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This article originally appeared in the
February 1999 issue of Scientific American. For a copy of it, please
contact Scientific American at 212.754.0550. We highly recommend a
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A Multifractal
Walk Down Wall Street |
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"The geometry that
describes the shape of coastlines and the patterns of galaxies also elucidates how stock
prices soar and plummet."
by Benoit B.
Mandelbrot |
Individual investors and
professional stock and currency traders know better than ever that prices quoted in any
financial market often change with heart-stopping swiftness. Fortunes are made and lost in
sudden bursts of activity when the market seems to speed up and the volatility soars. Last
September, for instance, the stock for Alcatel, A French telecommunications equipment
manufacturer, dropped about 40 percent one day and fell another 6 percent over the next
few days. In a reversal, the stock shot up 10 percent on the fourth day.
The classical financial models used for most of this
century predict that such precipitous events should never happen. A cornerstone of finance
is modern portfolio theory, which tries to maximize returns for a given level of risk. The
mathematics underlying portfolio theory handles extreme situations with benign neglect:
it regards large market shifts as too unlikely to matter or as impossible to take
into account. It is true that portfolio theory may account for what occurs 95
percent of the time in the market. But the picture it presents does not reflect reality,
if one agrees that major events are part of the remaining 5 percent. An inescapable
analogy is that of a sailor at sea. If the weather is moderate 95 percent of the time, can
the mariner afford to ignore the possibility of a typhoon?
The risk-reducing formulas behind portfolio theory rely on
a number of demanding and ultimately unfounded premises. First, they suggest that price
changes are statistically independent of one another: for example, that today’s price
has no influence on the changes between the current price and tomorrow’s. As a
result, predictions of future market movements become impossible. The second presumption
is that all price changes are distributed in a pattern that conforms to the standard bell
curve. The width of the bell shape (as measured by its sigma, or standard deviation)
depicts how far price changes diverge from the mean; events at the extremes are considered
extremely rare. Typhoons are, in effect, defined out of existence.
Do financial data neatly conform to such assumptions? Of
course, they never do. Charts of stock or currency changes over time do reveal a constant
background of small up and down price movements – but not as uniform as one would
expect if price changes fit the bell curve. These patterns, however, constitute only one
aspect of the graph. A substantial number of sudden large changes – spikes on the
chart that shoot up and down as with the Alcatel stock – stand out from the
background of more moderate perturbations. Moreover, the magnitude of price movements
(both large and small) may remain roughly constant for a year, and then suddenly the
variability may increase for an extended period. Big price jumps become more common as the
turbulence of the market grows – clusters of them appear on the chart.
According to portfolio theory, the probability of these
large fluctuations would be a few millionths of a millionth of a millionth of a millionth.
(The fluctuations are greater than 10 standard deviations.) But in fact, one observes
spikes on a regular basis – as often as every month – and their probability
amounts to a few hundredths. Granted, the bell curve is often described as normal –
or, more precisely, as the normal distribution. But should financial markets then be
described as abnormal? Of course not – they are what they are, and it is portfolio
theory that is flawed.
Modern portfolio theory poses a danger to those who
believe in it too strongly and is a powerful challenge for the theoretician. Though
sometimes acknowledging faults in the present body of thinking, its adherents suggest that
no other premises can be handled through mathematical modeling. This contention leads to
the question of whether a rigorous quantitative description of at least some features of
major financial upheavals can be developed. The bearish answer is that large market swings
are anomalies, individual "acts of God" that present no conceivable regularity.
Revisionists correct the questionable premises of modern portfolio theory through small
fixes that lack any guiding principle and do not improve matters sufficiently. My own work
– carried out over many years – takes a very different and decidedly bullish
position.
I claim that variations in financial prices can be
accounted for by a model derived from my work in fractal geometry. Fractals – or
their later elaboration, call multifractals – do not purport to predict the future
with certainty. But they do create a more realistic picture of market risks. Given the
recent troubles confronting the large investment pools call hedge funds, it would be
foolhardy not to investigate models providing more accurate estimates of risk.
Multifractals and
the Market
An extensive mathematical basis already exists for
fractals and multifractals. Fractal patterns appear not just in the price changes of
securities but in the distribution of galaxies throughout the cosmos, in the shape of
coastlines and in the decorative designs generated by innumerable computer programs.
A fractal is a geometric shape that can be separated into
parts, each of which is a reduced-scale version of the whole. In finance, this concept is
not a rootless abstraction but a theoretical reformulation of a down-to-earth bit of
market folklore – namely, that movements of a stock or currency all look alike when a
market chart is enlarged or reduced so that is fits the same time and price scale. An
observer then cannot tell which of the data concern prices that change from week to week,
day to day or hour to hour. This quality defines the charts as fractal curves and makes
available many powerful tools of mathematical and computer analysis.
