L’introduction de la loi de Pareto dans la modélisation financière

Pareto's distribution became part of financial modelling at the beginning of the 1960s due to the identification of two unidentified properties of this distribution: it is a scale invariant distribution and a limit distribution in probability. These two properties of Pareto's distribution were first introduced by Benoît Mandelbrot, who based Pareto's distribution on Paul Lévy's alpha-stable distributions in the modelling of cotton prices and then of financial assets. This incorporation of the ancient Pareto law into the modelling of the dynamics of the stock market changes its status and confers on it a new life in financial economics, by feeding power laws into the models of the stock market. After having recalled the mathematical properties of the Paretian framework, in particular the paradox of the expected value in a power law, we introduce the modelling debate between a mixture of models (Gaussian for average values and Paretian for extreme values) and a single model for all values (alpha-stable), by the introduction of the extreme value theory and the generalized Pareto distribution. Then we describe the arrival of Pareto's distribution in finance by using Mandelbrot's methodology, which enables us to present the concept of Paretian chance. Finally, we suggest considering the financial practices of private equity as "natural Paretian mathematics", a hypothesis that opens perspectives on the enlargement of rationality to Paretian environments.