Fractals in finance and risk
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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.

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Presentation Transcript

Why does it matter
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
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
Brownian Motion

  • http://classes.yale.edu/Fractals/RandFrac/Brownian/Brownian3.html

  • The Mathematical formula behind Brownian Motion:

    • |dYi| = (dti)1/2


Brownian motion and stock
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
A Problem Emerges….

  • However, when the graph is plotted by successive prices, differences emerge between stock and Brownian Motion.


An alternate method
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
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

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 1:Initiator and Generator

All 3 segments of the Cartoon

Satisfy the condition

|dYi| = (dti)1/2



Step 3 randomize
Step 3: Randomize

  • In order to make more realistic, introduce randomness in the direction of the linear segments



Money mimics cartoons
Money mimics Cartoons

  • When this new cartoon is placed alongside financial data, they are very similar in terms of large jumps and correlation.


Conclusion
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.


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