Science for Survival,
Pontificia Academia Scientiarum, MM
Fractal financial
fluctuations;
do they threaten
sustainability?
Benoit B. Mandelbrot
"Were all the fountains of the great deep broken up,
and the windows of heaven were opened. And the rain
was upon the earth forty days and forty nights"
Genesis: 6, 11-12
"There came seven years of great plenty throughout the
land of Egypt. And there shall arise after them seven
years of famine" Genesis: 41, 29-30.
The very diversity of the speakers at this meeting
suffices to show that sustainability is threatened
by very diverse obstacles. Several speakers
described world-wide trends whose continuation
would make sustainability impossible. My
presentation is, instead, concerned with instances
where the trends themselves might be tolerable
but severe deviations from the norm are not, in
fact, may well be unacceptable . For good reason,
and as witnessed by the quote from the Bible that
opens this text and will be commented upon
momentarily, a deep concern with those
deviations has an extremely long history. Until
recently, no scientific tools were available to begin
to face those deviations, not even to measure their
intensity. This paper will not discuss all the tools
now available, only those the speaker introduced
very recently.
Based on the two quotes from the Bible,
my terminology classifies the threatening
deviations from norms in two categories. Because
of the story of the Flood and Noah, I denote by
the term "Noah Effect" those major changes that
occur rapidly (even instantaneously) but have
strong and durable consequences. Because of the
story of the dream of Pharaoh and Joseph the son
of Jacob, I denote by the term "Joseph Effect"
those sequences of changes that need not seem
threatening when viewed individually but
become major in their cumulative effect.
The Noah and the Joseph obstacles to
sustainability range from the wholly natural
(earthquakes, volcanoes, non man-made climate
changes) to the wholly man-made. Both "effects"
have long characterized the phenomena I studied
very energetically. Many of those studies were
reported or summarized in my 1982 monograph,
or included in my Selecta books, of which the first
three have been published. These are big books
and it would be of little use to try to summarize
them in these Proceedings. Instead, I shall focus
on one topic and make minimal reference to the
other aspects of the Noah and Joseph Effects.
That topic is the variation of financial
prices, that is, of prices quoted on financial
markets that trade securities, commodities, and
exchange or interest rates. The wild volatility of
those markets has long been known but only
recently did it form an intimate link with such
issues as sustainable growth of developing or
other economically fragile countries.
The details of that intimate link belong to
political economy. They will be briefly touched
upon at the end of this text, but to keep in focus
and manage space, an extensive discussion will be
carried out elsewhere. The sole ambition of this
paper is to further the knowledge of the
underlying facts. Unfortunately and perhaps
surprisingly, the existing models of financial price
variability for good enough and new research is
needed. Such statements tend to be viewed as
self-serving - and an excuse for delaying urgent
action. But in the present instance skepticism
would be unwarranted. New research is keenly
needed, because the points that matter most in the
discussion of smooth sustainability coincide very
precisely with points that previous research had
deliberately set aside or disregarded. The text that
follows is a draft of Chapter 1 of a forthcoming
book of mine.
THIS BOOK TOUCHES MANY TOPICS, BUT ITS
MAIN AMBITION is to contribute to a better
understanding of price variation. Inevitably, it
criticizes previously held views on this topic,
particularly, the "coin tossing model," to be
described momentarily.
The point of departure is that financial
prices, including those of securities, commodities,
foreign exchange or interest rates, are largely
unpredictable. The best one can do is to evaluate
the odds for or against some desired or feared
outcomes, the most extreme being "ruin." Those
odds will also be used as inputs for decisions
concerning economic policy or changes in
institutional arrangements. To handle all those
issues, the first step - but far from the last! - is to
represent different instances of price variation by
suitable random processes.
The word "suitable" and the plural in
"processes" will surprise many readers. It is,
indeed, widely believed that "random change" is a
synonym for "prices that move up a bit or down a
bit following the toss of a coin." The technical
term is "simple random walk." It was made
popular by a book title that won the high
distinction of becoming a cliché., namely,
"random walk down the Street."
