Saturday, June 12, 2021

Technical analysis only "works" as a self-fulfilling prophesy

The Bitcoin price is apparently likely to exhibit the "death cross" within a week. It is a bearish sign that makes people expect a significant collapse of the Bitcoin price. If you click, you will learn that a death cross is the situation when the 50-day moving average is above the 200-day one but they are approaching one another and the "death cross" moment is when the short-term, 50-day moving average drops below the long-term, 200-day one.



It tells you that "the recent prices are getting lower than the historical prices" so we're going down. The opposite situation is a "golden cross" which would be a bullish sign, a sign expected to induce an increase in the price.

Is this rule correct? Does it follow from pure mathematics? From the pure laws of economics? From the laws of physics? Or is it a counterpart of astrology? Indeed, these methods are called "technical analysis" (and these astrologers – believing that you may predict the future of a graph just by seeing patterns in the past – are called "chartists") and the first paragraph of the Wikipedia page says that "many Academics consider technical analysis a pseudoscience". Well, of course, I largely endorse that group. But things are more subtle and I would be absolutely dissatisfied with a simple one-word judgement here.



There are lots of things to say about these "laws". We may discuss them as a whole but to have some truly technical, serious discussion, we need to discuss them separately, too. I may produce (vague quasi-) arguments why these "laws" should work; and arguments why they cannot work. But I can also produce more interesting things – namely why these "laws" are more interesting (and self-promoting) than generic similar "laws" of the same character (almost equivalently, why these laws have a mathematical beauty); and why the "laws" may ultimately work within a trading community with a certain mood and a strong enough belief system.



I will only talk about the "death cross" and the "Fibonacci retracement" that I have already mentioned in 2005. Great.

So first, why would the "death cross law" be convincing at all? Well, we are comparing the average price in the recent 200 days; and the average price in the recent 50 days. They are average prices of some intervals of time whose "center" sits at moments which are 200/2=100 days ago; and 50/2=25 days ago. If you imagine that the price changes linearly for a while (and it's true for linear regressions of any graphs), the fact that the 200-day average was lower than the 50-day average roughly means that "some blurred price assigned to 100 days ago" was lower than "some blurred prices assigned to 25 days ago" – which indeed represents a rising trend (the price goes up from minus 100 to minus 25 days). When this inequality reverses the direction, we get the opposite trend.

Needless to say, the values "200 days" and "50 days" are completely random (well, 200 days is supposed to be "how much a trader is looking to the future"; 50 days is some short enough interval that is enough to smooth out the short term wiggles, however). We could compare the moving averages over two periods that are arbitrarily long; we could even compute different averages, including weighted averages with some non-trivial extra weighting function, geometric averages, or we could invent an infinite-dimensional space of similar possible (but more contrived) laws which still morally convey "the switch from a rising trend to a dropping one". There exists no sensible proof (and it's clear that there can't exist any sensible proofs) why this popular "death cross 200,50 law" should be true or more true than all the competitors, so it must be a superstition or a "convention in a belief system".

The previous paragraph argued that the "law" must be a superstition because it's randomly selected from an infinite-dimensional space of morally similar but unequivalent "laws". But we may go further. We may present incomplete but persuasive arguments why the whole class of these candidate "laws" is superstitious – or even falsified. What is wrong with them? Well, the technical analysis as a whole should be impossible simply because it shouldn't be straightforward to make predictions about complex systems, especially those about their future. To strengthen the point, we may refer to the "efficient-market hypothesis" which I largely believe (in the context of capitalism that still works: so whether the hypothesis is right is pretty much tautologically decided by the question whether the markets are efficient and sometimes, they fail to be, like in the "subcommunity of traders" that pay huge amounts for nearly worthless Tesla stocks and others in the ARKK funds). The efficient-market hypothesis says that the prices fairly reflect the information known to everybody or the body of traders and when you perceive a deviation, it is just a deviation of your knowledge from the collective one and that should be classified as noise, especially if you are a random trader, and it must be expected to be uncorrelated to some other, similar noise in the past.

