when observing the tick price action in Forex, I was wondering whether the prices have any bias towards certain repetitive decimal levels ("00", for example).

General considerations

In plain language, I was asking myself: "are there more price ticks ending with the digit 0 than with the digit 5"? Or "are there more price ticks ending with the digits 50 than with the digits 75?"

Since all the securities' prices are expressed in the decimal numerical system, the basic information we are interested in here is to know what percentage of price ticks in a data set are ending with a particular decimal digit or a set of digits. This research shows there's no doubt the decimal symbol set (digits 0 to 9) repetition in the numerical price representation of quantity psychologically distorts the free price flow and causes the decimal bias.

The decimal bias simply means that the probability that the price will end with a certain decimal digit (or a certain set of decimal digits) changes with the digit (or set of digits). Also, according to the research results it seems that every security features different distribution of its decimal levels' probabilities creating a unique fingerprint of the security.

Now the numerical system symbol set iteration boundary psychological bias, as I call it in a more universal way (don't be scared, I won't use the term more than once ), emerges in a real life every time when we are dealing with some representation of quantity in some predetermined numerical system (see Numerical systems in Wikipedia). For the purpose of this statistics, I will be talking solely about the decimal symbol set psychological bias, or a decimal bias for short. Decimal bias is the most common variety, although one can develop other numerical system-based biases (I can imagine a computer programmer having a severe hexadecimal bias when ordering "1A" pizzas instead of decimal "26" ahead of his weekend coding spree, but that's another story ).

The decimal system

The most widely used numerical system for ordinary life purposes is the decimal system, the same as the one used for expressing all the securities' prices. Decimal system uses 10 symbols in a certain sequence to express any required quantity representation.

The reason why the decimal system is so widely popular is attributed to the fact that people have 10 fingers, so that they are better equipped to imagine the multiples of 10 than the multiples of some other quantity. By the way, the very English word "digit" meaning a number is derived from the Latin word "digitus" meaning a finger.

Other numerical systems used heavily in computer science today are the binary system (uses 2 symbols), hexadecimal system (16 symbols) and the octal system (8 symbols). We call the decimal symbols "digits" just as we call the alphabet symbols "letters". A "number" is then a set of one or more digits that expresses a quantity, just as a "word" is a set of one or more letters that expresses certain meaning in language.

The above mentioned "iteration boundary psychological bias" occures when you run out of symbols and thus you reach the boundary symbol in the numerical system. Remember, we only have 10 symbols in a decimal system - thence the name "decimal", meaning "ten". The boundary symbols for the decimal system are "0" and "9". I'll explain below how these two boundary symbols play an important role in how do we percieve the quantity subjectively.

Before running the statistics, I thought that the major distrortion of probability distribution would be seen around these boundary symbols. For one digit statistic, the boundaries are 0 and 9, respectively, while for two digits statistic, the boundaries are 00 and 99. The fact is that the results haven't confirmed this hypothesis - the various decimal levels show various distortions, regardless of the proximity to the boundary symbol. Since the best known decimal biases in real life occur around the boundary symbols (see below), I'll use the boundary symbol bias as an example to show how the decimal bias may develop in the first place, although the term decimal bias means "bias towards any decimal digit".

What we humans do internally in our mind when we are using the decimal (or any other numerical system) is that we convert the written symbols (digits) to the actual quantity in our head.

We have all learned how to deal with the decimal system and how to quickly convert between quantity and its decimal representation in a primary school. Being able to count is so important for a modern homo sapiens that we are bombarded with the decimal numerical system from our early childhood so that now we even don't realize this system is artificial and has nothing to do with the quantity itself.

To illustrate, let's use similarly artificial numerical system and let's say I told you to buy CMXLIV shares of something. We are not trained to convert "CMXLIV" to quantity just as fluently as we are with decimal symbols. It would certainly take you several seconds to realize I'm talking about Roman numerals and that the representation in decimal symbols is "944". Only then, upon converting the "CMXLIV" symbol string to "944" symbol string, you would be able to finally convert the "944" symbol string to the actual quantity in your head.

Now a million dollar question: which one of the two symbol strings represent the same quantity better?

- CMXLIV
- 944

Limited number of symbols

People strive to find ways to simplify things, so the development of numerical systems was inevitable since the dawn of civilization. Instead of saying "I want this this this this apple", it's more comfortable to say "I want four apples". The trouble with numerical systems is that to design a system that is simple enough to be broadly usable, there must be only a limited number of symbols in the system and you have to come up with some rules to shuffle those symbols around to arrive at the required quantity.

The major trouble with using a limited set of symbols is that you inevitably run out of symbols if there are less symbols than the values you want to represent. You can't have symbols for all individual values - imagine having one million symbols for all numbers from 1 to 1 million. Thus, numerical systems are always using a limited set of symbols and shuffle them around in some particular way to be able to describe all the values.

We are all doomed - the ubiquitous decimal bias

To make my point more clear, before we get down to the decimal bias in price action statistics, I'll show you several examples of how the decimal symbols psychologically distort people's decisions every single day.

The basic trouble is that we are so used to the decimal system that we treat the symbol boundaries (the symbol switch from "9" to "0") as the natural boundaries in the object's quantity almost as if this artificial boundary had some magical tie to the inner quantitative structure of all the objects.

