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http://www.aci.net/kalliste/Chaos5.htm (9 of 14) [12/04/2001 1:30:00]
Chaos and Fractals in Financial Markets, Part 5, by J. Orlin Grabbe

Now that weâ€™ve worked our way to the heart of the
matter, letâ€™s take a break from probability theory and
turn our attention again to dynamical systems. In
particular, letâ€™s look at our old friend the logistic
equation:
x(n+1) = k x(n) [1 â€“ x(n)],
where x(n) is the input variable, x(n+1) is the output
variable, and k is a constant.
In Part 1, we looked at a particular version of this
equation where k = 4. In general, k takes values 0 < k <=
4.
The dynamic behavior of this equation depends on the
value k, and also on the particular starting value or
starting point, x(0). Later in this series we will examine
how the behavior of this equation changes as we change
k. But not now.
Instead, we are going to look at this equation when we
substitute for x, which is a real variable, a complex
variable z:
z(n+1) = k z(n) [1 â€“ z(n)].
Complex numbers z have the form
z = x + i y,
where i is the square root of minus one. Complex
numbers are normally graphed in a plane, with x on the
horizontal ("real") axis, while y is on the vertical
("imaginary") axis.
That means when we iterate z, we actually iterate two
values: x in the horizontal direction, and y in the vertical
direction. The complex logistic equation is:
x + i y = k (x + i y) [ 1 â€“ (x + i y)].
(Note that I have dropped the notation x(n) and y(n) and
just used x and y, to make the equations easier to read.
But keep in mind that x and y on the left-hand side of the
equation represent output, while the x and y on the
right-hand side of the equation represent input.)
The output x, the real part of z, is composed of all the
terms that do not multiply i, while the output y, the

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Chaos and Fractals in Financial Markets, Part 5, by J. Orlin Grabbe

imaginary part of z, is made up of all the terms that
multiply i.
To complete the transformation of the logistic equation,
we let k be complex also, and write
k = A + B i,
giving as our final form:
x + i y = (A + B i) (x + i y) [ 1 â€“ (x + i y)].
Now we multiply this all out and collect terms. The
result is two equations in x and y:
x = A (x-x2+y2) + B (2xy-y)
y = B (x-x2+y2) - A (2xy-y).
As in the real version of the logistic equation, the
behavior of the equation depends on the multiplier k = A
+ B i (that is, on A and B), as well as the initial starting
value of z = x + i y (that is, on x(0) and y(0) ).
Julia Sets
Depending on k, some beginning values z(0) = x(0) + i
y(0) blow off to infinity after a certain number of
iterations. That is, the output values of z keep getting
larger and larger, diverging to infinity. As z is composed
of both an x term and a y term, we use as the criterion for
"getting large" the value of
x2 + y2.
The square root of this number is called the modulus of
z, and represents the length of a vector from the origin
(0,0) to the point z = (x,y). In the iterations we are about
to see, the criterion to determine if the equation is
diverging to infinity is
x2 + y2 > 4,
which implies the modulus of z is greater than 2.
When the equation is iterated, some starting values
diverge to infinity and some donâ€™t. The Julia set is the
set of starting values for z that remain finite under
iteration. That is, the Julia set is the set of all starting
values (x(0), y(0)) such that the equation output does not
blow off to infinity as the equation is iterated.

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Chaos and Fractals in Financial Markets, Part 5, by J. Orlin Grabbe

Each value for k will produce a different Julia set (i.e., a
different set of (x(0) ,y(0) ) values that do not diverge
under iteration).
Letâ€™s do an example. Let k = 1.678 + .95 i. That is, A =
1.678 and B = .95. We let starting values for x(0) range
from â€“.5 to 1.5, while letting starting values for y(0)
range from -.7 to +.7.
We keep k constant always, so we are graphing the Julia
set associated with k = 1.678 + .95 i.
We iterate the equation 256 times. If, at the end of 256
iterations, the modulus of z is not greater than 2, we
paint the starting point (x(0), y(0)) black. So the entire
Julia set in this example is colored black. If the
modulus of z exceeds 2 during the iterations, the starting
point (x(0), y(0)) is assigned a color depending on the
rate the equation is blowing off to infinity.
To see the demonstration, be sure Java is enabled on
We can create a plot that looks entirely different by
making a different color assignment. For the next
demonstration, we again iterate the dynamical system
256 times for different starting values of z(n). If, during
the iterations, the modulus of z exceeds 2, then we know
the iterations are diverging, so we plot the starting value
z(0) = (x(0), y(0)) black. Hence the black region of the
plot is made up of all the points not in the Julia set.
For the Julia set itself, we assign bright colors. The color
assigned depends on the value of z after 256 iterations.
For example, if the square of the modulus of z is greater
than .6, but less than .7, the point z(0) is assigned a light
red color. Hence the colors in the Julia set indicate the
value of the modulus of z at the end of 256 iterations.
To see the second demonstration of the same equation,
but with this alternative color mapping, be sure Java is

So, from the complex logistic equation, a dynamical
system, we have created a fractal. The border of the Julia
set is determined by k in the equation, and this border
was created in a working Euclidean space of 2
dimensions, has a topological dimension of 1, but has a
Hausdorff dimension that lies between these two

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Chaos and Fractals in Financial Markets, Part 5, by J. Orlin Grabbe

numbers.
Meanwhile, we have passed from mathematics to art. Or
maybe the art was there all along. We just had to learn
how to appreciate it.
Notes
[1] This is the stock market equivalent of the Interest
Parity Theorem that relates the forward price F(t+T) of a
currency, T-days in the future, to the current spot price
S(t). In the foreign exchange market, the relationship can
be written as:
F(t+T) = S(t) [1 + r (T/360)]/[1+r*(T/360)]
where r is the domestic interest rate (say the dollar
interest rate), and r* is the foreign interest rate (say the
interest rate on the euro). S is then the spot dollar price
of the euro, and F is the forward dollar price of the euro.
We can also use this equation to give us the forward
value F of a stock index in relation to its current value S,
in which case r* must be the dividend yield on the stock
index.
(A more precise calculation would disaggregate the
"dividend yield" into the actual days and amounts of
dividend payments.)
This relationship is explored at length in Chapter 5,
"Forwards, Swaps, and Interest Parity," in J. Orlin
Grabbe, International Financial Markets, 3rd edition,
Prentice-Hall, 1996.
Samorodnitsky and Murad S. Taqqu, Stable
Non-Gaussian Random Processes: Stochastic Models
with Infinite Variance, Chapman & Hall, New York,
1994.
[3] This is Theorem VI.1.1 in William Feller, An
Introduction to Probability Theory and Its Applications,
Vol 2, 2nd ed., Wiley, New York, 1971.