The belief that there is no alternative is
strengthened by the fact that coin tossing is
indeed, the oldest and by far the most widely
used model of price variation. The (unsaid) point
of departure of this book and this Survey is that
the term, random, has a far broader meaning,
allowing the coin tossing model to be replaced by
alternatives. Many of the alternatives are
"unsuitable," but it will be argued that the
alternative I put forward in this book, based on
"multifractals," is very suitable indeed.
The multifractal model does not belong to
esoteric mathematics and it must not be allowed
to remain part of pure science. Its practical
consequences are many and very serious. The first
but not last is in the spirit of the Hippocratic Oath,
"do no harm," which deserves to be generalized to
finance and is best expressed in nautical term.
When a ship was built to navigate placid lakes by
fair weather, to send it across the ocean in
typhoon season is to do serious harm. Similarly,
the "coin tossing" model of financial prices may
well be beloved by mathematicians, but it denies
the existence of hurricanes; therefore it is
dangerous.
The preceding nautical analogy will be
heavily used throughout this text, because it
resides at the very center of the present study. Not
only alternatives to the coin tossing model are
available, but the multifractal alternative differs in
"qualitative" ways that have immediate
consequences for finance and economic policies.
The coin tossing model exemplifies a form
of randomness (a "state of randomness," as I shall
argue) that can be called "mild." Had the evidence
agreed with this model - but it does not at all variability
in finance would be as easily
controllable as is variability in physics.
However, the coin tossing model must
not be criticized too hard. It is always best to start
with the simplest possible model and hold to it
until it has begun to bring more harm than value.
In its time, it played a fundamental and positive
role in creating awareness of the difficulty of even
the simplest forms of randomness. One can also
argue that for the "man in the street" coin tossing
is an adequate description of the facts. But policy
makers and the professionals in finance are (or
should be) far more demanding. It matters very
much for them that, as will be seen, coin tossing is
very far from accounting for some essential facts.
Once again, the history of price variation is filled
with "financial hurricanes" while we shall see that
coin tossing claims that they practically never
happen. Shipbuilders and ship owners cannot
predict the dates and destructiveness of the
hurricanes their vessel will encounter over its
lifetime. But the knowledge that hurricanes will
happen - and realistic evaluation of the
corresponding odds - permeates ship-building,
ship ownership and navigation.
This book argues that tools needed to
acquire some mastery of the intensities of
financial hurricanes are already available. They
are those of fractal and multifractal geometry, a
discipline better known as describing the shapes
of coastlines and clouds and the distribution of
galaxies and as having led to the discovery of the
Mandelbrot set. My claim is that it also describes
the growth and collapse of financial prices.
PICK THE FAKES
This book includes a multitude of words or
numbers and also of formulas and dry diagrams.
Without mastering them, my claims and
contributions cannot be fully understood and
appreciated. But to make the central point, as this
Survey proposes, words, formulas and diagrams
are not really necessary.
To explain my central point, the best and
quickest way is to encourage the reader to
participate in a test concerning Figures 1 and 2.
No one is asked to accept pictures as the sole or
final arbiter in scientific discourse, only as a useful
additional tool. Pictures are often used to delude,
but in this instance they deserve to be described
as uncovering a widespread delusion and
assisting in the selection of an improvement.
Among the many graphs in Figure 1,
some are "real." They follow the practice of
financial journals and trace the sequence of daily
closing of some price series such as security,
commodity, foreign exchange or interest rate. The
other graphs in Figure 1 will be called "forgeries."
They correspond to imitations of financial reality
provided by mathematical models that are fully
specified in quantitative
fashion, therefore can be sampled and illustrated
without resorting to unreported stretching and
reducing or other such manipulations.
Figure 2 plots price differences from one
day to next.
