Chapter 6
Fractals
In short, fractals are the geometric objects that are constructed by
repeating geometric patterns at a smaller and smaller scale. The
fractal theory is a beautiful theory that describes beautiful objects.
Development of the fractal theory and its financial applications has
been greatly influenced by Mandelbrot [1]. In this chapter, a short
introduction to the fractal theory relevant to financial applications is
given. In Section 6.1, the basic definitions of the fractal theory are
provided. Section 6.2 is devoted to the concept of multifractals that
has been receiving a lot of attention in the recent research of the
financial time series.
6.1 BASIC DEFINITIONS
Self-similarity is the defining property of fractals. This property
implies that the geometric patterns are isotropic, meaning shape
transformations along all coordinate axes are the same. If the geo-
metric patterns are not isotropic, say the object is contracted along
the y-axis with a scale different from that of along the x-axis, it is said
that the object is self-affine. The difference between self-similarity and
self-affinity is obvious for geometric objects. However, only self-
affinity is relevant for the graphs of financial time series [1]. Indeed,
since time and prices are measured with different units, their scaling
factors cannot be compared.
59
If the geometric pattern used in fractal design is deterministic, the
resulting object is named a deterministic fractal. Consider an example
in path (a) of Figure 6.1 where a square is repeatedly divided into nine
small squares and four of them that have even numbers are deleted
(the squares are numerated along rows). If four squares are deleted at
random, one obtains a random fractal (one of such fractals is depicted
in path (b) of Figure 6.1). While the deterministic and stochastic
fractals in Figure 6.1 look quite different, they have the same fractal
dimension. Let us outline the physical sense of this notion.
Consider a jagged line, such as a coastline. It is embedded into a
plane. Thus, its dimension is lower than two. Yet, the more zigzagged
the line is, the greater part of plane it covers. One may then expect
that the dimension of a coastline is higher than one and it depends on
a measure of jaggedness. Another widely used example is a crumpled
paper ball. It is embedded in three-dimensional space. Yet, the
(a)
(b)
Figure 6.1 Deterministic (a) and stochastic (b) fractals with the same
fractal dimension D ¼ ln(5)/ln(3) .
60 Fractals
volume of a paper ball depends on the sizes of its creases. Therefore,
its dimension is expected to be in the range of two to three. Thus, we
come to the notion of the fractal (non-integer) dimension for objects
that cannot be accurately described within the framework of Eucli-
dian geometry.
There are several technical definitions for the fractal dimension [2].
The most popular one is the box-counting dimension. It implies map-
ping the grid boxes of size h (e.g., squares and cubes for the two-
dimensional and the three-dimensional spaces, respectively) onto the
object of interest. The number of boxes that fill the object is
N(h) h
D
. The fractal dimension D is then the limit
D ¼ lim
h!0
[ ln N(h)= ln (1=h)] (6:1:1)
The box-counting dimension has another equivalent definition with
the fixed unit size of the grid box and varying object size L
D ¼ lim
L!1
[ ln N(L)= ln (L)] (6:1:2)
The fractal dimension for both deterministic and stochastic fractals in
Figure 6.1 equals D ¼ ln (5)= ln (3) 1:465. Random fractals exhibit
self-similarity only in a statistical sense. Therefore, the scale invari-
ance is a more appropriate concept for random fractals than self-
similarity.
The iterated function systems are commonly used for generating
fractals. The two-dimensional iterated function algorithm for N fixed
points can be presented as
X(k þ 1) ¼ rX(k) þ (1 r)X
F
(i)
Y(k þ 1) ¼ rY(k) þ (1 r)Y
F
(i) (6:1:3)
In (6.1.3), r is the scaling parameter; X
F
(i) and Y
F
(i) are the coordin-
ates of the fixed point i; i ¼ 1, 2, . . . N. The fixed point i is selected at
every iteration at random. A famous example with N ¼ 3, the Sier-
pinski triangle, is shown in Figure 6.2.
Now, let us turn to the random processes relevant to financial time
series. If a random process X(t) is self-affine, then it satisfies the
scaling rule
X(ct) ¼ c
H
X(t) (6:1:4)
Fractals 61

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