Louis Bachelier's Theory of Speculation (1900)

If asset prices in the short term show an identifiable pattern, won’t speculators find this pattern and exploit it, thereby eliminating it?

Colourized photograph of Louis Jean-Baptiste Alphonse Bachelier at the age of 20 in 1890, and Bachelier’s doctoral dissertation Théorie de la spéculation (1900).

Are financial markets inherently unpredictable? Let’s suppose they aren’t. Let’s suppose there are patterns that may be discerned by humans or computers, and acted upon. What would that do to the prices of those assets? Let’s think it over. The simplest case of benefitting from a predictable pattern is probably buying an asset prior to a predicted movement upward in such a way at least in theory, one benefits from the expected increase in price.

However, if the person buying the asset is expecting its price to rise according to a prediction about the future movements of the market (based on historical information) the buyer is not including in his/her expectation what his/her own buying of the asset will do to the price in the future. Having bought the asset, the buyer increases demand in such a way that the price for the asset incrementally rises, i.e:

Any intelligent economic agent will react to a forecast. That reaction will have to be anticipated when the forecast is made. The agent would take this into account, which would also have to be anticipated, and so on.

Acting on a potential opportunity one identifies in a predicted pattern indeed alters the predicted pattern in such a way that the opportunity, in the limit, vanishes. Reconceptualized by economist Oskar Morgenstern (1902-77) in 1928 using the plot and characters from Doyle’s The Memoirs of Sherlock Holmes: The Final Problem (1893) as a narrative device:

Sherlock Holmes, pursued by his opponent, Moriarty, leaves London for Dover. The train stops at a station on the way, and he alights there rather than traveling on to Dover. He has seen Moriarty at the railway station, recognizes that he is very clever and expects that Moriarty will take a faster special train in order to catch him in Dover. Holmes' anticipation turns out to be correct.

But what if Moriarty had been still more clever, had estimated Holmes' mental abilities better and had foreseen his actions accordingly? Then, obviously, he would have travelled to the intermediate station. Holmes, again, would have had to calculate that, and he himself would have decided to go on to Dover. Whereupon, Moriarty would again have "reacted" differently.

Because of so much thinking they might not have been able to act at all or the intellectually weaker of the two would have surrendered to the other in the Victoria Station, since the whole fight would have become unnecessary.

This week’s newsletter is about the person who first discovered this inherent limitation of financial markets, French mathematician Louis Jean-Baptiste Alphonse Bachelier and his doctoral dissertation Théorie de la spéculation.

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Louis Bachelier (1870-1946)

Louis Jean-Baptiste Alphonse Bachelier was born to bourgeois parents in Le Havre, Normandy in 1870. His father Alphonse was a wine merchant and amateur scientist, his mother Cecile a banker’s daughter. They both died just after Louis graduated from high school, forcing Louis to take over his father’s business at age 19, rather than continue on to universities studies (Courtault, 2000). He did so for two years from 1889-91 before enrolling at the Sorbonne (the University of Paris) in 1892. Following a B.Sc. degree in 1895 and a certificate in mathematical physics, he was accepted as a Ph.D. student working under Henri Poincaré (1854-1912) until submitting his dissertation in 1900.

Bachelier’s Thesis (1900)

Bachelier’s doctoral thesis was quite different from other dissertations in mathematics at the time, and perhaps even since. Rather than derive results related to a technical problem in e.g. topology (his supervisor Poincaré’s interest), Bachelier instead set to work on trying to accurately price stock options. For those unfamiliar, a stock option is a contract which gives an investor the right (but not obligation) to buy or sell an underlying asset (such as a stock) at a predetermined price on or before a given date.

Bachelier’s thesis opens with various financial preliminaries, such as a description of the various instruments (forward contracts, futures, options) available on the French stock market at the time, which were quite different from those available in the American market (Cootner, 1964). Having provided the necessary preliminaries (for his readers who were mathematicians) he proceeds with developing a mathematical model of stock price movements.