Now the "find-the-fakes test" can be
described: you are asked to separate reality and
forgery as completely as you can. For a perfect
score, you must rank the diagrams from "most
obviously a forgery" to "apparently real."
When the test relates to Figure 1, it is very
difficult to separate the real and forged records.
Indeed, all such records tend to look alike. This
Figure 1 A collection of Diagrams, illustrating ­ in no
particular order ­ the behavior in time of some actual
financial prices and of some mathematical models of
this behavior. It would be very difficult to pick the
fakes.
impression is confirmed by looking through the
financial press and the books on the mathematics
of finance. The optimist will rush to conclude that
coin tossing, which is represented by one of the
graphs in Figure 1, is perfectly acceptable. In fact,
Figure 1. A collection of diagrams, illustrating - in
no particular order - the behavior in time of
someactual financial prices and of some
mathematical models of this behavior. It would be
very difficult to pick the fakes.
we shall see momentarily that this optimism
would be seriously misplaced. The resemblance
between those curves is due to the fact that graphs
of prices themselves do not reveal or enhance
important differences, but hide them. In other
words, plots of prices are a very inefficient way of
presenting information. This is well-known to
students of the psychology of vision: position is
seen less accurately than change.
In sharp contrast, the lines in the stack of this
Figure 2 show striking differences between
Figure 2. A stack of diagrams, illustrating the
successive "daily" differences in some actual financial
prices and some mathematical models. Pick the fakes!
one another. The meaning of those differences
will be refined through this Survey, ending with
the solution of the test.
LARGE STOCK MARKET MOVEMENTS AND
THEIR ODDS
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. In
September 1998, for instance, the stock for Alcatel,
a French telecommunications equipment
manufacturer, dropped about 40 percent one day
and another 6 percent over the next few days.
Figure 3. Shapes of the Gaussian distribution and
of two "stable" distributions. The latter provide a
far better fit for many financial data, but the
multifractal model is even more satisfactory.
In a reversal, the stock shot up 10 percent on the
fourth day. On a longer time scale, most real price
changes behave like those in the lower portion of
Figure 2. However, not all lines at the bottom of
Figure 2 are real. (That is, I am not giving away
the test the reader is taking!)
The coin tossing model, which served as
foundation for the theory of finance used most
widely in this century, is represented by the top
line of Figure 2 (now, I am giving away part of the
test). We shall see in a moment that precipitous
events like the Alcatel debacle are given totally
negligible odds in that theory. Certainly, they
should never happen in the lifetime of this
generation and the next few. A cornerstone of
finance is modern portfolio theory, which tries to
maximize returns for a given level of risk. The
mathematics underlying portfolito ignore the
possibility of a typhoon?
This term, coin-tossing, is actually an
oversimplification, but the risk-reducing formulas
behind portfolio theory rely on a number of
demanding premises that are mathematically
attractive but rely on hope rather than reality.
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. This is the "efficient market"
hypothesis to ignore the possibility of a typhoon?
This term, coin-tossing, is actually an
oversimplification, but the risk-reducing formulas
behind portfolio theory rely on a number of
demanding premises that are mathematically
attractive but rely on hope rather than reality.
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. This is the "efficient market"
hypothesis of Louis Bachelier. As a result,
prediction of future market movements is never
possible.
The second assumption is that all price
changes are distributed in a pattern that conforms
to the standard bell curve. Of the three curves in
Figure 3, the bell curve is the flattest in the center.
The width of its bell (as measured by its "sigma,"
or standard deviation) depicts how far price
changes diverge from the mean. In this
perspective, 95% of all cases fall into the narrow
range between minus two sigmas and plus two
sigmas. As was already mentioned and will be
elaborated momentarily, the bell curve declares
extreme events to be extremely rare. Typhoons
are, in effect, defined out of existence.
