Prechter’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<------------ -------------------->3hHere 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 . So what Prechter is looking for is the ratio of h to g to be the same as h to the whole. However, the whole is simply h, so Prechter seeks g + h and g such that: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 = 1. (Note that by setting h = 1, we are choosing our hunit of measurement.) We then have
1. / (1+g)Multiplying the equation out we get
This gives two solutions:
^{0.5}] / 2 = 0.618033…, and
^{0.5}] / 2 = -1.618033…The first, positive solution golden mean. Using as our scale of measurement, then h = 1the golden meang, ,is the solution to the ratio
. h / (g+h)By contrast, if we use , we haveh
, which gives the equationh / (1+h)
Which gives the two solutions:
^{0.5}] / 2 = 1.618033…, and
^{0.5}] / 2 = -0.618033…Note that since the units of measurement are somewhat aribitrary, to being the solution to Prechter’s drum roll. Naturally, g and g are closely related:h
as the unit scale) = 1/ gusing g ( as the unit scale). hfor 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 t=0:<------------ -------------------->hThen we find that for = 1, we haveh-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 The same applies for = -0.618033, sinceh1 / -0.618033 = -0.618033/(1 – 0.618033), but in this case The golden mean , are found throughout the natural world. Numerous books have been devoted to the subject. These same ratios are found in financial markets. hSymmetric Stable Distributions and the Golden Mean Law In Part 5, we saw that symmetric stable distributions are a type of probability distribution that are fractal in nature: a sum of In the case of the normal or Gaussian distribution, the Hausdorff dimension a
In the case of the Cauchy distribution (Part 4), the Hausdorff dimension a
In general, 0 < a
<=2. This means that Interestingly, however, many financial variables are symmetric stable distributions with an a
parameter that hovers around the value of is the reciprocal of the golden mean h derived and discussed in the previous section. This implies that these market variables follow a time scale law of Tg^{1/a
} = T^{1/h} = T = T^{g}. ^{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
with a margin of error of plus or minus .04. You can’t get much closer than that to a
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 The Fibonacci Dynamical System The value of sequence of numbers. The Fibonacci sequence of numbers is a sequence in which each number is the sum of the previous two:Fibonacci1, 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 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 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 = 1.618033 …). Thus we haveh
^{0.5}] / 2 = 1.618033…, and
^{0.5}] / 2 = -0.618033…Inserting these into the solution for the Fibonacci system F(n), we get F(n) = [1/5 ^{n} – [-1]/ h ^{n} }, n = 1, 2, . . . Alternatively, writing the solution using the golden mean F(n) = [1/5 ^{-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
## Notes[1] See "Research Strategy in Empirical Work with Exchange Rate Distributions," in J. Orlin Grabbe, [2] There are two key papers by DuMouchel which yield the background needed for doing maximum likelihood estimates of a
, where a
DuMouchel, William H. (1973), "On the Asymptotic Normality of the Maximum Likelihood Estimate when Sampling from a Stable Distribution," DuMouchel, William H. (1975), "Stable Distributions in Statistical Inference: 2. Information from Stably Distributed Samples," [3] See, for example: Robert R. Prechter, Jr., Robert R. Prechter, Jr., [4] See 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 |