[Flojomojo]

The Normality Assumption and Fractal Nature of Asset Prices

Hi everyone, I’ve recently read the book “The (mis-)Behavior of Markets” by Benoit Mandelbrot. I found that some of his ideas matched the experience I have made so far. The key message of the book is that he basically says that all financial theory built so far on the assumption of a normal distribution are for the trash bin and we need a different approach. Especially his idea of “trading time” instead of “normal time” rang a bell in my head. Inspired by his 1963 study on cotton “The variation of certain speculative prices” I decided to test two assumptions myself on EUR/USD Forex rates:

1) Are Forex returns normally distributed?
2) Is there a fractal nature in Forex data?

In the next couple of paragraphs I want to share my findings and thoughts with you. Comments and corrections are always welcome! As I am a ‘hobby quant’ I do not claim the results to be set in concrete. If you have made a study on your own I’d be happy to hear from you.

The data I have used for this study is 47 weeks of IB minute OHLC data from the year 2007. I made a test on the following time intervals: 1, 5, 15, 30 minutes
For that I analyzed the open to close movement of each bar for the respective time intervals. That are 334012 data points for one minute and 11111 for the thirty minute bars…so don’t complain about too little data!

Let’s get started:

1) Are Forex returns normally distributed?

Are absolute changes [P(t+1)-P(t)] normally distributed?
I was always a bit irritated why academic literature always works with returns (absolute or logarithmic) instead of absolute changes. Sure, there are advantages to using them, but when sitting in front of the screen I look for absolute changes. To analyze whether the normality assumption holds I chose to plot the return data in a probability plot. The red line shows the points of the fitted normal distribution. If the return distribution is normal, all points would lie on or close to the red line. As you can see this is clearly not the case for any of the four time horizons! In fact the S shape of the blue data shows that the distribution is a lot more fat tailed than the normal distribution suggests.



Are logarithmic returns [ln(P(t+1)/P(t))] normally distributed?
When looking at logarithmic returns the same picture arises as seen on the probability plot for logarithmic one minute returns.

Therefore, however the returns are calculated, the normal distribution assumption is pretty far off reality. Beware if you are using it!

2) Is there a fractal nature in Forex data?

So if changes are not normal, do they follow a power law of the form a*x^n? In Mandelbrot’s book he states on page 151: “Such power laws are common in physics and are a form of what I call fractal scaling”.
To check the power law assumption one needs to create a histogram and plot its data in a loglog plot. As the absolute return distribution is of interest, the amount of occurrences in the negative histogram half were added to the positive side giving a distribution of absolute changes. Since the number of observations varies from timeframe to timeframe, the frequency was normalized in each case by dividing it by the total number of observations. If the data in the loglog plot follows a power law, the data points should form a straight line.

Before showing any pictures, there is one open question: Histograms count the number of occurrences in specific intervals (bins). How big should these bins be?

When trading, the first moves are critical as the position should show a profit as soon as possible. The first move that really matters is the move that overcomes the initial spread plus payable commissions. So assuming a one pip spread and two pip cost for opening and closing, then a short term trader is only interested in moves >3 pips in his direction. Traders using larger timeframes aim to catch larger moves and don’t watch every pip move. To them moves of for e.g. 10, 20, 30,… pips are more important. Due to this reasoning it makes sense to use larger bins for the histogram of larger timeframe data.

Here are the loglog plots of the different time frames for the 2007 data:



To the human eye they all look fairly similar. When fitting a power law into the tails one can see that in the tails they all follow approximately the same power law. This shows by all lines having approximately the same slope n. (I want to point out here that such a visual test is scientifically not sufficient to prove that there truly is a power law at work!)

Now you might say “Well, all results were calculated on the same sample data! The picture might look different if the different bar intervals were taken from different data sets!” To test his, I split up the 2007 data set into three parts that had the same length for bar intervals of 1, 5 and 15 minutes. In addition the bin size was chosen so that the number of data points in the histogram is equal. Here is the result with the data points being offset from one another to display them in one graph:



If I wouldn’t have labeled which data points belong to which timeframe, I bet neither you nor I would have been able to determine the timeframes correctly other than by luck!!!

With this finding I conclude that Forex data indeed shows fractal characteristics over different timeframes.

Now the really relevant question is: What are the implications?

Here are some points that come to my mind:
- Due to the self similarity, a good trading approach should work on multiple timeframes
- If a trend can be pinned down in one timeframe, entries can be timed in a lower timeframe
- If trading speeds up, change to a lower timeframe
- Accumulation, mark up, distribution, mark down cycles are present in all timeframes

If you actually managed to read until this line in this monster post…what comes to your mind?

Best regards and good trading,
Flojomojo

[JBWTrader]

Awell put together point of View Flowjomojo

I cannot fault it. A well put together analysis.

Well done. I agree with your conclusions.

