## Hazardous WorldMany things in life are random. They are governed by probability, by chance, by hazard, by accident, by the god Hermes, by fortune. So we measure them by probability—by our one-pound jar of jam. Places where there is more jam are more likely to happen, but the next outcome is uncertain. The next outcome might be a low probability event. Or it might be a high probability event, but there may be more than one of these. Radioactive decay is measured by probability. The timing of the spontaneous transformation of a nucleus (in which it emits radiation, loses electrons, or undergoes fission) cannot be predicted with any certainty. Some people don’t like this aspect of the world. They prefer to believe there are "hidden variables" which really determine radioactive decay, and if we only understood what these hidden variables were, it would all be precisely predictable, and we could return to the paradise of a Laplacian universe. Well, if there are hidden variables, I sure wish someone would identify them. If wishes were horses, David Bohm would ride.[1] Albert Einstein liked to say, "God doesn’t play dice." But if God wanted to play dice, he didn’t need Albert Einstein’s permission. It sounds to me like "hidden" is just another name for probability. "Was it an accident?" "No, it was caused by hidden forces." Hidden variable theorists all believe in conspiracy. But, guess what? People who believe God doesn’t play dice use probability theory just as much as everyone else. So, without further ado, let’s return to our discussion of probability. Coin Flips and Brownian Motion We can create a kind of Brownian motion (or Bachelier process) by flipping coins. We start with a variable x = 0. We flip a coin. If the coin comes up heads, we add 1 to x. If the coin comes up tails, we subtract 1 from x. If we denote the input x as x(n) and the output x as x(n+1), we get a dynamical system: x(n+1) = x(n) + 1, with probability p = ½ Here |

Much of finance is based on a simple probability model like this one. Later we will change this model by changing the way we measure probability, A Simple Stochastic Fractal Using probability, it is easy to create fractals. For example, here is a dynamical system which creates a Simple Stochastic Fractal. The system has two variables, x and y, as inputs and outputs: x(n+1) = - y(n) with probability p = ½ , but x(n+1) = 1 + 2.8*(x(n)-1)/(x(n)*x(n)-2*x(n)+2+y(n)*y(n)) with probability q = ½. We map x and y on a graph of two dimensions. If the coin flip comes up heads, we iterate the system by the first two equations. This iteration represents a simple 90-degree rotation about the origin (0,0). If the coin flip comes up tails, we iterate the system by the second two equations. This second type of iteration contracts or expands the current point with respect to (1,0). To see this Simple Stochastic Fractal system works in real time, be sure Java is enabled on your web browser, and click here. [2] Simple stochastic dynamical systems create simple fractals, like those we see in nature and in financial markets. But in order to get from Bachelier to Mandelbrot, which requires a change in the One we’ve learned to measure length, we’ll find that probability is jam on toast. Sierpinski and Cantor Revisited In Part 2, when we looked at Sierpinski carpet, we noted that a Sierpinski carpet has a Hausdorff dimension D = log 8/log 3 = 1.8927… So if we have a Sierpinski carpet with length 10 on each side, we get N = r smaller copies of the original. (For a nice round number, we can take 9 feet on a side, and get N = 9 So let’s ask the question: How much space (area) does Sierpinski carpet take up relative to ordinary carpet? We have 78.12 smaller copies of the original. So if we know how much area (in terms of ordinary carpet) each of these smaller copies takes up, we can multiply that number by 78.12 and get the answer. Hmmm. To calculate an answer this question, let’s take the same approach we did with Cantor dust. In the case of Cantor dust, we took a line of length one and began cutting holes in it. We divided it into three parts and cut out the middle third, like this:
That left 2/3 of the original length. Then we cut out the middle thirds of each of the two remaining lines, which left 2/3 of what was there before; that is, it left (2/3)(2/3), or (2/3) If we take the limit as Well. Now let’s do the same thing with Sierpinski carpet. We have an ordinary square and divide the sides into three parts (divide by a scale factor of 3), making 9 smaller squares. Then we throw out the middle square, leaving 8 smaller squares, as in the figure below: |

