Chaos and Fractals in Financial Markets

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 doesn’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 let’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

This gives two solutions:

g = [- 1 + 5^0.5] / 2 = 0.618033…, and

g = [- 1 - 5^0.5] / 2 = -1.618033…

The first, positive solution (g = 0.618033…) is called the golden mean. Using h = 1 as our scale of measurement, then g, the golden mean, is the solution to the ratio

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

Which gives the two solutions:

h = [ 1 + 5^0.5] / 2 = 1.618033…, and

h = [ 1 - 5^0.5] / 2 = -0.618033…

Note that since the units of measurement are somewhat aribitrary, h has as much claim as g to being the solution to Prechter’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:

1.618033… = 1/ 0.618033…

-0.618033… = 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, let’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 – 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.

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 n^{1/ a} , where a is the Hausdorff dimension of the given symmetric stable distribution.

In the case of the normal or Gaussian distribution, the Hausdorff dimension a = 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 T^1/a = T^1/2 law.

In the case of the Cauchy distribution (Part 4), the Hausdorff dimension a = 1, which is equivalent to the dimension of a line. A Cauchy process would be governed by a T^1/a = T^1/1 = T law.

In general, 0 < a <=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 (a = log 8/ log 3 = 1.8927….) or Koch curve (a = log 4/ log 3 = 1.2618….).

Interestingly, however, many financial variables are symmetric stable distributions with an a 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 T^1/a = T^1/h = T^g = T^0.618033... 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 a 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 a was calculated using maximum likelihood techniques [2]. The value I found was

a = 1.62

with a margin of error of plus or minus .04. You can’t get much closer than that to a = h = 1.618033…

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 T^g law.

The value of h = 1.618033… 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, …

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 generation—the second 1 in the sequence), then it has a baby rabbit in the following generation (giving 2 rabbits—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 system—the complex logistic equation—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 be solved to yield the solution:

F(n) = [1/5^0.5] { [(1+5^0.5)/2]ⁿ – [(1-5^0.5)/2]ⁿ }, n = 1, 2, . . .

The following table gives the value of F(n) for the first few values of n:

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 …). Thus we have

h = [ 1 + 5^0.5] / 2 = 1.618033…, and

- 1/ h = [ 1 - 5^0.5] / 2 = -0.618033…

Inserting these into the solution for the Fibonacci system F(n), we get

F(n) = [1/5^0.5] { [h]ⁿ – [-1/ h ]ⁿ }, n = 1, 2, . . .

Alternatively, writing the solution using the golden mean g, we have

F(n) = [1/5^0.5] { [g]^-n – [-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 a = h = 1.618033… 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 background needed for doing maximum likelihood estimates of a , where a < 2:

[3] See, for example:

[4] See R.N. Elliott’s Masterworks—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 .