Fractals

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Fractals
And the economy
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To begin...
For centuries, individuals have searched for some
way, whether it be a mathematical formula or
spiritual premonition, to predict the
fluctuations of the global markets. As years
passed, systems were created that had the ability
to predict market swings with the correct
amount of study. These systems, relying on yield
curves and the bell curve to gather and organize
their data, had been the teachings of business
teachers for decades.
Then came the 1980s..
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Along came the Fractal
During this decade of leg warmers And big hair,
the study of a new form of Math, known as Fractal
geometry, was booming. Benoit Mandelbrot,
discoverer of the famed Mandelbrot set, along
with other mathematicians, began to investigate
the inefficiencies of the current market
forecasters, and the potential fractal math had
in replacing these systems.
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What was the problem?
One may wonder, what exactly was the problem with
the old systems of predicting markets. These
systems, which had been in use for decades,
obviously worked to some extent if they were
considered valuable enough to be taught.
The answer, then, can be found by examining the
philosophy of the old methods and how they may
be less accurate than those based on the math of
fractals
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The old way
Many of the older methods of market
prediction are based around a belief that the
fluctuations of the stock market could be
categorized as utter chaos. In these systems, the
only way to forecast a large growth or decay is
to closely observe trends and percentages on a
larger picture scale to decide when to buy or
sell. Such methods are so non specific, however,
that the advice would be to buy a certain year as
opposed to a certain week.
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Chaotic or not?
So are the fluctuations of markets chaotic? This
is one of the many questions addressed in Benoit
Mandelbrots new book, The (Mis)behavior of
Market, in which the fractal response is
NO!!
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Why the old system is wrong.
Mandelbrot and others who claim that a possible
fractal system of market prediction is the best
bet, heavily rely on the idea that all market
fluctuations are dependent. Why do the markets
fluctuate? Do brokers, one day decide to panic
an sell as much stock as possible, causing the
markets to crash, without any rhyme or reason?
No, whether it is a hurricane over Cuba,
destroying sugar cane crops or the announcement
of a corporate scandal, the markets fluctuate as
a result of stimuli, not spontaneously
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Other Flaws
There are some very basic statistics which
display the gross inaccuracies found in the old
systems of market forecasting. To begin, market
fluctuations are believed to follow the Bell
Curve as pictured below, with zero fluctuation in
the center and large, outlier fluctuations at
the ends. The truth behind this is that the Bell
Curve is actually an extremely poor fit to the
economic swings of the last century. Lets look
at some numbers
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The following chart exhibits a given minimum
percentage of fluctuation for any given day, the
number of such fluctuations which would occur
over roughly the last century if the market
followed the Bell Curve, and then finally the
actually numbers of days with such fluctuation.
Minimum Percentage
Number for Bell Curve
Actual number
3.4 percent
58 days
1,001 days
4.5 percent
6 days
366 days
1 day every 300,000 years
7 percent
48 days
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Real Vs. Fake
The graphs to the right were presented in
Mandelbrots previously mentioned book on
fractal applications in the economy. Two of the
graphs are made from real data of economic
trends, one is from a system Mandelbrot created,
and another was generated by the random walk
model, one of the older systems of market
forecasting, used when dealing with irregular
growth. The question now posed is which two are
real and which are fake?
Another way of exposing the defects of the old
systems is through the study of real data
created by both a fractal system and a non
fractal based system.
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Impossible!
Dont think to hard because it is virtually
impossible, based on these charts, to detect
which is the fake. However, by observing a
different set of charts we shall be able to
easily identify the fraudulent graphs.
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1 down, 1 to go
The graphs displayed to the right are graphs of
the change in price, from moment to moment, as
opposed to the initial set of graphs which only
displayed the prices themselves.
BUT WAITWhich graph is the fractal model???
By observing these sets of data it becomes
apparent that the 2nd model simply does not fit
in. This model is indeed fake and was created by
the random walk model. Unlike the other graphs
which display a range of price changes, the 2nd
graph is completely unique, exhibiting a steady
period of similar change, failing to replicate
any real data.
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Here We Go
The fractal model was actually number 4,
however, like the initial set of graphs, it is
impossible to distinguish it from the real. This
graph was created by Mandelbrot with his newest
model for the working of financial markets. The
name of this model is Brownian motion in
multifractal time and simply has the ability to
replicate the chaos known as market fluctuations.
Now that we know why the old systems of the
economy need to be changed, we can explore some
of the fractal math which will be used in future
market forecasting.
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So here's the deal...
It is essential that when beginning our look into
the economic applications of fractals it is
understood that fractals will not and can not
predict the movement of individual stock
pricessorry, but you will not become rich from
watching this presentation. However, there are
many applications of fractals which, with the
correct amount of study, can make you bundles.
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The Big Deal
The biggest generator of excitement surrounding
this topic is the simple fact that fractals can
so easily imitate the behavior of markets. As
demonstrated in the earlier graphs, Mandelbrots
system, while it is not an exact copy of an
actual market fluctuation, is indistinguishable
in both price and price change charts from
actual market data. This is the first step.
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How to form systems
Mandelbrots latest system, Brownian motion in
multifractal time, however involved it may sound
is actually somewhat simple. When generalizing
different intricacies, the formation breaks down
to 4 basic parts.

