Chaos and Fractals in Financial Markets

Grow Brain

Many dynamical systems create solution paths, or trajectories, that look strange and complex. These solution plots are called "strange attractors".

Some strange attractors have a fractal structure. For example, we saw in Part 3 that it was easy to create a fractal called a Sierpinski Carpet by using a stochastic dynamical system (one in which the outcome at each step is partially determined by a random component that either selects among equations or forms part of at least one of the equations, or both).

Here is a dynamical system that I ran across while doing computer art. I labeled it "Grow Brain" because of its structure. To see Grow Brain in action, make sure Java is enabled on your browser (you can turn it off afterward) and click here. (The truly paranoid can, of course, compile their own applet, since I provide the source code, as usual.)

The trajectory of Grow Brain is amazingly complex. But is it a fractal? That is, at some larger or smaller scale, will similar structures repeat themselves? Unlike the case of the Sierpinski Carpet, the answer to this question is not obvious for Grow Brain.

Some dynamical systems create fractal structures in time (as Brownian motion does, in Part 2, or the Fibonacci-type systems of Part 6 do), while others create fractal structures in space (as in the aforementioned Sierpinski carpet).

And some systems are all wet. Or maybe not, as the case may be.

Hurst, Hydrology, and the Annual Flooding of the Nile

For centuries, perhaps millennia, the yearly flooding of the Nile was the basis of agriculture which supported much of known civilization. The annual overflowing of the river deposited rich top soil from the Ethiopian Highland along the river banks. The water and silt were distributed by irrigation, and the staple crops of wheat, barley, and flax were planted. The grain was harvested and stored in silos and granaries, where it was protected from rodents by guard cats, whom the Egyptians worshipped and turned into a cult (of the goddess Bast) because of their importance for survival of the grain, and hence for human survival.

The amount of Nile flooding was critical. A good flood meant a good harvest, while a low-water flood meant a poor harvest and possible food shortage. The flooding came (and still comes) from tropical rains in the Upper Nile Basin in Ethiopia (the Blue Nile) and in the East African Plateau (the White Nile). The river flooding would begin in the Sudan in April, and reach Aswan in Egypt by July. (This would occur about the time of the heliacal rising of the Dog-Star Sirus, or Sothis, around July 19 in the Julian calendar.) The waters would then continue to rise, peaking in mid-September in Aswan. Further down the river at Cairo, the peak wouldn’t occur until October. The waters would then fall rapidly in November and December, and continue to fall afterward, reaching their low point in the March to May period. Ancient Egypt had three seasons, all determined in reference to the river: akhet, the "inundation"; peret, the season when land emerged from the flood; and shomu, the time when water was low.

A British government bureaucrat named Hurst made a study of records of the Nile’s flooding and noticed something interesting. Harold Edwin Hurst was a poor Leicester boy who made good, eventually working his way into Oxford, and later became a British "civil servant" in Cairo in 1906. He got interested in the Nile. He looked at 800 years of records and noticed that there was a tendency for a good flood year to be followed by another good flood year, and for a bad (low) flood year to be followed by another bad flood year.

That is, there appeared to be non-random runs of good or bad years. Later Mandelbrot and Wallis [1] used the term Joseph effect to refer to any persistent phenomenon like this (alluding to the seven years of Egyptian plenty followed by the seven years of Egyptian famine in the biblical story of Joseph).

Of course, even if the yearly flows were independent, there still could be runs of good or bad years. So to pin this down, Hurst calculated a variable which is now called a Hurst exponent H. The expectation was that H = ½ if the yearly flood levels were independent of each other.

Calculating the Hurst Exponent

Let me give a specific example of Hurst exponent calculation which will illustrate the general rule. Suppose there are 99 yearly observations of the height h of the mid-September Nile water level at Aswan: h(1), h(2), . . ., h(99).

Calculate a location m and a scale c for h. If we assume in general that h has a finite variance, then m is simply the sample mean of the 99 observations, while c is the standard deviation.

The first thing is to remove any trend, any tendency over the century for h to rise or fall as a long-run phenomena. So we subtract m from each of the observations h, getting a new series x that has mean zero:

x(1) = h(1) - m,
x(2) = h(2) - m,
…
x(99) = h(99) - m .

The set of x’s are a set of variables with mean zero. Positive x’s represent those years when the river level is above average, while negative x’s represent those years when the river level is below average.

Next we form partial sums of these random variables, each partial sum Y(n) being the sum of all the years prior to year n:

Y(1) = x(1),
Y(2) = x(1) + x(2),
. . .
Y(n) = x(1) + x(2) + . . . + x(n),
. . .
Y(99) = x(1) + x(2) + x(3) + . . . + x(99).

