My model (which doesn't use golden mean as input parameters ) shows that numerically (as outputs) there are golden ratios between equilibrium levels of the market but so does a mere simple mesure of Haussdorf dimension that was done by Orlin grabbe:
http://orlingrabbe.com/chaos6.htm
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 T1/a = T1/2 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 T1/a = T1/1 = 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 = 1.8927â¦.) or Koch curve (a = log 4/ log 3 = 1.2618â¦.).
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 T1/a = 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 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 canât get much closer than that to a = h = 1.618033â¦
http://orlingrabbe.com/chaos6.htm
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 T1/a = T1/2 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 T1/a = T1/1 = 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 = 1.8927â¦.) or Koch curve (a = log 4/ log 3 = 1.2618â¦.).
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 T1/a = 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 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 canât get much closer than that to a = h = 1.618033â¦
