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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
PrechterвЂ™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 doesnвЂ™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 letвЂ™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 вЂ“ 1 = 0.
This gives two solutions:
g = [- 1 + 50.5] / 2 = 0.618033вЂ¦, and
g = [- 1 - 50.5] / 2 = -1.618033вЂ¦
The first, positive solution (g = 0.618033вЂ¦) 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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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 вЂ“ 1 = 0.
Which gives the two solutions:
h = [ 1 + 50.5] / 2 = 1.618033вЂ¦, and
h = [ 1 - 50.5] / 2 = -0.618033вЂ¦
Note that since the units of measurement are somewhat aribitrary,
h has as much claim as g to being the solution to PrechterвЂ™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:
1.618033вЂ¦ = 1/ 0.618033вЂ¦
-0.618033вЂ¦ = 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, letвЂ™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 вЂ“ 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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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/ О± , where О± is the Hausdorff
dimension of the given symmetric stable distribution.
In the case of the normal or Gaussian distribution, the Hausdorff
dimension О± = 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/О± = T1/2 law.
In the case of the Cauchy distribution (Part 4), the Hausdorff
dimension О± = 1, which is equivalent to the dimension of a line. A
Cauchy process would be governed by a T1/О± = T1/1 = T law.
In general, 0 < О± <=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 (О± =
log 8/ log 3 = 1.8927вЂ¦.) or Koch curve (О± = log 4/ log 3 =
1.2618вЂ¦.).
Interestingly, however, many financial variables are symmetric
stable distributions with an О± 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/О± = 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 О± 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.  (The time period was July 1973 to
June 1979.) The value of О± was calculated using maximum
likelihood techniques . The value I found was
О± = 1.62
with a margin of error of plus or minus .04. You canвЂ™t get much
closer than that to О± = h = 1.618033вЂ¦
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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The value of h = 1.618033вЂ¦ 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, вЂ¦
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
generationвЂ”the second 1 in the sequence), then it has a baby
rabbit in the following generation (giving 2 rabbitsвЂ”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 systemвЂ”the complex logistic equationвЂ”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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be solved to yield the solution:
F(n) = [1/50.5] { [(1+50.5)/2]n вЂ“ [(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 вЂ¦). Thus we have
h = [ 1 + 50.5] / 2 = 1.618033вЂ¦, and
- 1/ h = [ 1 - 50.5] / 2 = -0.618033вЂ¦
Inserting these into the solution for the Fibonacci system F(n), we
get
F(n) = [1/50.5] { [h]n вЂ“ [-1/ h ]n }, n = 1, 2, . . .
Alternatively, writing the solution using the golden mean g, we
have
F(n) = [1/50.5] { [g]-n вЂ“ [-g]n }, n = 1, 2, . . .
The use of Fibonacci relationships in financial markets has been
popularized by Robert Prechter  and his colleagues, following
the work of R. N. Elliott . The empirical evidence that the
Hausdorff dimension of some symmetric stable distributions
encountered in financial markets is approximately О± = h =
1.618033вЂ¦ indicates that this approach is based on a solid
empirical foundation.

Notes
 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.
 There are two key papers by DuMouchel which yield the

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background needed for doing maximum likelihood estimates of О±
, where О± < 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.
 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.
 See R.N. ElliottвЂ™s MasterworksвЂ”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 5, by J. Orlin Grabbe

Chaos and Fractals in Financial Markets
Part 5
by J. Orlin Grabbe
Louis Bachelier Visits the New York Stock Exchange
Louis Bachelier, resurrected for the moment, recently
visited the New York Stock Exchange at the end of May
1999. He was somewhat puzzled by all the hideous
concrete barriers around the building at the corner of
Broad and Wall Streets. For a moment he thought he was
in Washington, D.C., on Pennsylvania Avenue.
Bachelier was accompanied by an angelic guide named
Pete. "The concrete blocks are there because of Osama
bin Ladin," Pete explained. "HeвЂ™s a terrorist." Pete didnвЂ™t
bother to mention the blocks had been there for years. He
knew Bachelier wouldnвЂ™t know the difference.
"Terrorist?"
"You know, a ruffian, a scoundrel."
"Yes, and before that, there was Abu Nidal."
"Abu Nidal. The father of Nidal. Hey! Ladin is just Nidal
spelled backwards. So weвЂ™ve gone from the father of
Nidal to the son of backwards-Nidal?"
"Yes," Pete said cryptically. "The spooks are never too
creative when they are manufacturing the boogeyman of
the moment. If you want to understand all this, read
about вЂ˜GoldsteinвЂ™ and the daily scheduled вЂ˜Two Minutes
HateвЂ™ in George OrwellвЂ™s book 1984."
"1984? LetвЂ™s see, that was fifteen years ago," Bachelier
said. "A historical work?"
"Actually, itвЂ™s futuristic. But he who controls the present
controls the past, and he who controls the past controls
the future."
Bachelier was mystified by the entire conversation, but
once they got inside and he saw the trading floor, he felt

