Saturday, September 26, 2020

The Grand-Père Of Option Pricing Theory: Louis Bachelier

We've looked at the work of young Mr. Bachelier, most recently in February 2020's "Following Up on "Emanuel Derman: 'Trading Volatility'" which linked to a letter to the editor of Inference Review by gadabout theoretical physicist Jeremy Bernstein.
The letter begins:

On Louis Bachelier

To the editors:
In Emanuel Derman’s essay on the Black–Scholes Equation, the French mathematician Louis Jean-Baptiste Alphonse Bachelier is mentioned in passing. Published at the turn of the twentieth century, Bachelier’s PhD thesis was the foundation for the later work of Fischer Black, Myron Scholes, and Robert Merton.1 Scholes and Merton were jointly awarded the 1997 Nobel Prize in Economics for developing “a pioneering formula for the valuation of stock options.”2 Black had passed away in 1995. While brief sketches of Bachelier’s life can be found, no one, as far as I know, has written a full biography. Since Bachelier’s only lasting contribution to mathematics seems to have been his thesis, it is not clear what one would say in such a biography. It is a curious and rather sad story.

Bachelier was born in Le Havre on March 11, 1870.3 His father was a wine merchant. Bachelier would certainly have been headed for one of the grandes écoles, but in 1889 both his parents died. He took over the family wine business and then did his compulsory military service. It was not until 1892 that he was able to begin his studies at the Sorbonne. During this period, he may have worked at the Bourse, although the details remain elusive. In any event, he became interested in the question of how to predict the future price of a stock. It is not clear whether he had heard of Brownian motion, or if he simply invented the idea for himself. He assumed that, on its next move, it was equally likely for the price of a stock to go up or go down. This is Brownian motion. Upon hearing of it for the first time, a natural reaction is to inquire how the price goes anywhere. Of course, after the first move, it is as likely for a price to advance further as it is to go backwards. This results in the random walk that is characteristic of Brownian motion....MORE
And I forgot to point out that Bernstein's letter linked to both Bachelier's 1900 thesis:


Translated by D. May from Annales scientifiques de l’Ecole Normale Superieure

The influences which determine the movements of the Stock Exchange are innumerable. Events past, present or even anticipated, often showing no apparent connection with its fluctuations, yet have repercussions on its course.

Beside fluctuations from, as it were, natural causes, artificial causes are also involved. The Stock Exchange acts upon itself and its current movement is a function not only of earlier fluctuations, but also of the present market position.
The determination of these fluctuations is subject to an infinite number of factors: it is therefore impossible to expect a mathematically exact forecast.
Contradictory opinions in regard to these fluctuations are so divided that at the same instant buyers believe the market is rising and sellers that it is falling.

Undoubtedly, the Theory of Probability will never be applicable to the movements of quoted prices and the dynamics of the Stock Exchange will never be an exact science.
However, it is possible to study mathematically the static state of the market at a given instant, that is to say, to establish the probability law for the price fluctuations that the market admits at this instant. Indeed, while the market does not foresee fluctuations, it considers which of them are more or less probable, and this probability can be evaluated mathematically.

Up to the present day, no investigation into a formula for such an expression appears to have been published: that will be the object of this work....
....MUCH MORE (47 pages on Google Drive)

Now what Bachelier was about to do wasn't quite at the level of Leibniz or Newton inventing the calculus but it was orders of magnitude beyond the state of the art as practiced by Russell Sage, who amassed one of the greatest Wall Street fortunes of his day, $3 billion or so in 2020 dollars, based on simple put/call parity to evade New York state usury laws.

Here's the introduction to a 2012 post where I decided to go with Bachelier rather than Sage:  
The Guy Who Discovered Black-Scholes Before Mssrs. Black and Scholes
Okay, not all of B-S but enough that if anyone had been paying attention they could have made money off the young man's work.

I was going to do a post on the King of 19th century put and call brokers but you may find this more interesting.....
But back to Professor Bernstein's letter, he notes that the B-man's thesis advisor was Henri Poincaré, which is a pretty good start but additionally links to

On the centenary of Théorie de la Spéculation 
which begins:
Centenary of mathematical finance
The date March 29, 1900 should be considered as the birthdate of mathematical fi nance ....
Alrighty then, a pretty big deal. If interested see also:
"Pricing the Future: Finance, Physics, and the 300-Year Journey to the Black-Scholes Equation "

Variables: σ = volatility of returns of the underlying asset/commodity;
S = its spot (current) price; δ = rate of change; V = price of financial derivative;
r = risk-free interest rate; t = time.
Finally a cautionary tale from a very smart guy: 
"Volatility as the new Black-Scholes" (VIX; VXX; CVOL)