Fractals in Finance and Risk
By: Will Brennan
Why does it matter?
Finance played a crucial role in the
development of fractal theory
Fractals are used in finance to make
predictions as to the risk involved for
particular stocks.
The Current Model
Brownian Motion discovered
by Louis Bachalier in 1900
(Theorie de la speculation).
Theory was later developed by
Albert Einstein, Jean Perrin,
and Norbert Weiner
Brownian Motion
http://classes.yale.edu/Fractals/RandFra
c/Brownian/Brownian3.html
The Mathematical formula behind
Brownian Motion:
 |dYi| = (dti)1/2
Brownian Motion and Stock
A graph of IBM
stock vs. a graph of
Brownian Motion.
When stocks are
graphed price vs.
time, and thus look
similar to Brownian
Motion
A Problem Emerges….
However, when the graph is plotted by
successive prices, differences emerge
between stock and Brownian Motion.
An Alternate Method?
While the Brownian Motion model can be
adjusted to fit observed data, the BM model
is not useful in predicting data.
Another method is based on the observation
that stocks are statistically self-scaling. The
method is to input a simple algorithm that
provides the same scaling, and then observe
how many features follow automatically.
Cartoons aren’t just for kids
Because this method is designed with
no thought to the mechanics of the
stock market, it is called a cartoon.
Begins with an initiator and continues
with a generator
Step 1:Initiator and Generator
All 3 segments of the Cartoon
Satisfy the condition
|dYi| = (dti)1/2
dt1 = 4/9 - 0 = 4/9,
dY1 = 2/3 - 0 = 2/3,
so dY1 = (dt1)1/2
dt2 = 5/9 - 4/9 = 1/9, dY2 = 1/3 - 2/3 = -1/3, so -dY2 = (dt2)1/2
dt3 = 1 - 5/9 = 4/9, dY3 = 1 - 1/3 = 2/3,
so dY3 = (dt3)1/2
Step 2: Generate!
Step 3: Randomize
In order to make more realistic,
introduce randomness in the direction
of the linear segments
The Final Cartoon
Money mimics Cartoons
When this new
cartoon is placed
alongside financial
data, they are very
similar in terms of
large jumps and
correlation.
Conclusion
Through utilizing a cartoon, a sufficient
fractal model is able to make up for the
failings of the Brownian Motion model,
allowing for investors to predict
financial risk.