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right at home. Buying, selling, changing prices. The
chalk boards were now electric, he saw, and that made
the air much fresher.
"Look," Bachelier said, "the Dow Jones average is still
around!"
"Yes," nodded Pete, "but there are a lot of others also.
Like the S&P500 and the New York Stock Exchange
Composite Index."
"I want some numbers!" Bachelier exclaimed
enthusiastically. Before they left, they managed to con
someone into giving them the closing prices for the
NYSE index for the past 11 days.
"You can write a book," Pete said. "Call it Eleven Days
in May. Apocalyptic titles are all the rage these
daysвЂ”except in the stock market."
Bachelier didnвЂ™t pay him any mind. He had taken out a
pencil and paper and was attempting to calculate
logarithms through a series expansion. Pete watched in
silence for a while, before he took pity and pulled out a
pocket calculator.
"Let me show you a really neat invention," the angel
said.
BachelierвЂ™s Scale for Stock Prices
Here is BachelierвЂ™s data for eleven days in May. We
have the calendar date in the first column of the table;
the NYSE Composite Average, S(t), in the second
column; the log of S(t) in the third column; the change in
log prices, x(t) = log S(t) вЂ“ log S(t-1) in the fourth
column; and x(t)2 in the last column. The sum of the
variables in the last column is given at the bottom of the
table.

x(t)2
Date S(t) log S(t) x(t)

May 14 638.45 6.459043

May 17 636.92 6.456644 -.002399 .000005755

May 18 634.19 6.452348 -.004296 .000018456

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May 19 639.54 6.460749 .008401 .000070577

May 20 639.42 6.460561 -.000188 .000000035

May 21 636.87 6.456565 -.003996 .000015968

May 24 626.05 6.439430 -.017135 .000293608

May 25 617.34 6.425420 -.014010 .000196280

May 26 624.84 6.437495 .012075 .000145806

May 27 614.02 6.420027 -.017468 .000305131

May 28 622.26 6.433358 .013331 .000177716

sum of all x(t)2 = .001229332

What is the meaning of all this?
The variables x(t), which are the one-trading-day
changes in log prices, are the variables in which
Bachelier is interested for his theory of Brownian motion
as applied to the stock market:
x(t) = log S(t) вЂ“ log S(t-1).
Bachelier thinks these should have a normal distribution.
Recall from Part 4 that a normal distribution has a
location parameter m and a scale parameter c. So what
Bachelier is trying to do is to figure out what m and c
are, assuming that each dayвЂ™s m and c are the same as
any other dayвЂ™s.
The location parameter m is easy. It is zero, or pretty
close to zero.
In fact, it is not quite zero. Essentially there is a drift in
the movement of the stock index S(t), given by the
difference between the interest rate (such as the
broker-dealer loan rate) and the dividend yield on stocks
in the average. But this is tiny over our eleven trading
days (which gives us ten values for x(t)). So Bachelier
just assumes m is zero.

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So what Bachelier is doing with the data is trying to
estimate c.
Recall from Part 2 that if todayвЂ™s price is P, Bachelier
modeled the probability interval around the log of the
price change by
(log P вЂ“ a T0.5, log P + a T0.5), for some constant a.
But now, we are writing our stock index price as S, not
P; and the constant a is just our scale parameter c. So,
changing notation, Bachelier is interested in the
probability interval
(log S вЂ“ c T0.5, log S + c T0.5), for a given scale
parameter c.
One way of estimating the scale c (c is also called the
"standard deviation" in the context of the normal
distribution) is to add up all the squared values of x(t),
and take the average (by dividing by the number of
observations). This gives us an estimate of the variance,
or c2. Then we simply take the square root to get the
scale c itself. (This is called a maximum likelihood
estimator for the standard deviation.)
Adding up the terms in the right-hand column in the
table gives us a value of .001229332. And there are 10
observations. So we have
variance = c2 = .001229332/10 = .0001229332.
Taking the square root of this, we have
standard deviation = c = (.0001229332)0.5 = .0110875.
So BachelierвЂ™s changing probability interval for log S
becomes:
(log S вЂ“ .0110875 T0.5, log S + .0110875 T0.5).
To get the probability interval for the price S itself, we
just take exponentials (raise to the power exp = e =
2.718281вЂ¦), and get
( S exp(вЂ“ .0110875 T0.5), S exp(.0110875 T0.5) ).
Since the current price on May 28, from the table, is
622.26, this interval becomes:

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(622.26 exp(вЂ“ .0110875 T0.5), 622.26 exp(.0110875
T0.5) ).
"This expression for the probability interval tells us the
probability distribution over the next T days," Bachelier
explained to Pete. "Now I understand what you meant.
He who controls the present controls the past, because he
can obtain past data. While he who masters this past data
controls the future, because he can calculate future
probabilities!"
"Umm. That wasnвЂ™t what I meant," the angel replied.
"But never mind."
Over the next 10 trading days, we have T0.5 = 100.5 =
3.162277. So substituting that into the probability
interval for price, we get
(622.26 (.965545), 622.26 (1.035683)) = (600.82,
644.46).
This probability interval gives a price range for plus or
minus one scale parameter (in logs) c. For the normal
distribution, that corresponds to 68 percent probability.
With 68 percent probability, the price will lie between
600.82 and 644.46 at the end of 10 more trading days,
according to this calculation.
To get a 95 percent probability interval, we use plus or
minus 2c,
(622.26 exp(вЂ“ (2) .0110875 T0.5), 622.26 exp( (2)
.0110875 T0.5) ),
which gives us a price interval over 10 trading days of
(580.12, 667.46).
Volatility
In the financial markets, the scale parameter c is often
called "volatility". Since a normal distribution is usually
assumed, "volatility" refers to the standard deviation.
Here we have measured the scale c, or volatility, on a
basis of one trading day. The value of c we calculated, c
= .0110875, was calculated over 10 trading days, so it
would be called in the markets "a 10-day historical
volatility." If calculated over 30 past trading days, it
would be "a 30-day historical volatility."

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However, market custom would dictate two criteria by
which volatility is quoted:
1. quote volatility at an annual (not daily) rate;
2. quote volatility in percentage (not decimal) terms.
To change our daily volatility c = .0110875 into annual
the year. The square root of 256 is 16, so to change daily
volatility into annual volatility, we simply multiply it by
16:
annual c = 16 (daily c) = 16 (.0110875) = .1774.
Then we convert this to percent (by multiplying by 100
and calling the result "percent"):
annual c = 17.74 percent.
The New York Stock Exchange Composite Index had a
historical volatility of 17.74 percent over the sample
period during May.
Note that an annual volatility of 16 percent corresponds
to a daily volatility of 1 percent. This is a useful
relationship to remember, because we can look at a price
or index, mentally divide by 100, and say the price
change will fall in the range of plus or minus that amount
with 2/3 probability (approximately). For example, if the
current gold volatility is 16 percent, and the price is
\$260, we can say the coming dayвЂ™s price change will fall
in the range of plus or minus \$2.60 with about 2/3
probability.
Notice that 256 trading days give us a probability
interval that is only 16 times as large as the probability
interval for 1 day. This translates into a Hausdorff
dimension for time (in the probability calculation) as D =
log(16)/log(256) = ВЅ or 0.5, which is just the
Bachelier-Einstein square-root-of-T (T0.5) law.
The way we calculated the scale c is called "historical
volatility," because we used actual historical data to
estimate c. In the options markets, there is another
measure of volatility, called "implied volatility."
Implied volatility is found by back-solving an option
value (using a valuation formula) for the volatility, c,
that gives the current option price. Hence this volatility,

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which pertains to the future (specifically, to the future
life of the option) is implied by the price at which the
Fractal Sums of Random Variables
Now for the fun part. We have been looking at random
variables x(t) (representing changes in the log of price).
Under the assumption these random variables were
normal, we estimated a scale parameter c, which allows
us to do probability calculations.
In order to estimate c, we took the sums of random
variables (or, in this instance, the sums of squares of
x(t)).
Were our calculations proper and valid? Do they make
any sense? The answer to these questions depends on the
issue of the probability distribution of a sum of random
variables. How does the distribution of the sum relate to
the distributions of the individual random variables that
In answering this question we want to focus on ways we
can come up with a location parameter m, and a scale
parameter c. For the normal distribution, m is the mean,
but for the Cauchy distribution the mean doesnвЂ™t exist
("is infinite"). For the normal distribution, the scale
parameter c is the standard deviation, but for the Cauchy
distribution the standard deviation doesnвЂ™t exist.
Nevertheless, a location m and a scale c exist for the
Cauchy distribution. The maximum likelihood estimator
for c will not be the same in the case of the Cauchy
distribution as it was for the normal. We canвЂ™t take
squares if the x(t) have a Cauchy distribution.
Suppose we have n random variables Xi, all with the
same distribution, and we calculate their sum X:
X = X1 + X2 + вЂ¦ + Xn-1 + Xn.