A more specific technical term for the resemblance between
the parts and the whole is self-affinity. This property is related to the better-known
concept of fractals called self-similarity, in which every feature of a picture is reduced
or blown up by the same ratio – a process familiar to anyone who has ever ordered a
photographic enlargement. Financial market charts, however, are far from being
self-similar.
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Illustration
1 - THREE-PIECE-FRACTAL GENERATOR (top) can be interpolated
repeatedly into each piece of subsequent charts (bottom three diagrams).
The pattern that emerges icreasingly resembles market price oscillations. (The
interpolated generator is inverted for each descending piece.) |
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In a detail of a graphic in which the features are higher
than they are wide – as are the individual up-and-down price ticks of a stock –
the transformation from the whole to a part must reduce the horizontal axis more than the
vertical one. For a price chart, this transformation must shrink the time-scale (the
horizontal axis) more than the price scale (the vertical axis). The geometric relation of
the whole to its parts is said to be one of self-affinity.
The existence of unchanging properties is not given much
weight by most statisticians. But they are beloved of physicists and mathematicians like
myself, who call them invariances and are happiest with models that present an attractive
invariance property. A good idea of what I mean is provided by drawing a simple chart that
inserts price changes from time 0 to a later time 1 in successive steps. The intervals
themselves are chosen arbitrarily; they may represent a second, an hour, a day or a year.
The process begins with a price, represented by a straight
trend line (illustration 1). Next, a broken line called a generator is used to
create the pattern that corresponds to the up-and-down oscillations of a price quoted in
financial markets. The generator consists of three pieces that are inserted (interpolated)
along the straight trend line. (A generator with fewer than three pieces would not
simulate a price that can move up and down.) After delineating the initial generator, its
three pieces are interpolated by three shorter ones. Repeating these steps reproduces the
shape of the generator, or price curve, but at compressed scales. Both the
horizontal axis (timescale) and the vertical axis (price scale) are squeezed to fit the
horizontal and vertical boundaries of each piece of the generator.
Interpolations Forever
Only the first stages are shown in the illustration,
although the same process continues. In theory, it has no end, but in practice, it makes
no sense to interpolate down to time intervals shorter than those between trading
transactions, which may occur in less than a minute. Clearly, each piece ends up with a
shape roughly like the whole. That is, scale invariance is present simply because it was
built in. The novelty (and surprise) is that these self-affine fractal curves exhibit a
wealth of structure — a foundation of both fractal geometry and the theory of chaos.
A few selected generators yield so-called unifractal
curves that exhibit the relatively tranquil picture of the market encompassed by modern
portfolio theory. But tranquillity prevails only under extraordinarily special conditions
that are satisfied only by these special generators. The assumptions behind this
oversimplified model are one of the central mistakes of modern portfolio theory. It is
much like a theory of sea waves that forbids their swells to exceed six feet.
The beauty of fractal geometry is that it makes possible a
model general enough to reproduce the patterns that characterize portfolio theory’s
placid markets as well as the tumultuous trading conditions of recent months. The just
described method of creating a fractal price model can be altered to show how the activity
of markets speeds up and slows down — the essence of volatility. This variability is
the reason that the prefix "multi-" was added to the word "fractal."
To create a multifractal from a unifractal, the key step
is to lengthen or shorten the horizontal time axis so that the pieces of
the generator are either stretched or squeezed. At
the same time, the vertical price axis may remain untouched. In illustration 2, the
first piece of the unifractal generator is progressively shortened, which also provides
room to lengthen the second piece. After making these adjustments, the generators become
multifractal (M1 to M4). Market activity speeds up in the interval of time represented by
the first piece of the generator and slows in the interval that corresponds to the second
piece (illustration 3).
Such an alteration to the generator can produce a full
simulation of price fluctuations over a given period, using the process of interpolation
described earlier. Each time the first piece of the generator is further shortened —
and the process of successive interpolation is undertaken — it produces a chart that
increasingly resembles the characteristics of volatile markets (illustration 4).
The unifractal (U) chart shown here (before any
shortening) corresponds to the becalmed markets postulated in the portfolio
theorists’ model. Proceeding down the stack (M1 to M4), each chart diverges further
from that model, exhibiting the sharp, spiky price jumps and the persistently large
movements that resemble recent trading. To make these models of volatile markets achieve
the necessary realism, the three pieces of each generator were scrambled — a process
not shown in the illustrations. It works as follows: imagine a die on which each side
bears the image of one of the six permutations of the pieces of the generator. Before each
interpolation, the die is thrown, and then the permutation that comes up is selected.
