• Benoit B. Mandelbrot


A pragmatic attitude towards the slippery notion of randomness is described in Section 1. A premise of my work is that graphics must not be spurned; Section 2 argues that it remains indispensable even beyond the first stage of investigation.

A more pointed premise of my work is that the rules of price variation are not the same on all markets, hence a single statistical model may not describe every market without unacceptable complication. I agree that “it is better to be approximately right than certifiably wrong,” and worked, in succession or in parallel, with several distinct fractal models of finance of increasing generality. Adapting the style of reference used throughout this book, those models will be denoted by the letter M followed by the year of original publication or announcement.

The core of this chapter is made of Sections 5 to 9. A first glance at scaling is taken in Section 5. Section 6 introduces the M 1963 model, which deals with tail-driven variability and the “Noah” effect and is based on L-stable processes. Section 7 introduces the M 1965 model, which deals with dependence-driven variability and the “Joseph” effect and is based on fractional Brownian motion. Old-timers recall these models as centered on “cotton prices and the River Nile.” Section 8 introduces the M 1972 combined Noah-Joseph model, which this book expands beyond the original fleeting reference in M 1972j{N14}. That model is based on fractional Brownian motion of multifractal trading time.

The M 1965 and M 1972 models have a seemingly peculiar but essential feature: they account for the “bunching” of large price changes indirectly, by invoking unfamiliar forms of serial dependence with an infinite memory. All the alternative models (as sketched and critized in Section 4 of Chapter E2) follow what seems to be common sense, and seek to reach the same goal by familiar short memory fixes. Infinite memory and infinite variance generate many paradoxes that are discussed throughout this book, beginning with Section 8.3.

Here are the remaining topics of this chapter: Brownian motion (the 1900 model!) and martingales are discussed in Section 3. The inadequacies of Brownian motion are listed in Section 4. Section 9 gives fleeting indications on possible future directions of research. Finally, the notions of “creative model” and of “understanding without explanation” are the topics of Section 10.


Brownian Motion Price Change Fractional Brownian Motion Price Variation Spot Price 


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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Benoit B. Mandelbrot
    • 1
  1. 1.Mathematics DepartmentYale UniversityNew HavenUSA

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