Animated Attractor

Chaos on the trading floor

Economists and speculators would like to be able to predict the ups and downs of the financial and commodities markets. Could chaos theory help?

Robert Savit

It is a remarkable experience to visit the trading floor of a financial or commodities exchange. During an active period of trading, pandemonium reigns. Imagine the scene: hundreds of people waving their arms and shouting at the tops of their lungs trying to make the right transaction at exactly the right time, while trying to monitor the behaviour of their fellow traders and assimilate the new information assaulting them from every direction.

But underpinning this melee, there are some specific rules and motivations common to all the players. The primary goal is, of course, to make money , and to do it in the least risky way. Is there any rational description of this capitalist brawl, and of the capitalist economies in general, of which these markets are a reflection and a seminal part?

 One important feature that all financial markets, and the larger economies share, is that they are systems with many feedback and self-regulatory mechanisms. We know, for example, that if the price of an item rises too high, demand for the item will decrease and the price will drop. This is the simplest of many such self- regulatory mechanisms in economics and finance. But the existence of these kinds of inherent mechanisms has profound and surprising implications for the ways in which markets, prices and economies could behave.

 To see why this is so, consider a simple model of a system with self-regulating feedback. Suppose we have an extremely simple market with just one commodity, say, gold, for sale. The price of an ounce of gold during a particular week t is p(t). Suppose also that the gold dealers are rather greedy, and try to raise the price by a factor, A (bigger than 1), each week. Then the price during week t + 1 will be given by the simple equation, p(t+1)=Ap(t). But the consumers are also sensitive to price hikes, and as the price goes up, there are fewer buyers. We can encode this effect, qualitatively, in a simple way by subtracting from the right-hand side of the above equation some number that gets larger as p gets larger. If we subtract something of the form Bp(t) , then we have not really done anything to the equation. That is, we now have:

p(t+1) = A(t) - Bp(t) = (A - B)p(t)

If A - B is greater than 1 , this equation will still lead to continued growth, so that the negative feedback (the reduced demand by consumers for the product which is too expensive) will not be sufficient to stem inflation. On the other hand, if A - B is less than 1, the price will eventually go to zero: we have a depression (at least in the gold market).
  Clearly, a model of a market in which the only two choices are indefinitely rising prices or the absolute collapse of the market is not very satisfactory. To try to remedy this, we can consider the next simplest thing that will still al1ow us to mimic the effects of consumer resistance to high prices. Rather than subtracting a term from the right-hand side of the equation that is linear in p(t), we can subtract a term with a higher power of p(t) in it. The simplest choice is A2(t), so that our equation becomes :

p(t+l) = A(t) - A2(t).

This equation is the famous logistic map that previous articles on chaos have described . Despite its simplicity , the time series p(t) generated by this map can have a staggering variety of behaviour depending on the value of A . It turns out that if A is between 0 and 3, then after a long enough time p(t) will become constant. For somewhat larger values of A, the behaviour of p(t) is described as a "limit cycle" ; p(t) bounces among several values periodically (the precise number of values depends on the value of A). Finally, for still larger values of A the behaviour of p(t) becomes aperiodic; it never repeats itself, and for A =4 this aperiodicity becomes quite complicated, in fact, it becomes chaotic.
 One important feature of this chaotic behaviour is that it resembles randomness. First, if we simply look at this graph, we do not see any obvious repetitive structure-no periodicity. Secondly even if we apply some slightly more sophisticated statistical methods to this function , we are not assured of finding any structure. Many chaotic systems pass as random under common statistical tests.

  So, what have we learnt? We see that a simple equation such as the one above , which includes a self-regulating mechanism , can produce series that look quite random. But this equation is anything but random. If we know p(t) exactly, then, in principle, we can predict p(t+l) exactly. On the other hand, there is no predictability for a random process, by definition. Therefore, we can have two diametrically opposite underlying mechanisms, one random, and one deterministic, which produce price sequences that look more or less the same.
  Now, change gears for a moment and describe, generally, the way in which a market is supposed to work, according to some simple version of received economic wisdom. Taking again the gold market, suppose that the market for gold is highly liquid ; many people are buying and selling gold all the time, and the quantity of gold that changes hands is large. Furthermore, suppose that the larger economic environment surrounding the gold market is more or less stable: the international situation is stable, there are no revolutions or upheavals expected, production, energy supplies and international trade are all stable and normal, nothing untoward is happening or is expected to happen. Now suppose that in this idyllic world, one jewellery manufacturer decides to increase output and the demand for gold goes up. The market is very liquid, then you would expect it to respond quickly to this increased demand, and prices should rise.