What should a corporate treasurer, currency trader or
other market strategist conclude from all this? The discrepancies between the pictures
painted by modern portfolio theory and the actual movement of prices are obvious. Prices
do not vary continuously, and they oscillate wildly at all timescales. Volatility —
far from a static entity to be ignored or easily compensated for — is at the very
heart of what goes on in financial markets. In the past, money managers embraced the
continuity and constrained price movements of modern portfolio theory because of the
absence of strong alternatives. But a money manager need no longer accept the current
financial models at face value.
Instead multifractals can be put to work to
"stress-test" a portfolio. In this technique the rules underlying multifractals
attempt to create the same patterns of variability as do the unknown rules that govern
actual markets. Multifractals describe accurately the relation between the shape of the
generator and the patterns of up-and-down swings of prices to be found on charts of real
market data.
On a practical level, this finding suggests that a fractal
generator can be developed based on historical market data. The actual model used does not
simply inspect what the market did yesterday or last week. It is in fact a more realistic
depiction of market fluctuations, called fractional Brownian motion in multifractal
trading time. The charts created from the generators produced by this model can simulate
alternative scenarios based on previous market activity.
These techniques do not come closer to forecasting a price
drop or rise on a specific day on the basis of past records. But they provide estimates of
the probability of what the market might do and allow one to prepare for inevitable sea
changes. The new modeling techniques are designed to cast a light of order into the
seemingly impenetrable thicket of the financial markets. They also recognize the
mariner’s warning that, as recent events demonstrate, deserves to be heeded: On even
the calmest sea, a gale may be just over the horizon.
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Pick
the Fake
How do multifractals stand up
against actual records of changes in financial prices? To assess their performance, let us
compare several historical series of price changes with a few artificial models. The goal
of modeling the patterns of real markets is certainly not fulfilled by the first chart,
which is extremely monotonous and reduces to a static background of small price changes,
analogous to the static noise from a radio. Volatility stays uniform with no sudden jumps.
In a historical record of this kind, daily chapters would vary from one another, but all
the monthly chapters would read very much alike. The rather simple second chart is less
unrealistic, because is shows many spikes; however, these are isolated against an
unchanging background in which the overall variability of prices remains constant. The
third chart has interchanged strengths and failings, because it lacks any precipitous
jumps.
The eye tells us that these three diagrams
are unrealistically simple. Let us now reveal the sources. Chart 1 illustrates price
fluctuations in a model introduced in 1900 by French mathematician Louis Bachelier. The
changes in prices follow a "random walk" that conforms to the bell curve and
illustrates the model that underlies modern portfolio theory. Charts 2 and 3 are partial
improvements on Bachelier’s work: a model I proposed in 1963 (based on Levy
stable random processes) and one I published in 1965 (based on fractional Brownian
motion). These revisions, however, are inadequate, except under certain special market
conditions.
In the more important five lower diagrams of the graph, at
least one is a real record and at least another is a computer-generated sample of my
latest multifractal model. The reader is free to sort those five lines into the
appropriate categories. I hope the forgeries will be perceived as surprisingly effective.
In fact, only two are real graphs of market activity. Chart 5 refers to the changes in
price of IBM stock, and chart 6 shows price fluctuations for the dollar-deutsche mark,
exchange rate. The remaining charts (4, 7 and 8) bear a strong resemblance to their two
real-world predecessors. But they are completely artificial, having been generated through
a more refined form of my multifractal model. -B.B.M.
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The Author
Benoit B. Mandelbrot has contributed to numerous fields of
science and art. A mathematician by training, he has served since 1987 as Abraham Robinson
Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at
the Thomas J. Watson Reasearch Center in Yorktown Heights, N.Y., where he worked from 1958
to 1993. He is a fellow of the American Academy of Arts and Sciences and foreign associate
of the U.S. National Academy of Sciences and the Norwegian Academy. His awards include the
1993 Wolf Prize for physics, the Barnard, Franklin and Steinmetz medals, and the Science
for Art, Harvey, Humboldt and Honda prizes.
Further Reading
The Fractal Geometry of Nature. Benoit B. Mandelbrot.
W.H. Freeman and Company, 1982.
Fractals and Scaling in Finance: Discontinuity,
Concentration, Risk. Benoit B. Mandelbrot. Springer-Verlag, 1997.
"The Multifractal Model of Asset Returns."
Discussion papers of the Cowles Foundation for Economics, Nos. 114-1166. Laurent Calvert,
Adlai Fisher and Benoit B. Mandelbrot. Cowles Foundation, Yale University, 1997.
Multifractals and 1/F Noise: Wild Self-Affinity in
Physics. Benoit B. Mandelbrot, Springer-Verlag, 1999.
Continue to
Prechter's Letter to the
Editor
Scientific
Controversy Introduction -
Mandelbrot's Article - Prechter's
Letter to the Editor
Prechter's Response -
Follow-up Responses
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Socionomics |