Do financial data neatly conform to such
assumptions? Of course, they never do! This is
shown by a more attentive inspection of the
bottom portion of Figure 2. It is true that charts of
stock or currency changes over time reveal a
constant background of small up and down price
movements - though not as uniform as one would
expect of price changes that fit the bell curve.
Invariably, however, these patterns 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
Figure 3. Shapes of the Gaussian distribution and of
two "stable" distributions. The latter provide a far
better fit for many financial data, but the multifractal
model is even more satisfactory.
stock - stand out from the background of more
moderate perturbations. Moreover, it is typical of
the magnitude of price movements - both large
and small - to remain roughly constant for an
extended period and then suddenly unpredictably
increase for another extended period. Big price
jumps become more common as the turbulence of
the market grows -they cluster on the chart,
expressing an obvious amount of dependence.
According to the coin tossing model,
these large fluctuations often exceed ten sigmas,
meaning ten standard deviations. This value is so
huge that standard textbook tables of the
Gaussian fail to include it. But a good calculator
should yield their probability a few millionths of a
millionth of a millionth of a millionth, that is,
approximately one day out of ten million million
million years. If risks were so tiny, they would be
truly negligible, not worth even a passing
thought. But this tiny value grossly contradicts
the evidence. The real world of finance produces
"ten sigma" spikes on a regular basis - as often as
every month, to give an idea - and their
probability should be expected to be a few
hundredths.
The tiny probability mentioned a few
lines above illustrates that the Gaussian
practically vanishes near the left and right ends of
the graph. Had the horizontal axis been part of
Figure 3 (which it is not), it would have hidden
those insignificant tails.
Reality is far better represented by the
other two curves in Figure 3, both with more
peaked heads and fatter tails. These curves belong
to the "M 1963 model" produced by my first attack
on financial data, in 1963. Having revealed this
fact, it is best to narrow down the test the reader is
supposed to be taking. Price changes according to
the M 1963 model are the source of the second line
in Figure 2. This is clearly better than the top line,
to be sure, but far from being the last word.
COIN TOSSING NORMALITY VERSUS THE
FINANCIAL REALITY
The bell curve is often described as illustrating the
"normal" distribution. But does it follow that
financial markets should be described as
"abnormal or anomalous"? Of course not: they are
what they are, and it is the coin tossing model,
therefore portfolio theory, that are 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, the extreme bearish answer is
that there is no alternative: large market swings
are anomalies, individual "acts of God" that
present no conceivable regularity. Other
adherents suggest that coin tossing must be
maintained, "faute de mieux," because no other
premises can be handled through tough
mathematical modeling leading to a rigorous
quantitative description of at least some features
of major financial upheavals.
An increasingly wide agreement is being
Figure 4. Constructing a "pseudo-Brownian cartoon"
of the idealized coin-tossing model that underlies
modern portfolio theory. The construction starts with a
linear trend ("the initiator") and breaks it repeatedly
by following a prescribed "generator." The interpolated
generator is inverted for each descending piece. The
pattern that emerges increasingly resembles market
price oscillations.
reached, that the extreme bearish view is
untenable and that the coin tossing model must be
replaced by one that allows (near-) instantaneous
price changes and substantial temporal
dependence. This agreement marks a change of
mood in the "mainstream," bringing it toward the
views I have been campaigning for since 1963 and
1965, respectively. From this point on, however,
two general approaches are in conflict, leading to
what I shall call "micromanaged" and
"macromanaged" models.
Micromanaged models agree with my
diagnosis but not my follow-up. They proceed
through a series of "fixes." Each fix "patches" a
perceived defect of coin tossing, independently of
its other defects. The outcome is that this
approach accumulates a large number of
parameters and no property is present that was
not knowingly incorporated in the construction.
In the nautical analog, the fixes consist in
lengthening a small boat's keel, lengthening its
mast, reinforcing its engine, etc..., one by one.
