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 selfregulatory 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 selfregulating 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 righthand 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 righthand 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
A^{2}(t), so that our equation becomes :
p(t+l) = A(t)  A^{2}(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 structureno 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 selfregulating 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 longterm 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 selfregulatory
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 selfregulating 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 marketthe logistic wayThe logistic map in equation: p(t+1) = A(t)  A^{2}(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 limitcycle. In this case, it is called a 2cycle 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 8cycle). 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 randomlooking 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 selfregulating 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) selfregulating mechanisms.
There is a wonderful little example of the way in which feedback
and selfregulatory 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 roleplaying 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 systemlimit 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
15. 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 twodimensional 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
twodimensional 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 randomlooking 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 shortterm 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 itselfwith 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.
Minicrashes could 'vaccinate' the stock market

Further Reading
WHEN RANDOM IS NOT
RANDOM:AN INTRODUCTION TO CHAOS IN MARKET PRICES. Robert Savit.
J Future Markets Vol 8, p271,1988.

Chaos  Quantum  Logic  Cosmos  Conscious  Belief  Elect.  Art  Chem.  Maths 