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Fat fingers and the log-log law

The Power Law | Fat fingers and the log-log law Anthony Harrington

On Thursday May 6th the Dow had its biggest ever intraday fall and for 90 seconds the stock market went completely bananas. According to MSNBC, reporting on a blog from the Washington Post, some well known stocks were briefly worth zero, nada, nothing, during that 90 second period of mayhem, while auctioneers Sotherby’s enjoyed a brief stellar moment where their share price translated into market cap would have given them a $6 trillion value, or more than the combined GDP of the USA and China.

For some days thereafter the market mayhem was blamed on an unnamed trader hitting a “b” instead of an “m” (a “fat finger” error) and posting a $16 billion trade in Proctor and Gamble shares, instead of a $16 million trade. At that point, the theory went, algorithmic, machine-based trading systems were triggered and went into “mad sell” mode. At the time of writing, this hypothesis was unproven and, in the light of what we are going to go on to say, possibly irrelevant (as we will see, basically, any trigger could do it).

After a brief period of chaos, the market bounced almost all the way back up again. In the process, small investors in long positions with stop losses got their positions wiped out. The New York Stock Exchange decided to annul some trades that were clearly “erroneous” during that window of insanity. But that did not stop the howls of outrage and demands for better protection for small investors, while others muttered darkly of conspiracies and big dealers trying to manipulate the market.

There is another explanation, and it is one that the science writer John Gribbin outlines in his book Deep Simplicity: Bringing Order to Chaos and Complexity. It has to do with the log log law, also known as the “power law.” This is a law that is very simple to state and that provides deep insight into complex systems, specifically complex systems that have both some kind of feedback mechanism and an outside source of energy (it does not take genius to see that this is a description that applies rather well to stock markets).

Bear with me for a moment while I sketch out the power law, paraphrasing Gribbin. Charles Richter, who gave his name to the Richter scale for measuring earthquake intensities, applied a logarithmic scale to this. A magnitude 2 earthquake is 30 times more powerful than a magnitude 1 earthquake. A magnitude 3 earthquake is 30 times more powerful again, or 900 times more powerful than a magnitude 1 earthquake. That is the nature of a logarithmic scale.

Then Richter and a colleague, Beno Gutenberg, turned their attention to the frequency, rather than the size of earthquakes. Studying all the data for all the earthquakes they could get their hands on, they found an astonishing fact. Grouping earthquakes of various magnitudes each in their own bin, they found that if they then took the log of the number of earthquakes in each magnitude bin, and plotted it against magnitude (a log log plot), what they got was a straight line through all the data points. In other words, they had uncovered a power law that was always “there.”

As Gribbin puts it:

“For every 1000 earthquakes of magnitude 5, there are roughly 100 earthquakes of magnitude 6, 10 earthquakes of magnitude 7 and one earthquake of magnitude 8.”

Why does this matter?

It matters because, as Gribbin goes on to point out:

“… what it means in practical terms is that there is essentially no difference between a large earthquake and a small earthquake, except for their size. You do not need to invoke some special, rare and peculiar physical effect in order to explain why large earthquakes occur – they just do. Large earthquakes occur more rarely than small earthquakes but they are produced by essentially the same physical process as small earthquakes – and the power law tells you this even if you have no idea what that single physical cause of earthquakes is. This is quite different from the idea, still widely held by both lay people and even some geologists, that large earthquakes require big triggers.”

Essentially, in a complex system, even a tiny trigger can generate a massive event. Tiny triggers will, in general, generate vastly more tiny events than one big event (the power law tells you this too), but the big event is always possible—lurking, if you like. And having happened, there is nothing stopping it from happening again, in short order. (The fact that you have a 100 year storm does not mean that you are guaranteed 100 years of breathing space before the next 100 year storm hits).

So what do we take from this? First, systems that obey power laws are scale-free. If you compare a chart of a huge market movement with the subsection of an ordinary intraday market chart that has a good spike in it, blow the intraday chart up to the same scale and remove the numbering, you will be hard pressed to tell the two charts apart.

Big events simply happen and, if the power law holds, it will be interesting to see what any regulator could possibly come up with to prevent them. Bear this in mind when you hear President Obama speaking sternly of the Administration’s determination to make the market safe for small investors. The market is not safe and never will be… not as long as the power law applies.

Further reading in The Power Law and risk management

Tags: earthquakes , regulation , stocks and shares , trading , US
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