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,”[1] 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:^{[2]}

- 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

Fractal Cauliflower

** **

**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.

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What the fuck is going on here?

Those pictures make me want to kill someone.

Odd. Obviously you know at some point I’m goign to describe the current market dislocation in terms of a multi-fractal (more on multi-fractals later).

So, when you look at a chart of the Dow Jones, or the S&P, do you want to kill someone?

Wood:

With all due respect, your fucking brain is on drugs with this fractal shit.

The world is getting sucked into a fucking vortex and you are putting up pretty pictures of ferns.

If I didn’t know any better, I’d say this was some sort of blog, authored by some sort of homo in NZ.

God that cracks me up. I am from NZ and wrote a post about fractals. (Not a homo though, sorry to disappoint you Mr Fly). http://etfhq.com/blog/2010/09/30/fractal-adaptive-moving-average-frama/

Keep it up, Wood. I love fractals. I have fractal pics up all over my house.

Late at night, they talk to me.

Wood – Is corn on the cob fractal?

Fly, with all due respect, the answer is right in front of your face. The vortex you mention is simply a fractal.

For example, a smoothly flowing, placid river will have small eddies and small rapids. However, the same river, flooded, swollen, and violent, will still have eddies and rapids, formed from the same processes as the previously smaller eddies and rapids.

What is going on in the markets should not be at all surprising. It is simply the normal machinations, raised to a power, and magnified further by the fact that market time has been increased, probably exponentially, by forced selling, margin calls, and the need to process and adapt to years worth of data in only a few weeks or months.

More on this later.

Dr. Reid, I would not think it is, but I’m in no way an expert.

ROFL

Well played Wood. You were head of the debate team in Elementary school, no doubt.

I will now explore the importance of Cauliflower fractals, and how it pertains to my portfolio.

NOTE: Do not be alarmed if your blog is covered with Cauliflowers, in the near term.

Fly, have I scared you? What part bothers you the most? The standard distribution stuff, or the cauliflower? lmao…

Be patient. This is a multi-part series.

Wood,what in the fuck are smoking bra?I thought the shit in Hawaii was good.

Wood is smoking the Kiwi, if I may be so bold.

He needs to be esoteric, in order to prove his intelligence. Let him run with it.

Fly, I agree, I like thinking, and I like thinking about thinking, so I do sometimes need to be philosophical, but not esoteric.

This is not about proving my intelligence. What I am presenting has already been written about before. Its not original to me. I’m adapting it to the current market environment in hopes it will give people a different model for understanding how a market moves.

Wood, go one step beyond fractals to the concept of ‘self-organized criticality’. I believe you will find some concepts very pertinent to the market.

You will want and need to read this book, it was extremely interesting (very high praise from me).

http://www.amazon.com/How-Nature-Works-Self-Organized-Criticality/dp/038798738X/ref=sr_1_2?ie=UTF8&s=books&qid=1225163241&sr=1-2

cheers.

I know. Other have wrote about it. And, every time I read it, I want to rip the face off of the author.

Aside from that, I love fractals.

Wood-

Don’t listen to these heathens. Get excited (like I am) that mandelbrot and taleb are currently working on a risk-management theory involving fractals and paretian distributions.

its impossible to understand this stuff without a thorough knowkedge of statistics and mathematics that probably outreaches 99% of the readership of this blog

(too busy yellin for more hot dogs, ova heah!)

knowledge even. wow.

“The Fly Says:

What the fuck is going on here?

Those pictures make me want to kill someone.”

HA!

The Fly is God.

Canâ€™t wait to see how you tie this all together!

Fly,

Easy on the wood man. No one holds it against you for banking extreme coin using the Martingale system of investing. To each his own. 🙂

Oh, great site too. I think you really got something here.

Awake, I hope not. My goal is to get the point across without the math. I’m a crude statistician and not at all a mathematician.

As for Taleb and Mandelbrot…that would be out of this world.

I can’t wait. Taleb and Mandelbrot, OMG!?!

It can’t be. Best duo since Batman and Robin.

Boca, that books looks like something I would enjoy. You got a used copy for sale? 😉

Fly, Taleb would punch your mustache off.

You could probably take Mandelbrot.

Looking forward the Third Act.

Taleb is the coolest guy no one gives two fucks and a gay fiddle about.

I kid you not, I don’t know anyone in the business that listens to Taleb’s crap. It’s an internet investor phenomenon.

