Download Fractals in Financial Markets

Fractals in Financial
Markets
Stable Distributions
Levy skew alpha-stable distribution
Pareto distribution
Ivan Hristov
Probability and Statistics
Summer 2005
What is a Fractal?
•
•
A fractal is a geometrical structure that is
self-similar when scaled. A branch of a tree
is often used as an example. The branch is
similar to the whole tree, and if you break a
twig off the branch, the twig is similar to the
branch. In a true, mathematical, fractal, this
scaling goes on forever, but in all real
systems, there is a largest and smallest
scale which exhibits fractal behavior.
A fractal always has a fractal dimension. The
fractal dimension tells us what happens to
the length, area or volume of the fractal
when you enlarge it. Think of the length of a
jagged shore line. If you measure it on a
map, you may not see the small bays. At the
other extreme, if you try to measure it with a
ruler, you will see every stone at the shore.
The smaller the structures that you measure
are, the longer the shore line will seem. The
fractal dimension will tell you how much
longer it will become when you measure
smaller structures.
Louis Bachelier to Benoit Mandelbrot
•
•
•
•
Louis Bachelier was a French mathematician at the turn of the 20th
Century. He is credited with being the first person to model Brownian
motion, which was part of his PhD thesis The Theory of Speculation,
(published 1900). His thesis, which discussed the use of Brownian motion to
evaluate stock options, is historically the first paper to use advanced
mathematics in the study of finance.
Bachelier’s simplest model is: let Z(t) be the price of a stock at the end of
period t. Then is is assumed that successive differences of the form Z(t+T) –
Z(t) are independent, Gaussian random variables with zero mean and
variance proportional to the difference interval T2.
Benoit Mandelbrot however noted that the abundant data gathered by
empirical economists did not fit this model as the actual distributions looked
“too peaked” to be samples of a Gaussian population. In his momentous
paper titled “The Variation of Certain Speculative Prices” he went on to
replace Bachelier’s Gaussian distribution with a new family of probability
laws and introduced a general stable distribution model.
The normal distribution is a special case of a stable distribution, and it is the
only one that has finite variance.
Gaussian vs. Empirical Data
Why is Stable Distribution Fractal?
•
Definition: A random variable X is said to have a stable distribution if
for any n >= 2 (greater than or equal to 2), there is a positive number Cn
and a real number Dn such that :
X1 + X2 + … + Xn-1 + Xn ~ Cn X + Dn
•
•
where X1, X2, …, Xn are all independent copies of X.
Think of what this definition means. If their distribution is stable, then the
sum of n identically distributed random variables has the same distribution
as any one of them, except by multiplication by a scale factor Cn and a
further adjustment by a location Dn .
Does this remind you of fractals? Fractals are geometrical objects that
look the same at different scales. Here we have random variables whose
probability distributions look the same at different scales (except for the
add factor Dn).
Mandelbrot Quiz
• Of the different
graphs below, one is
Brownian motion, one
fractional Brownian
motion, one a Levy
process, two are real
financial data, and
three are multifractal
forgeries. Can you tell
which is which?
Answers:
•
•
•
•
•
•
•
•
1 is Brownian motion. Most of the values
form a band of approximately constant width
(independence), and the values outside this
band are mostly very small (large jumps are
rare).
2 is a Levy process. Most of the values form
a band of approximately constant width
(independence), but a considerable number
of values outside this band are large (large
jumps are not rare).
3 is fractional Brownian motion. Most of the
values form a band, but of varying width
(dependence), and the values outside this
band are mostly very small (large jumps are
rare)
.4 is a multifractal forgery.
5 is successive differences in IBM prices.
6 is successive differences in dollarDeutschmark exchange rates.
7 is a multifractal forgery.
8 is a multifractal forgery.
Levy skew alpha-stable distribution
•
•
A Lévy skew stable distribution is specified by scale
c, exponent α, shift μ and skewness parameter β.
The skewness parameter must lie in the range
[−1, 1] and when it is zero, the distribution is
symmetric and is referred to as a Lévy symmetric
alpha-stable distribution. The exponent α must lie
in the range (0, 2].
The Lévy skew stable probability distribution is
defined by the Fourier transform of its characteristic
function φ(t) :
•
where φ(t) is given by:
•
where Φ is given by
•
for all α except α = 1 in which case:
•
μ is the location of the peak of the distribution. β is a
measure of asymmetry, with β=0 yielding a
distribution symmetric about c. c is a scale factor
which is a measure of the width of the distribution
and α is the exponent or index of the distribution
and specifies the asymptotic behavior of the
distribution for α < 2. Note that this is only one of
the parameterization in use for stable distributions;
it is the most common but is not continuous in the
parameters.
Pareto distribution
• The Pareto distribution, named after the Italian economist Vilfredo
Pareto, is a power law probability distribution found in a large
number of real-world situations. This distribution is also known,
mostly outside economics, as the Bradford distribution.
• Pareto originally used this distribution to describe the allocation of
wealth among individuals since it seemed to show rather well the
way that a larger portion of the wealth of any society is owned by a
smaller percentage of the people in that society. This idea is
sometimes expressed more simply as the Pareto principle or the
"80-20 rule" which says that 20% of the population owns 80% of the
wealth.
•
It can be seen from the PDF graph on the right, that the
"probability" or fraction of the population p(x) that owns a small
amount of wealth per person (x ) is rather high, and then
decreases steadily as wealth increases. This distribution is not
just limited to describing wealth or income distribution, but to
many situations in which an equilibrium is found in the
distribution of the "small" to the "large". The following examples
are sometimes seen as approximately Pareto-distributed:
•
•
Frequencies of words in longer texts
The size of human settlements (few cities,
many hamlets/villages)
File size distribution of Internet traffic which
uses the TCP protocol (many smaller files, few
larger ones)
Clusters of Bose-Einstein condensate near
absolute zero
The value of oil reserves in oil fields (a few
large fields, many small fields)
The length distribution in jobs assigned
supercomputers (a few large ones, many small
ones)
The standardized price returns on individual
stocks
Size of sand particles
Size of meteorites
Number of species per genus (please note the
subjectivity involved: The tendency to divide a
genus into two or more increases with the
number of species in it)
Areas burnt in forest fires
•
•
•
•
•
•
•
•
•
Mathematically speaking, if X is a random
variable with a Pareto distribution, then
the probability distribution of X is
characterized by the statement
where x is any number greater than xm,
which is the (necessarily positive)
minimum possible value of X, and k is a
positive parameter. The family of Pareto
distributions is parameterized by two
quantities, xm and k.