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Inequality in price change distribution


Distribution of price change
Benoît Mandelbrot Mathematician
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Now these ideas were very much in the forefront of my mind in the late '50s and early '60s when I was working on this matter, and recently in my book on Fractals in Finance I made them sharper by making a distinction between what I call three states of randomness. One state of randomness is concentration in the small and the large, that is, concentration in groups made of two, one, two, three, or four add ins and groups made into thousands of add ins. The second group has a lack of concentration, that is the case of the Gaussian and exponential. Finally, a third group has; concentration is small, and lack of concentration is large, which is the case of the distribution called lognormal and of other distributions proposed in this context. Now this takes us very close to an essential feature. Given that the consequences of these different hypotheses are so sharply distinct, can we separate between them a priori on the basis of, how should I say, ordinary statistics? The answer is complicated and in fact disappointing. That is, for all these years different people have been applying different tests that were designed for other purposes and then applied them to this particular issue. Some people say it is lognormal, or close enough except perhaps for a few cases. Other people say it is power-law, unquestionably, except perhaps for some cases that belong to a different domain of understanding. Statistics has not been able to make us distinguish between these two contexts, and I think that when honourable, very skilled practitioners go on for a long time disagreeing on something, then one should not say that one is entirely right and the other is entirely wrong. One should say probably the tools they are using in their disagreement are not applicable to the problem. Therefore my main feeling in my study of inequality has been that the phenomenon of inequality is true for small assemblies and for large ones. The fact that the largest of add-ins to a certain ensemble is quite non-negligible compared to a sum of all the others, that this phenomenon invariably, forcibly, obliges us to consider distributions of a power-law form: probability of U exceeding u is equal to u the power (of) minus alpha (u-?), where alpha is a characteristic exponent that measures degree of inequality. This is a general statement that I would like to apply immediately to the problem of the distribution of price changes, therefore of finance. The contemporary, the near contemporary, of Pareto was Bachelier and he came up with the notion that, in a certain sense, there is nothing special about prices, that prices fluctuate, but price changes if plotted horizontally in time go up and down, up and down, up and down, and they fluctuate around a very well defined value, which is zero. There is an equilibrium; price doesn't change in the long run, and price changes are sometimes positive or negative, they're all independent, they are due to different causes, there is nothing that forces them to be related in any fashion, therefore they are independent and the price as a result becomes the sum of these independent price changes. Therefore it follows what is called random walk, or more precisely a Brownian motion. Now, if price variation were following Brownian motion, what consequences would we have? Look at a chart of price changes. Price changes would be again just up and down, up and down, more or less even over a long period. But look at actual records of price change. First of all, most of them do indeed concentrate in this kind of strip or serpent, if you wish, a snake, and this snake goes on but it is of very variable width. For long periods price variation is rather small, then becomes big, then becomes small again, then again becomes big. Besides, very often, one finds very large price changes. Not large in the sense that the Gaussian distribution can take large values, because for Gaussian distribution there is a measure of dispersion, standard deviation, and the probability of being twice, three times standard deviation is not negligible. But if you represent the middle of this strip by Gaussian distribution you find that the large prices changes are of the order of ten times sigma, twenty times sigma. The probability of a Gaussian distribution exceeding ten times sigma is 10-23. This should never happen. But it happens all the time. As a matter of fact it happens very, very often discontinuously. Price does not vary continuously, but very often jumps from one value to a very different value. How does the jump occur? Sometimes overnight. In the morning some news has come in that makes price go up and down. Even a very large company like IBM has ten percent, fifteen percent changes, quite often. Sometimes the news comes in the day, something happened, trading is stopped for a few minutes, maybe an hour, when it resumes; it resumes at a very different price.

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, 18 seconds

Date story recorded: May 1998

Date story went live: 29 September 2010