Cartoons of the Variation of Financial Prices and of Brownian Motions in Multifractal Time

This article describes a versatile family of functions increasingly roughened by successive interpolations. They provide models of the variation of financial prices. More importantly, they are helpful “cartoons” of Brownian motions in multifractal time, BMMT, which are better models described in the next article. Ordinary Brownian motion and two models the author proposed in the 1960s correspond to special cartoons. More general cartoons are richer in structure but (by choice) remain parsimonious and easily computed. Their outputs reproduce the main features of financial prices: continually varying volatility, discontinuity or concentration, and other events far outside the mildly behaving Brownian “norm.”