Scaling in financial prices: III. Cartoon Brownian motions in multifractal time
Quantitative Finance, 2001, vol. 1, issue 4, 427-440
This article describes a versatile family of functions that are increasingly roughened by successive interpolations. They reproduce, in the simplest way possible, the main features of financial prices: continually varying volatility, discontinuity or concentration, and the fact that many changes fall far outside the mildly behaving Brownian 'norm'. Being illuminating but distorted and incomplete, these constructions deserve to be called 'cartoons'. They address both the observed variation of financial prices and the generalized model the author introduced in 1997, namely, Brownian motion in multifractal time. Special cases of the same construction provide cartoons of the Bachelier model - the Wiener Brownian motion - or the two models the author proposed in the 1960s, namely, Levy stable and fractional Brownian motions. The cartoons are the embodiment of the author's 'principle of scaling in economics'. While rich in structure, they are unexpectedly parsimonious, easily computed, and easily compared to one another by being associated with points in a square 'phase diagram'.
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