Are markets fractal?
Started: November 22nd, 2013 (07:21 AM) by Big Mike Views / Replies: 4,443 / 46
Last Reply: July 27th, 2015 (03:49 AM) Attachments: 10

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Are markets fractal?

November 30th, 2013, 10:09 AM
#21 (permalink )
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xelaar
Check my previous 2 posts here.

Yup, saw those. Thanks.
You don’t trade the markets; you only trade your beliefs about the markets.
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November 30th, 2013, 11:08 AM
#22 (permalink )
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The metal is called Bismuth

DarkPoolTrading
Insightful. Do you plan on elaborating?

Im quite surprised to see the differing views on this topic. I always thought the dominant opinion was that markets are fractal, and the longer I trade, the more I believe it. But this is what makes a market,...differing opinions. Without them, their would be no markets. Nice discussion.

In my opinion, markets at their core are no different now than how they were 100 years ago. Yes technology has progressed beyond anything imaginable when markets first began. But their reason for being remains unchanged. Buyers and sellers coming together with differing opinions of both current and future value, and meeting in the middle so that a transaction can take place.

Whether you're an

HFT , long term investor, day trader, whatever,...you're all coming to the market because it is an auction place that facilitates trade. It is those differing time lines of market participants that ultimately causes the markets to be fractal. Repeating patterns of behavior reflected in the way markets operate across time.

Zoom in, it's an auction. Zoom out, it's an auction. Irrelevant of how the data is visually represented. Charts are just one representation of how the past auction took place. What underlies that data, the repeating human behavior across timeframes from the

scalper to longer term investor is what causes fractal behavior.

For those that feel that the markets are not fractal, I would be interested in hearing why. There have been several posts saying that they are not. But why not? Saying something is not true requires more than simply saying so. I have posted my reason for believing markets are fractal,....what are yours for believing they are not.

Maybe as with just about everything in trading, their is no one single right answer.

Again,...nice discussion.

Xelaar nailed it. The markets do have some self similarities which are important to note - even Paul Tudor used chart patterns from years prior to predict what was going to happen on a day to day basis . But with news and other events I can't say in whole that price resembles this statement "each part of which has the same statistical character as the whole. ". I'm not saying you can't use fractal geometry with trading to be profitable but to say the market is a fractal is false considering it's not a perfect fractal. I believe the market goes in and out of cycles which resemble both symmetry and chaos both attracting people and taking their money. I have seen near perfect Elliott Wave formations after a news release but I've probably seen countless asymmetric formations as well. That's the thing is that the eye is drawn to these patterns and will remember or spot them out more so because they make sense.
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Last edited by Itchymoku; November 30th, 2013 at 11:55 AM .

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November 30th, 2013, 11:36 AM
#23 (permalink )
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Itchymoku
with news and other events I can't say in whole that price resembles this statement "each part of which has the same statistical character as the whole.

News and events have always been part of the trading game, and what you may recognize as a fractal structure today is exactly a result of an interaction between mass psychology and market events occurred in the past. There's not a perfect symmetry, but the overall macro-pattern could be re-scaled and could show similarities more often than not.

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December 1st, 2013, 03:27 PM
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Anything can be a fractal to some degree but where do you draw the line? The word is very ambigious. However, I do believe fibs and waves work with better accuracy when you zoom out from the noise. The same probably goes for market and volume profile . When you zoom out you see more participants and therefor a larger aggregate of opinions through mass psychology at work. But, when you zoom out too far you may see where fomc or nfp announcements make for asymmetrical impacts that don't comply with the previous structure. So take it as you will.

Last edited by Itchymoku; December 1st, 2013 at 03:50 PM .

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December 1st, 2013, 07:50 PM
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The short answer to the question is YES, markets are fractal. The more liquid the market the more fractal it is. By "fractality" we don't mean they are the same on different timeframes, we are only talking about similarity of pattern/behaviour, which all comes down to the fact that the market actors operate on different timeframes - in fact many people believe that by using some exotic timeframe they will gain an advantage over others.
CAUSE: multitude of timeframes used; EFFECT: fractal structure of the markets.
I was trying to find definition of the "fractal dimension" when I stumbled upon this article:
Fractal market structure, fractals in stock market | Triple.net
So I'll spare myself an effort of rephrasing what fractal dimension is in reference to markets, here is it's full content:
Quoting

Fractals in Market

Fractal Financial Markets

Are financial markets fractals? Financial markets are often modeled with a series of small normally distributed (Gaussian) changes. Such a model has great theoretical appeal, and is the core of other models, e.g. for measuring risk and pricing options. There is only one problem – capital markets clearly are not Gaussian random walks. There are a large number of arguments both against the Gaussian assumption and against the random walk assumption. Some of those argument can be explained away, but others can’t. One of the most compelling arguments against a Gaussian random walk is that markets appear to have fractal structure.

