"A lively account of physicists in finance...An enjoyable debut appropriate for both specialists and general readers."

"Anyone interested in how markets work will appreciate this serious hypothesis."

"Fascinating history...Happily, the author has a gift for making complex concepts clear to lay readers."

James Owen Weatherall is a physicist, philosopher, and mathematician, currently working as assistant professor of logic and philosophy of science at the University of California, Irvine, where he is also a member of the Institute for Mathematical Behavioral Science. He lives in Irvine, CA with his wife and two daughters.

The Physics Of Wall Street

A Brief History Of Predicting The Unpredictable

Houghton Mifflin Harcourt Publishing Company

Contents

Primordial Seeds

La fin de siècle, la belle epoque. Paris was abuzz with progress. In the west, Gustave Eiffel's new tower — still considered a controversial eyesore by Parisians living in its shadow — shot up over the site of the 1889 World's Fair. In the north, at the foot of Montmartre, a new cabaret called the Moulin Rouge had just opened to such fanfare that the Prince of Wales came over from Britain to see the show. Closer to the center of town, word had begun to spread of certain unexplained accidents at the magnificent and still-new home of the city's opera, the Palais Garnier — accidents that would lead to at least one death when part of a chandelier fell. Rumor had it that a phantom haunted the building.

Just a few blocks east from the Palais Garnier lay the beating heart of the French empire: the Paris Bourse, the capital's principal financial exchange. It was housed in a palace built by Napoleon as a temple to money, the Palais Brongniart. Its outside steps were flanked by statues of its idols: Justice, Commerce, Agriculture, Industry. Majestic neoclassical columns guarded its doors. Inside, its cavernous main hall was large enough to fit hundreds of brokers and staff members. For an hour each day they met beneath ornately carved reliefs and a massive skylight to trade the permanent government bonds, called rentes, that had funded France's global ambitions for a century. Imperial and imposing, it was the center of the city at the center of the world.

Or so it would have seemed to Louis Bachelier as he approached it for the first time, in 1892. He was in his early twenties, an orphan from the provinces. He had just arrived in Paris, fresh from his mandatory military service, to resume his education at the University of Paris. He was determined to be a mathematician or a physicist, whatever the odds — and yet, he had a sister and a baby brother to support back home. He had recently sold the family business, which had provided sufficient money for the moment, but it wouldn't last forever. And so, while his classmates threw themselves into their studies, Bachelier would have to work. Fortunately, with a head for numbers and some hard-won business experience, he had been able to secure a position at the Bourse. He assured himself it was only temporary. Finance would have his days, but his nights were saved for physics. Nervously, Bachelier forced himself to walk up the stairs toward the columns of the Bourse.

Inside, it was total bedlam. The Bourse was based on an open outcry system for executing trades: traders and brokers would meet in the main hall of the Palais Brongniart and communicate information about orders to buy or sell by yelling or, when that failed, by using hand signals. The halls were filled with men running back and forth executing trades, transferring contracts and bills, bidding on and selling stocks and rentes. Bachelier knew the rudiments of the French financial system, but little more. The Bourse did not seem like the right place for a quiet boy, a mathematician with a scholar's temperament. But there was no turning back. It's just a game, he told himself. Bachelier had always been fascinated by probability theory, the mathematics of chance (and, by extension, gambling). If he could just imagine the French financial markets as a glorified casino, a game whose rules he was about to learn, it might not seem so scary.

He repeated the mantra — just an elaborate game of chance — as he pushed forward into the throng.

"Who is this guy?" Paul Samuelson asked himself, for the second time in as many minutes. He was sitting in his office, in the economics department at MIT. The year was 1955, or thereabouts. Laid out in front of him was a half-century-old PhD dissertation, written by a Frenchman whom Samuelson was quite sure he had never heard of. Bachelor, Bacheler. Something like that. He looked at the front of the document again. Louis Bachelier. It didn't ring any bells.