[4] If Y = X/n1/ Î± , then for n independent copies of Y,

Y1 + Y2 + â€¦ + Yn-1 + Yn ˜ n1/ Î± Y = n1/ Î± (X/n1/ Î± ) =
X.

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Chaos and Fractals in Financial Markets, Part 5, by J. Orlin Grabbe

J. Orlin Grabbe is the author of International Financial
Markets, and is an internationally recognized derivatives
expert. He has recently branched out into cryptology,
located at http://www.aci.net/kalliste/homepage.html .
-30-
from The Laissez Faire City Times, Vol 3, No 29, July
19, 1999

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Julia Plot of the Complex Logistic Equation

Julia Plot of the Complex Logistic Equation
If you can read this, then your browser is not set up for Java. . .
This applet plots the Julia set for the complex logistic equation: z(n+1) = k z(n)[1-z(n)], for k = 1.678 +
.95 i. The equation is iterated 256 times for different starting values of z(n). If the equation is not
diverging to infinity at the end of the 256 iterations, then the starting point z(0) is painted black. Hence
the black area of the plot represents the Julia set (except for some points on the outer corners that are
also painted black if they diverge on the first iteration). If, however, the equation begins diverging
(defined as occurring when the modulus of z(n+1) > 2), the iterations are terminated, and the point is
plotted with one of 16 colors. The iterations cycle through this set of colors, but terminate when the
equation diverges. Hence, the color plotted indicates the rate the equation output is blowing off to
infinity.
The Java source code.

http://www.aci.net/kalliste/logistic_julia.html [12/04/2001 1:30:03]
Alternative Julia Plot of the Complex Logistic Equation

Alternative Julia Plot of the Complex Logistic
Equation
If you can read this, then your browser is not set up for Java. . .
This applet plots the Julia set for the complex logistic equation: z(n+1) = k z(n)[1-z(n)], for k = 1.678 +
.95 i. The equation is iterated 256 times for different starting values of z(n). If the equation is not
diverging to infinity at the end of the 256 iterations, then the terminal value z(n+1) = z(256) is assigned a
color based on the value of the square of its modulus (i.e., on the value of x2+y2). Hence, the brightly
colored area of the plot consists of the Julia set. If, however, the equation begins diverging (defined as
occurring when the modulus of z(n+1) > 2), the iterations are terminated, and the point z(0) is painted
black. Hence the black area of the plot represents points that diverge and which are therefore not in
the Julia set.
The Java source code.

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Chaos and Fractals in Financial Markets, Part 6, by J. Orlin Grabbe

Chaos and Fractals in Financial Markets
Part 6
by J. Orlin Grabbe
Prechterâ€™s Drum Roll
Robert Prechter is a drummer. He faces the following problem. He
wants to strike his drum three times, creating two time intervals
which have a special ratio:
1<------------g-------------->2<--------------------h-------------------->3
Here is the time ratio he is looking for: he wants the ratio of the
first time interval to the second time interval to be the same as the
ratio of the second time interval to the entire time required for the
three strikes.
Let the first time internal (between strikes 1 and 2) be labeled g,
while the second time interval (between strikes 2 and 3) be labeled
h. So what Prechter is looking for is the ratio of g to h to be the
same as h to the whole. However, the whole is simply g + h, so
Prechter seeks g and h such that:
g / h = h / (g+h).
Now. Prechter is only looking for a particular ratio. He doesnâ€™t
care whether he plays his drum slow or fast. So h can be anything:
1 nano-second, 1 second, 1 minute, or whatever. So letâ€™s set h = 1.
(Note that by setting h = 1, we are choosing our unit of
measurement.) We then have
g / 1 = 1 / (1+g).
Multiplying the equation out we get
g2 + g â€“ 1 = 0.
This gives two solutions:
g = [- 1 + 50.5] / 2 = 0.618033â€¦, and
g = [- 1 - 50.5] / 2 = -1.618033â€¦
The first, positive solution (g = 0.618033â€¦) is called the golden
mean. Using h = 1 as our scale of measurement, then g, the
golden mean, is the solution to the ratio

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Chaos and Fractals in Financial Markets, Part 6, by J. Orlin Grabbe

g / h = h / (g+h).
By contrast, if we use g = 1 as our scale of measurement, and
solve for h, we have
1 / h = h / (1+h), which gives the equation
h2 - h â€“ 1 = 0.
Which gives the two solutions:
h = [ 1 + 50.5] / 2 = 1.618033â€¦, and
h = [ 1 - 50.5] / 2 = -0.618033â€¦
Note that since the units of measurement are somewhat aribitrary,
h has as much claim as g to being the solution to Prechterâ€™s drum
roll. Naturally, g and h are closely related:
h (using g as the unit scale) = 1/ g (using h as the unit scale).
for either the positive or negative solutions:
1.618033â€¦ = 1/ 0.618033â€¦
-0.618033â€¦ = 1/ -1.618033.
What is the meaning of the negative solutions? These also have a
physical meaning, depending on where we place our time origin.
For example, letâ€™s let the second strike of the drum be time t=0:
<------------g-------------->0<--------------------h-------------------->
Then we find that for g = -1.618033, h = 1, we have
-1.618033 /1 = 1/ [1 - 1.618033].
So the negative solutions tell us the same thing as the positive
solutions; but they correspond to a time origin of t = 0 for the
second strike of the drum.
The same applies for g = 1, h = -0.618033, since
1 / -0.618033 = -0.618033/(1 â€“ 0.618033),
but in this case time is running backwards, not forwards.
The golden mean g, or its reciprocal equivalent h, are found
throughout the natural world. Numerous books have been devoted
to the subject. These same ratios are found in financial markets.
Symmetric Stable Distributions and the Golden Mean Law

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Chaos and Fractals in Financial Markets, Part 6, by J. Orlin Grabbe