There is also a very general internal contradiction in the whole philosophy that "you may deduce the bullish trends or bearish trends from some inequalities between the functions of the past data". What's wrong with this whole idea? Well, if you believe the "death cross law" or any generalization of it, you basically believe that the price will go up (or down) because it has some inertia (you are psychologically a "momentum trader" which really means a "bad investor", Buffett would agree). You may determine the sign of the trend from an analysis of the past and it will apply to the future, you assume. However, the problem is that this very "law" is being used to deduce the change of the sign of the trend. But the very assumption that the trend sign is changing contradicts the other assumption that the sign of the trend for the future coincides with the sign determined from some recent past. That's morally bad. The conclusion is that any "death cross-like law" unavoidably says that the signs of the trends may change but they last for a very long time in order to be effectively trusted and there exists a precise delay after which you may determine the change. The internal contradiction isn't sharp but the tension is intense, all such hypotheses are stretched because they exaggerate the importance of both the stability of the trends (the very idea that one well-defined sign of a trend is important at every moment for a random-walk-like function just looks damn stupid to me!) and the change of these trends. None of these things is really useful for predictions, the efficient-market hypothesis says, and it seems even worse to believe that the cacophonous union of these mutually contradicting principles is very reliable for the predictions.

But in the real world, we have lots of traders who pay attention to the "death cross-like laws". In particular, a fraction of the Bitcoin anti-investors (hodlers etc.) are learning about the "death cross" these days, some become terrified, and they will sell the Bitcoin to protect themselves against the destiny – the dropping Bitcoin price. We will see whether this scenario materializes. The empirical data are obviously mixed (but in most of them, the evidence for "YES" seems stronger than expected from the efficient-market hypothesis). Some "death crosses" in the past indicated huge Bitcoin price drops; others, not so much, and you may even find some intense rising Bitcoin prices in the vicinity of a "death cross".

The more the community of traders is exposed to the memes and propaganda about the technical analysis, e.g. "death crosses", the more likely it is that the "death cross" will really work because many traders are actually behaving as if these predictions about the price were facts or likely facts! And I must add that if the technical analysis is true for some particular asset, it should be particularly true for things like the Bitcoin whose intrinsic value is zero – and therefore almost all the psychological processes leading to the decisions to buy or sell are determined by something like the technical analysis! In the cryptocurrency case, there are no fundamentals (except for Musk's tweets and governments bans of the Bitcoin or their adoption of the Bitcoin etc.). So everything is about the technical analysis-like astrology which is why its "laws" could work rather well!

The other "law" of the technical analysis that I would like to mention are the "Fibonacci retracement laws". The most notorious one is that if the price goes from a turning point (local extremum) \(A\) to another turning point \(B\) and then it moves back towards \(A\), there is a natural candidate for another turning point,\[ B - 0.618(B-A) \] where the number \(0.618\) should have infinitely many digits and it is the golden ratio one, namely \((\sqrt 5 - 1)/2\). So the lengths of the intervals between the turning point have an interesting ratio, the golden mean. Under the 2005 article, Robert said that such a "law" is interesting because we are interested in roots of simple polynomials with simple integer-valued coefficients. Many of these roots are too simple rational numbers. That's true for all the linear polynomials as well as the quadratic polynomials like \(x^2-x\). Rational numbers are alright but you don't sound esoteric enough if the parameters of your mysterious rules to predict things contain just dull and mundane rational numbers! Some quadratic polynomials can be easily factorized to linear ones (with rational coefficients), others have complex (non-real) roots and non-real numbers can't be the results of observations, and the polynomial \(x^2+x-1\) is pretty much the simplest nontrivial polynomial with small integer-valued coefficients that doesn't suffer from any of these problems. And that is indeed the polynomial with the "golden mean" roots.

The golden mean has been said to exist in many natural systems such as the "human body or its organs". I always had to laugh about the latter statement. A person whose aspect ratio is 1.618-to-1 is damn fat, indeed! And if your legs have this ratio, well, that's really terrible. If your body proportions are the golden mean and you live in Japan, you may hire an even fatter person to feel better. The company is called Debucari.

