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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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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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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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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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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

digital cash. His home page is located at

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

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. "Heâ€™s a terrorist." Pete didnâ€™t

bother to mention the blocks had been there for years. He

knew Bachelier wouldnâ€™t know the difference.

"Terrorist?"

"You know, a ruffian, a scoundrel."

"Oh," Bachelier mused. "Bin Ladin. The son of Ladin."

"Yes, and before that, there was Abu Nidal."

"Abu Nidal. The father of Nidal. Hey! Ladin is just Nidal

spelled backwards. So weâ€™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 â€˜Goldsteinâ€™ and the daily scheduled â€˜Two Minutes

Hateâ€™ in George Orwellâ€™s book 1984."

"1984? Letâ€™s see, that was fifteen years ago," Bachelier

said. "A historical work?"

"Actually, itâ€™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

daysâ€”except in the stock market."

Bachelier didnâ€™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.

Bachelierâ€™s Scale for Stock Prices

Here is Bachelierâ€™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) â€“ 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) â€“ 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 dayâ€™s m and c are the same as

any other dayâ€™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.[1] 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 todayâ€™s price is P, Bachelier

modeled the probability interval around the log of the

price change by

(log P â€“ 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 â€“ 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 Bachelierâ€™s changing probability interval for log S

becomes:

(log S â€“ .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 =

2.718281â€¦), and get

( S exp(â€“ .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(â€“ .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 wasnâ€™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(â€“ (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

terms, we note that there are about 256 trading days in

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 dayâ€™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) = Â½ 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

option is trading.

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

are added together?

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 doesnâ€™t exist

("is infinite"). For the normal distribution, the scale

parameter c is the standard deviation, but for the Cauchy

distribution the standard deviation doesnâ€™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 canâ€™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 + â€¦ + 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? Letâ€™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 + â€¦ + Xn-1 + Xn ˜ Cn X + Dn

where X1, X2, â€¦, 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

the add factor Dn).

Letâ€™s define two more terms.[2]

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 symmetricâ€”that is, if X and

-X have the same distribution.

The scale parameter Cn necessarily has the form [3]:

Cn = n1/ Î± , where 0< Î± <=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/ Î± times as large.

For the normal or Gaussian distribution, Î± = 2. So for n

independent copies of a normal distribution, their sum

has a scale that is n1/ Î± = n1/ 2 times as large.

For the Cauchy distribution, Î± = 1. So for n independent

copies of a Cauchy distribution, their sum has a scale

that is n1/ Î± = 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/ Î± , we get the probability equivalent [4] of N = n

copies of X/n1/ Î± . So the Hausdorff dimension is

D = log N/ log c = log n/ log(n1/ Î± ) = Î± .

This gives us a simple interpretation of Î± . The

parameter Î± 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 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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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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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

your web browser and click here.

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

enabled on your web browser and click here

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.

[2] The definitions here follow those in Gennady

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,

banking security, and digital cash. His home page is

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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