In Part 5, we saw that symmetric stable distributions are a type of
probability distribution that are fractal in nature: a sum of n
independent copies of a symmetric stable distribution is related to
each copy by a scale factor n1/ Î± , where Î± is the Hausdorff
dimension of the given symmetric stable distribution.
In the case of the normal or Gaussian distribution, the Hausdorff
dimension Î± = 2, which is equivalent to the dimension of a plane.
A Bachelier process, or Brownian motion (as first covered in Part
2), is governed by a T1/Î± = T1/2 law.
In the case of the Cauchy distribution (Part 4), the Hausdorff
dimension Î± = 1, which is equivalent to the dimension of a line. A
Cauchy process would be governed by a T1/Î± = T1/1 = T law.
In general, 0 < Î± <=2. This means that between the Cauchy and
the Normal are all sorts of interesting distributions, including ones
having the same Hausdorf dimension as a Sierpinski carpet (Î± =
log 8/ log 3 = 1.8927â€¦.) or Koch curve (Î± = log 4/ log 3 =
1.2618â€¦.).
Interestingly, however, many financial variables are symmetric
stable distributions with an Î± parameter that hovers around the
value of h = 1.618033, where h is the reciprocal of the golden
mean g derived and discussed in the previous section. This implies
that these market variables follow a time scale law of T1/Î± = T1/h =
Tg = T0.618033... That is, these variables following a
T-to-the-golden-mean power law, by contrast to Brownian
motion, which follows a T-to-the-one-half power law.
For example, I estimated Î± for daily changes in the
dollar/deutschemark exchange rate for the first six years following
the breakdown of the Bretton Woods Agreement of fixed
exchange rates in 1973. [1] (The time period was July 1973 to
June 1979.) The value of Î± was calculated using maximum
likelihood techniques [2]. The value I found was
Î± = 1.62
with a margin of error of plus or minus .04. You canâ€™t get much
closer than that to Î± = h = 1.618033â€¦
In this and other financial asset markets, it would seem that time
scales not according to the commonly assumed square-root-of-T
law, but rather to a Tg law.
The Fibonacci Dynamical System

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Chaos and Fractals in Financial Markets, Part 6, by J. Orlin Grabbe

The value of h = 1.618033â€¦ is closely related to the Fibonacci
sequence of numbers. The Fibonacci sequence of numbers is a
sequence in which each number is the sum of the previous two:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, â€¦
Notice the third number in the sequence, 2=1+1. The next number
3=2+1. The next number 5=3+2. And so on, each number being
the sum of the two previous numbers.
This mathematical sequence appeared 1202 A.D. in the book
Liber Abaci, written by the Italian mathematician Leonardo da
Pisa, who was popularly known as Fibonacci (son of Bonacci).
Fibonacci told a story about rabbits. These were mathematical
rabbits that live forever, take one generation to mature, and
always thereafter have one off-spring per generation. So if we start
with 1 rabbit (the first 1 in the Fibonaaci sequence), the rabbit
takes one generation to mature (so there is still 1 rabbit the next
generationâ€”the second 1 in the sequence), then it has a baby
rabbit in the following generation (giving 2 rabbitsâ€”the 2 in the
sequence), has another offspring the next generation (giving 3
rabbits); then, in the next generation, the first baby rabbit has
matured and also has a baby rabbit, so there are two offspring
(giving 5 rabbits in the sequence), and so on.
Now, the Fibonacci sequence represents the path of a dynamical
system. We introduced dynamical systems in Part 1 of this series.
(In Part 5, we discussed the concept of Julia Sets, and used a
particular dynamical systemâ€”the complex logistic equationâ€”to
create computer art in real time using Java applets. The Java
source code was also included.)
The Fibonacci dynamical system look like this:
F(n+2) = F(n+1) + F(n).
The number of rabbits in each generation (F(n+2)) is equal to the
sum of the rabbits in the previous two generations (represented by
F(n+1) and F(n)). This is an example of a more general dynamical
system that may be written as:
F(n+2) = p F(n+1) + q F(n),
where p and q are some numbers (parameters). The solution to the
system depends on the values of p and q, as well as the starting
values F(0) and F(1). For the Fibonacci system, we have the
simplification p = q = F(0) = F(1) = 1.
I will not go through the details here, but the Fibonacci system can

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Chaos and Fractals in Financial Markets, Part 6, by J. Orlin Grabbe

be solved to yield the solution:
F(n) = [1/50.5] { [(1+50.5)/2]n â€“ [(1-50.5)/2]n }, n = 1, 2, . . .
The following table gives the value of F(n) for the first few values
of n:

n 1 2 3 4 5

F(n) 1 1 2 3 5

And so on for the rest of the numbers in the Fibonacci sequence.
Notice that the general solution involves the two solution values
we previously calculated for h. To simplify, however, we will now
write everything in terms of the first of these values (namely, h =
1.618033 â€¦). Thus we have
h = [ 1 + 50.5] / 2 = 1.618033â€¦, and
- 1/ h = [ 1 - 50.5] / 2 = -0.618033â€¦
Inserting these into the solution for the Fibonacci system F(n), we
get
F(n) = [1/50.5] { [h]n â€“ [-1/ h ]n }, n = 1, 2, . . .
Alternatively, writing the solution using the golden mean g, we
have
F(n) = [1/50.5] { [g]-n â€“ [-g]n }, n = 1, 2, . . .
The use of Fibonacci relationships in financial markets has been
popularized by Robert Prechter [3] and his colleagues, following
the work of R. N. Elliott [4]. The empirical evidence that the
Hausdorff dimension of some symmetric stable distributions
encountered in financial markets is approximately Î± = h =
1.618033â€¦ indicates that this approach is based on a solid
empirical foundation.

Notes
[1] See "Research Strategy in Empirical Work with Exchange
Rate Distributions," in J. Orlin Grabbe, Three Essays in
International Finance, Ph.D. Thesis, Department of Economics,
Harvard University, 1981.
[2] There are two key papers by DuMouchel which yield the

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background needed for doing maximum likelihood estimates of Î±
, where Î± < 2:
DuMouchel, William H. (1973), "On the Asymptotic
Normality of the Maximum Likelihood Estimate
when Sampling from a Stable Distribution," Annals
of Statistics, 1, 948-57.
DuMouchel, William H. (1975), "Stable Distributions
in Statistical Inference: 2. Information from Stably
Distributed Samples," Journal of the American
Statistical Association, 70, 386-393.
[3] See, for example:
Robert R. Prechter, Jr., At the Crest of the Tidal
Wave, John Wiley & Sons, New York, 1995
Robert R. Prechter, Jr., The Wave Principle of Human
Social Behavior and the New Science of Socionomics,
New Classics Library, Gainesville, Georgia, 1999.
[4] See R.N. Elliottâ€™s Masterworksâ€”The Definitive Collection,
edited by Robert R. Prechter, Jr., Gainesville, Georgia, 1994.