 The price of gold should move from one fairly stable value, to a higher stable value in the short interval that it takes for the market to adjust to this new demand. Of course, during the adjustment period, the price could wobble up and down some as market participants try to take advantage of the change in demand to reap a profit. For example, a rumour might start that the jeweller is about to increase production even more than the jeweller actually intends. That would send the price up momentarily , but eventually, it would settle down to a new "correct" and stable value.

This story seems reasonable, and has behind it the basic ideas of the efficient market hypothesis. Generally in a public, liquid market, the prices respond quickly to unpredictable changing circumstances and there is, in some sense no possibility of consistent long-term profits. In particular, changes in the price of some commodity in a liquid market are generated by new information, which the market reacts to quickly. In its simplest version, the new price of gold just reflects the new demand for gold, and because nothing else in the environment is changing, this new price should be stable, and should not change unless new information enters the market. So in a stable economic environment, the price of a commodity should be the same , except possibly for small variations due to new pieces of information entering the market place. Because new information, by its definition cannot be anticipated, then these price movements are also not predictable, that is, they are random. If you look at real price movements in some financial markets, you will see changes in the price that do appear to be random.

The efficient market hypothesis
This is a very simple view of the efficient market hypothesis. There are more sophisticated versions, but they more or less share the philosophy expressed here. This view has been around for some 25 years, and has a good deal of credibility, at least in academic circle - although defectors from the efficient market camp have recently become more numerous. But contrast this model with the behaviour we would expect in a market with nonlinear self-regulatory mechanisms. Suppose that the jeweller's increased demand induced a rise in the price. The price could not rise indefinitely; at some point, gold would be too expensive and people would invest their money in other commodities. The self-regulating mechanisms of the market would act to limit the increase in price, and would do so in a nonlinear way.

Mapping out chaos in the the gold market-the logistic way

The logistic map in equation: p(t+1) = A(t) - A2(t) is just the simple parabola, familiar from high school mathematics shown in Figure 1. As we increase A, the parabola gets steeper, but its form does not change very much. Nevertheless, the series, p(t), generated by the logistic map has dramatically different behaviour depending on the precise value of A. In Figure2, (A = 2.5), we see that p(t) eventually settles down to a value of p(t) = 2.6. For a somewhat larger value of A , (A = 3.15), p(t) bounces indefinitely between two values as shown in Figure 3.

This is the simplest example of a limit-cycle. In this case, it is called a 2-cycle because the series bounces between two values. For still larger values of A , P(t) oscillates among more values: For 3<A<3.5699, p(t) will oscillate between some number of values which is a multiple of 2. For example, in Figure 4, we show p(t) for A = 3.55, where p(t) oscillates indefinitely between eight values (an 8-cycle). For still larger values of A , the series p(t) takes on many different kinds of unusual behaviour. One particularly interesting case is A =4, shown in Figure 5, for which the series p(t) is chaotic. In this Figure , p(t) never repeats itself, unlike the behaviour in Figures 2 to 5. This behaviour looks much more "random" than the other graphs, and could easily be mistaken for a random sequence.


As we saw in our example of the logistic map, depending on the value of A , the price might, indeed, settle down to a new stable value, but it might also do something different: the nonlinear regulatory mechanisms could create all kinds of interesting price movements as a function of time, even random-looking ones. It may be that in a nonlinear market, the price movements may not be solely due to the new information affecting the market but may result partly from the nonlinear dynamics of the market itself. In fact, we ought to expect that some of the underlying structure of the dynamics of the market will be reflected in the price movements. There are also many sources of noise and new unpredictable information in the markets. In addition, whatever self-regulating mechanisms exist are vastly more complicated than that of the simple logistic map.

To complicate matters still more , the environment in which a market exists is not static. Changes take place in societies and economies on all time scales, from seconds to millennia. A financial market, coupled to other markets and to the society at large will, in some enormously complex way, reflect in its prices all of these changes (as well as the anticipation of changes) over all time scales, incorporating new information, and expressing, the effects of its own underlying (generally nonlinear) self-regulating mechanisms.

