My Math Forum Some thoughts on Fibronacci Numbers and Stocks

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October 5th, 2010, 12:20 AM   #1
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Some thoughts on Fibronacci Numbers and Stocks

Just brain storming at the moment:

Some thoughts

People in technical analysis often look at successive points were trend lines change. If you take the difference of these two points for two successive trend lines then supposedly the ratio should be close to the ratio of a Fibonacci number.

Here are some of the ratios:

21/34 = 0.61764 ~ 0.618 34/21 = 1.61904 ~ 1.619
21/55 = 0.38181 ~ 0.382 55/21 = 2.61904 ~ 2.619
21/89 = 0.23595 ~ 0.236 89/21 = 4.23809 ~ 4.238

http://theforexbooks.com/index.php/D...rk-Deaton.html

For this to be fractal like it should be somewhat scale independent.

However, in long time scales the fundamental of the company should dominate over short term emotion so the question is how to separate these two signals?

My hypothesis would be that in the long term the average growth rate should match the fundamentals. This should give some relationship between how amplitude scales with time

Stocks generally follow exponential growth curves. The fabroci sequence is also exponential as is any linear time independent finite difference equation.

Quote:
 :$F\left(n\right)= {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}={{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}\, ,$ where $\varphi= \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,$ is the golden ratio
http://en.wikipedia.org/wiki/Fibonac...e_golden_ratio

Where I would like to go with this

- Explore how various fabroci ratio's scale with time
- Explore how the ratio's are influenced by the fundamentals of the stock.
- Find the distribution of fabroci ratio's for a given time scale

Anyway, if their is anything two these fabroci numbers then it would be nice to include them in noise models such as those used to estimate the valuation of options. I think I'm quite short of tying this into any strong theoretical framework at the moment. For instance I would be curious of how the noise is based on human factors vs market factors. I would also like to know why fabonacii ratio's are important or are they only important because people think they are important.

 October 5th, 2010, 01:06 AM #2 Member   Joined: Sep 2010 Posts: 63 Thanks: 0 Re: Some thoughts on Fibronacci Numbers and Stocks Some more thoughts from the above we can get the form of the difference equation which may describe the noise: The ratio of the difference of successive trends is: (x1-x2)/(x2-x3)=(+/-)Rn x1-x2=(+/-)Rn(x2-x3) Rn(x1-(1-Rn)x2)=(+/-)x3 where Rn is a Fibronacci ratio: Now how might one combine equations of this form for various Fibronacci ratios to get some kind of noise model. And still get the same ratio properties for trend lines.
October 5th, 2010, 01:45 AM   #3
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Re: Some thoughts on Fibronacci Numbers and Stocks

Quote:
 Originally Posted by John Creighton Some more thoughts from the above we can get the form of the difference equation which may describe the noise: The ratio of the difference of successive trends is: (x1-x2)/(x2-x3)=(+/-)Rn x1-x2=(+/-)Rn(x2-x3) Rn(x1-(1-Rn)x2)=(+/-)x3 where Rn is a Fibronacci ratio: Now how might one combine equations of this form for various Fibronacci ratios to get some kind of noise model. And still get the same ratio properties for trend lines.
Let phi be the golden ratio, then consider a random process were a quantity only changes by multiples of integer powers of phi. Or more specifically consider the a random process where the quantity either changes only by multiplying or dividing by phi and which operation we do is chosen at random. Then we have something like a random walk only we are multiplying instead of adding. This has the property were the ratio of any two terms in the series will be a power of the golden ratio.

These are Fibronacci ratios. Now this doesn`t relate to trend lines but perhaps gives some of the answers. I wonder what sort of noise properties this type of random process would have.

October 5th, 2010, 05:06 AM   #4
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Re: Some thoughts on Fibronacci Numbers and Stocks

Quote:
 Originally Posted by John Creighton I wonder what sort of noise properties this type of random process would have.
In the limit, as the exponents become large, it would behave as Brownian motion, I think. This is a common model for stocks; Google it.

October 5th, 2010, 01:25 PM   #5
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Re: Some thoughts on Fibronacci Numbers and Stocks

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by John Creighton I wonder what sort of noise properties this type of random process would have.
In the limit, as the exponents become large, it would behave as Brownian motion, I think. This is a common model for stocks; Google it.
What is your motivation for this conjection?

Quote:
 Brownian motion is among the simplest of the continuous-time stochastic (or probabilistic) processes, and it is a limit of both simpler and more complicated stochastic processes (see random walk and Donsker's theorem). This universality is closely related to the universality of the normal distribution. In both cases, it is often mathematical convenience rather than the accuracy of the models that motivates their use
A random walk converges but this isn't a random walk as the noise is multiplicative instead of additive. However, I suppose the log of the noise would behave like a random walk and in the limit behave like Brownian motion. Therefore I suppose you could derive the distribution of the random process I proposed from the distribution of brownian motion.

On another note, I haven't thought of any reasons why the trends should be related by fibronacci numbers but I thought a bit about retracements.

I gave a noise model where the next value takes on one of two values

y(n)=y(n-1)*a
y(n)=y(n-1)/a

each with separate probabilities. Now say their was some target value
y*(n-1) based on what people perceived the fundamentals were.

If their was some kind of oscillatory restoring force (like a feedback loop) with momentum (similar to a spring system) that that would give retracements and because this would act as a linear filter then the same ratio's between successive values would hold.

October 5th, 2010, 05:33 PM   #6
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Re: Some thoughts on Fibronacci Numbers and Stocks

Quote:
 Originally Posted by John Creighton A random walk converges but this isn't a random walk as the noise is multiplicative instead of additive. However, I suppose the log of the noise would behave like a random walk and in the limit behave like Brownian motion.
I neglected to mention this; that's always how it's done with stocks, which are multiplicative creatures. (Negative stock prices are meaningless.)

 March 18th, 2011, 11:33 AM #7 Member   Joined: Sep 2010 Posts: 63 Thanks: 0 Re: Some thoughts on Fibronacci Numbers and Stocks Here's what I was looking for: http://en.wikipedia.org/wiki/Elliott_wave_principle I'll start a new thread on this shortly.

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