## Monday, April 18, 2011

### The possibility of cognition

The most fundamental question raised by “The Blank Swan” may be that of the level of cognition of the market an individual can acquire, and the usefulness of such a cognition if it is, at all, possible. Such a matter is obviously paramount to the validity of Technical Analysis. The untotalization of possibilities Elie Ayache shows with regard to financial markets, seems to invalidate most of the current attempts at thinking this market in explicit terms, as most, if not all, of these attempts are ultimately based on probabilities computation (and therefore on unwarranted, even false, assumptions about the totalization of possible states), and this is indeed the case for Technical Analysis, though I believe that the fractal analysis I have endeavored to develop in this blog provides for an untotalization by means of an implicit multifractal model, where Hurst exponent keeps being recomputed (I however start thinking this model still falls short of being efficient at a theoretical point of view). In this post, I therefore intend to examine, from the standpoint of such a critic of probability theory, whether some kind of cognition is still possible as to what the market is going to be.

I-READING A BOOK BEFORE IT IS WRITTEN

The best way to read a book before it is written is to write it, and that is, to some extent, what Elie Ayache is proposing us to do in The turning. There he shows how the market can be dealt with, not by predicting it by computing some probabilities artificially attached to possible states of the world, but rather by writing contingency, i.e. writing contingent claims. However, the book of the market is not written by any single individual (or even any single intentional entity), as is clearly said on page 43:

“The place of the contingent claim is nobody’s place in particular. It falls to no subject to assign a price to the contingent claim or to reflect it in his mind.”

Writing a contingent claim, therefore, does not quite amount to write the book of the market. It does amount, however, to protect one’s financial interest from the uncertainty of the market, from its contingency. In this sense of one’s direct financial interest, as being under the threat of contingency, writing of contingent claims indeed appears as the means to “mediate contingency”. The question which interests me, at the level of Technical Analysis, is whether we can mediate contingency beyond this direct financial interest, and still do that in a speculative manner (in the philosophical sense of the term “speculative”), in other terms, can we speculate (financially) speculatively?
As to read the book of the market before it is written, it obviously is not possible, as such a thing would clearly come down to write it, and as such, it would make it redundant, and therefore destroy it. If the book of the market was to be written by one subject (or if its writing could be seen as being the work of one subject), it would immediately cease to be a market, as a market can only be a place of exchange, necessarily supposing the presence of at least two independent subjects.
Nonetheless, speculative knowledge is not perfect knowledge of the phenomenon under inquiry, on the contrary, speculative knowledge is precisely imperfect, partial, fragmentary, as such a knowledge is rooted in the necessity of contingency, which implies the knowledge that perfect knowledge is illusory (not in an epistemological sense but in an ontological one).
As a consequence, we will not be able to read the book of the market before it is written, we will not be able to predict it in a deterministic way, nor will we be able to predict it in a probabilistic way, what we could endeavor to know however is the language of the market, and from knowing its grammar, we may be able to infer something about the market and its dynamics, just like a knowledge of a natural language allows us to expect a verb after a subject (or the reverse, depending on the language we consider). Such a knowledge may not be enough to diminish the absolute contingency of the market, but it should be sufficient to provide a basis for a speculative speculation, or, as Nishida calls it, an action-like intuition.

II-ACTION-LIKE INTUITION (行為的直感, KOUITEKI CHOKKAN)

Robert Wilkinson presents the concept of Action-like Intuition, that he calls Action-Intuition, in the following manner:

“We must experience the world in order to act on it, and we learn to perceive the world better by acting on it. Just as he [Nishida] insists that practical reason is more profound than the theoretical, so he insists that our natural mode of being-in-the-world is not contemplative but active, an aspect of the constant mutual interaction between individual and the world. The idea that experience is a passive reflection of the world he regards as entirely false: ‘intuition, separated from action, is either merely an abstract idea, or mere illusion’(Intelligibility and the Philosophy of Consciousness, p.208). Action-intuition, like any other form of action in Nishida’s late thought, is a mutual relation of forming and being-formed: ‘Action-intuition means our forming of objects, while we are formed by the objects. Action-intuition means the unity of the opposites of seeing and acting.’(ibid, p.191)
[…], the philosophy of pure experience leads Nishida to take a view of concept formation diametrically opposed to that to be found, for example, in the classic empiricists, according to whom concepts are arrived at by some process of abstraction based on noting common elements in numerically disctinct perceptions. Concepts are not formed in this way in Nishida’s view. We form concepts in the course of action-intuition: ‘Conceiving something through action-intuition means: seeing it through formation, comprehending it through poiesis.’(ibid, p.210)
The basic thesis of the philosophy of pure experience is that the world is a construction from such pure experience, and manifestly such construction has to have some method: action-intuition is the basic formative operation by means of which this construction is carried out. […]. Cognition has to be understood as a form of dynamic, reciprocal expression”

[Nishida and Western Philosophy, Robert Wilkinson(2009),p.120-121]