All the Best
John

[Minetoo]

Flo:
You may be interested in Nichols' stuff here, http://www.fractalmarketreport.com/index.php
Your avatar should be a piece of broccoli.

[aediaz1]

Thank you for doing this research. I just got confirmed, from an academic point of view, some of my believes regarding the FX market. Great post !

[Yorkid]

Flojo -
Here is another implication: social change is fractally ordered.
I think you must be disappointed that you have not more responses to your excellent post.
If I were you I would submit your work to the Socionomics Institute for publication in their online journal. It is exactly the kind of independent corroboration they are looking for.
If you want to get in touch by email I'd be interested. I have ideas that my mathematical ability is not up to pursuing and I'd be grateful for your help.
Y.

[Flojomojo]

Originally Posted by Yorkid »

Flojo -
I think you must be disappointed that you have not more responses to your excellent post.

Thats true. I was hoping for some of the big boys in this forum to comment a bit on my work. But to be honest there already has been research in this field, especially from Fama, a former doctoral student of Mandelbrot. Their work is a lot more detailed than my, in scientific terms "sloppy" analysis. So the study is not that new, but as far as I am aware of, the academic literature does not touch the practical implications for traders.

Since this post I have taken the results a lot further with explaining the fat tail problem and implications for risk management. Not sure yet whether I make a post out of it though.

Feel free to post your ideas here and I'll see how I can comment on them.

[Yorkid]

Hi Flojo -
Referring specifically to the implications of the fractal nature of markets for traders, Elliott wave theory is the largest relevant body of work. Elliott is credited by Prechter with discovering fractals before Mandelbrot. I think the evidence bears out his claim.
I should have given you a link for the Socionomics Institute (although I'm sure you already found it): http://www.socionomics.net
Although I trade on a daily basis, and I take the fractal view of markets as something of a given, I was interested in your post because of its application to wider social theory, which is something that would prabably be of little interest for a traders' discussion site.
I was particularly struck by the way you focus on the change from one time interval to the next, thus laying emphasis on the dynamic aspect of the market rather than the static pattern-related aspect. The patterns are evidence of the fractal nature (Elliott's work) of the way in which they are themselves generated (your work): it is exploring the possibility of the existence of an algorithm which generates this fractal dynamic which interests me. I don't know if this is ground that Fama has covered, but I suspect that if it is then his mathematics would be too difficult for me to follow. It was with a view to getting some digestible mathematical input that I responded to your post.
When you say "sloppy", I think you are being too hard on yourself because there is an inherent barrier in this investigation: it is the irreducibly inductive and reflexive nature of thinking about patterns which we ourselves constitute.
My email address is leslie[at]weydale.freeserve.co.uk

[Logic]

I've been researching and studying pure price with a view of their fractal nature for over 15 years and have come up with a few objective conclusions with regards to applying them inside price charting environments.

1. Applying any study to variable chart environments (all time, tick or range based charts are inherently variable in nature) will result in inaccurate and inconsistent conclusions.

2. Applying any study to the FOREX Markets (pairs) is a futile effort. FOREX is based on separate intra-bank (6) data feeds each of which are producing only time or tick data. None of these feeds release volume data, either in whole or in groups because this is how these individual or groups of banks manipulate the data they release. These banks have repeatedly broken negotiations with GLOBEX when trying to join these feeds into a single stream of information. The only point in time where FOREX will be a reasonable and fair environment to trade will be when all of these feeds are joined and their data and volume related inforamtion is validated.

3. Constant (capped) Volume/Share Bar charting is the only way to accurately, objectively and consistently view the effect of fractals on pure price action (movement). This is because this charting environment is based on the fact that each bar is equally weighted and thus contains no variable aspect. Having the bars of any chart we are making are trading or investment decisions from being grounded in equally weighted bars not only levels the risk associated with that chart but tilts it in our favor.

Mandelbrot & Elliott were on the right track but neither of them had access to the tools to take the information to the next level and that was to smooth out the information that price gives us in its pure movement. Elliott saw price move in oscillations (waves) but erroneously concluded those waves were predictable. Yes, price moved in oscillations (waves) but there is no consistent number of waves inside the longer term extreme moves in price action. Mandelbrot saw how fractals made up everything on the face of the earth but couldn't accurately define the exact effect they had on price action.

A clearer definition to the "fractal effect" on price is "Fractionals" not fractals.

I've been lobbying some of the software companies since 1995 to produce Constant (capped) Volume/Share bars but it took Ensign in 2003 to first offer them. Since then a few more charting companies have offered them (I use MultiCharts) but due to their complete difference to the established way charts are currently built a lot of companies refuse to redo their complex code to offer them. The charting companies approach their products with blinders and feel they only need to do the minimum expected and do not see a need to be cutting edge on any of their products and services. This is a real shame.