Category Archives: Fractals
The Father of Fractal geometry has passed away, due to cancer. Mandelbrot has had a significant influence on my thinking in ways that have influenced both my trading system design and my life in general. I do not believe that he was ever given the respect that he deserved, and I’m very sorry to see him pass on without receiving it.
A couple of years ago I started a series on fractals, which I never finished. Nonetheless, you may find the posts I did complete to be of some interest.
I’ve been stymied with a busy work and personal schedule, but intend to resume my series On Fractals and Market Crashes this week.
In the meantime, PBS just released a 1 hour special on fractals. It is able to be viewed online, and is divided into 5 parts to make for easy viewing. The link is below…
Enjoy, and feel free to leave any thoughts generated from the special in the comments section.
In Part 1, I established that financial markets do not conform to a standard distribution. Thus, as most of the tools used for analysis of the markets require a standard distribution, they must be disposed of. Now is the time to change the way we think about, analyze, describe, and understand the financial markets. I am convinced that fractals hold the most promise for developing new tools with which to model financial markets.
Side Note: Michael Stokes from MarketSci Blog just published a similar, but better articulated piece on standard distributions and the markets. Fat Tails, Normal Distributions, Random Walks, and all that Jazz. Since his thoughts mirror mine, and are therefore somewhat self-similar and recursive, I thought it would be neat and geeky to link back in his blog in this post about fractals.
A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity. The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractus meaning “broken” or “fractured.”
A fractal often has the following features:
- It has a fine structure at arbitrarily small scales.
- It is too irregular to be easily described in traditional Euclidean geometric language.
- It is self-similar
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes.
Fractals in Nature (from Wikipedia Fractals)
Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels. Coastlines may be loosely considered fractal in nature.
Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples. A branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature.
A fractal that models the surface of a mountain (animation)
A fractal fern computed using an Iterated function system
The Mandelbrot Set
Perhaps the most famous of all fractals are the computer generated ones often found in art, screen savers, album covers, and t-shrits. These fractals are typically generated by the Mandelbrot Set, named after Benoit Mandelbrot. The Mandelbrot Set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition.
Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.
What Does This All Have to Do With Market Crashes?
At this point, I’m sure fractals seem to have very little relationship to markets and market crashes. While I had hoped to have covered fractals in one article, it appears I’m going to have to flesh out fractals in 2 parts.
For now, the two most important things to remember about fractals (and the financial markets):
- They are self-similar.
- They are infinitely complex and complicated, yet can be described with simple definitions.
A good portion of this piece is from various Wikipedia articles. I believe I have linked to the appropriate pages to give credit where it is due.
For the ultimate web-based guide to fractals, one must visit the site of Michael Frame, Benoit Mandelbrot, and Nial Neger Fractal, housed at Yale. This is a fantastic resource.
1. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company.
2. Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd., xxv.
“Lastly, the cotton story shows the strange liaison among different branches of the economy, and between economics and nature. That cotton prices should vary the way income does; that income variations should look like Swedish fire-insurance claims; that these, in turn, are in the same mathematical family as formulae describing the way we speak, or how earthquakes happen–this is, truly, the greatest mystery of all.” -Benoit Mandelbrot 2004
If there are any economists, investors, or scientists who still believe the movements of the financial markets follow the standard (Gaussian) distribution, 2008 has likely shaken their foundations and fractured their paradigms.
A quick calculation of the Standard Deviation of one day’s close to the following day’s close (in percentage terms), from October 1, 1928 to October 24, 2008, is below. The calculations show that the average daily change on the Dow Jones has been .024%, with one standard deviation being 1.15%. Already, we see the daily change can vary greatly from the average change.
Above is the confidence intervals (orange background) for various standard deviations beyond the mean. If a data distribution is approximately normal then about 68% of the values are within 1 standard deviation of the mean, about 95% of the values are within two standard deviations and about 99.7% lie within 3 standard deviations.
In October alone, there have been 3 days with close-to-close changes greater than 6 standard deviations beyond the mean. The change on October 13th was 11.08%, better than 9 standard deviations from the mean. The crash in October 1987 would be greater than 18 standard deviations.
These large variances in the Dow Jones data should not be present, unless the day-to-day changes are not normally distributed.
For the rest of this article, I will then assume, as I believe it to be true, that the financial markets cannot be described within a Gaussian distribution.
Of course, to know that financial markets do not conform to a standard distribution is to understand that the Capital Asset Pricing Model, along with Value At Risk and Beta; Black-Scholes Formula and the Modern Portfolio Theory, are hopelessly and inherently flawed.
Indeed, the recent bear market has proven the financial markets to be much more volatile and risky, and anyone using the above models and theories to manage risk are likely finding themselves having to re-work all their models and risk-management formulae. Should they continue to rely on a standard distribution, they will forever be re-working their models, assuming they are not first ruined by them.
Since I have thrown out the idea that the market’s movements will be contained within X standard deviations from the mean, it is then crucial to digest and internalize this:
There is nothing to stop the markets from experiencing large and devastating losses, of a magnitude never before witnessed.
Part 2 of this series will describe fractals and how understanding them may be the key to understanding the recent market dislocations.
Part 3 will synthesize parts 1 and 2 in a discussion of why and how markets crash.
Benoit Mandelbrot is the expert on this topic. His latest book, The (Mis) Behavior of Markets is a must read for anyone interested in fractals and the financial markets.