While Nishida obviously considers these remarks to apply to the whole of reality, and while such a stance may be argued against, I believe there is not much argument as to the relevance of his remarks when it comes to the market. Cognition, in this domain, can only “be understood as a form of dynamic, reciprocal expression”, and concepts formation, according to Nishida, can only occur within a poietic attitude, that is an active one, and not a detached, analytical one. This dimension is well-established by Elie Ayache in “The Blank Swan” with regards to the writing of contingent claims, and particularly with the logic of inverting dynamic replication with the view of implying volatility. When it comes to Technical Analysis, what Nishida is saying, also has an interesting consequence, in that it tells us, that, in order to grasp the market, we must grasp the grasping itself. We therefore need a Technical Analysis tool that is essentially self-referential, there is however a difficulty in understanding this sentence, that lies in the difference of velocity between the processes in historical reality, which are the ones Nishida is treating, and the processes in the market which are the ones interesting us.
The remarkable characteristic of the market is its proximity to the virtual (wherein speed is infinite), a consequence of this proximity is its very high speed, and its emancipation from causality. This high speed also accounts for the absence of a subject-object distinction because such a duality does not have the time to accrete. We are therefore confined, within the market, in a relatively unfriendly environment when it comes to scientific investigation (even a probabilistic one). In this context, self-reference itself becomes an ill-defined notion, since we don’t even know on which entity to apply such a self-reference. Of course, we may say that the market is self-referential, in some sense, but since we don’t know what the market is, since we can’t reduce it to a subject or an object, we have no direct way to comprehend such a self-referentiality in order to translate it in a cognition (be it a partial one) of how the market may evolve. This ambiguity is enough to invalidate a TA tool that would simply be self-referential since such a tool could only be efficient if every market-actors were to use this specific tool, which is obviously impossible. What we need is a tool that is self-referential in the way the market (whatever it is) is self-referential, we therefore need a TA tool that accounts for the very grammar the market is writing itself in.

III-THE GRAMMAR OF THE MARKET

What I call the grammar of the market, extending the analogy made by Elie Ayache between the market and a book, asks for a little precision here. As said above, the velocity of the virtual is infinite (because the virtual is not situated in time), and the market inherits some of this velocity more directly than history, as such it appears much faster than history and mundane life (this high speed also contaminates real history and accelerates it in some way, this is particularly visible in recent times). Natural languages also happens in history and as such, their grammar seems relatively constant to us, nonetheless, natural languages change, and so do their grammar, we must therefore expect the grammar of the market to change faster than the pace we are accustomed to with natural grammar.
In order to elucidate what we can know of this grammar (that can only amounts to some structure of it, and therefore to a meta-grammar), we must first look at the market globally and that leads us to recognize that it has fractal features. This, in itself, is already a very interesting finding, one from which I have tried to develop some TA tools , but many unknowns remain, such as the adequate period for calculation, the real meaning of fractal dimension, the scope of the probabilistic model (Fractional Brownian Motion) used,…,and the mathematics that sprang from the fractal theory seem relatively limited to clarify these unknowns. The holistic approach of Fractal Theory only provides a very global view of the price dynamics, and Mandelbrot himself even excluded its possible application either to investing or to trading; in his view, Fractal Theory only served to invalidate probabilistic and statistical inference from the market.
However, to obtain a model that would provide a higher interest in building TA tools, we need to start considering a reductionist approach at some level. Again here, I must insist, it would be absurd to look forward obtaining a precise account of the working of the market, when I am talking of reductionism, it must be clear that I mean a very partial one, that will inevitably fall short of elucidating the processes of the market. Reductionism may indeed not be the right word, what I am intending to look at, is something in between holism and reductionism. Despite such reserves, I believe there may be something valuable to find and to explicit about the market, and that this something may lead to a deeper understanding of the whole reality.

I said earlier that the fundamental properties I wish to look at are to be found at a topological level. One way to study such properties is to find a space homeomorphic to the one we wish to investigate, and that is simpler to manipulate.
When it comes to self-similar fractals, which are typically build by IFS (Iterated Function Systems), it is known that we can find a map ψ so as to assert the homeomorphism of some self-similar fractals with a space of p-adic integers:

$\psi _{b}:\mathbb{Z}_{p}\rightarrow [0,1]$
$\psi _{b}(\sum _{i\geq 0}a_{i}p^{i})=\frac{b-1}{p-1}\sum _{i\geq 0}\frac{a_{i}}{b^{i+1}}$

From this map, we can obtain the fractal dimension of the constructed self-similar set:

$D=\frac{log(p)}{log(b)}$

For b=3 and p=2, we get:

$\psi _{3}:\sum_{i\geq 0}a_{i}2^{i}\mapsto \sum_{i\geq 0}\frac{2a_{i}}{3^{i+1}}$

Where this homeomorphism is actually mapping the ring of 2-adic integers onto the Cantor Set
Alain M. Robert provides a more detailed discussion of these maps in "A course in p-adic Analysis"(pp.8-17)

