Scaling in financial prices: IV. Multifractal concentration

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Abstract
In the Brownian model, even the largest of N successive daily price increments contributes negligibly to the overall sample variance. The resulting 'absent' concentration justifies the role of variance in measuring Brownian volatility. Mandelbrot introduced in 1963 an alternative 'mesofractal model', in which the population variance is infinite. A significant proportion of the overall sample variance comes from an absolutely small number of large contributions, expressing a 'hard' form of concentration. To achieve a prescribed proportion of the overall measured variance, those 1900 and 1963 models require numbers of days of the order of N 1 and N 0, respectively. This paper shows that an intermediate possibility exists: a new and very flexible 'soft' form of concentration is provided by the 'multifractal' model Mandelbrot introduced in 1997. The standard 'extreme values' theory applies to mesofractals but multifractals behave very differently. The single largest contribution to sample variance is asymptotically negligible; however, an arbitrarily high proportion of the overall variance is contributed by a number of days of the order of N D, where 0<DD, a fractal dimension, is a consequence of scaling. It allows 'softness' to be modulated between the unrealistic extremes N 1 and N 0. As N increases, so does the absolute number N D, but the relative number N D/N decreases to zero. As a result, the bulk of the significant effects concentrates in a small proportion of cases. (This is a finite approximation of a set of measure zero, but mathematical refinements do not matter in this paper.)

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