What is the relative importance of
private information, investor trading strategies, and pure whim in predicting
the market? What is the relative importance of conventional economic news
(interest rates, budget deficits, accounting scandals, and trade balances),
popular culture fads (in sports, movies, fashions), and germane political and
military events (terrorism, elections, war) too disparate even to categorize? If
we were to carefully define the problem, predicting the market with any
precision is probably what mathematicians call a universal problem, meaning that
a complete solution to it would lead immediately to solutions for a large class
of other problems. It is, in other words, as hard a problem in social prediction
as there is.
Certainly, too little notice is taken
of the complicated connections among these variables, even the more clearly
defined economic ones. Interest rates, for example, have an impact on
unemployment rates, which in turn influence revenues; budget deficits affect
trade deficits, which sway interest rates and exchange rates; corporate fraud
influences consumer confidence, which may depress the stock market and alter
other indices; natural business cycles of various periods are superimposed on
one another; an increase in some quantity or index positively (or negatively)
feeds back on another, reinforcing or weakening it and being reinforced or
weakened in turn.
Few of these associations are
accurately described by a straight-line graph and so they bring to a
mathematician’s mind the subject of nonlinear dynamics, more popularly known as
chaos theory. The subject doesn’t deal with anarchist treatises or surrealist
manifestoes but with the behavior of so-called nonlinear systems. For our
purposes these may be thought of as any collection of parts whose interactions
and connections are described by nonlinear rules or equations. That is to say,
the equations’ variables may be multiplied together, raised to powers, and so
on. As a consequence the system’s parts are not necessarily linked in a
proportional manner as they are, for example, in a bathroom scale or a
thermometer; doubling the magnitude of one part will not double that of
another—nor will outputs be proportional to inputs. Not surprisingly, trying to
predict the precise long-term behavior of such systems is often
futile.
Let me, in place of a technical
definition of such nonlinear systems, describe instead a particular physical
instance of one. Picture before you a billiards table. Imagine that
approximately twenty-five round obstacles are securely fastened to its surface
in some haphazard arrangement. You hire the best pool player you can find and
ask him to place the ball at a particular spot on the table and take a shot
toward one of the round obstacles. After he’s done so, his challenge is to make
exactly the same shot from the same spot with another ball. Even if his angle on
this second shot is off by the merest fraction of a degree, the trajectories of
these two balls will very soon diverge considerably. An infinitesimal difference
in the angle of impact will be magnified by successive hits of the obstacles.
Soon one of the balls will hit an obstacle that the other misses entirely, at
which point all similarity between the two trajectories ends.
The sensitivity of the billiard balls’
paths to minuscule variations in their initial angles is characteristic of
nonlinear systems. The divergence of the billiard balls is not unlike the
disproportionate effect of seemingly inconsequential events, the missed planes,
serendipitous meetings, and odd mistakes and links that shape and reshape our
lives.
This sensitive dependence of nonlinear
systems on even tiny differences in initial conditions is, I repeat, relevant to
various aspects of the stock market in general, in particular its sometimes
wildly disproportionate responses to seemingly small stimuli such as companies’
falling a penny short of earnings estimates. Sometimes, of course, the
differences are more substantial. Witness the notoriously large discrepancies
between government economic figures on the size of budget surpluses and
corporate accounting statements of earnings and the “real” numbers.
Aspects of investor behavior too can
no doubt be better modeled by a nonlinear system than a linear one. This is so
despite the fact that linear systems and models are much more robust, with small
differences in initial conditions leading only to small differences in final
outcomes. They’re also easier to predict mathematically, and this is why they’re
so often employed whether their application is appropriate or not. The chestnut
about the economist looking for his lost car keys under the street lamp comes to
mind. “You probably lost them near the car,” his companion remonstrates, to
which the economist responds, “I know, but the light is better over
here.”
The “butterfly effect” is the term
often used for the sensitive dependence of nonlinear systems, a characteristic
that has been noted in phenomena ranging from fluid flow and heart fibrillations
to epilepsy and price fluctuations. The name comes from the idea that a
butterfly flapping its wings someplace in South America might be sufficient to
change future weather systems, helping to bring about, say, a tornado in
Oklahoma that would otherwise not have occurred. It also explains why long-range
precise prediction of nonlinear systems isn’t generally possible. This
non-predictability is the result not of randomness but of complexity too great
to fathom.
Yet another reason to suspect that
parts of the market may be better modeled by nonlinear systems is that such
systems’ “trajectories” often follow a fractal course. The trajectories of these
systems, of which the stock price movements may be considered a proxy, turn out
to be aperiodic and unpredictable and, when examined closely, evince even more
intricacy. Still closer inspection of the system’s trajectories reveals yet
smaller vortices and complications of the same general kind.
In general, fractals are curves,
surfaces, or higher dimensional objects that contain more, but similar,
complexity the closer one looks. A shoreline, to cite a classic example, has a
characteristic jagged shape at whatever scale we draw it; that is, whether we
use satellite photos to sketch the whole coast, map it on a fine scale by
walking along some small section of it, or examine a few inches of it through a
magnifying glass. The surface of the mountain looks roughly the same whether
seen from a height of 200 feet by a giant or close up by an insect. The
branching of a tree appears the same to us as it does to birds, or even to worms
or fungi in the idealized limiting case of infinite branching.
As the mathematician Benoit
Mandelbrot, the discoverer of fractals, has famously written, “Clouds are not
spheres, mountains are not cones, coastlines are not circles, and bark is not
smooth, nor does lightning travel in a straight line.” These and many other
shapes in nature are near fractals, having characteristic zigzags, push-pulls,
bump-dents at almost every size scale, greater magnification yielding similar
but ever more complicated convolutions.
And the bottom line, or, in this case,
the bottom fractal, for stocks? By starting with the basic up-down-up and
down-up-down patterns of a stock’s possible movements, continually replacing
each of these patterns’ three segments with smaller versions of one of the basic
patterns chosen at random, and then altering the spikiness of the patterns to
reflect changes in the stock’s volatility, Mandelbrot has constructed what he
calls multifractal “forgeries.” The forgeries are patterns of price movement
whose general look is indistinguishable from that of real stock price movements.
In contrast, more conventional assumptions about price movements, say those of a
strict random-walk theorist, lead to patterns that are noticeably different from
real price movements.
These multifractal patterns are so far
merely descriptive, not predictive of specific price changes. In their modesty,
as well as in their mathematical sophistication, they differ from the Elliott
waves mentioned in chapter 3.
Even this does not prove that chaos
(in the mathematical sense) reigns in (part of) the market, but it is clearly a
bit more than suggestive. The occasional surges of extreme volatility that have
always been a part of the market are not as nicely accounted for by traditional
approaches to finance, approaches Mandelbrot compares to “theories of sea waves
that forbid their swells to exceed six feet.”
ไม่มีความคิดเห็น:
แสดงความคิดเห็น