Sorry, I sold my copy a few months ago after reading it. I highly recommend it though after seeing the direction you are heading in via this post. Worth every penny.

Fly, business has been bad, no?

Maybe they should have listened…

Ritholtz seems to be a fan of Taleb.

Fly, Taleb made a billion dollars on his personal account in one day (heard it was sometime in 1987).

more options ova heah?

Honestly I think a lot of people don’t uderstand Taleb because he pulls concepts together from many different inntellectual disciplines … gives a very fresh view to the markets, but it helps to be very widely read to get all his references.

All this talk of fractals and mandelbrot sets makes me want to listen to Blue Man Group while pouring over solutions for differential equations and Fourier series, as they relate to the cyclicality of the market…..all within the confines of my economically insulated cave, of course.

Ouch, the mention of Diff EQ and For-yay series gave me nasty headache.

Taleb is right on. Dow is headed to 225 – two hundred twenty five to be exact.

Fractals can also determine if someone is gay.

Sounds like a nice cave, Alpha!

yeah it’s fantastic stuff, but it’s fucking hard to model. how do you map these to the financial markets? mandelbrot’s handwaving with mapping to time in his book is total garbage, because this can happen irrespective of time.

the closest attempt i’ve seen has been stephen wolfram’s attempt in A New Kind of Science, in which he tries to use cellular atomata to map to the markets in one of the chapters.

So i,

I agree the time issue is troubling.

I have not heard of Wolfram. Thanks for the mention of him and his book. I’ll have to take a look at it.

here’s the relevant section:

http://www.wolframscience.com/nksonline/section-8.8

nice article with great pics, the controversy above implies that it’s quite an interesting subject in many ways, eh?

So i,

I had googled Wolfram and found his book and was ready to post the link. Thanks!

“What the fuck is going on here?

Those pictures make me want to kill someone.”

Fly made me laugh, so I +1ed everyone!

Pics don’t make me want to kill people, but that picture makes me want to punch someone in the ear repeatedly, while they’re strapped to the floor listening to some bullshit bagpipe music until I get their ear to resemble that cauliflower shit!

I say fucks em

Wood, this is a great series and well-written, too. The fact that it irritates Fly is just a bonus. Of course, the real challenge and triumph of the exercise was in getting Boca to post again.

You have earned fractal-patterned pocket protector for that alone.

Here’s an interview with Taleb and Uncle Mandelbrot about our current global economic problems. Less fractal than fucked-all, but it seems on point.

http://www.pbs.org/newshour/bb/business/july-dec08/psolman_10-21.html

Actually this makes more sense than it looks like.

I do agree, though the art direction of this post is a bit odd – I felt like I had stumbled into a web page from the mid 90’s.

Anyways – some mathematicians did a study and found that right before a major crash – the markets begin auto-correlating (read self-similar) higher than normal much the same way small tremors feed off themselves until you end up with the whole shelf doing the spasm dance.

Point being – standard distributions are for ‘normal’ markets (sort of – I do agree with Taleb’s skepticism of the normal distribution but the sad thing is that it works 65% of the time) and for the rest – watch out!

These days it’s more important to look at what’s correlating in the past three to five days than what was correlating 30 days to six months ago.

I’m interested to see what will happen when the three to five days becomes one to three days.

Anton, at this time of year it’s very busy in my business, leaving me no time or energy to trade. Let’s hope sales will be good giving me enough of a stake to get back into trading more after January.

I still try to read as much as I can here, will try to post a bit more.

Apparently, Armageddon has been cancelled for today, wishing you all good luck in your trades.

Amazing stuff!!! Reminds me of my college days. Fractal concepts should also touch the discussions on the elliot waves eh?

Cuervo, it has been adapted as art, but it is still a mathematical construct.

As for self-similarity, you are exactly right, and that is one major point that I intend to try and make.

As for Taleb’s skepticism, imagine if boat builders designed their boats to only withstand storms of up to 3 standards above the mean?

Pinoy, good to see you here.

Elliot Waves and fractals do seem to have some obvious similarities. Mandelbrot touches upon this, but I recall he is very skeptical of Elliot Wave Theory. I’ll see if I can touch on that in the future.

Wood – Excellent Fucking Series.

There was a great book I read many years ago on Chaos Theory, by James Gleick. It introduced me to Mandelbrot’s work.

I had not thought about Mandelbrot again until I got into Taleb’s work. Now it looks like I’ll grab up Mandelbrot’s book and complete the circle.

Keep posting. This is the type of stuff that makes my mind dance.