Its author's anonymity notwithstanding, the document open on Samuelson's desk was astounding. Here, fifty-five years previously, Bachelier had laid out the mathematics of financial markets. Samuelson's first thought was that his own work on the subject over the past several years — the work that was supposed to form one of his students' dissertation — had lost its claim to originality. But it was more striking even than that. By 1900, this Bachelier character had apparently worked out much of the mathematics that Samuelson and his students were only now adapting for use in economics — mathematics that Samuelson thought had been developed far more recently, by mathematicians whose names Samuelson knew by heart because they were tied to the concepts they had supposedly invented. Weiner processes. Kolmogorov's equations. Doob's martingales. Samuelson thought this was cutting-edge stuff, twenty years old at the most. But there it all was, in Bachelier's thesis. How come Samuelson had never heard of him?

Samuelson's interest in Bachelier had begun a few days before, when he received a postcard from his friend Leonard "Jimmie" Savage, then a professor of statistics at the University of Chicago. Savage had just finished writing a textbook on probability and statistics and had developed an interest in the history of probability theory along the way. He had been poking around the university library for early-twentieth- century work on probability when he came across a textbook from 1914 that he had never seen before. When he flipped through it, Savage realized that, in addition to some pioneering work on probability, the book had a few chapters dedicated to what the author called "speculation" — literally, probability theory as applied to market speculation. Savage guessed (correctly) that if he had never come across this work before, his friends in economics departments likely hadn't either, and so he sent out a series of postcards asking if anyone knew of Bachelier.

Samuelson had never heard the name. But he was interested in mathematical finance — a field he believed he was in the process of inventing — and so he was curious to see what this Frenchman had done. MIT's mathematics library, despite its enormous holdings, did not have a copy of the obscure 1914 textbook. But Samuelson did find something else by Bachelier that piqued his interest: Bachelier's dissertation, published under the title A Theory of Speculation. He checked it out of the library and brought it back to his office.

Bachelier was not, of course, the first person to take a mathematical interest in games of chance. That distinction goes to the Italian Renaissance man Gerolamo Cardano. Born in Milan around the turn of the sixteenth century, Cardano was the most accomplished physician of his day, with popes and kings clamoring for his medical advice. He authored hundreds of essays on topics ranging from medicine to mathematics to mysticism. But his real passion was gambling. He gambled constantly, on dice, cards, and chess — indeed, in his autobiography he admitted to passing years in which he gambled every day. Gambling during the Middle Ages and the Renaissance was built around a rough notion of odds and payoffs, similar to how modern horseraces are constructed. If you were a bookie offering someone a bet, you might advertise odds in the form of a pair of numbers, such as "10 to 1" or "3 to 2," which would reflect how unlikely the thing you were betting on was. (Odds of 10 to 1 would mean that if you bet 1 dollar, or pound, or guilder, and you won, you would receive 10 dollars, pounds, or guilders in winnings, plus your original bet; if you lost, you would lose the dollar, etc.) But these numbers were based largely on a bookie's gut feeling about how the bet would turn out. Cardano believed there was a more rigorous way to understand betting, at least for some simple games. In the spirit of his times, he wanted to bring modern mathematics to bear on his favorite subject.

In 1526, while still in his twenties, Cardano wrote a book that outlined the first attempts at a systematic theory of probability. He focused on games involving dice. His basic insight was that, if one assumed a die was just as likely to land with one face showing as another, one could work out the precise likelihoods of all sorts of combinations occurring, essentially by counting. So, for instance, there are six possible outcomes of rolling a standard die; there is precisely one way in which to yield the number 5. So the mathematical odds of yielding a 5 are 1 in 6 (corresponding to betting odds of 5 to 1). But what about yielding a sum of 10 if you roll two dice? There are 6 × 6 = 36 possible outcomes, of which 3 correspond to a sum of 10. So the odds of yielding a sum of 10 are 3 in 36 (corresponding to betting odds of 33 to 3). The calculations seem elementary now, and even in the sixteenth century the results would have been unsurprising — anyone who spent enough time gambling developed an intuitive sense for the odds in dice games — but Cardano was the first person to give a mathematical account of why the odds were what everyone already knew them to be.