So we have left 8/9 of the original area. Next, we divide up each of the smaller squares and throw out the centers. Each of them now has 8/9 of its original area, so the area of the big square has been reduced to (8/9)(8/9) of its original size, or to (8/9) What? This seems properly outrageous. The 78.12 smaller copies of the original Sierpinski carpet that measured 10 x 10 (or 64 smaller copies of an original Sierpinski carpet that measured 9 x 9), actually take up zero area. By this argument, at least. By this way of measuring things. We can see what is happening, if we look at the Sierpinski carpet construction again. Note in the graphic above that the Next note that the border of the first Now, with respect to Cantor dust, we said we had an infinite number of disconnected points, each with Hmm. Your eyebrows raise. Previously, in Part 2, I said Sierpinski carpet had an ordinary (or topological) dimension of 2 . That was because we started with a 10 by 10 square room we wanted to cover with carpet. So, intuitively, the dimension we were working in was 2. The confusion lies in the phrase "topological or ordinary" dimension. These are not the same. Or, better, we need more precision. In the case of Sierpinski carpet, we started in a context of two-dimensional floor space. Let’s call this a Thus we now have three different dimensions for Sierpinski carpet: a Euclidean dimension (E) of 2, a topological dimension (T) of 1, and a Hausdorff dimension (D) of 1.8927… Similarly, to create Cantor dust, we start with a line of one dimension. Our So here are three different ways [4] of looking at the same thing: the Euclidean dimension (E), the topological dimension (T), and the Hausdorff dimension (D). Which way is best? Blob Measures Are No Good Somewhere (I can’t find the reference) I read about a primitive tribe that had a counting system that went: 1, 2, 3, many. There were no names for numbers beyond 3. Anything numbered beyond three was referred to as "many". "We’re being invaded by foreigners!" "How many of them are there?" "Many!" It’s not a very good number system, since it can’t distinguish between an invading force of five and an invading force of fifty. (Of course, if the enemy was in sight, one could get around the lack of numbers. Each individual from the local tribe could pair himself with a invader, until there were no unpaired invaders left, and the result would be an opposing force that matched in number the invading force. George Cantor, the troublemaker who invented set theory, would call this a "Many." A blob. Two other blob measures are: We get a little more information if we know that Cantor dust has a topological dimension of zero, while a Sierpinski carpet has a topological dimension of one. But topology often conceals more than it reveals. The topological dimension of If we have a circle, for example, it is fairly easy to measure its length. In fact, we can just measure the radius L = C = 2 p r where p
= 3.141592653… is known accurately to millions of decimal places. But suppose we attempt to measure the length of a Sierpinski carpet? After all, we just said a Sierpinski carpet has topological dimension of one, like a To measure the Sierpinski carpet we began measuring smaller and smaller squares, so we keep having to make our measuring rod smaller and smaller. But as the squares get smaller, there are more and more of them. If we actually try to do the measurement, we discover the length goes to infinity. (I’ve measured my Sierpinski carpet; haven’t you measured yours yet?) Infinity. A blob. "How long is it?" "Many!" Coastlines and Koch Curves If you look in the official surveys of the length of borders between countries, such as that between Spain and Portugal, or between Belgium and The Netherlands, you will find they can differ by as much as 20 percent. [5] Why is this? Because they used measuring rods that were of different lengths. Consider: one way to measure the length of something is to take a measuring rod of length L = m N (where "m N" means "m times N"). For example, suppose we are measuring things in feet, and we have a yardstick ( L = 3 (100) = 300 feet. And, if instead of using a yardstick, we used a smaller measuring rod—say a ruler that is one foot long, we would still get the same answer. Using the ruler, Portugal is a smaller country than Spain, so naturally it used a measuring rod of shorter length. And it came up with an estimate of the length of the mutual border that was longer than Spain’s estimate. We can see why if we imagine measuring, say, the coastline of Britain. If we take out a map, lay a string around the west coast of Britain, and then multiply it by the map scale, we’ll get an estimate of the "length" of the western coastline. But if we come down from our satellite view and actually visit the coast in person, then we will see that there are a lot of ins and outs and crooked jags in the area where the ocean meets the land. The smaller the measuring rod we use, the longer will our measure become, because we capture more of the length of the irregularities. The difference between a coastline and the side of a football field is the coastline is fractal and the side of the football field isn’t. To see the principles involved, let’s play with something called a Koch curve. First we will construct it. Then we will measure its length. You can think of a Koch curve as being as being a section of coastline. We take a line segment. For future reference, let’s say its length L is L = 1. Now we divide it into three parts (each of length 1/3), and remove the middle third. But we replace the middle third with |

