The variations in prices during the day is a source of wonderment to all who study them. For example the price of 1 comes up so frequently as to excite the admiration for its fortitude and staying power. Of 26 markets on my screen with a total of 81 digits among them, 30 of them have/are the digit one. Indeed, the proverbial battle during the day between the ensemble of markets and the bulls and the bears might well be considered as a battle among the prices themselves for replication and survivability.

From similar observations in the field of evolution Richard Dawkins came up with the theory of Selfish Genes. He pointed out that evolution works by copying genes. The genes themselves, without any motivation on their part, are in a battle to be passed on. They don't care about the interests of the organism that they are part of. His book based on this theory is considered one of the two most influential books of science of the last 50 years, and has sold more than 1 million copies. It explains and illuminates many phenomena that the traditional view of organisms competing at the level of the phenotype in a struggle for survival of the fittest find hard to explain — particularly altruism, deception, kinship, acting against interest, vivid and startling coloration (green beards).

The time has come to apply this theory to prices themselves. They are the units of variation that try to reproduce that control markets, not the other way around as is so frequently posited. Let's start with the battle of the price 0 to extend itself. Using daily prices, we see the Dow crossing from above 10000 to below 10000 three times during the last two years and crossing from below 10000 to above 10000 on two occasions. The 0 in 10000 gets to express itself four times while in all other prices of recent vintage it is only expressed three times so that 10000 is a particularly noteworthy price to achieve.

In addition, it's a green beard that attracts other prices at 0. When the Dow hits 10000 every financial media is likely to have a headline that the magic number has been broken. Other zeroes in other markets such as the Nikkei at 10000, gold at 10000, oil at 100, the yen at 1000, and the S&P at  1000, soybeans at $10.00 are sure to note the price and copy it. The zeroes in 10000 while acting essentially selfishly benefit other zeroes in other market that have the intellect to recognize what is happening in the Dow. The transmission of these effects in the media magnifies what has been called "green bearding" by Dawkins in the concept of the selfish gene.

a green beardOf course, if recognition plays a part in the propagation of prices, so does deceit. The same way that butterflies mimic wasps, markets may pretend to be going to a recognized number like 10000 but stop right before it as fast moving operations like the specialists or the high frequency traders step in to beat out those who have been deceived by the path. Such activities lead to the well known phenomena that highs below the round number and lows just above it happen much too frequently to be explained by chance in individual stocks and the major market averages.   

As a first crack at systematizing the theory of the selfish price, I calculated the closing 10 digit of the S&P unadjusted futures for the last three years, 743 observations in all.

Battle of selfish opening and closing prices

             opening price        closing price
0                  71                  88
1                  76                  69
2                  64                  55
3                  74                  79
4                  77                  70
5                  84                  79
6                  77                  76
7                  68                  65
8                  68                  76
9                  84                  70

One notes that the digit of 2 is losing the investment table, some 5 standard errors away from expectation, while the old faithful of 0 is winning the ultimate battle closing 88 times, 3 standard errors above expectation from its 74 expectancy. There are other wonderful and noteworthy phenomena revealed in this table, and its extensions, and many beautiful aspects of the struggle for existence, the mutualism, and antagonism of the prices for one another, and always their tendency to be in a positive feedback system with the growth of the market organism itself, which I will not gainsay the reader the jubilation of ascertaining for himself.

It is well known that genes often work together with each for the greater good of each other. For example, there could be a gene to make disease less likely under certain circumstances, and a gene for long life. A typical example of a gene that is beneficial to other genes but not to itself is a gene in birds for calling out loudly and clearly in situations of danger. The gene helps all the other genes survive in its kin, but not necessarily itself as it calls attention to itself. Genes tend to  work together to make for a greater likelihood that the whole organism and all its genes will survive and reproduce. The cost benefit function of a given gene may be y expressed as pb  versus c  where b is the benefit a gene gives to another gene, c is the cost, and p is the increase in probability that the other gene will provide to it.

The cost benefit function creates a situation where the genes come to be represented according to their net contribution to their ability to be reproduced in successive generations, including their cumulative impact on all other genes in the genome. The opposite situation which occurs must less frequently is called intragenomic conflict, and the classic example is referred to as segregation distorter genes which act to crowd out other genes that are beneficial to fertility. Egbert Leigh expresses this unlikelihood as follows: The genes act as "a parliament of genes, each acting in its own interests, but if it acts hurt the others, they will combine together to suppress it."                            