Does the distribution of the sum X have a simple form?
In particular, can we relate the distribution of X to the
common distribution of the Xi? LetвЂ™s be even more
specific. We have looked at the normal (Gaussian) and
Cauchy distributions, both of which were parameterized
with a location m and scale c. If each of the Xi has a

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location m and scale c, whether normal or Cauchy, can
that information be translated into a location and a scale
for the sum X?
The answer to all these questions is yes, for a class of
distributions called stable distributions. (They are also
sometimes called "Levy stable", "Pareto-Levy", or
"stable Paretian" distributions.) Both the normal and the
Cauchy are stable distributions. But there are many
more.
We will use the notation "˜" as shorthand for "has the
same distribution as." For example,
X1 ˜ X2

means X1 and X2 have the same distribution. We now
use "˜" in the following definition of stable distributions:
Definition: A random variable X is said to have a stable
distribution if for any n >= 2 (greater than or equal to
2), there is a positive number Cn and a real number Dn
such that
X1 + X2 + вЂ¦ + Xn-1 + Xn ˜ Cn X + Dn

where X1, X2, вЂ¦, Xn are all independent copies of X.

Think of what this definition means. If their distribution
is stable, then the sum of n identically distributed
random variables has the same distribution as any one of
them, except by multiplication by a scale factor Cn and a
further adjustment by a location Dn .

Does this remind you of fractals? Fractals are
geometrical objects that look the same at different scales.
Here we have random variables whose probability
distributions look the same at different scales (except for

LetвЂ™s define two more terms.
Definition: A stable random variable X is strictly stable
if Dn = 0.

So strictly stable distributions are clearly fractal in
nature, because the sum of n independent copies of the
underlying distribution looks exactly the same as the

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underlying distribution itself, once adjust by the scale
factor Cn. One type of strictly stable distributions are
symmetric stable distributions.
Definition: A stable random variable X is symmetric
stable if its distribution is symmetricвЂ”that is, if X and
-X have the same distribution.
The scale parameter Cn necessarily has the form :

Cn = n1/ О± , where 0< О± <=2.

So if we have n independent copies of a symmetric
stable distribution, their sum has the same distribution
with a scale that is n1/ О± times as large.
For the normal or Gaussian distribution, О± = 2. So for n
independent copies of a normal distribution, their sum
has a scale that is n1/ О± = n1/ 2 times as large.
For the Cauchy distribution, О± = 1. So for n independent
copies of a Cauchy distribution, their sum has a scale
that is n1/ О± = n1/ 1 = n times as large.
Thus if, for example, Brownian particles had a Cauchy
distribution, they would scale not according to a T0.5
law, but rather according to a T law!
Notice that we can also calculate a Hausdorff dimension
for symmetric stable distributions. If we divide a
symmetric stable random variable X by a scale factor of
c = n1/ О± , we get the probability equivalent  of N = n
copies of X/n1/ О± . So the Hausdorff dimension is

D = log N/ log c = log n/ log(n1/ О± ) = О± .
This gives us a simple interpretation of О± . The
parameter О± is simply the Hausdorff dimension of a
symmetric stable distribution. For the normal, the
Hausdorff dimension is equal to 2, equivalent to that of a
plane. For the Cauchy, the Hausdorff dimension is equal
to 1, equivalent to that of a line. In between is a full
range of values, including symmetric stable distributions
with Hausdorff dimensions equivalent to the Koch Curve
(log 4/log 3) and the Sierpinski Carpet (log 8/log3).
Some Fun with Logistic Art

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

Now that weвЂ™ve worked our way to the heart of the
matter, letвЂ™s take a break from probability theory and
turn our attention again to dynamical systems. In
particular, letвЂ™s look at our old friend the logistic
equation:
x(n+1) = k x(n) [1 вЂ“ 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 вЂ“ 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 вЂ“ (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 вЂ“ (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 donвЂ™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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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).
LetвЂ™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 вЂ“.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
 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.
 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.

 If Y = X/n1/ О± , then for n independent copies of Y,

Y1 + Y2 + вЂ¦ + Yn-1 + Yn ˜ n1/ О± Y = n1/ О± (X/n1/ О± ) =
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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