J. Orlin Grabbe is the author of International Financial Markets,
and is an internationally recognized derivatives expert. He has
recently branched out into cryptology, banking security, and
http://www.aci.net/kalliste/homepage.html .
-30-
from The Laissez Faire City Times, Vol 3, No 35, September 6,
1999

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Chaos and Fractals in Financial Markets, Part 7, by J. Orlin Grabbe

Chaos and Fractals in Financial Markets
Part 7
by J. Orlin Grabbe
Grow Brain
Many dynamical systems create solution paths, or
trajectories, that look strange and complex. These
solution plots are called "strange attractors".
Some strange attractors have a fractal structure. For
example, we saw in Part 3 that it was easy to create a
fractal called a Sierpinski Carpet by using a stochastic
dynamical system (one in which the outcome at each
step is partially determined by a random component that
either selects among equations or forms part of at least
one of the equations, or both).
Here is a dynamical system that I ran across while doing
computer art. I labeled it "Grow Brain" because of its
structure. To see Grow Brain in action, make sure Java is
enabled on your browser (you can turn it off afterward)
compile their own applet, since I provide the source
code, as usual.)
The trajectory of Grow Brain is amazingly complex. But
is it a fractal? That is, at some larger or smaller scale,
will similar structures repeat themselves? Unlike the case
of the Sierpinski Carpet, the answer to this question is
not obvious for Grow Brain.
Some dynamical systems create fractal structures in time
(as Brownian motion does, in Part 2, or the
Fibonacci-type systems of Part 6 do), while others create
fractal structures in space (as in the aforementioned
Sierpinski carpet).
And some systems are all wet. Or maybe not, as the case
may be.
Hurst, Hydrology, and the Annual Flooding of the
Nile

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Chaos and Fractals in Financial Markets, Part 7, by J. Orlin Grabbe

For centuries, perhaps millennia, the yearly flooding of
the Nile was the basis of agriculture which supported
much of known civilization. The annual overflowing of
the river deposited rich top soil from the Ethiopian
Highland along the river banks. The water and silt were
distributed by irrigation, and the staple crops of wheat,
barley, and flax were planted. The grain was harvested
and stored in silos and granaries, where it was protected
from rodents by guard cats, whom the Egyptians
worshipped and turned into a cult (of the goddess Bast)
because of their importance for survival of the grain, and
hence for human survival.
The amount of Nile flooding was critical. A good flood
meant a good harvest, while a low-water flood meant a
poor harvest and possible food shortage. The flooding
came (and still comes) from tropical rains in the Upper
Nile Basin in Ethiopia (the Blue Nile) and in the East
African Plateau (the White Nile). The river flooding
would begin in the Sudan in April, and reach Aswan in
Egypt by July. (This would occur about the time of the
heliacal rising of the Dog-Star Sirus, or Sothis, around
July 19 in the Julian calendar.) The waters would then
continue to rise, peaking in mid-September in Aswan.
Further down the river at Cairo, the peak wouldnâ€™t occur
until October. The waters would then fall rapidly in
November and December, and continue to fall afterward,
reaching their low point in the March to May period.
Ancient Egypt had three seasons, all determined in
reference to the river: akhet, the "inundation"; peret, the
season when land emerged from the flood; and shomu,
the time when water was low.
A British government bureaucrat named Hurst made a
study of records of the Nileâ€™s flooding and noticed
something interesting. Harold Edwin Hurst was a poor
Leicester boy who made good, eventually working his
way into Oxford, and later became a British "civil
servant" in Cairo in 1906. He got interested in the Nile.
He looked at 800 years of records and noticed that there
was a tendency for a good flood year to be followed by
another good flood year, and for a bad (low) flood year
to be followed by another bad flood year.
That is, there appeared to be non-random runs of good or
bad years. Later Mandelbrot and Wallis [1] used the term
Joseph effect to refer to any persistent phenomenon like

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this (alluding to the seven years of Egyptian plenty
followed by the seven years of Egyptian famine in the
biblical story of Joseph).
Of course, even if the yearly flows were independent,
there still could be runs of good or bad years. So to pin
this down, Hurst calculated a variable which is now
called a Hurst exponent H. The expectation was that H =
Â½ if the yearly flood levels were independent of each
other.
Calculating the Hurst Exponent
Let me give a specific example of Hurst exponent
calculation which will illustrate the general rule.
Suppose there are 99 yearly observations of the height h
of the mid-September Nile water level at Aswan: h(1),
h(2), . . ., h(99).
Calculate a location m and a scale c for h. If we assume
in general that h has a finite variance, then m is simply
the sample mean of the 99 observations, while c is the
standard deviation.
The first thing is to remove any trend, any tendency over
the century for h to rise or fall as a long-run phenomena.
So we subtract m from each of the observations h,
getting a new series x that has mean zero:
x(1) = h(1) - m,
x(2) = h(2) - m,
â€¦
x(99) = h(99) - m .
The set of xâ€™s are a set of variables with mean zero.
Positive xâ€™s represent those years when the river level is
above average, while negative xâ€™s represent those years
when the river level is below average.
Next we form partial sums of these random variables,
each partial sum Y(n) being the sum of all the years prior
to year n:
Y(1) = x(1),
Y(2) = x(1) + x(2),
...
Y(n) = x(1) + x(2) + . . . + x(n),
...
Y(99) = x(1) + x(2) + x(3) + . . . + x(99).

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Since the Yâ€™s are a sum of mean-zero random variables
x, they will be positive if they have a preponderance of
positive xâ€™s and negative if they have a preponderance of
negative xâ€™s. In general, the set of Yâ€™s
{Y(1), Y(2), â€¦. , Y(99)}
will have a maximum and a minimum: max Y and min
Y, respectively. The difference between these two is
called the range R:
R = max Y - min Y
If we adjust R by the scale parameter c, we get the
rescaled range:
rescaled range = R/c .
Now, the probability theorist William Feller [2] had
proven that if a series of random variables like the xâ€™s 1)
had finite variance, and 2) were independent, then the
rescaled range formed over n observations would be
equal to:
R/c = k n 1/2
where k is a constant (in particular, k = (Ï€ /2)1/2 ) . That
is, the rescaled range would increase much like the
partial sums of independent variables (with finite
variance) we looked at in Part 5ï£§ namely, the partial
sums would increase by a factor of n1/2.
In particular, for n = 99 in our hypothetical data, the
result would be:
R/c = k 991/2 .
Now, the latter equation implies log(R/c) = log k + Â½ log
99. So if you ran a regression of log(R/c) against log(n)
[for a number of rescaled ranges (R/c) and their
associated number of years n] so as to estimate an
intercept a and a slope b,
log(R/c) = a + b log(n),
you should find that b is statistically indistinguishable
from Â½.
But that wasnâ€™t what Hurst found. Instead, he found that
in general the rescaled range was governed by a power