Thus, we are foolishly:

- celebrating anniversaries that are multiples of decimal symbol boundary, like 30, 50 etc. Why are we not celebrating anniversaries that are multiples of 8? Is eight a worse quantity than ten?
- drawing borders between salaries that are at a decimal symbol boundary, like "up to $100k" and "over $100k" (why not "$98723"?) as if the symbol boundary symbols of "zero zero" possessed some magical power
- treating psychologically the quantity (for example price) that is close to the decimal symbol boundary differently. "$99.99" is seemingly much lower than the price "$100" just because there's no symbol added to the left of the leftmost "9" and our decimal-trained brain is telling us "this quantity is order of magnitude lower!" (actually, prices ending with "95" or "99" digits as a psychological weapon were invented in 1922 by my Czech countryman, Tomas Bata, who established the biggest shoe making company in the world to this time)
- getting quantity discounts for buying 10, 50 or 100 items. You never get quantity discount for buying 11, 47 or 83 items. In fact, this is a Bata prices ($99.99) phenomenon reversed: 100 items is an order of magnitude higher order than a 99 items order in the eye of the vendor
- .. I'm sure you can come up with many other examples of decimal bias from your own experience

Mining for the decimal bias data with C#

I wrote a data-mining code in C#.NET that went thru the tick database for these 15 major FOREX symbols in a sample period of about 8 months ending June 2006: AUDJPY, AUDUSD, CHFJPY, EURAUD, EURCHF, EURGBP, EURJPY, EURUSD, GBPCHF, GBPJPY, GBPUSD, NZDUSD, USDCAD, USDCHF, USDJPY.

Thus, the total amount of data used for conduting the research was huge. About 22 million price ticks were examined (uh, you see? I didn't say "about 21,827,645 ticks" - as stated above, the decimal bias is ubiquitous, we are all doomed ). This means that the results are statistically significant and are not victim to the non-representative data sample selection.

I performed the statistics for two decimal orders:

- all prices ending with one decimal digit
- all prices ending with a combination of two decimal digits

Example - one decimal digit:

- 1.2746 = returns value of 6
- 1.2349 = returns value of 9
- 1.7523 = returns value of 3

- 1.2438 = returns value of 38
- 1.7324 = returns value of 24
- 1.9467 = returns value of 67

- for one digit stats: if xx_number of price ticks in the sample were found to be ending with "9", then the value of xx_number has been assigned to the "ending decimal digit: 9" row in the statistics
- for two digits stats: if yy_number of price ticks in the sample were found to be ending with "26", then the value of yy_number has been assigned to the "ending decimal digits: 26" row in the statistics

If I was to perform the stats for some higher order like the last three digits (actually I did), then certain combinations of ending decimal digits would be so tinily represented in the sample that it would distort the statistical validity of the particular test, so I stayed with one and two ending digits statistics only.

How to make a living trading decimal bias

Now what can we conclude from the results for practical trading? Clearly, the distribution of ending decimal digits in examined price tick history data is not even.

Let's have a look at this EURGBP two digits statistic example:

You can see that the "48" decimal level (all price ticks that ended with "48") was occuring only in 0.74% of total cases (price ticks). On the other hand, this "48" level lies in the valley, surrounded by higher values. The decimal level "31" (all price ticks that ended with "31") has been present in 1.36% of cases, which is almost double the value for the "48" level. And the "62" decimal level shows that even 1.57% of all price ticks ended with the digits "62".

Now let's ask: what drives the prices to stay at or leave certain decimal level? Is it a weather? Is it a political climate? Is it a day's time?

No. The answer is simple: the prices are driven by the human traders. And, in turn, one of the forces that drive traders is the decimal bias. The above chart shows that the traders don't feel comfortably trading EURGBP at or around the market prices that end with "48". But they love to trade EURGBP at prices ending "31" and around. Still better, the absolutely most cherished two-digit ending level for traders is "62". They feel comfortable trading EURGBP at "62" and around. Why the "48" level is so uncomfortable for traders? I dont' know really, but I know that it is so and I can take advantage of it easily as a FOREX trader:

- the hot 48 level in EURGBP means that no one wants to trade at that level
- if the price goes from our leisure 31 level up to the hot 48, I know that it will either:
- turn back (no one wants to stay at the hot level!)
- or very probably go all the way up to the super relaxed 62 level. Then I can take my profits at 62 (I'm not talking age here ) since the majority of traders are very probably going to take a nap at that price level, too

- the same applies to the price action in the opposite direction, too
- when the prices are at the "uneasy" level - in this case the hot 48, i may place a straddle with stop buy at 52 and a stop sell at 44 with limits of say 8 pips each, knowing that the price will go either one or the other way all the way up/down to the nearest comfort level

Decimal bias statistics - one ending digit

All the statistics charts for one ending digit are set to the same vertical scale to be able to compare the absolute percentage distribution accross the symbols (charts). It seems the statistics for only one ending digit are not much usable for practical trading and are more of an "academic nature" only:

Decimal bias statistics - two ending digits

All the statistics charts for two ending digits are set to the same vertical scale to be able to compare the absolute percentage distribution accross the symbols (charts):

I'd love to hear your comments

This was just an example analysis. Many other interesting conclusions can be drawn from the statistics (including those I've overlooked) . Below I am attaching the zip file with all the statistics in excel.

So tell me, will you join me in taking advantage of other traders' decimal bias?

Have a nice day all,

Michal