Mathematics 1900

The Theory of Speculation

Louis Bachelier

Stock prices wander at random — and Bachelier was first to write down the mathematics of that wandering.

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In depth · the introduction

In 1900, a young Frenchman asked a question nobody thought was mathematics — how does the price of a stock move? — and ended up inventing the maths of randomness.

The big idea

Bachelier's insight was that you cannot predict whether a price will go up or down next, but you can describe the odds. He pictured a price as a speck of dust jostled by countless tiny, independent shoves from buyers and sellers. Add up enough random shoves and a clean regularity appears: the price is as likely to drift up as down, and the typical distance it strays grows with the square root of the time that has passed. Wait four times as long, and the price wanders only twice as far.

How it came about

Louis Bachelier studied at the Sorbonne and worked around the Paris stock exchange, where he watched bond and option prices flicker all day. For his 1900 doctoral thesis he turned that flickering into mathematics, supervised by the great Henri Poincaré. He arrived five years before Einstein famously explained the identical jiggling of pollen grains in water — Brownian motion — yet his examiners, baffled that anyone would do mathematics about the stock market, gave the thesis only an "honourable" grade. Bachelier spent much of his career in obscure teaching posts, and his work was nearly lost until economists rediscovered it in the 1950s.

Why it mattered

Almost everything in modern finance grew from this thesis. The idea that markets move unpredictably but with measurable odds underlies the famous Black–Scholes option formula, the way banks price risk, and the "random walk" that markets are so often said to follow. And the very same √t law of spreading describes ink in water, heat in metal, and Einstein's pollen — Bachelier had found one of nature's deepest patterns by staring at a price ticker.

A way to picture it

Imagine a tipsy walker leaving a lamppost at night, taking one random step every second — sometimes forward, sometimes back. After 100 seconds he is not 100 steps away; he is, typically, only about 10 steps from the lamppost — the square root of 100. A price is exactly this drunkard's walk: each tick is a coin-flip step, and the crowd of possible prices fans out slowly, as the square root of time rather than in step with it.

Where it sits

Bachelier stands at a crossroads in the Library. His random walk is the same phenomenon Einstein (1905) explained for pollen grains, and the probability theory that Kolmogorov (1933) later put on rigorous foundations; downstream, the line runs straight to Black–Scholes option pricing. He took a puzzle off the trading floor and handed it to physics and mathematics at once.

The original document

Original source text

L. Bachelier · Théorie de la spéculation · Annales scientifiques de l'É.N.S., 3e série, t. 17 (1900): 21–86 · thesis defended at the Sorbonne, 29 March 1900

Bachelier opens by separating two kinds of cause that move a price: those already known to the market, which are priced in, and the unknowable future, which is not. From this he draws his founding principle — that at any instant a buyer and a seller judge a fair price to be as likely to rise as to fall.

« L'espérance mathématique du spéculateur est nulle. » — "The mathematical expectation of the speculator is zero."

He then builds the probability that a price has moved by a given amount after a given time. In the section he titles « Rayonnement de la probabilité » — the radiation of probability — he argues that probability flows from each price to its neighbours in proportion to their difference. The reasoning yields, in modern terms, the heat (diffusion) equation, and a Gaussian whose width grows as the square root of the elapsed time.

[ … ]

The remainder applies this propagator to the instruments traded on the Paris Bourse, deriving prices for options ("primes") and the chance that they expire in the money, and comparing the formulas against quoted market prices. The complete 86-page text (in French) is at the source below.