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A
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by Benoit B. Mandelbrot
sional 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 mar-
ket seems to speed up and the volatility soars. Last Septem-
ber, for instance, the stock for Alcatel, a French telecommuni-
cations 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 indepen-
dent of one another: for example, that today’s price has no
inﬂuence on the changes between the current price and to-
morrow’s. As a result, predictions of future market move-
ments 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 mea-
I
ndividual investors and profes-
sured by its sigma, or standard de-
viation) depicts how far price changes
diverge from the mean; events at the ex-
tremes 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 chang-
es 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, how-
ever, 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
Modern portfolio theory poses a danger to
those who believe in it too strongly and is
a powerful challenge for the theoretician.
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 mil-
lionth of a millionth. (The ﬂuctuations are greater than 10
standard deviations.) But in fact, one observes spikes on a reg-
ular 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 distri-
bution. But should ﬁnancial markets then be described as ab-
normal? Of course not
—
they are what they are, and it is port-
folio theory that is ﬂawed.
Modern portfolio theory poses a danger to those who be-
lieve in it too strongly and is a powerful challenge for the the-
oretician. Though sometimes acknowledging faults in the
present body of thinking, its adherents suggest that no other
70
Scientific American
February 1999
A Multifractal Walk down Wall Street
Copyright 1999 Scientific American, Inc.
The geometry that describes the shape of
coastlines and the patterns of galaxies
also elucidates how stock prices
soar and plummet
premises can be handled through math-
ematical 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 con-
ceivable 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, called multifractals—do
not purport to predict the future with certainty. But they do
create a more realistic picture of market risks. Given the re-
cent troubles confronting the large investment pools called
hedge funds, it would be foolhardy not to investigate models
providing more accurate estimates of risk.
chart, this transforma-
tion must shrink the time-
scale (the horizontal axis)
more than the price scale (the ver-
tical 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 arbi-
trarily; 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 gener-
1
THREE-PIECE FRACTAL GENERATOR (
top
) can be in-
terpolated repeatedly into each piece of subsequent charts
(
bottom three diagrams
). The pattern that emerges increas-
ingly resembles market price oscillations. (The interpolated
generator is inverted for each descending piece.)
1
TREND LINE
GENERATOR
Multifractals and the Market
0
0
TIME
1
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 dec-
orative 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 the-
oretical reformulation of a down-to-earth bit of market folk-
lore
—
namely, that movements of a stock or currency all look
alike when a market chart is enlarged or reduced so that it
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 be-
tween 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.
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 re-
duce the horizontal axis more than the vertical one. For a price
INTERPOLATED
GENERATOR
A Multifractal Walk down Wall Street
Scientific American
February 1999 71
Copyright 1999 Scientific American, Inc.
A
n extensive mathematical basis already exists for fractals
The new modeling techniques are designed
to cast a light of order into the seemingly
impenetrable thicket of the financial markets.
The unifractal (U) chart shown here (before any shortening)
corresponds to the becalmed markets postulated in the port-
folio 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 move-
ments 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
ator 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 (inter-
polated) 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 hori-
zontal and vertical boundaries of each piece of the generator.
2
MOVING A PIECE of the fractal generator to the left ...
1
GENERATOR
2
/
3
Interpolations Forever
1
/
3
U
= Unifractal
M1 = Multifractal 1
M2 = Multifractal 2
M3 = Multifractal 3
M4 = Multifractal 4
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 oc-
cur 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 en-
compassed by modern portfolio theory. But tranquillity pre-
vails only under extraordinarily special conditions that are
satisfied only by these special generators. The assumptions be-
hind 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 charac-
terize portfolio theory’s placid markets as well as the tumul-
tuous 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 short-
ened, which also provides room to lengthen the second piece.
After making these adjustments, the generators become multi-
fractal (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 simula-
tion 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 in-
terpolation is undertaken
—
it produces a chart that increasingly
resembles the characteristics of volatile markets (
illustration 4
) .
0
0
4
/
9
5
/
9
1
TIME
... causes the same amount of market activity in a shorter time in-
terval for the ﬁrst piece of the generator and the same amount in a
longer interval for the second piece ...
4
/
9
5
/
9
1
1
5
/
9
4
/
9
U
M1
M2
M3
M4
0
0
1
TIME
4
... Movement of the generator to the left causes market
activity to become increasingly volatile.
U
M1
M2
M3
M4
72
Scientific American
February 1999
Copyright 1999 Scientific American, Inc.
O
nly the ﬁrst stages are shown in the illustration, although
3
Pick the Fake
ﬁnancial prices? To assess their performance, let us compare sev-
eral 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 back-
ground 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 chap-
ters would vary from one another, but all
the monthly chapters would read very
much alike. The rather simple second
chart is less unrealistic, because it shows
many spikes; however, these are isolated
against an unchanging background in
which the overall variability of prices re-
mains constant. The third chart has inter-
changed strengths and failings, because it
lacks any precipitous jumps.
The eye tells us that these three dia-
grams are unrealistically simple. Let us
now reveal the sources. Chart 1 illus-
trates price ﬂuctuations in a model intro-
duced 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 Lévy stable random processes) and one I published in 1965
(based on fractional Brownian motion).
These revisions, however, are inade-
quate, except under certain special mar-
ket conditions.
In the more important ﬁve lower dia-
grams 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 com-
pletely artiﬁcial, having been generated
through a more refined form of my multi-
fractal model.
1
2
3
4
5
6
7
8
—
B.B.M.
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 oth-
er 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 port-
folio 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 un-
known rules that govern actual markets. Multifractals de-
scribe accurately the relation between the shape of the genera-
tor 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 gen-
erator 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 depic-
tion of market fluctuations, called fractional Brownian mo-
tion 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 or-
der into the seemingly impenetrable thicket of the ﬁnancial
markets. They also recognize the mariner’s warning that, as re-
cent events demonstrate, deserves to be heeded: On even the
calmest sea, a gale may be just over the horizon.
SA
The Author
Further Reading
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 Uni-
versity and IBM Fellow Emeritus (Physics) at the Thomas J. Watson
Research 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 Sci-
ences 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.
The Fractal Geometry of Nature. Benoit B. Mandelbrot.
W. H. Freeman and Company, 1982.
Fractals and Scaling in Finance: Discontinuity, Concen-
tration, Risk. Benoit B. Mandelbrot. Springer-Verlag, 1997.
The Multifractal Model of Asset Returns. Discussion pa-
pers of the Cowles Foundation for Economics, Nos. 1164–1166.
Laurent Calvet, 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.
A Multifractal Walk down Wall Street
Scientific American
February 1999 73
Copyright 1999 Scientific American, Inc.
H
ow do multifractals stand up against actual records of changes in
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