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

Benoît Mandelbrot, the father of Fractal Geometry, developed a multifractal 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 mathematics involved into multifractals is presented; Section 3 is addresses to the extension of multifractality towards stochastic processes, introducing the crucial concept of local H\"older exponent of a function. Finally, Section 4 deeply analyze the mathematical properties of the scaling function which drives the ``wildeness'' of the process. The proof of Theorem 4.4 is unpublished and the generalization of a Mandelbrot's result, which highlights a possible alternative motivation for the presence of heavy tails and a connection with the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.