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law
R/c = k nH
where the Hurst exponent H was greater than Â½ (Hurst
found H â‰… .7)
This implied that succeeding xâ€™s were not independent of
each other: x(t) had some sticky, persistent effect on
x(t+1). This was what Hurst had observed in the data,
and his calculation showed H to be a good bit above Â½
[3].
That this would be true in general for H > Â½ , of course,
needs to be proven. Nevertheless, to summarize, for
reference, for the Hurst exponent H:
H = Â½ : the flood level deviations from the
mean are independent, random; the xâ€™s are
independent and correspond to a random
walk;
Â½ < H <=1: the flood level deviations are
persistentï£§ high flood levels tend to be
followed by high flood levels, and low
flood levels by low flood levels; x(t+1)
tends to deviate from the mean the same
way x(t) did; the probability that x(t+1)
deviates from the mean in the same
direction as x(t) increases as H approaches
1;
0<=H< 1/2: the flood level deviations are
anti-persistentï£§ the xâ€™s are
mean-reverting; high flood levels have a
tendency to be followed by low flood
levels, and vice-versa; the probability that
x(t+1) deviates from the mean in the
opposite direction from x(t) increases as H
approaches 0.
A Misunderstanding to Avoid
Recall that Bachelier had noted that the probability range
of the log of a stock price would increase with the square
root of time T. The probability range, starting at log S,
would grow with T according to:
(log S â€“ k T1/2 , log S + k T1/2),

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where k is the scale (in his case, the standard deviation)
c, k = c. But, more generally, the symmetric stable
distributions of Part 5, increase with T raised to the
reciprocal power of the Hausdorff dimension Î± (Î± <=2):

(log S â€“ k T1/Î± , log S + k T1/Î± ).
Hurst similarly said the rescaled range of the flood level
varied according to (setting n = T):
R/c = k TH .
So it is tempting to equate the Hurst exponent H with the
reciprocal of the Hausdorff dimension D, to equate H
with 1/D = 1/Î± . But we must be careful.
Recall that symmetric stable distributions, with Î± < 2,
have infinite variance (for them, variance is a blob
measure that is not meaningful). However, here in
discussing the Hurst exponent we are assuming that the
variance, and standard deviation (the scale c), are finite,
and hence Î± =2. The role of the Hurst exponent is to
inform us whether the yearly flood deviations are
independent or persistent. H is not related to the need for
a different scale measure. The variance and the standard
deviation are well defined for these latter processes.
Nevertheless, the formal equation H = 1/D or D = 1/H
yields the correct exponent for T in the case Â½<= H <=1.
Even though Î± =2, the calculation of the Hausdorf
dimension D yields D<2 if the increments are not
independent. Hence D can take a minimum value of 1, D
= 1/H = 1/1 = 1 when H=1, so that the process
accumulates variation (rescaled range) much like a
Cauchy sequence (TH = T); or a maximum of 2, D = 1/H
= 1/Â½ = 2 when H=Â½, so that the process accumulates
variation (rescaled range) like a Gaussian sequence (TH
= T1/2), or ordinary Brownian motion. [4]
Mandelbrot called these types of processes where Î± =2,
but where H â‰  Â½, fractional Brownian motion. (I will not
here elaborate the case H < Â½.)
Bull and Bear Markets
We are, of course, used to the idea of persistent
phenomena in the stock market and foreign exchange

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markets. The NASD rises relentlessly for a period of
time. Then it falls just as persistently. There are bull and
bear markets, implying the price rise or decline is a
persistent phenomena, and not just an accidental
accumulation of random variables in one direction.
The US dollar rises relentless for a period of years, then
(as it is doing now) begins a relentless decline for
another period of years. In the case of the Nile, the
patterns of rising and falling are partly governed by the
weather patterns in the green rain forest of the Ethiopian
highlands. In the case of the US dollar, the patterns of
rising and falling are partly governed by the span of
Green in the Washington D.C. lowlands.
Notes
[1] B.B. Mandelbrot & J. R. Wallis, "Noah, Joseph, and
Operational Hydrology." Water Resources Research 4,
909-918, (1968).
[2] W. Feller, "The asymptotic distribution of the range
of sums of independent random variables." Annals of
Mathematical Statistics 22, 427 (1951).
[3] H. E. Hurst, "Long-term storage capacity of
reservoirs." Tr. of the American Society of Civil
Engineers 116, 770-808 (1951).
Mandelbrot, The Fractal Geometry of Nature. W.H.
Freeman and Company, New York, 1983.

J. Orlin Grabbe is the author of International Financial
Markets, and is an internationally recognized derivatives
expert. He has recently branched out into cryptology,
located at http://orlingrabbe.org/ .

-30-
from The Laissez Faire City Times, Vol 5, No 3,
January 15, 2001

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Iterative Plot of Grow Brain

Iterative Plot of Grow Brain
If you can read this, then your browser is not set up for Java. . .
Wait a minute for the applet to run. This amazingly complex structure is created by the simple iteration
of the two equations:
X = Y - (X/|X|)*sqrt(|aX|)
Y=b-X
(Here "sqrt" means to take the square root.)
The first series of iterations here starts with the initial values X = .9 and Y = 0. The two equations are
then iterated 10,000 times, plotting each (X,Y) point of the iteration. Then the starting value for Y is
increased slightly, and another 10,000 plots are plotted (see the Java source code for details).
By varying the parameter a in the first equation above (aX is set at 1.53 X for the applet supplied here),
and also the rate at which the starting value for Y is increased (set Y = .025*col, say, instead of Y =
.047*col), different structures can be obtained.

The Java source code.

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Chaos and Fractals in Financial Markets, Part 2, by J. Orlin Grabbe

Chaos and Fractals in Financial Markets
Part 2
by J. Orlin Grabbe
The French Gambler and the Pollen
Grains
In 1827 an English botanist, Robert Brown got his hands
on some new technology: a microscope "made for me by
Mr. Dolland, . . . of which the three lenses that I have
generally used, are of a 40th, 60th, and 70th of an inch
focus."
Right away, Brown noticed how pollen grains suspended
in water jiggled around in a furious, but random, fashion.
To see what Brown saw under his microscope, make

What was going on was a puzzle. Many people
wondered: Were these tiny bits of organic matter
somehow alive? Luckily, Hollywood wasnâ€™t around at
the time, or John Carpenter might have made his
wonderful horror film They Live! about pollen grains
rather than about the infiltration of society by liberal
control-freaks.
Robert Brown himself said he didnâ€™t think the movement
had anything to do with tiny currents in the water, nor
was it produced by evaporation. He explained his
observations in the following terms:
"That extremely minute particles of solid
matter, whether obtained from organic or
inorganic substances, when suspended in
pure water, or in some other aqueous fluids,
exhibit motions for which I am unable to
account, and from which their irregularity
and seeming independence resemble in a
remarkable degree the less rapid motions of
some of the simplest animalcules of
infusions. That the smallest moving
particles observed, and which I have termed
Active Molecules, appear to be spherical, or
nearly so, and to be between 1-20,0000dth
and 1-30,000dth of an inch in diameter; and

