From an article in the New York Times today:

More recently, executives have blamed very unusual events — known to experts as 25-standard deviation moves, things expected only every 100,000 years — for the disruptions that computers could not predict.

Um, according to my calculation, a 25 standard deviation move on a normal distribution has a chance of occuring which is about [tex]$6 times 10^{-138}$[/tex]. This means then that if the above statement is correct (100,000 years equals one 25 standard deviation move), then financial transactions occur at a rate of one transaction every [tex]$10^{-117}$[/tex] seconds. This is, you know, only [tex]$10^{-73}$[/tex] times shorter than the Planck time relevant to a quantum theory of gravity.

Which is great if your a physicist! Forget about building the Large Hadron Collider, just use the financial markets to test your theory of quantum gravity! Maybe the recent credit crunch is evidence for the Higgs boson or a selectron? I mean, seriously, we already have huge numbers of physicists working in the financial sector. Maybe they were on to something we didn’t notice and they’re really doing fundamental physics using this incredible financial transaction speed (and even making money while they’re poor thesis advisors slave away in tenured at a state institute land 🙂 )

More seriously, I wonder if one could predict the future behavior of a financial instrument by examining the incidences of mathematical jargon in the instruments literature and the percentage of times the statement actually makes sense (would you invest if you found a clarifier about 25 standard deviation moves in a hedge funds plan?)

Yet more seriously, Chebyshev’s inequality is your friend. If a standard deviation exists at all, the probability of a k standard deviation fluctuation from the mean is 1/k^2, no matter what the distribution. With k = 25, this gives 1/625. This is small, but not crazy-small, or even once every 100,000 years small, unless the time-scale for financial market transactions is on the order of a century.

My tummyache just got a little better.

Dave – “at most” is quite right. (Proof-reading is an even better friend than Chebyshev.)

How dare you suggest that a normal distribution isn’t the correct distribution to be considering 😉 But even more seriously shouldn’t that read “the probability of k standard deviation fluctions from the mean is

at most1/k^2, no matter what the distribution”?Lets see: The shape of the distribution is key for the valuation of options, futures and all that, so most probably it is a “cooking secret” of each financial institution, even if they all agree that large events are more probable than in the gaussian case. So by telling us his interpretation of 25-sigmas, this unknown expert is leaking secret information about the fit they use.

FWIW, this discussion reminds me of something I read a while back, concerning the dynamics of geology in different places.

Basically, you can plot the processes that create topography on a line measuring “drasticness”, with tiny changes (a single sand grain rolling) on one end, and humongous ones (Krakatoa, Chicxulub) on the other. You can then pick a place & time interval, and plot a

distributionof these events — how often do tiny events happen, and how often do drastic ones happen.The interesting conclusion that this article came to is that Eastern and Western North America have very different topography-creating distributions. In the East, tiny periodic processes (e.g., yearly erosion due to rainfall) dominate… whereas the topography of the West is dominated by rare, catastrophic events (e.g., the collapse of glacial lake Missoula).

This leads to some fundamental problems for human society. For instance, the Army Corps of Engineers plans their dams & levees around a “100-year flood”. In the East, where the distribution of flood volumes has exponential tails, the probability of a flood

twiceas big as the 100YF is basically zilch… but in the West, where drasticness distributions can have polynomial tails, it’s possible that such a flood could occur every 400 years. Which is bad news if you have 400 dams…When I read something like “things expected only every 100,000 years” I like to imagine a big clock somewhere that counts down 100,000 years, and when it gets close to zero everyone gathers around to see the 25-standard deviation event.

I hope the physicsts can come up with a thesis to quadraple profits in stocks and shares and minimise the risk by a quarter.

I have developed a probability distribution that describes the returns statistics of the stock market. It has stretched exponential tails, so the tails are quite fat.It goes 20 sigmas each side. So I find the 25 sigs as a bit strange.This distribution works well with both high frequency data and low.