Stock prices and the shape of randomness

I remember an illustration in an article by Heinz von Foerster in the 1960s. In the first picture, a set of tiny cubes were represented, piled up in a random manner. The reader was told that these were steel cubes magnetized on one of their sides. The second picture showed what had happened after the cubes were put in a container that was then thoroughly shaken for a while: the cubes had aligned to create a complex structure with perpendicular ridges shooting out and closing up. There was still some randomness in the arrangement but it was so delicate and apparently purposeful that it was difficult admitting there had been no design of its final appearance. It looked like the structure had, so to say, emerged through the action of an “invisible hand.”

A termites’ mound is a complex structure that can be up to 18 feet tall. Within the shell is a maze of galleries and several “cathedral” type chambers. Building such a structure would – or so it seems – require sophisticated planning. This is however not the case. Worker termites that built the structure were following a very simple pattern: they would discharge the mud pellet they were carrying wherever a heap would already be present (the initial dropping down would take place on pre-existent irregularities). Columns would thus emerge little by little. The lack of precision in letting go the mud pellet would mean that, having reached some height, a column would begin to slant until it is bending so widely that it touches a neighboring column and an arch is therefore created. A “cathedral” chamber would result from multiple similar arches. In retrospect it would very much look like an “invisible hand’ had guided the insects. Nothing of the kind had actually happened: the chambers resulted simply from random variation in building the individual columns.

In both cases, whether the magnetized cubes or the termites’ mound, the end product’s structure has not resulted from goal-oriented organization but from what we call “self-organization,” meaning by that that the structure obtained is nothing more than the “shape” that randomness has taken due to the constraints proper to the building blocks (e.g. the cubes) or to the builders (e.g. the termite workers) in their interactions.

I’ve mentioned the “invisible hand” intentionally as the phrase is apt for describing “self-organization” by contrast with goal-oriented organization. The “invisible hand” was of course mentioned in 1776 by Adam Smith in his “Inquiry into the Nature and Causes of the Wealth of Nations.” In a recent paper of mine (*) I drew the attention on the resemblance between actual price curves, such as for stocks, and the curve obtained through generating a “price” curve through flipping a coin. Here again, the semblance of “organization” amounts essentially to “the “shape” that randomness has taken due to the constraints proper to the building blocks” e.g. that the process is generated by the encounter of two “populations”, one of buyers and one of sellers.

In price formation, such as with the price of a stock, organization is minimal: prices essentially oscillate although not always in a totally foolproof manner as the price may occasionally crash. Back in 1990 when a trader of futures, I had noticed the similarity between price formation and the oscillations in population in a simple predator / prey system. I had built accordingly an automated trading system based on the Lotka-Volterra equations. The similarity between the way a price on the stock market and a two-population predator / prey system evolves undermines the view that the markets are structured because of some of our virtues such as our industry, astuteness, etc. as the same degree of organization can thus be observed in absolutely unfettered natural processes.

In a correspondence, Didier Sornette, the author of Why Stock Markets Crash: Critical Events in Complex Financial Systems, Princeton University Press (2004) called my attention to the fact that when a larger number of species is involved, stability in population can obtain: “One point to make it stronger: in ecology (predator-prey), we find cycles ONLY where there are just one or very few species of predators and preys (a famous example is the Lynx-Caribou system in the North of the Northern American continent). In all other balanced ecological systems, there are several predator species and many prey species and this gives rise to much more smooth dynamics.” And he added: “It is interesting to think that the business cycles and financial cycles could be indeed the result of a poverty of diversity.”

A surprising conclusion I reached when observing the behavior of a multi-agent simulation I made of the stock market (*) was that when the agents (who could alternatively be buyers or sellers) were developing trading strategies in any amount, this would raise the likelihood of a crash. That outcome was actually logical as attaining a perfectly counterbalancing effect between trend-following strategies (positive feedback) and contrarian strategies (negative feedback) was more difficult to obtain than allowing the empirical “law of averages” to operate when the decision to buy or to sell was as if with the toss of a coin. I determined that it is in fact beyond the ability of any buyer or seller to predict correctly whether the market price is about to go up or down. The probability for him or her to be right about the next price move being up or down never departs therefore significantly from 50%, and this ensures that the stock market experiences some degree of stability.

An emergent structure may arise from the spontaneous interaction of multiple agents. We may call that structure “self-organizing.” That “self-organization” is often nothing more than the “shape” randomness takes when a large number of interacting elements are present, i.e. a purely “statistical” outcome – however “organized” the end-product may look like (von Foerster’s magnetized cubes; a termites’ mound). This applies as well to the markets where the action of the “invisible hand” in price formation is no more remarkable than when seen at work in the way termites build their mound.

(*) Adam Smith’s “Invisible Hand” Revisited, Proceedings of the 1st World Conference on Simulation of Social Systems, Kyoto, August 2006, Vol. I, Springer Verlag: 247-254

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