2012/01/24

Markets as complex systems

Markets are driven by people (and, lately, algorithms). Their decisions (driven by their motives) drive prices. However, economic theory has had little to say about how these interactions ‘add up’ to give the aggregate market dynamics we observe. It is a convenient excuse to say that markets are efficient, and so what we observe must be because of news events, which people immediately react on and incorporate in prices. This seems a little fanciful. We may consider instead what some have called the interacting agents hypothesis, which says that we can explain (inefficient) market behaviour by looking at the aggregation of individual interactions. I recently examined a class of models based on dynamical systems theory that does just this.

What we are trying to explain

In virtually all markets a number of stylised facts have been found. The ones that contradict the standard model of financial markets (the random walk model) have often been called anomalies, as if they are aberrant and infrequent. They are not. It is the standard model that is aberrant. Here are the things I mean

Unit roots

Standard tests of whether markets follow a random walk (whether they have a unit root) are unable to reject it. This would, on its own, seem to confirm the standard model. However, there is ample other evidence of market behaviour to reject it. That these standard tests thus still give this result is very interesting.

Fat tails

Market returns exhibit fat tails. Technically, it means that they have a distribution with a high kurtosis. This means that very large price movements as well as negligible price movements occur more frequently than the standard model would predict. A very interesting feature of the returns distribution is that the fat tails are observed for daily returns, but monthly and yearly returns appear to be approximately normal. One could argue that even daily returns are made of thousands of intra-daily returns – if these followed the standard model, the daily returns would have a normal distribution by the central limit theorem. Why is it not applying, or why is it acting so slowly? The interacting agents models attempt to explain it by including dependencies between agents (remember that the central limit theorem needs independent observations).

Volatility clustering

The market has periods of relative calm interspersed with periods of highly volatile prices. The autocorrelations absolute returns (and to lesser extent squared returns) is high. Markets are well-behaved most of the time, but sometimes something happens....

Dynamical systems

Dynamical systems theory, also known as chaos theory is a branch of mathematics that looks at the interactions of many particles. Non-linear interactions (even if each particle behaves according to a relatively simple set of rules) can result in surprisingly rich behaviour of the system as a whole, know as emergent properties. I have mentioned chaos theory before because it naturally leads to power law behaviour. These systems may have infrequent but sudden transitions, corresponding to critical points (or singularities) where the system makes a transition from disorder to order. Such properties may explain the relative calmness of markets most of the time, interspersed with periods of volatility.

The models

The models I have looked at use two kinds of agents, chartists and fundamentalists. The former are technical analysts, trend followers. They are supposed to exaggerate market movements. The latter believe the market will return to its fundamental price. They are supposed to have a stabilising effect on prices as they buy when prices are below fundamental value and sell when they are above. I have written about these investment philosophies before.

The basic mechanism of these models is imitation. Chartists may be optimistic (buyers) or pessimistic (sellers). This mood (of optimism or pessimism) can spread from trader to trader like a virus. As such it can be called contagion, infection, or herding. This is a simple way of modelling trader psychology. There is a force that results in people getting on the bandwagon. And the more people there are on already, the stronger the force. This is not necessarily irrational. It makes sense to look at the opinions of your peers as this provides information (especially if you do not have other information). For traders it may be very important not to underperform the rest of the market (as this may get them fired). The surest way to prevent this from happening is to follow the crowd.

As long as people have different opinions, i.e. no one group of traders dominates, the market is in a state of disorder. Participants' actions tend to counteract each other and the market is stable. When any one group dominates order is created in the market. Many traders agree and take the same action. Their actions reinforce each other and in the case of a dominance of optimistic (or pessimistic) noise traders, exaggerate market movements away from the fundamental value. It is the actions of fundamental traders, who then act on the mispricing, that drives the prices back toward fundamental value. These models thus explain intermittent waves of optimism and pessimism. The data generated appear to fit the stylized facts.

Caveat

Dynamical systems models are analytically intractable. We need to be happy with either grossly oversimplified models or else only approximate or simulated results. Even the more complicated models are a drastic oversimplification of reality. They may provide a useful explanatory tool, but applying them in a useful way, may still be a way off. Although I have heard of trading being done with chaos theory models, I do not know what form they take. There is certainly a lot interesting research that can still be done.

One important point that must be made is that these models do not (and cannot) prove that the efficient markets hypothesis (EMH) is incorrect. In fact, none of the stylized facts I mentioned contradict the EMH (they do however contradict the far stronger notion that markets follow a random walk). Human behaviour, at least, is partly observable (maybe even quantifiable) whereas the supposed news process driving fundamental prices is far more ethereal. If the source of market turmoil is primarily from trader behaviour, then we may hope to curtail it by appropriate policies and education. However, very little can be done if it follows from fundamental processes.

Disclaimer

I do not fully understand the models I have written about myself. I do not (yet) have the necessary mathematical knowledge. If you want to know more please see the essay I wrote which gives a more in depth discussion and references.

References

Please see my essay on this topic.

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