OK, so I think that lots of the claims about the appearance of the golden mean are pure nonsense but it may appear in some cases because the quadratic equation with these roots may be extracted from rather natural conditions. Why would the "Fibonacci retracement law" work for the turning points of prices? Let me tell you why the golden mean may emerge from something that sounds like a saner "law".

Look at a price graph and identify three turning points, \(A,B,C\), which are three values of the prices. There are intervals between them or the moves whose lengths are\[ C-A,\quad C-B, \quad B-A. \] I have ordered the values to have \(A\lt B \lt C\) which means that the three differences above are positive. Now, we may postulate a rather natural condition. These three lengths have ratios which are integral powers of a universal constant \(Q\). It means\[ \{C-A,\quad C-B, \quad B-A\} = \{D\cdot Q^K,D\cdot Q^L,D\cdot Q^M\} \] for some \(Q\in\RR^+\) and \(K,L,M\in\ZZ\). A constraining fact is that integers such as \(K,L,M\) cannot be continuously adjusted. We may only continuously adjust \(Q\) and we must obey two continuous conditions, namely that two of the three lengths above are particular multiples of the third length. And indeed, the simplest "Fibonacci retracement law" follows from some of the smallest, most natural integer choices of \(K,L,M\), namely \((K,L,M)=(0,1,2)\). Why? The longest interval \(C-A=D\) while the upper difference \(C-B=0.618\cdot D\) while the remaining distance \(B-A = (1-0.618)D = 0.382D = 0.618^2 \cdot D\). The magic is that the squared (small) golden mean is exactly the same number as the difference "one minus this small golden mean". Needless to say, this is just another equivalent way to describe the "golden ratio" quadratic equation! The prettiest form of the equation is arguably \(1/0.618 = 1+0.618\).

So with the golden means choice, the ratios of the "moves" (distances between the turning point prices) are powers of the universal constant (the golden mean) no matter which of the two "moves" (among the three candidates) are picked and compared! In this sense, the "Fibonacci retracement law" is a "mathematically nicer" or "more robust" law than a similar law where 0.618 would be replaced with a random number! If I really try to use some theoretical-physics-like jargon, I would say that unlike its competitors with different values of parameters, the "Fibonacci retracement law" brags an \(S_3\) (permutation) symmetry; it is invariant under the permutation of the three turning points! "Triality" of \(SO(8)\) is a more amazing example of the enhancement of a \(\ZZ_2\) to an \(S_3\). In this case and many others, the mathematical beauty isn't enough. But to say the least, it is enough to inspire the traders and make the "Fibonacci retracement law" (often) true through its being a self-fulfilling prophesy. Many traders believe that the calculated price is exactly where the next turning point is going to be and that is why they sell above the price, buy below the price, and they are therefore driving the whole price chart to the "predicted" scenario!

While I surely appreciate the mathematical beauty of (otherwise irrational) laws such as the "Fibonacci retracement law" (in spite of the fact that the beauty is less than 0.0001% of the beauty of string theory), it changes nothing about my statement that this "subdiscipline of economics" is on par with astrology. Why? They are totally analogous because astrology was often mathematically beautiful, too! I am especially thinking about Kepler's map linking the planets to the Platonic polyhedra. Platonic polyhedra are mathematically special and robust, much like the golden mean, and it is mathematically nice to associate them with objects in the real world (planets), much like it is nice to hire the golden mean to play a role in the financial markets (to determine the future turning points)!

But mathematically pretty laws are not necessarily "fundamentally true". You need some empirical data (which are mixed in the case of technical analysis) or some truly positive argument why these laws should be true (ideally a full proof!). Without real evidence (or proof), such "laws" remain unfounded. But I would also emphasize that before it is decided whether similar mathematically pretty laws are true or false, it is legitimate to assume that the mathematical beauty and robustness makes it more likely that the proof of "YES" will exist! ;-) In the case of the financial markets without traders manipulated by the self-fulfilling prophesy, I think that the correct proof ends with the conclusion "NO", however.

And that's the memo.

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