Of course the fractals we wish to investigate in Finance are not as simple as those built by IFS, in particular, the self-similarity is not strictly true. Nonetheless, I think such a direction may lead to some interesting results. The ideal objective would be to establish a general procedure to find a map between a set of arbitrary fractal dimension and a subset of the space of p-adic numbers. I believe such a question is still an open one, and I am not sure of the advancement of research in this area (or even whether there are any), as I am just starting to look at this question.
The fields of p-adic numbers also present another interesting feature when it comes to account for the process of decision-making at an atomic level. The market is clearly the product of multiple decision-making processes, and as such they are all, individually, rooted in a valuation of reality. While we are well-acquainted with the classical absolute value that leads to the intuitive definition of distance (metric), p-adic fields are equipped with an ultrametric that satisfies the strong triangle inequality.
Whereas a metric satisfies the following triangle inequality:

$d(x,z)\leq d(x,y)+d(y,z)$

An ultrametric satisfies the following:

$d(x,z)\leq max(d(x,y),d(y,z))$

Such a feature leads to rather counter-intuitive results, when we try to visualize them in geometric terms, such as the following formula, known as "The strongest wins":

$\left |x \right |> \left | y \right |\Rightarrow \left | x+y \right |= \left | x \right |$

However, if we think in terms of decision making, we will indeed tend to ignore menial parameters to base our decision on the one parameter we consider as the most relevant. In that, we seem to be closer to an “ultrametric mode” of thinking.

These considerations are still far from exploitable intuitions, and I myself am not very sure whether they will lead anywhere. Once again, I am only in the process of learning about this problematic, and anybody is welcome to criticize or comment, either positively or negatively, on such ideas.

Anonymous said...

I could refer to the the Incompleteness Theorem of Kurt Godel. In the technical analysis there are some things (axioms) that we just cannot prove.

On the other hand most of this knowledge was obtained as gnosis knowledge.

John Last

Jean-Philippe said...

Hi John,

As for Godel, I am not sure how his theorems may apply to technical analysis. Godel provided results for mathematical axiomatic systems, in which axioms articulate a theory in a very precise and specific way. When considering "axioms" with regard to TA, I am not sure his work will apply with the same clarity.
TA indeed uses "principles" but even those are difficult to define clearly.

The problem at the heart of the main post and maybe at the heart of this whole blog is indeed the status of the knowledge that is provided by TA.

Relative to this main problem, a question that first comes to mind is the criticism of the "self-fulfilling prophecy": TA is efficient only insofar that TA is widely practised and that is easily understood from the way offer and demand articulates price variation, For instance, if everybody follows a Bollinger Bands strategy, then Bollinger Bands become a relatively efficient indicator. Is that all there is in TA?
It may be, but if so, TA becomes part of the market itself, and only a part since a part of the market is made up by non-speculators, who are exchanging a product because they need it for some other reason. So TA is therefore place in a rather awkward position whereby it is assumed to be both a part and a representation of the market.
It is this problematic that is at the heart of my thinking these days, and it is not unaffiliated to some other big questions at the heart of science, such as quantum theory or social sciences, wherein the separation between the analyst and the object of analysis cannot be posited in terms as strong as is usually thought in classic sciences.

Cheers

Jean-Philippe

Anonymous said...

Hi Jean-Philippe,

As I'm not a well versed mathematician, is it possible for you to post a visual example of what you mean. Greatly appreciated!

All the best,
CamaRon

Anonymous said...

One more thing: I believe you are on the right track in terms of an understanding of the market, when going deeper into the application of the fractal bands. I have been, and am using them extensively. Not in the traditional "Bollinger" sense, but instead to see contraction and expansion of price; i.e. where a move starts, and where it ends. Also, I can easily use them for estimating what is normally referred to as support/resistance, or supply and demand. See for yourself! The settings I use are: 15/10/3.0

Best,
CamaRon

Jean-Philippe said...

Hi CamaRon,

I am not sure which part you want me to post an example of. If it is about the ultrametric understanding, as I said in the post, it is difficult to get an intuition of it using visual geometry as it is so different from the usual metric.
Anyway, you can check on wikipedia: http://en.wikipedia.org/wiki/Ultrametric_space

There also exist several resources for a basic study of p-adic numbers on the net such as the followings (but you can find others with a google search):

For a more thorough study, I also referred, in the original post, to the book of Alain M. Robert.

Anyway, I might develop these matters more in detail in future posts, so far, they are still at the level of intuition.
A concrete exemple is maybe to think of the price elaboration from fundamental news: if we consider the EUR/USD pair for example, it is well-known it often displays a positive correlation with the Dow Jones. We can however imagine a situation wherein the Dow Jones is up, and some news are out reporting some uncertainties as to the sovereign debt of some EU countries. In such a case, we'll have two opposite fundamental influences, the DJ theoretically pushing the EUR up and the sovereign debt pushing it down, by considering an ultrametric logic ("the strongest wins"), I simply mean that one of these influences will merely disappeared and only one of them will be effective, instead of offsetting each other.

Hope this clarifies a bit.

And thanks for your encouragement and for sharing on your setup, I will try it out.

Cheers

Jean-Philippe

Anonymous said...

Okat, I'll check that out.
Best,
CamaRon

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