## Paul Samuelson and the Obscure Origins of the Financial Crisis

Some time in the early 1950’s the late Paul Samuelson received a post card from L.J. “Jimmie” Savage, a noted mathematical statistician. It was one of several he had sent out at about the same time. Savage’s post card to Samuelson, and probably the others, said that it was essential that Samuelson read *Théorie de la Spéculation*, the Ph.D. thesis of the French mathematician Louis Jean Baptiste Alphonse Bachelier. Samuelson had never heard of Bachelier so he did not know that the thesis had first been published in 1900. Reading the thesis changed the course of Samuelson’s work. He improved Bachelier’s mathematics and used it to study the prices of warrants—options to buy, at a future date, stock issued by a company. These methods were passed on to his students. But for some of them, Bachelier’s ideas provided inspiration for a theory of financial engineering—the use of complex mathematical models to make risky investments that, taken to extremes (which Samuelson himself never did), nearly caused the collapse of our financial system in the fall of 2008.

Bachelier was born in Le Havre on the March 11, 1870. His father was a wine merchant and his mother was the daughter of a banker. Bachelier was surely headed for one of those careers in France that start with attending one of the *grandes écoles*—until both his parents died in 1889. Bachelier became the head of the family business but soon left to complete his compulsory military service. It was not until 1892 that he could begin his studies at the Sorbonne. One of the lecturers was Henri Poincaré, one of the greatest mathematicians of his time who later became Bachelier’s thesis advisor. Bachelier was not an outstanding student; he had had no practice taking examinations. He worked at the Paris Bourse, the stock exchange, during this time, which presumably gave him the idea for his thesis.

Among the commodities sold on the bourse were various government bonds called *rentes*. But one could also buy options on these *rentes*—a form of “derivative.” You bought the right to purchase such a bond at a later time at the price for which it sold at the time you bought the option. Until then you did not own the bond but just the option to buy it at a later time. If the price had gone up you would be in the money and if it went down you lost whatever it had cost you to buy the option. Unless you were a clairvoyant you would have to guess at the future price of the bond.

What Bachelier wanted to do was to replace clairvoyance by mathematics. To do this he needed to make some assumption of how stock prices evolve. He decided that at any given time it was as likely for a stock to go up as down. You might at first think that this means that a stock price would never get anywhere. But after a first up, say, the stock has a fifty-fifty chance of going up as down and thus moving further away from its starting point. In short the price of the stock, Bachelier decided, takes a “random walk.”

In his thesis Bachelier presents mathematics of this, along with examples. One of the things he shows is that the price evolves away from its initial price as the square root of the time elapsed (days, say)—not an obvious result. It is like the random walk of a drunk whose distance from his starting point increases with the square root of the time elapsed except here the object taking the random walk is the stock price.

What he did not know was that he had solved an outstanding physics problem. Early in the nineteenth century a Scottish botanist named Robert Brown had observed that microscopic particles suspended in liquids had a jiggling movement which we now call Brownian motion. It was Einstein in one of his great 1905 papers who presented the theory of this movement as a random walk induced by the bombardment of the suspended particles by the molecules of the liquid. The equations Einstein arrived at are identical to the ones in Bachelier’s thesis, which he had never heard of.

Only Poincaré, his thesis advisor, understood what Bachelier had done and was not put off by his humble educational background, although he graded the thesis as *honorable* rather than *trés honorable*. The only job that Bachelier could get was as an unpaid lecturer at the Sorbonne. He may have gone back to work at the bourse. He served in the French army from 1914 until 1918 and then got a few minor university positions. Poincaré died in 1912 so he could not help him. It was not until 1927 that Bachelier got a permanent position at the University of Besançon, where he remained until his retirement ten years later. He died on April 28, 1946. By this time the mathematical community understood the value of what he had done and his work was widely recognized by mathematicians. It was Samuelson who introduced it to economists.

Bachelier’s thesis had a profound influence on Samuelson’s work. The idea of using “stochastic” methods—of which the random walk is an example—to analyze things in economics like the movement of the stock market was novel.

Not only did this become part of his own work but he transmitted it to the brilliant young students and associates who had come to MIT to work with him. Among them were Robert Merton and Myron Scholes. Merton on his own and Scholes in collaboration with Fischer Black, who was working in Cambridge, extended Bachelier’s work on options to include “hedging.” You borrow just enough money to make your options investment less risky. Providing the foundations of modern financial engineering were the Black-Scholes and Merton equations, used by many hedge funds and investment houses to make the highly leveraged bets that precipitated the financial crisis.

Samuelson had a different background from these young people. He had been born in 1915 and attended the University of Chicago at the height of the Great Depression, the experience of which never left him. He became a convinced Keynesian and I think had a more detached view of the efficacy of mathematical models than his students. He understood that they were models and that things could go wrong. Keynes once noted that the market could remain irrational longer than you can remain solvent. I never had a chance to meet Samuelson but in a way I almost felt that I had. In 1948 as a Harvard sophomore I decided to take the beginning course in economics. I have no idea why. Our text was Samuelson’s *Economics: An Introductory Analysis*, which had just been published. It is a marvelous book which in its many editions and forty translations has sold over four million copies. In 1970 he became the first American to win the Nobel Prize for economics. A number of his students and associates, such as Merton and Scholes, later won theirs. (Black had died by the time Scholes and Merton won, in 1997—a year before the spectacular collapse of the hedge fund Long Term Capital Management, on whose board both men sat.) From time to time there were pictures of Samuelson in his eighties playing tennis and even in his nineties his interviews were always lucid and relevant. He had a full and long life. He died on December 13, 2009 at the age of ninety-four.