• Draw a square and in it draw a diagonal, either
  representing a positively
• or negatively moving system.

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2. Chose a generator, such as the squiggle below,
in the general direction Of your initial line and
iterate until you end up with a a very choppy
looking Graph.
Above is one such image, a choppy, yet extremely
predictable model of market movement..which is
why we are only half done.
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Oh geeze...
Step 3 is where I get a little confused but I
will try to explain as much as I can with the
help of Mr. Mandelbrot to the right.
3. This part of the model creation involves the
integration of a power Law. You are most likely
familiar with power laws, fitting data to
them In numerous applications, however the
power laws used in this situation is denoted by
H and is derived from The square root laws of
Brownian Motion. This H, which is created based
on data of whatever system you may be studying,
is mingled with the existing graph to alter the
predictable shape.
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Finished!!
4. The final step to the creation of a model is
the addition of any severe vertical jumps if
needed. These jumps would be added into a graph
if the model failed to recognize a certain jump,
but continued to follow the expected
fluctuation Patterns. Below is an example of a
finished model.
Now just think, you can make your own fractal
model of the exchange rate of the Yen!
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Good News
Fortunately, even if you cant draw your own
model, there is another aspect of market
predicting, involving much of the same math
taught in an introductory level class, which
many will have the ability to do. We will now
explore a method to determine how quickly a
model with diverge from the actual path the
real data will take.
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Lya- what?
Through the use of fractal math a simple equation
can be used to measure the rate two neighboring
graphs or trajectories diverge, with the use of
the Lyapunov Exponent. Knowing how large of an
impact small differences in initial conditions
can make in the fractal world, it is extremely
important, when attempting to replicate the
behavior of a certain data set, to know how long
it takes for ones model to move a certain
distance away from the real information.
In the following example, the lyapunov Exponent,
which is different depending on the system used,
will be for the logistic function. This example
is credited to J. Orlin Grabbe.
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The Lyapunov exponent is expressed in the
following equation where R is plus or minus
around a reference trajectory E is
the difference between two initial conditions
e is the natural log is
the Lyapunov exponent n is the step
or iteration number of the point of major
divergence, n is usually the
variable for which we solve.
EXAMPLE
There are two systems modeling the fluctuation of
mutual funds for the next 20 years using the
logistic functions. One model has a constant path
of .75 while the other starts at .7499 then
moves in its orbit. Knowing that for the
logistic function the Lyapunov Exponent is log 2
or .693147, at what value of n will the path of
the second model leave the interval (.74,.76)
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The answer is...
In the problem, you know your value for R is .01,
the distance or from the reference
trajectory of .75. Also, the value of E is known,
with the difference between the two initial
values being .0001. This provides us with a value
for all variables except n, what we are solving
for. The filled in equation is
Now with a little knowledge of powers, which
anyone studying fractals would of course posses,
the above equation can be easily solved with the
answer being n approximately 6.64. By simply
rounding this number to the next largest whole
number you have find the number of iterations it
takes before the paths of the two Trajectories
have diverged out of the specific range.
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Wrap up
Now, with a basic knowledge of the reasons
behind fractal exploration in the financial
markets, as well as a slight background in the
math behind these fractal applications, there is
little more for anyone to do but wait. Fractals
and their connections to the economy are
blossoming areas of mathematical and economic
study. This topic has a bright future with many
brilliant minds, such as Mandelbrot, making new
discoveries daily in hopes of finding order
in..
The chaos of the markets!