Assuming that the market formulates conditional expectations about future prices based on past information, he hypothesizes that prices evolve as (what is now known as) a continuous Markov process which is homogeneous in time and space (Courtault, 2000). That is, Bachelier argues that stocks are ‘memoryless’ in that the probability of each movement depends only on the state the stock is currently in (rather than its previous states). Bachelier goes on to show how the density of one-dimensional distributions of this process satisfiy the relation which would later be known as the Chapman-Kolmogorov equation, noting that the Gaussian distribution (bell curve) with linearly increasing variance solves this identity. He arrives at the same conclusion by considering price movements as random walks. A random walk describes a path that consists of random steps which may be described as the integral of a white noise signal:

Bachelier’s conclusion was that l’espérance mathématique du spéculateur est nulle (“the mathematical expectation of the speculator must be zero”).

The key insight of Bachalier’s dissertation is thus the realization that:

If asset prices in the short term show an identifiable pattern, speculators will find this pattern and exploit it, thereby eliminating it.

Poincaré’s Report

The notes from Bachelier’s dissertation committee, co-signed by Poincaré, describe the topic of his thesis in the following way:

One could imagine combinations of prices on which one could bet with certainty. The author cites some examples. It is clear that such combinations are never produced, or that if they are produced they will not persist. The buyer believes in a probable rise, without which he would not buy, but if he buys, there is someone who sells to him and this seller believes a fall to be probable. From this it follows that the market taken as a whole considers the mathematical expectation of all transactions and all combinations of transactions to be null.

Bachelier’s dissertation received the grade of honorable, “the highest note which could be awarded for a thesis that was essentially outside mathematics and that had a number of arguments far from being rigerous” (Courtault, 2000). It was accepted and published (at Poincaré’s recommendation) in the prestiguous Annales Scientifiques de l’École Normale Supérieure 3 (17), 21-86, and in the same year published as a book of the same name published by Gauthier-Villars.

Impact

Around 1956, American mathematician Leonard Jimmie Savage (1917–1971) discovered Bachelier’s book and brought it to the attention of economist and future Nobel Laureate Paul Samuelson (1915-2009) at MIT.

“Rotting away in the library of the University of Paris”

Samuelson later described:

“When I opened it up, it was as if a whole new world was laid out before me. In fact, I arranged to get a translation in English because I really wanted every precious pearl to be understood”

Samuelson’s colleague at MIT Paul Cootner (1930-78) did the translation. Samuelson developed Bachelier’s ideas further and in 1965 publish two (now) momentous papers in finance on the random nature of price movements:

• Samuelson, P. 1965a. Proof That Properly Anticipated Prices Fluctuate Randomly. Industrial Management Review 5(2), 41-49.

• Samuelson, P. 1965b. Rational Theory of Warrant Pricing. Industrial Management Review 6, 13-31.

In these papers, Samuleson proposes a model of option pricing closely related to the key result of Bachelier’s work, citing his influence. Samuelson also provides the mathematical foundation for what has later become known as the efficient market hypothesis (EMH), the conjecture that asset prices fully reflect all available information in the market.

“In competitive markets there is a buyer for every seller. If one could be sure that a price will rise, it would have already risen”

The model was later adapted by economists Sheen Kassouf (1928-2005) and Ed Thorp (1932-) before being used by Fischer Black (1938-95) and Myron Scholes (1941-) to build what is now known as the Black-Scholes model, simulating of the dynamics of financial markets with derivative instruments.

There is a BBC Horizon documentary about Louis Bachelier, Samuelson, Black and Scholes entitled ‘The Midas Fomula‘ (1999) about the spectacular crash of Long Term Capital Management (LTCM), a U.S-based hedge fund.

The documentary is available below. Bachelier’s work is discussed starting at 12:06.

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I hope you enjoyed this week’s newsletter! Do you think stock prices move at random? Join the discussion:

P.S. I was recently interviewed for Substack’s weekly ‘What to Read’ newsletter. I talk about Privatdozent, future plans, my love of Nash and Gödel and various other topics:

What To Read: Jørgen Veisdal is obsessed with genius, eccentricity and mathematics

Thank you, as always, for being a subscriber. Have a great weekend!

Sincerely,

Jørgen

Related Privatdozent Essays

• Oskar Morgenstern’s Transformation, August 27th 2021

• The Poincaré Conjecture, August 1st 2021

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References

• Cootner, P. 1964. The Random Character of Stock Market Prices. Cambridge, Massachusetts, MIT Press.

• Courtault, J-M., Kabanov, Y., Bru, B., Crépel, P., Lebon, I. & Le Marchand, A. 2000. Louis Bachelier. On the Centenary of Théorie de la Spéculation. Mathematical Finance 10 (3), 341-353.