Apparently the price units of selection in markets do not act to suppress their neighbors. During the last 2600 days for example in the  S&P, 2530 days in the S&P 24 hour futures, 2412 of them have allowed each of the ten separate 10 digits, 0 to 9 to appear. In other words the 24 hour range has been more than 10 on more than 95% of all days. Apparently it keeps all the individual prices healthy to exercise each of its competitors on almost all days.

Here is a good reference on this Selfish Price theory which I posit in all seriousity.

Rocky Humbert notes:

The paucity of "2" as described by the Chair is a persistent phenomenon. For the 12,143 trading days between 1955 to 2003 (when the S&P first went over 1,000), the digit "2" occurred (as a tens) only 5.1% of the time.

Perhaps some of this may be explained by number theory — i.e. index calculation effects due to stocks trading in eighths and quarters, and that may also explain the increase in the "2" in the Chair's data post decimalization. (He found "2" rose to 8% from the 5.1% over the longer period.)

One further notes that on most QWERTY keyboards, the lowly "@" sits above the "2". Prior to email, the @ was slowly facing extinction– only to be resurrected to prominence contemporaneous with AAPL stock. Hence I believe it's premature to put the "2" in the Peabody Museum diorama that also houses the Dodo Bird and Pig-footed Bandicoot. 

Marion Dreyfus comments:

There is apparently a marker gene for how many times a person sneezes when he or she sneezes daily–This might be a signal to alert noticers of the individual patterning of investment thinking or individual behavior. As some people always sneeze thrice, and only thrice, or twice if the gene for twice is embedded in the coding ''parliament'' of the genome sequencing, perhaps we also have an idiosyncratic pattern of investing that has hitherto gone unnoticed. Can this be mapped, one wonders. And if so, can one be thus invested with more knowledge of the other's "hand," as in playing poker with someone whose "tell" you know, so you can conserve bets for when a hand/bet/risk is most propitious…

Pete Earle writes:

morpholinoOne of the tools used in determining genetic action– or, more aptly, interaction– is the morpholino, a short, targeted nucleotide sequence which blocks ("knocks down") expression of one gene among two or more to see if, or how, the ultimate expression of said genes changes. My partner is involved in exactly this sort of research daily. Once she targets a gene– in this example, trying to determine the interaction of two genes in producing a specified outcome (gene A + gene B = expression C)–she then conducts subsequent experiments in which she varies the amount of the morpholino between 0% (no morpholino, the control group) and increments up to and including full strength (complete knock-down of gene A, 100%). This is to determine which gene, if any, is more important to a given expression than the other; and to see if a gene interaction is of the simply "on/off" type or if expressions take place along a spectrum of outcomes.

I suspect that with respect to Vic's Selfish Price Theory, we might look at morpholino-equivalent testing with a comparison of periods within which a given market approached a certain number-expressing level, and compare those with others, looking for volume superlatives; one would expect the day or week of the arrival of Dow 8888, 10000, and 11111 to be of higher volumes to a statistically more significant extent than, say, those when Dow hit 12345 or 9876. This could be broadened to look at random snapshots of days where, across a number of indices or index-constituting stocks– even, and perhaps especially, in the absence of such aesthetically pleasing prices as 10,000 or 55 and such– we would look for higher-than-expected volumes when and where there noteworthy appearances by a particular number across a spate of closing prices. 

Pitt Maner III writes:

My dentist last week mentioned to me that he was studying the latest papers (within one day of publication) on gene "crosstalk" so as to help his daughter in college who is doing an honors thesis on the subject (and how it relates to drug interactions with cancer cells). Cancer cells evidently have a means of (and this is over my head—cell experts please jump in) of dampening the effects of anti-cancer drugs through cellular cross-talk genes. Therefore drug manufacturer have a need to knock out the cancer cells through a series of steps to weaken these defense/signaling channel mechanisms.

Any underlying, as yet undefined, step-like mechanisms and pathways would seem to skew number distributions.

