Simply Complexity (19 page)

It turns out that
this random walk represents the standard model of how financial markets move
. And so it is this model which most finance professionals use to try to work magic with our savings. Most importantly, we can see that the main assumption underlying this random walk model of the markets is that the next price movement is best described by the toss of a coin. Heads gives an up movement of the price by a certain amount and tails gives a corresponding down movement.

But that immediately sets off alarm bells in our heads, based on what we have seen in this book. A financial market is a Complex System, and the output of a Complex System will not generally be a random walk. In particular, we discussed in
chapter 3
that one of the emergent phenomena in real-world Complex Systems is that they tend to be characterized by a walk that is neither too disordered nor too ordered – more specifically, the patterns that are observed tend to have values of the parameter
a
which are not equal to the random walk value of 0.5.

We are right to be worried. There is now a significant body of research which confirms that the price-series produced by real-world financial markets are characterized by values of
a
which differ significantly from 0.5. And here comes the big shock. Not only does the
a
value differ from 0.5 for any particular market, but the
way
in which it differs from 0.5 tends to be the same
irrespective
of where that market is located. This provides a wonderful example of how emergent phenomena from a Complex System can have universal properties – but more of that in a moment.

At this stage, your emotions are probably mixed. On the one hand, it is fascinating stuff from the point of view of the science of Complexity. But on the other hand, it sounds like terrible news for our pensions in that we cannot trust the standard model of finance.

So the standard model that most of the finance world uses to calculate how markets move is not accurate. It assumes that the price
wanders around according to a coin being flipped – or equivalently, a drunkard walking. However, the actual walk that markets follow is much more subtle. In particular, the market produces values of the parameter
a
– or equivalently the fractal dimension
D
discussed in
chapter 3
– which are
not
equal to the values for a random walk. (The random walk values are
a
= 0.5 and hence
D
= 2, since
D = 1/a
.) And this discrepancy makes sense: a random walk has no feedback, which is why it can be produced by something as simple as a coin or a drunkard with no memory of the past, whilst financial markets are truly complex since they are riddled with such feedback.

In
chapter 3
we saw that fractals have a value of
a
which differs from the random walk value of 0.5. Fractals appear in the music of Bach and modern jazz, and the coastlines and mountain ranges in our everyday world (see
figure 3.3
). The reason that such patterns can also appear in the output price-series of markets, is because the markets are continually moving around between disorder and order – just like any archetypal Complex System. And like all Complex Systems, financial markets are able to make occasional forays toward ordered behavior such as a crash, or disordered behavior where no pattern at all occurs, all by themselves. This in turn is entirely in keeping with them being made up of a collection of interacting traders who feed off of global information about past price movements. Financial markets are Complex Systems and they cannot be described accurately by anything other than a theory of Complex Systems. Standard finance theory may therefore appear to work for a while but it will eventually fail, for example in moments where strong movements appear in the market as a result of crowd behavior. And this is far from being a minor flaw since it is precisely these moments when your money is most at risk.

So how do the major financial markets that we all need to worry about, actually behave? Recent research says that the average price-change over
t
timesteps in most stock markets follows
t
^a
with the value of
a
being larger than 0.5. In other words, the price in most stock markets follows a walk which is more persistent – more positive feedback if you like – than a random walk. Remarkably, it turns out that many major stock markets throughout the
world have similar values of
a
– typically around 0.7 – despite the fact that these markets are on different continents, have very different sizes and compositions, and have very different net values of trades each day. They may even have very different rules governing when trades can and cannot be made. For example, some smaller markets close at lunchtime, while others do not.

But why should stock markets in such different locations, with different sizes and rules, behave so similarly? In other words, why does there seem to be a universal pattern in the way in which stock markets move? Let’s think – the only thing they seem to have in common is that they are dealing with stocks. And that is the key. They are all made up of collections of decision-making objects (i.e. traders) which are continually feeding off past information about price-movements in order to make their next decisions. Whether it is a group of traders in China, New York or London, makes essentially no difference. Instead, it is the way in which these people make decisions based on past information which is important. It is not the actual decisions themselves, but the
way
in which they make them.

Just think back to our model of competing agents in
chapter 4
. The fact that they all fed off of the same information led to the emergence of crowds and anticrowds – and this same phenomenon is found to emerge irrespective of where that common information is coming from. So it could describe traders in the Shanghai market looking at past Shanghai prices, or traders in the New York market looking at past New York prices. It doesn’t matter, as long as the people involved are competing freely, based on the common information which is being fed back to them. The detailed interplay between the crowds and anticrowds will then dictate what is observed in terms of the actual price (i.e. the system’s output). In particular, the size of the price fluctuations will depend on the extent to which these opposing groups cancel each other out, and this in turn will vary in time as a result of the feedback in the system.

The fact that we see the same type of price behaviors for stock markets across the world also supports our earlier claim that the overall behaviors of different groups of people can be far more
similar than their individual members’ traits would have suggested. There is therefore reason to believe that Complexity Science is onto a winner in attempting to describe the collective behavior of human systems.

The bar attendance problem discussed in
chapters 1
and
4
can be fairly easily turned into a model of financial market movements which reproduces the behavior seen in real financial markets. In particular, the fact that the system can move itself between ordered and disordered behavior as a result of feedback allows it to exhibit a movement in prices which has the same values of
a
that are observed in real financial markets. This whole topic is discussed in our book
Financial Market Complexity
(Oxford University Press, 2003). Here I will just indicate the issues that need to be addressed in building such a realistic market model.

