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

Swimming Upstream 
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when he plotted actual stock prices. the upshot of this analysis was 
that it’s the rate of return that undergoes a random walk, and not the 
price. this observation corrects an immediate, damning problem with 
Bachelier’s model. If stock prices are normally distributed, with the 
width of the distribution determined by time, then Bachelier’s model 
predicts that after a sufficiently long period of time, there would al-
ways be a chance that any given stock’s price would become negative. 
But this is impossible: a stockholder cannot lose more than he or she 
initially invested. osborne’s model doesn’t have this problem. no mat-
ter how negative the rate of return on a stock becomes, the price itself 
never becomes negative — it just gets closer and closer to zero.
osborne had another reason for believing that the rate of return, 
not the price itself, should undergo a random walk. He argued that 
investors don’t really care about the absolute movement of stocks. In-
stead, they care about the percentage change. Imagine that you have a 
stock that is worth $10, and it goes up by $1. You’ve just made 10%. now 
imagine the stock is worth $100. If it goes up by $1, you’re happy — but 
not as happy, since you’ve made only 1%, even though you’ve made a 
dollar in both cases. If the stock starts at $100, it has to go all the way 
up to $110 for an investor to be as pleased as if the $10 stock went up 
figure 2: osborne argued that rates of return, not prices, are normally distributed. 
Since price and rate of return are related by a logarithm, osborne’s model implies that 
prices should be log-normally distributed. these plots show what these two distribu-
tions look like at some time in the future, for a stock whose price is $10 now. Plot (a) is 
an example of a normal distribution over rates of return, and plot (b) is the associated 
log-normal distribution for the prices, given those probabilities for rates of return. 
note that on this model, rates of return can be negative, but prices never are.

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to $11. And logarithms respect this relativized valuation: they have the 
nice property that the difference between log(10) and log(11) is equal 
to the difference between log(100) and log(110). In other words, the 
rate of return is the same for a stock that begins at $10 and goes up to 
$11 as for a stock that begins at $100 and goes up to $110. Statisticians 
would say that the logarithm of price has an “equal interval” property: 
the difference between the logarithms of two prices corresponds to the 
difference in psychological sensation of gain or loss corresponding to 
the two prices.
You might notice that the argument in the last paragraph, which 
is just the argument osborne gave in “Brownian Motion in the Stock 
Market,” has a slightly surprising feature: it says that we should be in-
terested in the logarithms of prices because logarithms of prices better 
reflect how investors feel about their gains and losses. In other words, it’s 
not the objective value of the change in a stock price that matters, it’s 
how an investor reacts to the price change. In fact, osborne’s motiva-
tion for choosing logarithms of price as his primary variable was a 
psychological principle known as the Weber-fechner law. the Weber-
fechner law was developed by nineteenth-century psychologists ernst 
Weber and Gustav fechner to explain how subjects react to different 
physical stimuli. In a series of experiments, Weber asked blindfolded 
men to hold weights. He would gradually add more weight to the 
weights the men were already holding, and the men were supposed 
to say when they felt an increase. It turned out that if a subject started 
out holding a small weight — just a few grams — he could tell when a 
few more grams were added. But if the subject started out with a larger 
weight, a few more grams wouldn’t be noticed. It turned out that the 
smallest noticeable change was proportional to the starting weight. In 
other words, the psychological effect of a change in stimulus isn’t de-
termined by the absolute magnitude of the change, but rather by its 
change relative to the starting point.
So, as osborne saw it, the fact that investors seem to care about 
percentage change rather than absolute change reflected a general 
psychological fact. More recently, people have criticized mathemati-
cal modeling of financial markets using methods from physics on the 
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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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