There is a wonderful little example of the way in which feedback and self-regulatory mechanisms can operate, and can produce unexpected results. It is called the beer distribution game. The Sloane School of Management at the Massachusetts Institute of Technology has used it for many years to introduce business students to concepts of economic dynamics [I bet they have! -LB]. The game is a role-playing game in which players take the role of either a retailer, wholesaler, distributor, or brewery. The game sets up rules whereby the retailers can place orders to the wholesaler, the wholesaler to the distributor, and the distributor to the brewery for a number of cases of beer for each time step -called a week. The retailer is told, each week how many cases of beer consumers are buying that week. Players try to have on hand enough cases to satisfy the demands of their customers, without overstocking. They are penalised for overstocking and understocking the beer. The retailer will try to order just enough beer from the wholesaler to cover the anticipated demand for the next week, the wholesaler will try to do the same by ordering from the distributor , and the distributor will try to order enough beer from the brewery to cover the anticipated demand from the wholesaler. Similarly , the brewery will try to produce just enough beer to meet the demands of the distributor.

The game begins with a constant demand on the retailer by the customers of, say, four cases of beer per week. After a few weeks, retail demand jumps to eight cases per week, and stays there for the rest of the game. You would suppose that the players all the way down the line would smoothly adjust to this increased demand by quickly doubling their orders (or, in the case of the brewery production) of beer. But that does not happen. Instead, the weekly orders and inventories of the players typically undergo huge oscillations. For example, by about the 30th week, it is quite common to see distributors ordering 40 cases of beer per week from the brewery, even though consumers are buying a total of only eight cases of beer per week. Clearly , the feedback and regulatory dynamics of this extremely simple system produce highly unexpected and erratic behaviour.

Randomness and chaos in gold prices.Compare the random scatter of points in Figure a with the chaotic parabolic plot in b

Economists have studied other versions of this experimental game with different rules intended to mimic the dynamics of various aspects of the economy. Generally , such games produce outcomes that show the typical features expected in a nonlinear system-limit cycles, multiple periodic phenomena and chaos.

These games are, of course, much simpler than any real economy. It is true that they contain some of the kinds of feedback elements we expect in a real economy. But because real economies are so much more complex, it is possible that the chaotic effects we see in the games may simply not be present in a real economy. The real system may have so many different things going on that all the interesting deterministic effects are just averaged away. We may find ourselves back in a situation in which the economy can be described by linear processes with a lot of noise thrown in. To understand whether this happens, we have to go directly to the real economic data. Unfortunately , real economic data are very difficult to work with. The data sets are fairly short, and there is in any case a lot of noise. Furthermore, many of the statistical methods that economists usually apply will fail to pick up the kinds of nonlinear effects we are looking for. In the past few years, however, new techniques of analysis have been invented that are much more sensitive to the presence of underlying nonlinearities, and can distinguish between certain kinds of randomness and chaos. These new methods, developed by William Brock at the University of Wisconsin and his collaborators, and by some other research groups, including our group at the University of Michigan, are based on the idea that chaotic systems often reveal their structure most clearly when viewed in higher dimensions.

Take, for example, the logistic map discussed in the in Figures 1-5. Looking at the price of gold p(t) in the chaotic regime (see Figure 5), it is certainly not clear whether the values p(t) are random or whether there is some underlying structure and predictability. However, suppose we construct a two-dimensional graph of p(t) in which we plot p(t+1) versus p(t) [Such techniques have been used to predict heart failure -LB]. That is, we take two consecutive numbers from the list p(t).These two  numbers are the x and y coordinates of a point in a plane. We plot that point. Then we shift down the list p(t) by one step and take the next pair, and consider those two values to be the x and y coordinates of another point in the plane. We continue this way until we go through the whole series, p(t).

Now, if the values p(t) were really random, then our two-dimensional plot would look just like a scatter of points, as shown in Figure (a). On the other hand, in Figure (b), we show the results of the plot constructed from the logistic map. We see very clearly a parabola, which is strikingly different from the scatter of points in Figure (a). The existence of an object with some structure in some dimension, as in Figure (b), is an indication that underlying the random-looking series is a deterministic process. Of course, this example is particularly simple. In more realistic cases, one may have to view the series in more than two dimensions to see any structure at all, and even then, if (as is usually the case) there is noise in the system, the structure may be fuzzed out, and difficult to distinguish from the scatter of points that characterises randomness. It is at this point that the new techniques of statistical analysis are very useful. By properly sampling the higher dimensional space in which we view the series p(t),it is possible to identify the existence of an underlying deterministic process even in the presence of a good deal of noise.

