All that Technical Analysis is about is to reveal the underlying signal that is nested in the time series. In a narrow sense (for a specific kind of time series), TA is a form of signal processing. There is however a fundamental difference with classical signal processing; when we process a signal, we know beforehand that the signal exists and our goal is to exhibit it free of noise, to clean it in order to enjoy thoroughly its meaningful content, and to possibly alter it. A contradiction is then becoming apparent in TA since we assume the existence of such a signal while all the signal processing we perform seems to reveal its absence.

This obviously applies to the fractals-derived tools we may use to analyse the time series, as indeed, what this analysis is telling us is that the time serie keeps evading its nature as a fractal, since the Hurst exponent (and therefore the Fractal Dimension) keeps changing. These perpetual changes mean that the time serie cannot be identified as a fractal, at least not via the methodology we are employing, we basically failed to identify (or even to detect) the signal we intended to study. Nevertheless, we ignore this conclusion and go on with our assumption of an existing signal in order to make a decision to enter, modify or exit a trade.

That the time serie is not a fractal is not a surprise, since fractals, after all, are mathematical objects that are not to be found in nature, they are just models that may be helpful to describe natural phenomena. Even the most classical examples of fractals are not genuine fractals, for instance the coast of Brittany is clearly not a fractal since its “fractalisation” stops at best at a molecular level, it does not go on infinitely as is the case for a mathematical fractal. But even, in this sense, a financial time serie does not qualify as a phenomenon that can usefully be modelised as a fractal, since its parameters keep changing.

So is there any sense in which the market could be said to be fractal, given that the time serie does not seem to be satisfyingly modelised by a fractal?

In “The Blank Swan”, Elie Ayache provides a very interesting element to answer this question at the page 295:

“If it were known what the price of some traded asset would be in the next instant, and different from the present price, it would immediately become the present price because buyers or sellers would immediately want to hold the asset at that price. The price function catsches up with its virtual before it becomes actualized. Another, more regressive virtual is therefore needed.

The result of this doubly-composed breaking and differentiating is the infinitely broken line, whose simplest instance is Brownian motion. It is essentially fractal. There is no scale at which the differentiating can settle and the function look smooth. Let us call it the line of the market. From it, the whole world of derivatives unfolds, which will give us, retro-actively, the point of the market.

The point of the market does not just stop at noting that the underlying must follow Brownian motion (or some other, more complex and jagged trajectory), in order that its next movement may always be unpredictable. The abyss of differentiation, opening at every point, must not concern the price of the underlying alone but the price of any 'virtual' derivative that might be written, right there and right then, in that pit (for there is only one pit), on that underlying. That means that all the potential coefficients, not only of the 'initial' Brownian motion (or any other more complex process) but of all the following processes that will complicate that Brownian motion, differentiating it even further, should themselves never settle but perpetually differentiate. If there were a stage at which the coeffcicients settled, the price of the corresponding derivative would become a deterministic function of the preceding prices and would no longer admit a market, that is to say, an unsettlement, an unpredictability and a differentiation of its own.” [EA10, p.295]

From this extract, we can see that the fractal behaviour of the market is not to be identified with the time serie as a fractal, rather the fractality applies to the existence of the market as a whole, including the infinitely many virtual derivatives. At best therefore, the time serie would appear as a truncated version of the 'fractal of the market'. What this fractality implies cannot be detected at the level of the time serie, it may however be possible to account for this fractality, but this is not going to be done by considering the process of price formation, rather I believe its implication must be found at a topological level.

Without presuming what a topological study, which I am not yet equipped mathematically to conduct, would reveal, I may try to precise a bit what I expect to find from it. For that, I would use an analogy ( as all analogies, this one is limited and should not be carried forward too much): It seems to me that the classical approaches (TA or Mathematical Finance), by assuming the exitence of a signal or a stochastic process, assumes that we are exploring an unknown land by following an already built road. We don't know where the road is going, and we can only detect it by walking it, so we don't even see where it is leading us, we're just happy to follow it. Unfortunately, the more we are walking on the supposed road, the more we are told that there is no road at all (this is the absence of signal or of process). A topological study should not tell us anything about a road, but rather it may inform us of the topography of the land we are travelling through and help us, in some limited way, to chose the direction where we want to take the next step.