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Fractals in Finance and
Risk
By: Will Brennan
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Why does it matter?
Finance played a crucial role in thedevelopment of fractal
theory
Fractals are used in nance tomake predictions as to the risk
involved for particular stocks.
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The Current Model
Bronian Motiondiscovered !y "ouis
Bachalier in #$%% &Theoriede la speculation'.
Theory as later
developed !y (l!ert)instein* +ean ,errin* and-or!ert Weiner
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Bronian Motion
http:classes.yale.eduFractalsRandFrac
BronianBronian/.html
The Mathematical formula !ehind
Bronian Motion: 0d1i0 2 &dti'#.3
http://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.htmlhttp://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.html
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( ,ro!lem )mer5es7.
8oever* hen the 5raph isplotted !y successive prices*
di9erences emer5e !eteen stockand Bronian Motion.
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(n (lternate Method?
While the Bronian Motion model can !eadusted to t o!served data*
the BM modelis not useful in predictin5 data.
(nother method is !ased on the o!servationthat stocks are
statistically self;scalin5. Themethod is to input a simple
al5orithm thatprovides the same scalin5* and then o!serveho many
features follo automatically.
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Cartoons aren
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4tep #:6nitiator and
=enerator
dt1 = 4/9 - 0 = 4/9, dY
1 = 2/3 - 0 = 2/3, so dY
1 = (dt
1)1/2
dt2 = 5/9 - 4/9 = 1/9, dY
2 = 1/3 - 2/3 = -1/3, so -dY
2 = (dt
2)1/2
dt3 = 1 - 5/9 = 4/9, dY3 = 1 - 1/3 = 2/3, so
dY3 = (dt3)1/2
All 3 segments of the Cartoon
Satsf! the "ondton
#dY# = (dt
)1/2
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4tep 3: =enerate>
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4tep /: Randomie
6n order to make more realistic*introduce randomness in
thedirection of the linear se5ments
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The Final Cartoon
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Money mimics Cartoons
When this necartoon is placed
alon5sidenancial data*they are very
similar in termsof lar5e umpsand correlation.
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Conclusion
Throu5h utiliin5 a cartoon* asu@cient fractal model is
a!le to
make up for the failin5s of theBronian Motion model* alloin5for
investors to predict nancial
risk.