A fractal is a geometrical structure that is self-similar when scaled. A branch of a tree is often used as an example. The branch is similar to the whole tree, and if you break a twig off the branch, the twig is similar to the branch. In a true, mathematical, fractal, this

scaling goes on forever, but in all real systems, there is a largest and smallest

scale which exhibits fractal behavior.

A fractal always has a fractal dimension. The fractal dimension tells us what happens to the length, area or volume of the fractal when you enlarge it. Think of the length of a jagged shore line. If you measure it on a map, you may not see the small bays. At the other extreme, if you try to measure it with a ruler, you will see every stone at the shore. The smaller the structures that you measure are, the longer the shore line will seem. The fractal dimension will tell you how much longer it will become when you measure smaller structures.

A straight line, or a smooth curve have a fractal dimension of one. A filled circle has a fractal dimension of two. A fractal that you can draw on a piece of paper has a fractal dimension between one and two. (To be completely truthful, it could also have a fractal dimension below one, but that is beyond our purposes for now.)

A Gaussian random walk has a fractal dimension of exactly 1.5. And here comes the interesting part � capital market data tend to have a fractal dimension of 1.4. This result does not tell us much about what capital markets look like, but it does tell us that they are not Gaussian random walks. They could still be Gaussian or they could still be random walks, but they can�t be both.

The fractal dimension of a financial time series can be measured by constructing a bar chart and measuring the area of all the bars in the bar chart. This area in itself does not tell us anything. We are interested in how the area scales when we construct bar charts with different time frames, such as daily, weekly, monthly and yearly bar charts. If we had a Gaussian random walk, we would expect the average height of the bars to scale according to the square root of the time frame. In practice, the bars grow faster than that when we increase the scale. The scaling factor is about the same for different time frames � this is typical of a fractal.

In order to be able to explain the fractal scaling we will have to give up either the Gaussian hypothesis or the random walk hypothesis. Either risks in financial markets are much larger than would be expected by the Gaussian hypothesis or the market is prone to trending. In practice, it is hard to tell which alternative is true.

If we give up the Gaussian hypothesis, we almost certainly end up with something called stable probability distributions. The normal distribution is a special case of a stable distribution, and it is the only one that has finite variance. All the other stable distributions have infinite variance. Often, risk is equated with variance, and to say we have infinite variance would mean the risk would be unlimited. This is of course not true. With stable distributions, the variance may be infinite, but the average size of, say, a daily move is not. But what happens when we don�t have a normal distribution is that large moves become more likely, and very large moves become very much more likely. What would also happen is that theories such as CAPM would become complete non-sense, as the variance is unlimited.

We know that large moves in the capital markets are more common than would be suggested by a Gaussian random walk. This could be explained in other ways than unlimited variance, such as increased volatility or even strong trends. However, the fact that the large moves do exist as would be expected if they were to explain the fractal dimension is a strong indication that variance in the capital markets is indeed unlimited. (As always in practice, the fractal characteristic is only evident within certain limits. If we were to look at a sufficiently long financial time series, hundreds or even thousands of years perhaps, we might find that variance on that scale would be limited.)

The other possibility to explain the fractal dimension is to give up the random walk assumptions. Under a random walk, all changes are independent. At every point in time, the market is equally likely to move up as to move down, regardless of how it has moved before. A fractal dimension below 1.5 could be explained if the market where prone to trending (a straight line, the ultimate trend, would have a fractal dimension of one). A trend is not a random walk, as prices are more likely to move with the trend than against. But the trending would have to be fractal � there would have to be trends at every time scale, and those trends could potentially go in different directions.

We can test the trend hypothesis by taking the changes in a financial time series, e.g. the changes in a stock market index. We randomly shuffle the time series of changes and then create a new time series by adding them together again. This should eliminate the trends, and therefore should increase the fractal dimension to 1.5. In practice, we find that the fractal dimension does indeed increase, but not all the way to 1.5. However, if stable distributions were the only explanation for the low fractal dimension, we wouldn�t expect the fractal dimension of the shuffled time series to change at all. There may, however, be other explanations than trending for the change in fractal dimension, such as structure in the volatility of the time series for example. So the results are inconclusive.

There are some excellent books written by Edgar E. Peters that argue the case of fractal structure in capital markets. They are a very good introduction to chaos theory applied to capital markets, but I would recommend anyone reading them to check out the math for himself. I am sure that he didn�t make up the results. It is just that some of them require methods, assumptions and lines of reasoning that I do not think hold up to scrutiny. I am especially skeptical about the R/S-analysis that he is employing. While it might be good for certain purposes, I do not think that it is an appropriate general purpose tool.

Anyway, Peters has this theory, that in the intermediate run, the fractal structure is explained by what he calls fractal noise, i.e. stable probability distributions and infinite variance. In the long run the fractal structure is explained by what he calls noisy chaos, i.e. fractal trends with noise. He might be right. In the long run, capital markets should be determined by the economy, and to a large extent, the economy is deterministic, although very unpredictable, just as it should be in order to drive a fractal system. And there is certainly random noise affecting the economy all the time.

