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

From Coastlines to Cotton Prices 
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later, Lévy would recognize his own mistake and apologize to Bach-
elier. Part of what made Lévy return to Bachelier’s work was a renewed 
interest in random walk processes and probability distributions. Ironi-
cally, this later work of Lévy’s received far less attention than his earlier 
work, leaving Lévy alienated and obscure at the twilight of his career.
Lévy’s work on random processes had led him to study a class of 
probability distribution now called Lévy-stable distributions. the nor-
mal and cauchy distributions are both examples of Lévy-stable dis-
tributions, but Lévy showed that there is a spectrum of randomness, 
ranging between the two. (In fact, there are even wilder varieties of 
randomness than the cauchy distribution.) Wildness can be captured 
by a number, usually called alpha, that characterizes the tails of a Lévy-
stable distribution (see figure 4). normal distributions have an alpha 
of 2; cauchy distributions have an alpha of 1. the lower the number, 
the more wildly random the process (and the fatter the tails). distribu-
tions that have alpha of 1 or less don’t satisfy the law of large numbers
— in fact, it isn’t possible to even define the average value for a quantity 
that wild. distributions with alpha between 1 and 2, meanwhile, have 
average values, but they don’t have a well-defined average variability
— what statisticians call volatility or variance — which means it can be 
very hard to calculate an average value from empirical data, even when 
the average exists.
Houthakker, trained as an economist, likely knew very little about 
Lévy’s late work. But Mandelbrot had been a disciple of Lévy’s. And so 
when he saw the detailed data from Houthakker, something clicked. 
Houthakker was right that cotton prices didn’t follow a normal distri-
bution — but they also didn’t follow a cauchy distribution. they were 
somewhere in between, with an alpha of 1.7. cotton prices were ran-
dom, all right — far more wildly random than Bachelier or osborne 
could have imagined.
cotton markets were the first place that Mandelbrot found evidence 
of Lévy-stable distributions. But if cotton prices varied wildly, he won-
dered, why should other markets be different? Mandelbrot quickly 
began collecting data on markets of all sorts: other commodities (like 
gold or oil), stocks, bonds. In every case he found the same thing: the 

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alphas associated with these markets were less than 2, often substan-
tially so. this meant that Bachelier’s and osborne’s theories of random 
walks and normal distributions faced a big problem. 
Mandelbrot made the connection between Pareto distributions and 
Lévy-stable distributions in 1960, the year after osborne’s first paper; 
he published the extension of this work to cotton prices in 1963, early 
enough that Paul cootner, the MIt economist who edited the collec-
tion of essays the included Bachelier’s and osborne’s work, was able to 
include a paper by Mandelbrot outlining his alternative theory. this 
meant that the volume that brought Bachelier’s and osborne’s work 
to the wider community of economists and financial theorists already 
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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
figure 4: normal distributions and cauchy distributions are two extreme cases of a 
class of distributions called Lévy-stable distributions. Lévy-stable distributions are 
characterized by a parameter called alpha. If alpha = 2, the distribution is a normal dis-
tribution; if alpha = 1, it is a cauchy distribution. Mandelbrot argued that real market 
returns are governed by Lévy-stable distributions with alpha between 1 and 2, which 
means that returns are more wildly random than osborne had thought, though not as 
wild as a drunken firing squad. this figure shows three Lévy-stable distributions. As 
in figure 3, the solid line corresponds to a cauchy distribution and the dotted line is a 
normal distribution. But the third curve is a Lévy-stable distribution with alpha = 3/2. 
It’s a little taller and a little narrower than a normal distribution, and its tails are a little 
fatter, but it’s not so extreme as a cauchy distribution.

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