The main issue concerns the feedback of information. In particular, what is the global information in a financial market? There are clearly many types of information available to traders: for example, the price histories of assets, histories of traded volumes, dividend yields, market capitalization, recent items in the media, gossip, company reports. Any number of these information sources may actually be useful in making an investment decision for a particular asset. However, it is not our interest here to work out which of these information sources is actually useful; instead, we need to know which sources financial market agents tend to make most use of, since we are essentially modeling the population of traders. So if we take a moment to think about what we ourselves see most of, in relation to financial market assets such as stock, the answer has to be its price. The media is full of charts showing recent price movements up and down. Such charts also fill the majority of traders’ screens on trading floors. Hence it is reasonable to take the source of global information upon which the traders act, to be based on the past history of prices for the particular market of interest.

Next we have to decide on a method of “encoding” this past history of price movements in a simple way. Inspired by the
binary decision games discussed earlier, the simplest alphabet we can use is the binary alphabet of 0’s and 1’s – just as in the bar problem. A downward movement in price generally occurs when there is a large excess of sellers. Likewise an upward movement in price results from a large excess of buyers. In a similar way to the bar problem, where overcrowding and undercrowding can be represented as a 0 and 1 respectively, we can therefore encode the past history of prices by assigning a 0 to a price movement which is smaller than a given “comfort limit”
L
(i.e. the market is overcrowded in terms of the number of sellers), and a 1 for a price movement which is larger than
L
(i.e. the market is undercrowded in terms of the number of sellers). In the context of a financial market, the “comfort limit”
L
could represent a number of financial or economic variables, which could be either endogenous to the market or exogenous. An endogenous example would be to choose
L
according to the average market index. An exogenous example would be to choose
L
according to whether the news of the day was judged to be good or bad for that market. The comfort limit
L
could also be used to mimic a changing external environment due to some macroeconomic effect: for example, if interest rates are low people may be tempted to put their money into the stock market. Conversely, if interest rates become high then people may prefer to use a risk-free bank account. In other words,
L
indicates some measure of the attractiveness of stock, or the stock market as a whole, just as it indicates the attractiveness of the bar in the bar attendance problem.

There are, of course, no end of additional details that you could add in. For example: how do financial market agents actually win in the short and long term? What about the fact that each trader or investor will only have a finite amount of money to play with? And how about the fact that some people trade daily, some weekly, and some monthly? It turns out that many of these subtleties, when added into the model, tend to cancel each other out. In particular, the fact that the basic bar model produces a price-series with a similar value of
a
to real markets, remains robust to these additional bells and whistles.

This brings me to my philosophy for building models of financial markets, and for building models of real-world Complex
Systems in general, which can be summed up in terms of building a paper plane. As we all know, folding a piece of paper into a paper plane can give something that flies, and which therefore captures the essential ingredients of flight, i.e. the uplift cancels the downward pull due to gravity. In short, a paper plane flies for the same reasons that a big commercial jet does. A paper plane is an example of a great model since it is minimal, and yet captures the key ingredient of flight. However some people would not agree – after all, there is no frequent flyer program or meal service. To get such a frequent flyer program we would need to add passengers, and to get a meal service we would need to add flight attendants. But people weigh a lot, they need to sit down, and they have baggage. So we would end up having to add seats, a galley, and hence large jet engines and lots of fuel to help lift it off the ground. In effect, we would end up with such a realistic model that it would actually be nothing less than a full-size commercial jet. So we would have learned how to build an exact working replica of a commercial jet, but we wouldn’t have learned anything extra in terms of what it takes for something to fly. For that, we should have simply stuck with our original paper plane and explored different designs. In fact we would probably have learned much more about the intricacies of flight that way. I believe that a similar argument applies to the building of models of financial markets and real-world Complex Systems in general.

Given the paper plane example, it is not so surprising that in order to faithfully reproduce the quantitative features of real market price movements we can use a model as simple as a modified version of the bar attendance problem discussed above. After all, it captures several key ingredients of a Complex System in that it consists of a collection of objects which interact through common information and feedback, and which compete for the best price.

One important emergent phenomenon that the bar attendance model can reproduce is that of financial crashes. These are moments where the market tends to head downward for an
extended period. In our Complexity language, this is a classic example of a pocket of order appearing spontaneously out of relative disorder, and serves to confirm why financial markets can be thought of as inhabiting the ground between order and disorder like any other Complex System.

It turns out that there is one simple modification to the bar attendance problem that needs to be made in order to produce realistic crashes. We need to add the feature whereby the decision-making objects (i.e. traders) will only trade if their strategies have been sufficiently successful in making predictions in the recent past. This makes perfect sense – real traders and investors do not trade all the time. Instead, they are continually watching the markets, mentally checking what their strategies would have predicted, and waiting until they are confident enough about their prediction in order to enter the market and make a trade. This simple addition not only allows the traders who are at that moment in the market to form crowds and anticrowds, just as in
chapter 4
, but it also leads to traders moving in and out of the market in groups. Consequently there can be a rush of traders into, or out of, the market at any one moment. Then once they are in the market, they will tend to either join a crowd or anticrowd. Whenever traders happen to rush into the market in this way, there will tend to be a large increase in the number of trades – hence we say that the market has become more liquid. Conversely, whenever traders are pulling out of the market – which in our model occurs when they become insufficiently confident about their strategies – there tend to be less trades and hence we say that the market becomes more illiquid. Remarkably, and in contrast to common wisdom about markets, our bar model shows that there are several different species of crash. In other words, all crashes are not the same. Instead they fall into different classes – like a taxonomy of crashes.