Henrik Andersson comments:

Benford's lawThis seems somewhat related to Benford's law which predicts the probability of digits, for example the probability that a stock index of stock price will start with a '1' is slightly above 30%. A funny side note is that this theory of frequency of numbers in nature can be checked using Google searches.

Victor Niederhoffer responds:

I don't think it applies here, especially for the second, third and fourth digits.

Henrik Andersson replies:

Yes, it probably only over powers other forces in the market for the first digit.

Kim Zussman writes in:

The SP500 is Benfordian:

Using daily closes SP500 1950-present, counted days which closed with the first digit = 1. eg, {1XXX.XX, 1XX.XX, 1X.XX} (there were no 1.XX yet}.

Of 15135 total days, 5514 had 1 as the first digit.

Alston Mabry writes:

And to relate that chart to genetics: If volatility = selection pressure, then when volatility/selection pressure is low, variability in digit frequency/phenotype expression is high; but when volatility/selection pressure is high, variability in digit frequency/phenotype expression is low.

And different species have different time intervals, i.e., lifespans.

Peter Earle responds to Henrik Andersson's comment:

At risk of torturing the analogy a bit– but worth mentioning: "Yes it probably only 'over powers' other forces in the market for the first digit. "Let's discuss those "other powers", as they are germane to Vic's theory. It's appropriate to at this point bring up one of the hot topics that my partner, again, is working on: epigenetics. In short, it's the imposition of hard-coding changes on DNA (via methylation) by environmental effects. While still not fully understood, one example is depicted by rat experiments in which the pups of profoundly overweight mothers (exposed to high levels of interuterine glucose) switched at birth with skinnier rat mothers show a statistically significant greater chance, thereafter, of becoming obese, even setting aside "lifestyle" and dietary settings. (See the "Barker Hypothesis" for another example of this phenomenon.)

With respect to Vic's Selfish Price theory, we might quantitatively express these variations from expected (Benford's Law) vs. actually expressed frequencies of prices/digits as an epigenetic effect: 'environmental' effects whereby the impact of market participants and economic influences -forces and memes - push toward or away from predicted, anticipated baselines.To that end, tracking the ebb and flow in expressed, realized prices from what which the Law predicts over time could provide one way– no doubt an incomplete way, but a way nonetheless - of quantifying the ever-changing cycles. 

Alston Mabry says:

Back to the tens digit, this time in the S&P cash. Starting with January, 2004, I calculated a 250-tday rolling total for each digit, e.g., in the past 250 tdays, how many times has the tens digit of the S&P Close been 0 or 1 or 2, etc. Then calculated the gap between the most frequent and least frequent digit, e.g., if 6 was the most frequent in a given 250-tday period, occurring 48 times, and 3 was the least frequent, occurring 21 times, then the max-min gap would be 48-21 = 27.

Then for I calculated the SD for each 250-tday period, too, as a measure of volatility. The attached graph shows the two series. What can be seen is how the max-min gap is higher when volatility is low, but compresses into a narrow range when volatility increases. This seems intuitively sensible if one thinks of a more volatile S&P moving quickly through various values and thus being more "random" at the tens digit. Whereas, when volatility is low, the S&P would be "stickier", hanging around longer at certain tens digits, thus creating a wider max-min gap.

Of course, an underlying factor is the arbitrary nature of choosing a unit of time such as a trading day. If one zoomed in and out, using different lengths of time to create ecah "Close", then one would probably see a clear relationship between volatility and digits on different time scales.

One more take (esoteric, but I really like the chart): For each S&P day from 1990 to present, calculate the distribution of the tens digit in the S&P for the 250-tday period ending that day: 41 zeros, 24 ones, 23 twos, etc. Then get the SD for this distribution. Example:

41 0's
24 1's
23 2's
17 3's
26 4's
20 5's
20 6's
21 7's
19 8's
39 9's

SD: 8.33

Then calculate for the same 250-tday period the SD of the daily change in points of the S&P - points rather than percent because we are relating index point movement to digit distribution.

So, for each 250-tday period, we have a measure of the volatility of the index and the variability of the tens digit. Sort all the 250-tday periods by the S&P volatility value, high to low, and graph the result - see attached graph .

Nice inverse relationship between the S&P point volatility and the variability in the tens digit.


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