A fair number of financial and economic data series have been analysed using these methods. Although not all time series show these effects, taken all together, there is significant evidence for the existence of underlying nonlinear processes in economics and finance. Furthermore, we have recently devised methods for learning more about the details of the underlying dynamics, which hold promise for the problem of short-term forecasting and prediction in certain kinds of chaotic systems. Using these methods, we have studied a variety of mathematical examples of chaotic systems, as well as some real financial data. We have been able to ascertain systematically some of the deterministic structure of the underlying process and, in some examples, make relatively accurate predictions of the next term in a time series, even if there is random "noise". Of course, this is a subject of immense interest in many fields including the physical and biological sciences, as well as the financial markets.

Finally , to return to the scene on the trading floor of an exchange; all those traders acting individually, jockeying for position and profit, are both the observers of the market and the phenomenon of the market, both the audience and the actors. They anticipate changes, assimilate information, make their best guess about the direction of the market, using different techniques, and with different motivations (but all wanting to make a profit), and place their bets. They act, and they watch, and they act again. If things get too far out of line they usually (but not always) act in concert to bring prices back into line. Remarkably, the system, for the most part, regulates itself-with a little judicious regulatory help from the government. These markets are enormously complex. Their self regulatory mechanisms are exquisitely intricate, reflecting the effects of human psychology and social behaviour. It is hopeless to try to model such a system in detail. But the prospect of gaining deeper insight into the behaviour of such systems by recognising their intrinsic nonlinear structure is enormously exciting and promising.

Robert Savit is a professor of physics at the University of Michigan.His research is in theoretical physics as well as other applications of nonlinear dynamics.

Mini-crashes could 'vaccinate' the stock market
As Wall Street's sneezing fit continues, and the rest of the world can't shake off its cold, researchers are considering preventative medicine. Their idea is to "immunise" the stock market with a dose of minor downturns. If it works, a few jabs at the right time and place could make devastating crashes a thing of the past. Michael Hart and his colleagues at Oxford University have made a computer simulation of the stock market, mimicking a group of traders who try to profit by buying when the majority wants to sell and selling when the majority wants to buy. Each time the model is run, the virtual traders make different decisions and the stock market index traces a different path. To discover the most likely future for the stock market in the model, Hart calculated the possible future scenarios over a certain period of time. Some paths just wobbled up and down with no major fluctuations, while others eventually dived into a catastrophic crash. "All the model runs can be thought of as different parallel world lines running into the future," explains Hart.
Up and up
The researchers found that paths that eventually crashed tended to have an above-average number of upward movements in their history before the crash, while paths that did not crash had experienced more downward movements over a similar period. It seems that small drops take tension out of the system and allow it to ride through the period of market instability, says Hart. To spot when a crash was looming, the researchers ran their parallel world lines into the future. When many of the possible lines ended in disaster, they found that a crash was almost inevitable. "The fact that all the lines are following the same path means that the event is highly probable," explains Hart. The team hopes the model could be applied to the real world to spot major dips in the stock market and smooth them out before they happen. They envisage a regulator who would monitor the future stock market based on the parallel-worlds analysis. "If they see the warning signal of converging paths, then they step in and 'immunise' the market," says Hart.
'Sounds crackers'
To do this, the regulator could sell small amounts of stock - enough to force the market down but not make it crash. Another, less likely control would be laws to force major market movers to adjust their positions in the market by buying or selling stock. The team has discussed its ideas with the Bank of England, but the model is far from ready to simulate the real stock market. The bank declined to comment on how seriously it is taking the approach. One concern is how to gather data about the real world. Ton Coolen, an applied mathematician from King's College London, says applying the model to the stock market might be going too far. "Computing all the possible future world lines for the stock market would require getting inside the heads of the traders to see how they think. As far as I can see, this would be impossible," he says.
Stacks of money
The logistics would also be daunting. "A regulator would need stacks of money and a large buffer stock to change the direction of the stock market," says a spokesperson for the Financial Services Authority. "It sounds crackers to me." But Walter Kemmsies, head of global sector strategy at UBS Warburg, thinks gentle intervention in the stock market to prevent crashes would be a great idea, if it could be made to work: "Buying shares would become safer and people could be confident that the share price was not based on dreams." However, one city trader still sees a downside to the idea. "It would make the markets terribly boring," he says.
Kate Ravilious, Oxford
[New Scientist 07 August 02]

Further Reading

THE ECONOMY AS AN EVOLVING COMPLEX SYSTEM, Vol V Santa Fe Institute Studies in the Sciences of Complexity. P.Anderson, K.Arrow and D.Pines (eds) Addison -Wesley, Redwood City,1988

Material World 56    

What kind of bang was the big bang?

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