It is clearly preferable to design a large
boat from scratch. Similarly, my experience of
successful modeling in other fields has fostered
deep a priori doubts about the chances of
micromanaged modeling in finance. But personal
prejudices would not have mattered if a posteriori
modeling had been effective. I think it is not.
My own work - carried out over many
years - takes a very different and decidedly
bullish position. I claim that a financial model can
be redesigned following an approach that is
macromanaged by being guided by a principle of
fractal invariance to which we shall come soon.
The outcome, as I propose to show, is that the
variation of financial prices can be accounted for
by a model derived from my work in fractal
geometry. Once again, fractals - or their later
elaboration, called multifractals - do not claim the
ability 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 called hedge funds, it
would be foolhardy not to investigate models
providing more accurate estimates of risk.
FRACTALS, MULTIFRACTALS AND THE
MARKET
An extensive mathematical basis already exists for
fractals and multifractals. Fractal patterns do not
only appear in the price changes of securities but
also 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 reducedscale
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, the notion that movements of a
stock or currency all look alike when a market
chart is enlarged or reduced so that it fits some
prescribed time and price scales. This implies that
an observer 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 many powerful tools of
mathematical and computer analysis become
available.
A technical term for this form of close
resemblance between the parts and the whole is
self-affinity. This property is related to the betterknown
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 ordered a photographic enlargement
or a xerox copy. Financial market charts, however,
are far from
being self-similar. If we simply focus on a detail of
a graph, the features become increasingly higher
than they are wide &dash as are the individual
up-and-down price ticks of a stock. Hence, the
transformation from the whole to the parts must
shrink the time scale (the horizontal axis) more
than the price scale (the vertical axis). This task
can be performed by copiers using lasers. The
geometric relation of the whole to its parts is said
to be one of self-affinity.
Unchanging properties are 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
(interpolates) 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 called
"initiator," shown in the top panel of Figure 4.
Next, a broken line called "generator" is used to
create the pattern that corresponds to a slow upand-down
price oscillation. It is obviously
essential for the number and positions of the
pieces of the generator to be completely specified.
If it is not, or if one allows oneself the right to
fiddle with the generator during the construction,
no prediction could be made. In Figure 4, the
generator consists of three pieces that are inserted
(interpolated) along the straight trend line. (A
generator with fewer than three pieces could not
simulate a price that must be able to move up and
down.) Then, each of the generator's three pieces
is interpolated by three shorter ones. Repeating
these steps reproduces the shape of the generator,
or price curve, but at increasingly compressed
scales. Both the horizontal axis (time scale) and
the vertical axis (price scale) are squeezed to fit
the horizontal and vertical boundaries of each
piece of the generator.
INTERPOLATIONS CONTINUED (NOT
QUITE) FOREVER
Only four construction stages are shown in Figure
4, but 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 be
of the order of a minute. The fact that each piece
ends up with a shape like the whole is not a
surprise: this scale invariance is present simply
because it was built in. The novelty (and surprise)
is that these very simply defined self-affine fractal
curves exhibit a wealth of structure. The beauty of
fractal geometry is that it does not consist in
micromanaged models in which everything of
interest has been inputed separately. Fractals
involve only macromanaged instructions, yet
make possible a model general enough to
reproduce the patterns that characterize portfolio
theory's placid markets as well as the tumultuous
trading conditions of real markets.
Indeed, the construction's outcome, if
plotted as in Figure 2, is very sensitive to the exact
shape of the generator.
For example, Figure 4 uses a very special
generator that - according to a theory I developed
- will produce a behavior that is pseudoBrownian,
that is, close to the relatively tranquil,
"mildly random," picture of the market ruled by
coin tossing . But this level of tranquillity prevails
only under extraordinarily special conditions that
are satisfied only by equally special generators.
Figure 3 satisfies those conditions because each
generator segment's height - namely, 2/3, 1/3 or
2/3 - was made equal to the square root of the
corresponding width - namely, 4/9, 1/9 or 4/9.
