The Physics of Wall Street: a brief History of Predicting the Unpredictable

From Coastlines to Cotton Prices 
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pointed out that this wasn’t quite right; really, you expect the prices to 
change by some fixed percentage each time God flips his coin, rather 
than some fixed amount. this modification led to the observations 
that rates of return should be normally distributed and prices should 
be log-normally distributed.
the normal distribution shows up in all sorts of places in nature. 
If you took the heights of all of the men in a given part of the world 
and plotted how many of them were 5 feet 6 inches, how many 5 feet 
7 inches, and so on, you would get a normal distribution. If you used 
a thousand thermometers and tried to take your temperature with 
each of them, the results would look like a normal distribution. If you 
played a coin-flipping game in which you got a dollar every time the 
coin landed heads, and you lost a dollar every time it landed tails, the 
probabilities governing your profits after many plays would look like 
a normal distribution. this is convenient: normal distributions are 
easy to understand and to work with. for instance, if something is 
normally distributed and your sample is large enough, the sample’s 
average value tends to converge to a particular number; white men, on 
average, are about 5 feet 9 inches, and unless you are ill, the thousand 
thermometers’ readings will average 98.6 degrees fahrenheit. Your av-
erage profits in the coin-tossing game will converge to zero.
this rule can be thought of as the law of large numbers for prob-
ability distributions — a generalization of the principle discovered 
by Bernoulli, linking probabilities to the long-run frequencies with 
which events occur. It says that if something is governed by certain 
probability distributions, as men’s heights are governed by a normal 
distribution, then once you have a large enough sample, new instances 
aren’t going to affect the average value very much. once you have mea-
sured many men’s heights in a given region of the world, measuring 
one more man won’t change the average height by much.
not all probability distributions satisfy the law of large numbers, 
however. the location of the drunken vacationer in cancun does — he 
is taking a random walk, so on average, he will stay right where he 
started, just as the average profits from a coin-tossing game converge 
to zero. But what if instead of a drunk trying to walk to his hotel, you 
had a drunken firing squad? each member stands, rifle in hand, facing 

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a wall. (for argument’s sake, assume the wall is infinitely long.) Just 
like the drunk walking, the drunks on the firing squad are equally li-
able to stumble one way as another. When each one steadies himself to 
shoot the rifle, he could be pointing in any direction at all. the bullet 
might hit the wall directly in front of him, or it might hit the wall 100 
feet to his right (or it might go off in the entirely opposite direction, 
missing the wall completely).
Suppose the group engages in target practice, firing a few thousand 
shots. If you make a note of where each bullet hits the wall (counting 
only the ones that hit), you can use this information to come up with 
a distribution that corresponds to the probability that any given bullet 
will hit any given part of the wall. When you compare this distribu-
tion to the plain old normal distribution, you’ll notice that it’s quite 
different. the drunken firing squad’s bullets hit the middle part of the 
wall most of the time — more often, in fact, than the normal distribu-
tion would have predicted. But the bullets also hit very distant parts of 
the wall surprisingly often — much, much more often than the normal 
distribution would have predicted.
this probability distribution is called a cauchy distribution. Be-
cause the left and right sides of the distribution don’t go to zero as 
quickly as in a normal distribution (because bullets hit distant parts 
of the wall quite often), a cauchy distribution is said to have “fat tails.” 
(You can see what the cauchy distribution looks like in figure 3.)
one of the most striking features of the cauchy distribution is that 
it doesn’t obey the law of large numbers: the average location of the fir-
ing squad’s bullets never converges to any fixed number. If your firing 
squad has fired a thousand times, you can take all of the places their 
bullets hit and come up with an average value — just as you can aver-
age your winnings if you’re playing the coin-flip game. But this average 
value is highly unstable. It’s possible for one of the squad members to 
get so turned around that when he fires next, the bullet goes almost 
parallel with the wall. It could travel a hundred miles (these are very 
powerful guns) — far enough, in fact, that when you add this newest 
result to the others, the average is totally different from what it was be-
fore. Because of the distribution’s fat tails, even the long-term average 
location of a drunken firing squad’s bullets is unpredictable.
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t h e p h y s i c s o f wa l l s t r e e t

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From Coastlines to Cotton Prices 
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61
As Mandelbrot described it, the war, especially during the first two 
years under vichy rule, left huge swaths of france unaffected for long 
periods. But then “a storm” would come through and wreak havoc, 
followed by another period of calm. So perhaps it is no surprise that 
Mandelbrot was fascinated by these bursts, by random processes that 
couldn’t be tamed like a casino game. He called events that obeyed 
a cauchy distribution wildly random, to distinguish them from the 
plain, mild randomness of the random walk, and he devoted much of 
his career to studying them. When Mandelbrot began his career, most 
statisticians assumed that the world is filled with normally distributed 
figure 3: the location of a drunken vacationer trying to find his hotel room on a 
long corridor is governed by a normal distribution. But not all random processes are 
governed by normal distributions. Where the bullets fired by a drunken firing squad 
will land is determined by a different sort of distribution, called a cauchy distribution. 
(note that the angle at which the members of the drunken firing squad fire will be 
governed by a normal distribution; it’s the location on the wall that they are trying to 
hit that is governed by the cauchy distribution!) cauchy distributions (the solid line in 
this figure) are thinner and taller than normal distributions (the dashed line) around 
their central values, but their tails drop off more slowly — which means that events far 
from the center of the distribution are more likely than a normal distribution would 
predict. for this reason, cauchy distributions are called “fat-tailed” distributions. 
Mandelbrot called phenomena governed by fat-tailed distributions “wildly random” 
because they experience many more extreme events.

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events; though cauchy and other fat-tailed distributions might show 
up sometimes, they were believed to be the exception. More than any-
one else, Mandelbrot showed just how many of these so-called excep-
tions there are.
think back to the coastline of Britain. Suppose you want to figure 
out the average size of a promontory, or any outcropping of land. You 
might start by looking at boulders and jetties, things of a manageable 
size. You take the average size of all of these. But you aren’t done, be-
cause you realize that these jetties and outcroppings are themselves