Implications of Fractal Structure in Capital Markets

The two most obvious areas where fractal structure in capital markets would be of interest is in option pricing and risk management.

Option Pricing

Option pricing are traditionally priced using Black-Scholes formula, which is a special case of arbitrage theory. Arbitrage theory makes use of the fact that the fractal dimension is exactly 1.5 to create perfect hedges for options and other derivatives. Unfortunately, this is not possible with other fractal dimensions. Mathematically, the It� formula will not work. Conceptually, this means that it is still possible to hedge away the risk of small changes, but the risk of large moves will be impossible to hedge away. This is also an experience of the arbitrage community. So called

delta hedging of written options is perceived as much more risky than it should be according to normal arbitrage theory.

It is difficult to generalize arbitrage theory to fractal markets. One way would be to try to create close to perfect hedges for carefully selected portfolios of options rather than for individual options. That process would arrive at pricing relationships between options, rather that at a fixed price for each option. Another way would be to use an expected current value as a proxy for the option price. Under arbitrage theory, the price of an option is the expected value of the option at expiry, assuming that there is no compensation for risk for the option (or the underlying stock). It would be possible to generalize this to the fractal case, but assuming an arbitrary market price for risk. In most cases, the price of the options would not be very much affected by the size of the market price for risk (assuming reasonable values) so this would be a pretty stable pricing model. In addition, it is intuitively appealing.

One general observation about option pricing could be made � far out-of-the-money options should be much more expensive than predicted by the Black-Scholes formula. This is also the case in practice.

Risk Management

As mentioned before, risks are often associated with variance or volatility. In a fractal market, variance and volatility are unlimited. This means that risks are much larger than would be expected by standard theory.

One way around the problem with unlimited volatility would be to use the c parameter in the stable distributions as a measure of risk instead of volatility. For the Gaussian distribution, the c parameter is actually equivalent to volatility (with a scale factor). If all assets in a portfolio have the same fractal dimension, optimal portfolio selections exists, that are similar to the Gaussian case.

The most important part of risk management is to select the amount of risk that should be assumed given expected returns. One way to do this is to use a utility function and optimize the expected utility (in practice, position sizes much less than the optimal sizes must be used, as we only have estimates of the parameters used). With fractal markets, some unbounded utility functions could easily produce unreasonably large risks. Therefore, with fractal markets, it becomes much more important to pay attention to the utility function. If unlimited utility functions are used, the implications must be carefully considered.

I've found fractal dimension indicator (for NT) in the download section (I haven't used it yet):
https://futures.io/download/ninjatrader-7/indicators/865-download.html?view

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December 3rd, 2013, 09:35 PM
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Big Mike
I think this deserves its own discussion thread.
Are markets fractal?
Why?
Mike

Yes.
Because the guy who coined the term says so. He wrote some great books on it including:
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I like it a lot, it is one of my favorites on my shelf, but fair warning, he treats the markets as totally random. He doesn't take into account the idea of intuition, etc. in trading.
It also dovetails nicely with one of my other top shelf picks, Nassim Taleb's Antifragile.

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December 6th, 2013, 02:52 AM
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I was thinking on this, and I wanted to add that in the book above, Mandelbrot introduces the concept of "Market time" which has a fractal relationship to clock time. Not a hard concept to explain to anyone who has traded through fast and slow markets.
It explains why I like this indicator too: Tape Momentum | ThinkScripter
As practical matter the market/clock temporal relationship is quite spiky, at least on daytrading scales, so I set it to ring a bell when there are a lot of trades in a small timeframe.
Gotta write something similar for NT "one of these days"
Finally, I'm packing up my library and saw this other book he wrote. More technical, but when's the last time you saw a natural fractal literally made of gold?
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December 6th, 2013, 03:32 AM
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I don't know if I'd consider it a fractal, but it definitely doesn't look very random. I think that's what people are hung up on is the fact that it doesn't look like your average white noise and they feel compelled to label it anything.
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December 6th, 2013, 02:58 PM
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It explains why I like this indicator too:

Tape Momentum | ThinkScripter
As practical matter the market/clock temporal relationship is quite spiky, at least on daytrading scales, so I set it to ring a bell when there are a lot of

trades in a small timeframe.

Gotta write something similar for NT "one of these days"

@bob7123 Do you by any chance have a code or the algorithm for the tape momentum indicator? I would like to have it coded in NT as tape reading is definitely my weak spot.
EDIT: Is Tape Momentum the same as Pace of Tape as in:
http://www.tradewiththeflow.com/2010/05/17/pace-of-tape-indicator/

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December 7th, 2013, 02:33 AM
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gregid
I'll have to have a look at that, thanks for the link.
Also I wanted to add that volume charts are a second way to display "market time" on a longer timeframe.
Greater minds than mine have already figured this out. FT71 's "bread and butter" chart for the ES is a 2500 volume chart.

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