Bachelier's studies, published in 1900 in a doctoral thesis, had nothing to do with the erratic movements of particles suspended in water. Instead, he focused on the apparent random fluctuations of prices of stocks and bonds on the Paris Stock Exchange.
Because of its context, perhaps it's not surprising that physicists and mathematicians ignored or didn't even notice Bachelier's work. Those who did notice tended to dismiss his "theory of speculation" as unimportant.
Mathematician Henri Poincaré (1854-1912), for one, in reviewing Bachelier's thesis, observed that "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating."
Instead of receiving the highest award, mention très honorable, which would have assured Bachelier a job in the academic community, he merited a mere mention honorable for his thesis.
Nonetheless, Bachelier anticipated many of the mathematical discoveries later made by Wiener and others, and he correctly foretold the importance of such ideas in today's financial markets. Bachelier himself believed in the importance of his theory, insisting that "it is evident that the present theory solves the majority of problems in the study of speculation by the calculus of probability."
In examining the role that Brownian motion may play in finance, one place to start is with the link between one-dimensional random walks and gambling.
Imagine a walker starting at position zero at time zero, with coin flips deciding the direction of each step to the left or right. If the track has barriers at its two ends, which swallow up the walker to end the walk, this random process serves as an illuminating model of a famous betting dilemma called the gambler's ruin.
Suppose that Ernie starts $8 and Bert with $11. The two players repeatedly flip a coin. For each head, Bert gives Ernie $1 and for each tail Ernie gives Bert $1. The game ends when either player runs out of money.
Working out the probability that Ernie will win and the probability that Bert will win is equivalent to determining the probability of a walk ending at either barrier of a track that extends eight units in one direction from the origin and eleven units in the other.
In the example given, the probabilities of winning are simply 8/19 for Ernie and 11/19 for Bert (the original capital of each player divided by the total number of dollars held by both players).
Now, what if one barrier is removed and the track in that direction goes to infinity? If the walk continues long enough, it is certain to end at the remaining barrier. Thus, in betting, if Ernie plays against an opponent with an unlimited supply of capital, he will eventually be ruined.
That is certainly bad news for the compulsive gambler, who, even at fair odds, faces an opponent—the entire gambling world—with virtually unlimited funds.
It's possible to plot a player's cumulative total, showing the wins and losses as a line that fluctuates up and down as the number of coin tosses increases (see "Rolls and Flips" and "The Long Run"). For increasingly long sequences of data, the line looks more and more like one-dimensional Brownian motion.
The cumulative results of honest coin tosses serve as a model of a one-dimensional random walk. Heads represents a gain of 1 and tails a loss of 1. Plotting the running total of a sequence of tosses against the number of tosses generates a fluctuating curve. As the number of tosses increases, the plots become increasingly erratic and the line appears to jiggle, much like the track of a tiny particle suspended in a liquid and undergoing Brownian motion.
To Bachelier, it also resembled the day-to-day variations in bond prices, which suggests that the market has a strongly random character. Probability, he concluded, could diffuse in the same manner as heat.
Next: Money Matters II
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