This "square root rule" is a characteristic of a
process physicists call "simple diffusion"
Adherence to the assumptions behind this
oversimplified model is one of the central
mistakes of modern portfolio theory. It is much
like a theory of sea waves that forbids the swells
to exceed six feet.
A first and very important generalization
of Figure 4 yields models that are non Brownian
but can be called "unifractal." It consists in
continuing to require that the height of every
segment of the generator be linked to its width by
the same relation in the form of a power H .
Previously, we set H = 1/2 , but a different value
of H can be chosen, as long as it is positive and
less than 1. Taking H = .7 suffices to change the
Figure 5. This open cube illustrates related generators:
The "right wall" shows an oscillating generator in
trading time. This is the pseudo-Brownian (unifractal)
generator of Figure? The "back wall" shows four
multifractal oscillating generators in clock time. The
"floor" shows the generators that relate the clock time
to trading time. Each is an increasing function of the
other. Moving a piece of the fractal generator to the left
causes the same amount of market activity in a shorter
time interval for the first piece of the generator and the
same amount in a longer interval for the second piece.
top line of Figure 2 into its third line. On the
corresponding graphs in Figure 1, the place of
tranquility and mildness is taken by movements
that are non-periodic but described by everyone
as "cyclic," with many apparent cycle lengths,
ranging from very small up to "about three cycles
in a sample." (This last rule is a remarkable
observation that cannot be elaborated here.) Here,
cyclic behavior is present in the output without
having been incorporated in the input. This is
lovely, but large spikes of variation were lost and
must be reinstated.
There is a second and far more drastic
generalization of Figure 3. So far, market activity
was assumed constant but one can allow it to
speed up and slow down. This variability is the
essence of volatility, in fact, practical people
describe the diverse lines at the bottom of Figure 2
as proceeding at many different speeds at
different times. This is why models that allow for
variability add the prefix "multi" before the word
"fractal."
To define "activity" is beyond our concern
and not necessary. The key idea is that the market
does not follow the physical time that procedes
with the relentless regularity of a clock, but
Figure 6. The underlying pattern is as in Figure 5, but
limited to the left-most generator, and the generators
are replaced by the curve obtained by repeating it
recursively as done in Figure 3. To make the picture
clearer, the back and right wall are moved away from
the floor.
instead a subjective time that flows slowly during
some periods and fast during others.
In this spirit, the theory provides a neat
"transmutation" from uni - to multifractal. The key
step shown in Figure 5 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. As seen on the "back" wall of Figure 5,
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). As seen on the "floor," of
Figure 6, 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.
When the generators in Figure 5 are used
recursively, one obtains the patterns shown in
Figure 6. Recall that those patterns do not pretend
to exhaust all the possibilities offered by either
theory or the facts. Their sole aim is to show the
power of the very simplest fractal models. Such
an alteration to the generator can produce a full
Figure 7. Randomized multifractal "price increments"
that correspond to the five multifractal generators in
Figure 5, On top a pseudo-Brownian sequence of "price
increments". A gradual displacement of the generator
to the left causes market activity to increase gradually,
becoming more and more volatile.
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. The process of
successive interpolation produces a chart that
increasingly resembles the characteristics of
volatile markets (Figure 7).
Once again, the unifractal (U) chart that
prevails before any shortening corresponds to the
becalmed markets postulated in the portfolio
theorists' model. Proceeding down the stack (Ml
to M4), each chart diverges further from that
model, exhibiting the sharp, spiky price jumps
and the persistently large movements that
characterize financial trading.
An important detail has not been
mentioned yet 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. Altogether, the three pieces of the
generator allow the following six permutations:
1,2,3; 1,3,2; 2,1,3; 2,3,1; 3,1,2 and 3,2,1.
Conveniently enough, a die has six sides;
imagine that each bears the image of one of the six
permutations. Before each interpolation, the die is
thrown and the permutation that comes up is
selected.
