Multifractality in Finance: A deep understanding and review of Mandelbrot's MMAR

Benoit Mandelbrot , the father of Fractal Geometry, developed a multifract al model for describing price changes. Despite the commonly used models , such as the Brownian motion, the Mutifractal Model of Asset Return (MMAR) takes into account scale-consistency, long-range dependence and heavy tails, thus having a great flexibility in depicting the real-market peculiarities . In section 2 a review of the ma thematics involved into multifractals is presented; Section 3 is addressed to the extension of multifractality towards stochastic processes , introducing the crucial concept of local Hölder exponent of a function. Finally , Section 4 deeply analyzes the mathematical propertie s of the scaling function which drives the "wilderness" of the process. The proof of Theorem 4.4 is unpublished and the generalization of Mandelbrot's results, which highlights a possible alternative motivation for the presence of heavy tails and a connection with the Extreme Value Theory. Section 5 is devoted to the analysis of the connection between the scaling function, Multifractal Formalism and Large Deviation Theory, suggesting possible ways in order to estimate the quantities involved. Finally in Section 6 the MMAR is presented, listing all the theorems that make it a suitable model for financial modelling.