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Once in a while I see the golden ratio / Fibonacci numbers appears in the construction of technical indicators. (More specifically about Fibonnaci retracements, see here for example - "For unknown reasons, these Fibonacci ratios seem to play a role in the stock market, just as they do in nature."). I am not here to discuss the usefulness of TA as a whole. But I am curious about that specific definition / usage.

Is there any theoretical basis that would justify the usage of the golden ratio / Fibonnaci numbers when looking at stock price patterns ?

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    $\begingroup$ What ''theory'' should that be? A general equilibrium model with a representative household that has recursive utility? I'd be hard-pressed to believe that any sensible economic theory would arrive at Fibonacci retracements. At best, you could argue that markets are informationally inefficient and TA identifies resulting psychological patterns. But I doubt there's any serious argument why you ought to use 61.8% and 38.2% instead of 60% and 40% or other (random) numbers. As often with TA, it may be a self-fulfilling prophecy (if it works at all). $\endgroup$
    – Kevin
    Nov 14 '20 at 14:53
  • $\begingroup$ @Kevin While I doubt the authenticity of most of TA, I do believe Fibonacci numbers have a place in analysis. I mean it’s present everywhere inspite of the tendency of nature to be most entropic, somehow the sequence finds a way to beat that. And it’s not just that, Mandelbrot wrote his whole fractal theory in finance based on their presence in nature too. $\endgroup$ Nov 14 '20 at 16:00
  • $\begingroup$ Many of the “models” per se are based on underlying real life scenarios, take genetic algorithms or neural networks, while in depth they are very different from the workings in nature on what they’re based on, the fundamental thesis of them is very much in line with what happens in reality. $\endgroup$ Nov 14 '20 at 16:05
  • $\begingroup$ @Kevin: that is exactly what I am asking... $\endgroup$
    – lcrmorin
    Nov 14 '20 at 16:44
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    $\begingroup$ @DhruvMahajan: having read Mandelbrot myself there is still a significant gap between observing coastlines / trees or flour particle movement and using $\phi-1$ as a trading signal. $\endgroup$
    – lcrmorin
    Nov 14 '20 at 16:47
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In most literature on the use of Fibonacci numbers in technical indicators, it's referred to as a "phenomenon", which should be enough to tell you that there's no scientific or mathematical proof for its utility in market pricing.

Since the Fibonacci sequence does sometimes appear in nature, there is a leap of faith needed to say that stock market price action would also follow those natural patterns. Its usefulness is subjective and unproven.

With that said, if you're a believer in probability and mean-reversion in stock prices, the Golden Ratio of 61.8% is very close to a standard deviation (68%) of a Gaussian (normal) distribution, which may explain why statisticians find it to be a familiar and comfortable number to use.

If you're not a believer in mean-reversion, it's just an arbitrary number that can be used as a reference point.

Here's some research, trying to make the case: A computational exploration of the efficacy of Fibonacci Sequences in Technical analysis and trading

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