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that other particles of considerably greater
and various size, and either of similar or of
very different figure, also present analogous
motions in like circumstances.
"I have formerly stated my belief that these
motions of the particles neither arose from
currents in the fluid containing them, nor
depended on that intestine motion which
may be supposed to accompany its
evaporation."[1]
observations in special cases. For example, a Dr. James
Drummond had observed this fishy, erratic motion in
fish eyes:
"In 1814 Dr. James Drummond, of Belfast,
published in the 7th Volume of the
Transactions of the Royal Society of
Edinburgh, a valuable Paper, entitled â€˜On
certain Appearances observed in the
Dissection of the Eyes of Fishes.â€™
"In this Essay, which I regret I was entirely
unacquainted with when I printed the
account of my Observations, the author
gives an account of the very remarkable
motions of the spicula which form the
silvery part of the choroid coat of the eyes
of fishes."
Today, we know that this motion, called Brownian
motion in honor of Robert Brown, was due to random
fluctuations in the number of water molecules
bombarding the pollen grains from different directions.
Experiments showed that particles moved further in a
given time interval if you raised the temperature, or
reduced the size of a particle, or reduced the "viscosity"
[2] of the fluid. In 1905, in a celebrated treatise entitled
The Theory of the Brownian Movement [3], Albert
Einstein developed a mathematical description which
explained Brownian motion in terms of particle size,
fluid viscosity, and temperature. Later, in 1923, Norbert
Wiener gave a mathematically rigorous description of
what is now referred to as a "stochastic process." Since
that time, Brownian motion has been called a Wiener
process, as well as a "diffusion process", a "random
walk", and so on.
But Einstein wasnâ€™t the first to give a mathematical

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description of Brownian motion. That honor belonged to
a French graduate student who loved to gamble. His
name was Louis Bachelier. Like many people, he sought
to combine duty with pleasure, and in 1900 in Paris
presented his doctoral thesis, entitled ThÃ©orie de la
What interested Bachelier were not pollen grains and
fish eyes. Instead, he wanted to know why the prices of
stocks and bonds jiggled around on the Paris bourse. He
was particularly intrigued by bonds known as rentes sur
lâ€™Ã©tatâ€” perpetual bonds issued by the French
government. What were the laws of this jiggle?
Bachelier wondered. He thought the answer lay in the
prices being bombarded by small bits of news. ("The
British are coming, hammer the prices down!")
The Square Root of Time
Among other things, Bachelier observed that the
probability intervals into which prices fall seemed to
increased or decreased with the square-root of time
(T0.5). This was a key insight.
By "probability interval" we mean a given probability
for a range of prices. For example, prices might fall
within a certain price range with 65 percent probability
over a time period of one year. But over two years, the
same price range that will occur with 65 percent
probability will be larger than for one year. How much
larger? Bachelier said the change in the price range was
proportional to the square root of time.
Let P be the current price. After a time T, the prices will
(with a given probability) fall in the range
(P â€“a T0.5, P + a T0.5), for some constant a.
For example, if T represents one year (T=1), then the last
equation simplifies to
(P â€“a , P + a), for some constant a.
The price variation over two years (T=2) would be
a T0.5 = a(2)0.5 = 1.4142 a
or 1.4142 times the variation over one year. By contrast,
the variation over a half-year (T=0.5) would be
a T0.5 = a(0.5) 0.5 = .7071 a
or about 71 percent of the variation over a full year. That
is, after 0.5 years, the price (with a given probability)

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would be in the range
(P â€“.7071a , P + .7071a ).
Here the constant a has to be determined, but one
supposes it will be different for different types of prices:
a may be bigger for silver prices than for gold prices, for
example. It may be bigger for a share of Yahoo stock
than for a share of IBM.
The range of prices for a given probability, then,
depends on the constant a, and on the square root of
time (T0.5). This was Bachelierâ€™s insight.
Normal Versus Lognormal
Now, to be sure, Bachelier made a financial mistake.
Remember (from Part 1 of this series) that in finance we
always take logarithms of prices. This is for many
reasons. Most changes in most economic variables are
proportional to their current level. For example, it is
plausible to think that the variation in gold prices is
proportional to the level of gold prices: \$800 dollar gold
varies in greater increments than does gold at \$260.
The change in price, âˆ†P, as a proportion of the current
price P, can be written as:
âˆ†P/P .
But this is approximately the same as the change in the
log of the price:
âˆ†P/P â‰ˆ âˆ† (log P) .
What this means is that Bachelier should have written his
equation:
(log P â€“a T0.5, log P + a T0.5), for some constant a.
However, keep in mind that Bachelier was making
innovations in both finance and in the mathematical
theory of Brownian motion, so he had a hard enough
time getting across the basic idea, without worrying
about fleshing out all the correct details for a
non-existent reading audience. And, to be sure, almost
no one read Bachelierâ€™s PhD thesis, except the celebrated
mathematician Henri PoincarÃ©, one of his instructors.
The range of prices for a given probability, then, depends
on the constant a, and on the square root of time (T0.5),
as well as the current price level P.
To see why this is true, note that the probability range

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for the log of the price
(log P â€“a T0.5, log P + a T0.5)
translates into a probability range for the price itself as
( P exp(- a T0.5), P exp( a T0.5 ) ) .
(Here "exp" means exponential, remember? For
example, exp(-.7) = e -.7 = 2.718281-.7 = .4966. )
Rather than adding a plus or minus something to the
current price P, we multiply something by the current
price P. So the answer depends on the level of P. For a
(P â€“.7071a , P + .7071a )
we get
( P exp(- .7071 a ), P exp( .7071 a ) ) .
The first interval has a constant width of 1.4142 a, no
matter what the level of P (because P + .7071 a - (P
-.7071 a) = 1.4142 a). But the width of the second
interval varies as P varies. If we double the price P, the
width of the interval doubles also.
Bachelier allowed the price range to depend on the
constant a and on the square root of time (T0.5), but
omitted the requirement that the range should also
depend on the current price level P.
The difference in the two approaches is that if price
increments (âˆ†P) are independent, and have a finite
variance, then the price P has a normal (Gaussian
distribution). But if increments in the log of the price (âˆ†
log P) are independent, and have a finite variance, then
the price P has a lognormal distribution.
Here is a picture of a normal or Gaussian distribution:

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The left-hand tail never becomes zero. No matter where
we center the distribution (place the mean), there is
always positive probability of negative numbers.
Here is a picture of a lognormal distribution:

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The left-hand tail of a lognormal distribution becomes
zero at zero. No matter where we center the distribution
(place the mean), there is zero probability of negative
numbers.
A lognormal distribution assigns zero probability to
negative prices. This makes us happy because most
businesses donâ€™t charge negative prices. (However, US
Treasury bills paid negative interest rates on certain
occasions in the 1930s.) But a normal distribution
assigns positive probability to negative prices. We donâ€™t
want that.
So, at this point, we have seen Bachelierâ€™s key insight
that probability intervals for prices change proportional
to the square root of time (that is, the probability interval
around the current price P changes by a T0.5), and have
modified it slightly to say that probability intervals for
the log of prices change proportional to the square root
of time (that is, the probability interval around log P
changes by a T0.5).
How Big Is It?
Now we are going to take a break from price
distributions, and pursue the question of how we
measure things. How we measure length, area, volume,
or time. (This will lead us from Bachelier to
Mandelbrot.)
Usually, when we measure things, we use everyday
dimensions (or at least the ones we are familiar with
from elementary plain geometry). A point has zero
dimension. A line has one dimension. A plane or a
square has two dimensions. A cube has three
dimensions. These basic, common-sense type
dimensions are sometimes referred to as topological
dimensions.
We say a room is so-many "square feet" in size. In this
case, we are using the two-dimensional concept of area.
We say land is so-many "acres" in size. Here, again, we
are using a two-dimensional concept of area, but with
different units (an "acre" being 43,560 "square feet").
We say a tank holds so-many "gallons". Here we are
using a measure of volume (a "gallon" being 231 "cubic
inches" in the U.S., or .1337 "cubic feet").
Suppose you have a room that is 10 feet by 10 feet, or
100 square feet. How much carpet does it take to cover
the room? Well, you say, a 100 square feet of carpet, of
course. And that is true, for ordinary carpet.

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Letâ€™s take a square and divide it into smaller pieces.
Letâ€™s divide each side by 10:

We get 100 pieces. That is, if we divide by a scale factor of 10, we get 100 smaller
squares, all of which look like the big square. If we multiply any one of the smaller
squares by 10, we get the original big square.
Letâ€™s calculate a dimension for this square. We use the same formula as we used for
the Sierpinski carpet:
N = rD .
Taking logs, we have log N = D log r, or D = log N/ log r.
We have N = 100 pieces, and r = 10, so we get the dimension D as
D = log(100)/log(10) = 2.
(We are using "log" to mean the natural log, but notice for this calculation, which
involves the ratio of two logs, that it doesnâ€™t matter what base we use. You can use
logs to the base 10, if you wish, and do the calculation in your head.)
We called the dimension D calculated in this way (namely, by comparing the
number of similar objects N we got at different scales to the scale factor r) a
Hausdorff dimension. In this case, the Hausdorff dimension 2 is the same as the
ordinary or topological dimension 2.

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So, in any case, the dimension is 2, just as you suspected all along. But suppose you
covered the floor with Sierpinski carpet. How much carpet do you need then?
We saw (in Part 1) that the Sierpinski carpet had a Hausdorff dimension D = 1.8927â€¦
A Sierpinski carpet which is 10 feet on each side would only have N = 101.8927 =
78.12 square feet of material in it.
Why doesnâ€™t a Sierpinski carpet with 10 feet on each side take 100 square feet of
material? Because the Sierpinski carpet has holes in it, of course.
Remember than when we divided the side of a Sierpinski carpet by 3, we got only 8
copies of the original because we threw out the center square. So it had a Hausdorff
dimension of D = log 8/ log 3 = 1.8927. Then we divided each of the 8 copies by 3
again , threw out the center squares once more, leaving 64 copies of the original.
Dividing by 3 twice is the same as dividing by 9, so, recalculating our dimension, we
get D = log 64/ log 9 = 1.8927.
An ordinary carpet has a Hausdorff dimension of 2 and a topological (ordinary)
dimension of 2. A Sierpinski carpet has a Hausdorff dimension of 1.8927 and a
topological dimension of 2. [4]
Benoit Mandelbrot defined a fractal as an object whose Hausdorff dimension is
different from its topological dimension. So a Sierpinski carpet is a fractal. An
ordinary carpet isnâ€™t.
Fractals are cheap and sexy. A Sierpinski carpet needs only 78.12 square feet of
material to cover 100 square feet of floor space. Needing less material, a Sierpinski
carpet costs less. Sure it has holes in it. But the holes form a really neat pattern. So a
Sierpinski carpet is sexy. Cheap and sexy. You canâ€™t beat that.
Historyâ€™s First Fractal
Letâ€™s see if we have this fractal stuff straight. Letâ€™s look at the first known fractal,
created in 1870 by the mathematical troublemaker George Cantor.
Remember that we create a fractal by forming similar patterns at different scales, as
we did with the Sierpinski carpet. Itâ€™s a holey endeavor. In order to get a carpet whose
Hausdorff dimension was less than 2, we created a pattern of holes in the carpet. So
we ended up with an object whose Hausdorff dimension D (which compares the
number N of different, but similar, objects at different scales r, N = rD ) was more than
1 but less than 2. That made the Sierpinski carpet a fractal, because its Hausdorff
dimension was different from its topological dimension.
What George Cantor created was an object whose dimension was more than 0 but less
than 1. That is, a holey object that was more than a point (with 0 dimensions) but less
than a line (with 1 dimension). Itâ€™s called Cantor dust. When the Cantor wind blows,
the dust gets in your lungs and you canâ€™t breathe.
To create Cantor dust, draw a line and cut out the middle third:

0________________________________________________________1

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0__________________1/3 2/3_________________1

Now cut out the middle thirds of each of the two remaining pieces:

0____1/9 2/9____ 1/3 2/3____7/9 8/9 ____1

Now cut out the middle thirds of each of the remaining four pieces, and proceed in this
manner for an infinite number of steps, as indicated in the following graphic.

