8-Stochastic Integrals Page 231 Wednesday, February 4, 2004 12:50 PM
Stochastic Integrals 231The fractal dimension is a concept that measures quantitatively how a
geometric object occupies space. A straight line has fractal dimension
one, a plane has fractal dimension two, and so on. Fractal objects might
also have intermediate dimensions. This is the case, for example of the
path of a Brownian motion which is so jagged that, in a sense, it occu-
pies more space than a straight line.
The fractal nature of Brownian motion paths implies that each path is
a self-similar object. This property can be illustrated graphically. If we
generate random walks with different time steps, we obtain jagged paths.
If we allow paths to be graphically magnified, all paths look alike regard-
less of the time step with which they have been generated. In Exhibit 8.2,
samples paths are generated with different time steps and then portions of
the paths are magnified. Note that they all look perfectly similar.
This property was first observed by Benoit Mandelbrot in sequences
of cotton prices in the 1960s. In general, if one looks at asset or com-
modity price time series, it is difficult to recognize their time scale. ForEXHIBIT 8.2 Illustration of the Fractal Properties of the Paths of a Brownian Motionaa Five paths of a Brownian motion are generated as random walks with different time
steps and then magnified.