Since the Y’s are a sum of mean-zero random variables x, they will be positive if they have a preponderance of positive x’s and negative if they have a preponderance of negative x’s. In general, the set of Y’s

{Y(1), Y(2), …. , Y(99)}

will have a maximum and a minimum: max Y and min Y, respectively. The difference between these two is called the range R:

R = max Y - min Y

If we adjust R by the scale parameter c, we get the rescaled range:

rescaled range = R/c .

Now, the probability theorist William Feller [2] had proven that if a series of random variables like the x’s 1) had finite variance, and 2) were independent, then the rescaled range formed over n observations would be equal to:

R/c = k n ^1/2

where k is a constant (in particular, k = (p /2)^1/2 ) . That is, the rescaled range would increase much like the partial sums of independent variables (with finite variance) we looked at in Part 5¾ namely, the partial sums would increase by a factor of n^1/2.

In particular, for n = 99 in our hypothetical data, the result would be:

R/c = k 99^1/2 .

Now, the latter equation implies log(R/c) = log k + ½ log 99. So if you ran a regression of log(R/c) against log(n) [for a number of rescaled ranges (R/c) and their associated number of years n] so as to estimate an intercept a and a slope b,

log(R/c) = a + b log(n),

you should find that b is statistically indistinguishable from ½.

But that wasn’t what Hurst found. Instead, he found that in general the rescaled range was governed by a power law

R/c = k n^H

where the Hurst exponent H was greater than ½ (Hurst found H @ .7)

This implied that succeeding x’s were not independent of each other: x(t) had some sticky, persistent effect on x(t+1). This was what Hurst had observed in the data, and his calculation showed H to be a good bit above ½ [3].

That this would be true in general for H > ½ , of course, needs to be proven. Nevertheless, to summarize, for reference, for the Hurst exponent H:

A Misunderstanding to Avoid

Recall that Bachelier had noted that the probability range of the log of a stock price would increase with the square root of time T. The probability range, starting at log S, would grow with T according to:

(log S – k T^1/2 , log S + k T^1/2),

where k is the scale (in his case, the standard deviation) c, k = c. But, more generally, the symmetric stable distributions of Part 5, increase with T raised to the reciprocal power of the Hausdorff dimension a (a <=2):

(log S – k T^1/a , log S + k T^1/a ).

Hurst similarly said the rescaled range of the flood level varied according to (setting n = T):

R/c = k T^H .

So it is tempting to equate the Hurst exponent H with the reciprocal of the Hausdorff dimension D, to equate H with 1/D = 1/a . But we must be careful.

Recall that symmetric stable distributions, with a < 2, have infinite variance (for them, variance is a blob measure that is not meaningful). However, here in discussing the Hurst exponent we are assuming that the variance, and standard deviation (the scale c), are finite, and hence a =2. The role of the Hurst exponent is to inform us whether the yearly flood deviations are independent or persistent. H is not related to the need for a different scale measure. The variance and the standard deviation are well defined for these latter processes.

Nevertheless, the formal equation H = 1/D or D = 1/H yields the correct exponent for T in the case ½<= H <=1. Even though a =2, the calculation of the Hausdorf dimension D yields D<2 if the increments are not independent. Hence D can take a minimum value of 1, D = 1/H = 1/1 = 1 when H=1, so that the process accumulates variation (rescaled range) much like a Cauchy sequence (T^H = T); or a maximum of 2, D = 1/H = 1/½ = 2 when H=½, so that the process accumulates variation (rescaled range) like a Gaussian sequence (T^H = T^1/2), or ordinary Brownian motion. [4]

Mandelbrot called these types of processes where a =2, but where H ¹ ½, fractional Brownian motion. (I will not here elaborate the case H < ½.)

Bull and Bear Markets

We are, of course, used to the idea of persistent phenomena in the stock market and foreign exchange markets. The NASD rises relentlessly for a period of time. Then it falls just as persistently. There are bull and bear markets, implying the price rise or decline is a persistent phenomena, and not just an accidental accumulation of random variables in one direction.

The US dollar rises relentless for a period of years, then (as it is doing now) begins a relentless decline for another period of years. In the case of the Nile, the patterns of rising and falling are partly governed by the weather patterns in the green rain forest of the Ethiopian highlands. In the case of the US dollar, the patterns of rising and falling are partly governed by the span of Green in the Washington D.C. lowlands.

Notes

[1] B.B. Mandelbrot & J. R. Wallis, "Noah, Joseph, and Operational Hydrology." Water Resources Research 4, 909-918, (1968).

[2] W. Feller, "The asymptotic distribution of the range of sums of independent random variables." Annals of Mathematical Statistics 22, 427 (1951).

[3] H. E. Hurst, "Long-term storage capacity of reservoirs." Tr. of the American Society of Civil Engineers 116, 770-808 (1951).

[4] See also the discussion on pages 251-2 in Benoit B. Mandelbrot, The Fractal Geometry of Nature. W.H. Freeman and Company, New York, 1983.

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://orlingrabbe.org/ .