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What's left over after all the cutting is Cantor dust.
At each step we changed scale by r = 3, because we
divided each remaining part into 3 pieces. (Each of these
pieces had 1/3 the length of the original part.) Then we
threw away the middle piece. (Thatâ€™s how we created the
holes.) That left 2 pieces. At the next step there were 4
pieces, then 8, and so on. At each step the number of
pieces increased by a factor of N = 2. Thus the Hausdorff
dimension for Cantor dust is:
D = log 2 / log 3 = .6309.
Is Cantor dust a fractal? Yes, as long as the topological
dimension is different from .6309, which it surely is.
Butâ€”what is the topological dimension of Cantor dust?
We can answer this by seeing how much of the original
line (with length 1) we cut out in the process of making
holes.
At the first step we cut out the middle third, or a length
of 1/3. The next step we cut out the middle thirds of the
two remaining pieces, or a length of 2(1/3)(1/3). And so
on. The total length cut out is then:
1/3 + 2(1/32) + 4(1/33) + 8(1/34) + . . . = 1.
We cut out all of the length of the line (even though we
left an infinite number of points), so the Cantor dust
that's left over has length zero. Its topological dimension
is zero. Cantor dust is a fractal with a Hausdorff
dimension of .6309 and a topological dimension of 0.
Now, the subhead refers to Cantor dust as "historyâ€™s first
fractal". That a little anthropocentric. Because nature has
been creating fractals for millions of years. In fact, most
things in nature are not circles, squares, and lines.
Instead they are fractals, and the creation of these
fractals are usually determined by chaos equations.
Chaos and fractal beauty are built into the nature of
reality. Get used to it.
Today, there are roughly of order 103
recognized fractal systems in nature, though
a decade ago when Mandelbrot's classic
Fractal Geometry of Nature was written,
many of these systems were not known to
be fractal. [5]
Fractal Time
So far weâ€™ve seen that measuring things is a complicated

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business. Not every length can be measured with a tape
measure, nor the square footage of material in every
carpet measured by squaring the side of the carpet.
Many things in life are fractal, and follow power laws
just like the D of the Hausdorff dimension. For example,
the "loudness" L of noise as heard by most humans is
proportional to the sound intensity I raised to the
fractional power 0.3:
L = a I0.3 .
Doubling the loudness at a rock concert requires
increasing the power output by a factor of ten, because
a (10 I)0.3 = 2 a I0.3 = 2 L .
In financial markets, another subjective domain, "time"
is fractal. Time does not always move with the rhythms
of a pendulum. Sometimes time is less than that. In fact,
weâ€™ve already encounted fractal time with the Bachelier
process, where the log of probability moved according to
a T0.5 .
Bachelier observed that if the time interval was
multiplied by 4, the probability interval only increased
by 2. In other words, at a scale of r = 4, the number N of
similar probability units was N = 2. So the Hausdorff
dimension for time was:
D = log N/ log r = log 2/ log 4 = 0.5 .
In going from Bachelier to Mandelbrot, then, the
innovation is not in the observation that time is fractal:
that was Bachelierâ€™s contribution. Instead the question is:
What is the correct fractal dimension for time in
speculative markets? Is the Hausdorff dimension really
D = 0.5, or does it take other values? And if the
Hausdorff dimension of time takes other values, whatâ€™s
the big deal, anyway?
The way in which Mandlebrot formulated the problem
provides a starting point:
Despite the fundamental importance of
Bachelier's process, which has come to be
called "Brownian motion," it is now
obvious that it does not account for the
abundant data accumulated since 1900 by
empirical economists, simply because the
empirical distributions of price changes are
usually too "peaked" to be relative to

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samples from Gaussian populations. [6]
What does Mandelbrot mean by "peaked"? Itâ€™s now time
for a discussion of probability.
Probability is a One-Pound Jar of Jelly
Probability is a one-pound jar of jelly. You take the jelly
and smear it all over the real line. The places where you
smear more jelly have more probability, while the places
where you smear less jelly have less probability. Some
spots may get no jelly. They have no probability at
allâ€”their probability is zero.
The key is that you only have one pound of jelly. So if
you smear more jelly (probability) at one location, you
have to smear less jelly at another location.
Here is a picture of jelly smeared in the form of a
bell-shaped curve:

The jelly is smeared between the horizontal (real) line all
the way up to the curve, with a uniform thickness. The
result is called a "standard normal distribution".
("Standard" because its mean is 0, and the standard
deviation is 1.) In this picture, the point where the
vertical line is and surrounding points have the jelly
piled highâ€”hence they are more probable.
As we observed previously, for the normal distribution
jelly gets smeared on the real (horizontal) line all the
way to plus or minus infinity. There may not be much
jelly on the distant tails, but there is always some.
does Mandelbrot mean by the distribution of price
changes being "too peaked" to come from a normal

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distribution?
Does Mandelbrotâ€™s statement make any sense? If we
smear more jelly at the center of the bell curve, to make
it taller, we can only do so by taking jelly from some
other place. Suppose we take jelly out of the tails and
intermediate parts of the distribution and pile it on the
center. The distribution is now "more peaked". It is more
centered in one place. It has a smaller standard
deviationâ€”or smaller dispersion around the mean.
Butâ€”it could well be still normal.
So whatâ€™s with Mandelbrot, anyway? What does he
mean? Weâ€™ll discover this in Part 3 of this series.

The material therein should be helpful in solving
Problem 2.

Meanwhile, here are two new problems for eager
students:
Problem 3: Suppose you create a Cantor dust using a
different procedure. Draw a line. Then divide the line
into 5 pieces, and throw out the second and fourth
pieces. Repeat this procedure for each of the remaining
pieces, and so on, for an infinite number of times. What
is the fractal dimension of the Cantor dust created this
way? What is its topological dimension? Did you create
a new fractal?
Problem 4: Suppose we write all the numbers between 0
and 1 in ternary. (Ternary uses powers of 3, and the
numbers 0, 1, 2. The ternary number .1202, for example,
stands for 1 x 1/3 + 2 x 1/9 + 0 x 1/27 + 2 x 1/81.) Show
the Cantor dust we created here in Part 2 (with a
Hausdorff dimension of .6309) can be created by taking
all numbers between 0 and 1, and eliminating those
numbers whose ternary expansion contains a 1. (In other
words, what is left over are all those numbers whose
ternary expansions only have 0s and 2s.)
And enjoy the fractal:

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Notes
[1] Robert Brown, "Additional Remarks on Active
Molecules," 1829.
[2] Viscosity is a fluidâ€™s stickiness: honey is more
viscous than water, for example. "Honey donâ€™t jiggle so
much."
[3] I am using the English title of the well-known Dover
reprint: Investigations on the Theory of the Brownian
Movement, Edited by R. Furth, translated by A.D.
Cowpter, London, 1926. The original article was in
German and titled somewhat differently.
[4] I am admittingly laying a subtle trap here, because of
the undefined nature of "topological dimension". This is
partially clarified in the discussion of Cantor dust, and
further discussed in Part 3.
[5] H. Eugene Stanley, Fractals and Multifractals, 1991
[6] Benoit Mandelbrot, "The Variation of Certain
Speculative Prices," Journal of Business, 36(4), 394-419,
1963.

J. Orlin Grabbe is the author of International Financial
Markets, and is an internationally recognized derivatives
expert. He has recently branched out into cryptology,
located at http://www.aci.net/kalliste/homepage.html .

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-30-
from The Laissez Faire City Times, Vol 3, No 24, June
14, 1999

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