BACK TO THE GAME OF PICKING THE
FAKES
How do simulations of the multifractal model
stand up against actual records of changes in
financial prices? To respond, let us return to
Figure 2, which is a composite of several historical
series of price changes with a few outputs of
artificial models.
As we have already observed 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 it 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 recall 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: one
is the "M 1963" model I proposed in 1963 (based
on Levy stable random processes). The other is
the "M 1965" model I published in 1965 (based on
fractional Brownian motion). These revisions of
coin tossing are inadequate, except under certain
special market conditions.
By now, the test around which this
Survey is structured has been reduced to a careful
inspection of the more important five lower
diagrams of the graph. Let me now add a piece of
information that was withheld until now: at least
one record is real and at least one is a computergenerated
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 dollardeutsche
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.
Two technical points must be mentioned
before moving on to conclusions. The recursive
constructions in the body of the paper were
nothing but "cartoons". The artificial charts 4, 7
and 8 were, instead, generated through a refined
form of my multifractal model, called "fractional
Brownian motion in multifractal trading time."
Secondly, this introductory Survey necessarily
emphasizes graphics, but - once again - the theory
of multifractals is endowed with full numerical
tools of analysis.
VERY TENTATIVE CONCLUSIONS:
DIVERSIFICATION AND REINSURANCE
What conclusions should be drawn from all this?
Does this matter to a corporate treasurer, currency
trader or other market strategists? Does this
matter to the central banker and others concerned
with overall financial and economic policy? Does
this matter to the economist who seeks to explain
the workings of the economy and concedes that
his task may be helped by an accurate description
of part of what is to be explained?
All those questions arise because the
discrepancies between coin-tossing and the actual
movement of prices have become too obvious to
be ignored much longer. Prices do not vary
continuously, and they are subjected to wild
fluctuations at all time scales. 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, nearly everyone
embraced the modern portfolio theory because of
the absence of strong alternatives. But one need
no longer accept it at face value.
However, the multifractal alternative is
very young and very far from being fully
developed. It deserves to draw attention (and
criticism). By contrast, modern portfolio theory
was formulated twenty-four years ago and was
energetically developed ever since. Moreover,
wild variability is a new notion endowed with
little inherited capital. Modern portfolio theory
inherited a large accumulated capital of
techniques that statisticians designed to deal with
mild Gaussian variability. The challenge was to
adapt them to the context of financial prices.
Therefore, it is necessary, as we near a
conclusion, to separate thoughts concerning the
near future from thoughts concerning the longer
range. Multifractals can immediately be put to
work to "stress-test" portfolios, in particular, from
the viewpoint of a quantity called "value at risk,"
whose definition is unfortunately beyond the
scope of this text. stress-testing begins by
questioning how a portfolio would have
performed if it had been designed a while ago.
That is, the simplest stress test merely uses
historical data. But the actual market test will not
come in the past, rather in the future, and a future
that simply repeats the past is only one of many
alternatives, and not a very likely one. The goal of
every model of price variation (coin-tossing not
being an exception) is to use the past to create the
same patterns of variability as do the unknown
rules that govern actual markets. This attempt
should yield a collection of alternative scenarios
for the future, and Stress-testing should include
tests based on many such alternatives.
According to the coin-tossing model, the
differences between those alternatives are
comparatively slight. Not so with the multifractal
models. They describe the past market
fluctuations more realistically and the scenarios
they propose for the future include a quota of
extreme events that will really stress a portfolio.
This is all that can be said on this subject at this
point.
Of greatest interest, at least to me, are
problems that the multifractal model confronts on
broader institutional, temporal and spatial
horizons. They are more important than any
detail and question the worth of the widespread
faith in the power of diversification and other
forms of lumping and averaging. Here an
enlightening analogy and powerful guidance for
the future is provided by a distinction between
different levels of insurance that relates to my
distinction between mild and wild "states" of
chance.