At this point we have 4 smaller segments, each of length 1/3, so the total length is 4(1/3) = 4/3. Next we repeat this process for each of the 4 smaller line segments. This is stage three (c) in the graphic above. This gives us 16 At the However, the Koch curve is continuous, because we can imagine taking a pencil and tracing its (infinite) length from one end to the other. So, from the topological point of view, the Koch curve has a dimension of one, just like the original line. Or, as a topologist would put it, we can deform (stretch) the original line segment into a Koch curve without tearing or breaking the original line at any point, so the result is still a "line", and has a topological dimension T = 1. To calculate a Hausdorff dimension, we note that at each stage of the construction, we replace each line segment with N = 4 segments, after dividing the original line segment by a scale factor r = 3. So its Hausdorff dimension D = log 4/log 3 = 1.2618… Finally, when we constructed the Koch curve, we did so by viewing it in a Euclidean plane of two dimensions. (We imagined replacing each middle line segment with the other two sides of an equilateral triangle—which is a figure of 2 dimensions.) So our working space is the Euclidean dimension E = 2. But here is the key point: as our measuring rod got smaller and smaller (through repeated divisions by 3), the measured length of the line got larger and larger. Just like a coastline. (And just like the path of Brownian motion.) The total length (4/3) L = m N = (1/3) Well, there’s something wrong with measuring length this way. Because it gives us a blob measure. Infinity. "Many." Which is longer, the coast of Britain or the coast of France? Can’t say. They are both infinity. Or maybe they have the same length: namely, infinity. They are both "many" long. Well, how long is the coastline of Maui? Exactly the same. Infinity. Maui is many long too. (Do you feel like a primitive tribe trying to count yet?) Using a Hausdorff Measure The problem lies in our measuring rod L = m N , let’s adjust L = m This changes our way of measuring length L, because only when If we do this, replace d that are too small, L still goes to infinity. For values of d that are too large, L goes to zero. Blob measures. There is only one value of d that is just right: namely, the Hausdorff dimension d = D. So our measure of length becomes:L = m How does this work for the Koch curve? We saw that for a Koch curve the number of line segments at stage L = m Success. The Hausdorff dimension D is a natural measure associated with our measuring rod To make sure we understand how this works, let’s calculate the length of a Sierpinski carpet constructed from a square with a starting length of 1 on each side. For the Sierpinski carpet, N gets multiplied by 8 at each stage, while the measure rod gets divided by 3. So the length at stage L = m Hey! We’ve just destroyed the blob again! We have a finite length. It’s not zero and it’s not infinity. Under this measure, as we go from the original square to the ultimate Sierpinski carpet, the length stays the same. The [Note that if we use a If instead we have a Sierpinski carpet that is 9 on each side, then to calculate the "area", we note that the number of Sierpinski copies of the initial square which has a side of length 1 is (dividing each side into r = 9 parts) N = r ^{D} = 9^{1.8927…} = 64. A Sierpinski carpet with 10 on each side has an "area" of N = 10^{1.8927…} = 78.12. And so on.The Hausdorff dimension, D = 1.8927…, is closer to 2 than to 1, so having an "area" of 78.12 (which is in the region of 10 This way of looking at things lets us avoid having to say of two Sierpinski carpets (one of side 9 and the other of side 1): "Oh, they’re exactly the same. They both have Indeed do "many" things come to pass. To see a Sierpinski Carpet Fractal created in real time, using probability, be sure Java is enabled on your web browser, and click here. Jam Session One of the important points of the discussion above is that the Unfortunately, people who measure things using the wrong dimension often think they are saying something other than "many." They think their measurements mean something. They are self-deluded. Many empirical and other results in finance are an exercise in self-delusion, because the wrong dimension has been used in the calculations. When Louis Bachelier gave the first mathematical description of Brownian motion in 1900, he said the probability of the price distribution changes with the square root of time. We modified this to say that the probability of the The issue we want to consider is whether the appropriate dimension for time is D = ½. In order to calculate probability should we use T This was what Mandelbrot was talking about when he said the empirical distribution of price changes was "too peaked" to come from a normal distribution. Because the dimension D = ½ is only appropriate in the context of a normal distribution, which arises from simple Brownian motion. We will explore this issue in Part 4. Notes [1] David Bohm’s hidden-variable interpretation of the quantum pilot wave (which obeys the rules of quantum probability) is discussed in John Gribbin, [2] If your computer monitor has much greater precision than assumed here, you can see much more of the fractal detail by using a larger area than 400 pixels by 400 pixels. Just replace "200" in the Java program by one-half of your larger pixel width, and recompile the applet. [3] Note that in Part 2, we measured the length of the line segments that we [4] This three-fold classification corresponds to that in Benoit B. Mandelbrot, [5] L. F. Richardson, "The problem of contiguity: an appendix of statistics of deadly quarrels," [6] Whether one refers to the resulting carpet as "1 square Sierpinski" or just "1 Sierpinski" or just "a carpet with a side length of 1" is basically a matter of taste and semantic convenience. 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 26, June 28, 1999 |