Extreme values and fat tails of multifractal fluctuations

Abstract

In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framemork that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed ``inverse cubic law'' of the return pdf tails.

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• Non stationary multifractality in stock returns
• Goodness-of-Fit tests with Dependent Observations
• Extreme value statistics and return intervals in long-range correlated uniform deviates
• Linearization effect in multifractal analysis: Insights from the Random Energy Model
• Log-Normal continuous cascades: aggregation properties and estimation. Application to financial time-series
• Fluctuations in multiplicative systems with jumps
• Directed random walks on hierarchic trees with continuous branching: a renormalization group approach
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• The Markov Switching Multi-fractal models as a new class of REM-like models in 1-dimensional space
• The calculation of multi-fractal properties of directed random walks on hierarchic trees with continuous branching
• Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models
• On the existence of a glass transition in a Random Energy Model
• Evidence of Self-Organization in Time Series of Capital Markets