quantity which is of great significance does not have a well defined typical value, but a very, very broad distribution of values; such properties would be quite unthinkable in a system in which they would coexist with Brownian motion that has no inequality whatsoever. Short of having a well constructed system in which each part affects the others, it is very important to characterise the economy, including finance, by a few rules which catch as much as possible of its substance, and then take those rules as building blocks on which an understanding of other phenomena might be based. Now if one sees variation of prices as they occur, there is a naive view of prices, which Bachelier certainly had very strongly and that many people follow, which is that each price change follows from a different cause, or causes. Now if that were the case it would be very difficult to imagine that any system other than the simplest one, namely other than Brownian motion or the Bachelier, would be applicable. Bachelier, in a certain sense, says that causes are very, very small. They are so small that when they add up together each of them becomes invisible. I don't think that is the case. Causes are not very small. On the other hand, I proposed a model of price variation, which is very ambitious. It says that on the basis on some invariances, of some simple rules and some, how to say, almost theological thinking, variation of prices has certain invariances, which means that irrespective of what happens outside or dependent upon it, it has certain fluctuations. Now, I don't think that this is compatible with the idea that each price change corresponds to a well-defined outside, exogenous influence. I think that it conforms better to the idea that prices represent, how should I say, the resonances, the vibrations of a certain system that we have, which is the economic system, exchange system or even without an exchange as such, but an economic system which is extraordinarily well organised and extraordinarily interrelated, which, if provoked constantly by outside impulses, creates not simply a mirror image or an amplification of outside impulses, but produces its own vibrations, it's own way. Now this is purely hypothetical, however, and it is one of the two conjectures one can have concerning variation of prices. But by the very fact of obtaining a model which is often valid, whereas nothing at all suggests that the causes should follow any pattern whatsoever, one very much favours the idea that the rules of price variation are the rules of the market as well as the rules of what makes the market vary.

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: Bernard Sapoval Daniel Zajdenweber

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.

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

Duration: 6 minutes, 12 seconds

Date story recorded: May 1998

Date story went live: 29 September 2010