Most life, automobile, or homeowner
risks are mild. Very much like the coin-tossing
model of price change, they fall within a narrow
range and are mutually independent. Even when
a portfolio of insurance contracts is small, the
wonders of diversification (due to the law of large
numbers and related mathematics) can be trusted
to create a risk of ruin that is sufficiently small to
be profitable even for an insurance company with
limited reserves. To play safe and to insure the
occasional higher risk, the insurer of mild risks
will seek reinsurance - which will seldom be
needed, therefore will not be expensive. When a
tornado defeats diversification of homeowner
policies, the reinsurer is likely to be an entity that
collected no premiums, namely an agency of a
state.
However, many other risks seeking to be
insured are wild, very much like in my
multifractal model of price change. They involve
the equivalents of the notorious "ten sigma" price
changes that were discussed earlier in this Survey.
Ordinary diversification would be defeated by
such risks, even if the number of cases had
sufficed for a law of large numbers to apply. More
precisely, the odds of those wild risks, if included
in the usual calculations, would imply reserves
that are clearly unreasonable. However, such risks
become insurable if they are immediately shared
with reinsurers (or almost directly with
competitors, as is apparently the case in the
shipping industry).
The key fact is that insurers cannot
survive by only considering the "fair weather"
95% of the claims, which would have easily been
diversified. Not only the 5% of large "foul
weather" claims cannot be ignored, but their odds
are non-negligible and are an essential input of
planned and carefully priced reinsurance.
Once again, theories based on cointossing
legislate this "foul weather" out of
existence, but it is evident that many features of
the real world are best understood as designed to
tackle comparatively rare but potentially
disastrous situations. It is indeed filled with state
or private institutions and informal or ad-hoc
arrangements whose purpose can be viewed as
that of reinsurance. A central part of my thinking
in finance is that those arrangements may have
worked in the past but cannot be relied upon in
the future. As to institutions, their role deserves a
fresh examination.
As a result of globalization, the relevance
of the preceding comments on insurance is bound
to increase. Under the coin-tossing model, the
effects of globalization are limited. But the actual
behavior of financial prices confirms what
intuition suggests: the larger the markets, the
greater the attention demanded by the potentially
disastrous effects of financial storms.
To conclude no overall mathematical
technique comes close to forecasting a price drop
or rise on a specific day on the basis of past
records. Multifractals do not do any better. But
they provide estimates of the probability of what
the market might do in the future 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.
BIOGRAPHY
Benoit Mandelbrot is unusual in having left a
permanent mark on numerous fields of science
and art. He is presently the Abraham Robinson
Professor of Mathematical Sciences at Yale
University. In 1958 he had joined IBM's Thomas J.
Watson Research Center, where he served as IBM
Fellow in Physics, 1974-1993. He belongs to the
National Academy of Sciences of the USA, the
American Academy of Arts and Sciences, and the
Norwegian Academy of Science and Letters.
Having attended Ecole Polytechnique and earned
the Ph.D. from the University of Paris, he also
holds numerous honorary doctorates from
institutions throughout the world. He received
the 1993 Wolf Prize for Physics and also (among
others) the Barnard, Franklin and Steinmetz
Medals and the Harvey, Humboldt, Nevada,
Honda and Arts and Science Prizes.
BIBLIOGRAPHY
B. MANDELBROT, Fractals and Scaling in
Finance: Discontinuity, Concentration, Risk,
Springer-Verlag, 1997.
B. MANDELBROT, Multifractals and 1/f Noise:
Wild Self-Affinity in Physics, Springer-Verlag,
1999.
B. MANDELBROT, L. CALVET & A. FISHER, The
Multifractal Model of Asset Returns, three reports
of the Cowles Foundation for
Economic Research of Yale University. Web
address of the first:
http://papers.ssrn.com/sol3/paper.taf?/ABSTR
ACT_ID=78588.
Addresses of the other two: same beginning and
the endings _ID=78608 and _ID=78628.