Mandelbrot's Rough Rules, in The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward

by Corry Shores
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Benoit Mandelbrot & Richard Hudson

The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward

Chapter VII

Studies in Roughness:

A Fractal Primer

The Rules of Roughness

The pattern can take many forms. It can be a concrete shape that repeats on successively smaller scales, as with the fern or cauliflower. It can be an abstract, statistical pattern – for instance, the probability that a particular square in a grid will be black or white, or that a point in space will be occupied by a star or by vacuum. The pattern can scale up, scale down, and get squeezed, twisted – or both. They way the pattern gets used can be strictly defined by a precise, deterministic rule; or it can be left entirely to chance.

The construction of the simplest fractals starts with a classical geometric object: a triangle, a straight line, a solid ball. That is called the initiator. In the last chapter’s financial cartoon of the Bachelier model, the initiator was the straight, rising trend line [Image obtained gratefully from Mukul PAL's TIME triads.]

Then comes the generator, or template from which the fractal will be made. That is generally a simple geometric pattern: A zigzag line, a crinkly curve, or – in financial charts – a sequence of prices up $2 last week, down 37 cents today, and up $1.50 the next month. Then comes the process for building the fractal; it is called a rule of recursion. For instance, recall how we built the financial Bachelier cartoon. Start with the straight line initiator, squeeze the zigzag generator uniformly in each direction (without turning it) so that its end points coincide with those of the initiator, and then repeat indefinitely. Wherever a straight line appears in each diagram, replace it with a suitable scaled-down copy of the generator. (126b-is)

...

The variety of fractals is immense. But all have a few common traits. First, they scale up or down by a specific amount – that is, the parts echo the whole in accordance with a precise, measurable formula.

The simplest fractals scale the same way in all direction, hence are called self-similar. They are like high-quality zoom lenses that expand or shrink everything in the frame by the same degree; what they see at one focal length will be similar to what they see at another. But the Bachelier cartoon scales more in one direction than another and the same will be the case with other cartoons of price variations to be introduced in later chapters. Such fractals are called self-affine. They are like an office laser photocopier set to shrink a document’s image more cross-wise than length-wise. If the fractals scale in many different ways at different points, they are multifractal– and their mathematical properties become intricate and powerful. (127b.c)

Mandelbrot, Benoit B., & Richard L. Hudson. The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004.

Image obtained gratefully from:

http://timetriads.com/article/225