Cardano never published his book — after all, why give your best gambling tips away? — but the manuscript was found among his papers when he died and ultimately was published over a century after it was written, in 1663. By that time, others had made independent advances toward a full-fledged theory of probability. The most notable of these came at the behest of another gambler, a French writer who went by the name of Chevalier de Méré (an affectation, as he was not a nobleman). De Méré was interested in a number of questions, the most pressing of which concerned his strategy in a dice game he liked to play. The game involved throwing dice several times in a row. The player would bet on how the rolls would come out. For instance, you might bet that if you rolled a single die four times, you would get a 6 at least one of those times. The received wisdom had it that this was an even bet, that the game came down to pure luck. But de Méré had an instinct that if you bet that a 6 would get rolled, and you made this bet every time you played the game, over time you would tend to win slightly more often than you lost. This was the basis for de Méré's gambling strategy, and it had made him a considerable amount of money. However, de Méré also had a second strategy that he thought should be just as good, but for some reason had only given him grief. This second strategy was to always bet that a double 6 would get rolled at least once, if you rolled two dice twenty-four times. But this strategy didn't seem to work, and de Méré wanted to know why.

As a writer, de Méré was a regular at the Paris salons, fashionable meetings of the French intelligentsia that fell somewhere between cocktail parties and academic conferences. The salons drew educated Parisians of all stripes, including mathematicians. And so, de Méré began to ask the mathematicians he met socially about his problem. No one had an answer, or much interest in looking for one, until de Méré tried his problem out on Blaise Pascal. Pascal had been a child prodigy, working out most of classical geometry on his own by drawing pictures as a child. By his late teens he was a regular at the most important salon, run by a Jesuit priest named Marin Mersenne, and it was here that de Méré and Pascal met. Pascal didn't know the answer, but he was intrigued. In particular, he agreed with de Méré's appraisal that the problem should have a mathematical solution.

Pascal began to work on de Méré's problem. He enlisted the help of another mathematician, Pierre de Fermat. Fermat was a lawyer and polymath, fluent in a half-dozen languages and one of the most capable mathematicians of his day. Fermat lived about four hundred miles south of Paris, in Toulouse, and so Pascal didn't know him directly, but he had heard of him through his connections at Mersenne's salon. Over the course of the year 1654, in a long series of letters, Pascal and Fermat worked out a solution to de Méré's problem. Along the way, they established the foundations of the modern theory of probability.

One of the things that Pascal and Fermat's correspondence produced was a way of precisely calculating the odds of winning dice bets of the sort that gave de Méré trouble. (Cardano's system also accounted for this kind of dice game, but no one knew about it when de Méré became interested in these questions.) They were able to show that de Méré's first strategy was good because the chance that you would roll a 6 if you rolled a die four times was slightly better than 50% — more like 51.7747%. De Méré's second strategy, though, wasn't so great because the chance that you would roll a pair of 6s if you rolled two dice twenty- four times was only about 49.14%, less than 50%. This meant that the second strategy was slightly less likely to win than to lose, whereas de Méré's first strategy was slightly more likely to win. De Méré was thrilled to incorporate the insights of the two great mathematicians, and from then on he stuck with his first strategy.

The interpretation of Pascal and Fermat's argument was obvious, at least from de Méré's perspective. But what do these numbers really mean? Most people have a good intuitive idea of what it means for an event to have a given probability, but there's actually a deep philosophical question at stake. Suppose I say that the odds of getting heads when I flip a coin are 50%. Roughly, this means that if I flip a coin over and over again, I will get heads about half the time. But it doesn't mean I am guaranteed to get heads exactly half the time. If I flip a coin 100 times, I might get heads 51 times, or 75 times, or all 100 times. Any number of heads is possible. So why should de Méré have paid any attention to Pascal and Fermat's calculations? They didn't guarantee that even his first strategy would be successful; de Méré could go the rest of his life betting that a 6 would show up every time someone rolled a die four times in a row and never win again, despite the probability calculation. This might sound outlandish, but nothing in the theory of probability (or physics) rules it out.

So what do probabilities tell us, if they don't guarantee anything about how often something is going to happen? If de Méré had thought to ask this question, he would have had to wait a long time for an answer. Half a century, in fact. The first person who figured out how to think about the relationship between probabilities and the frequency of events was a Swiss mathematician named Jacob Bernoulli, shortly before his death in 1705. What Bernoulli showed was that if the probability of getting heads is 50%, then the probability that the percentage of heads you actually got would differ from 50% by any given amount got smaller and smaller the more times you flipped the coin. You were more likely to get 50% heads if you flipped the coin 100 times than if you flipped it just twice. There's something fishy about this answer, though, since it uses ideas from probability to say what probabilities mean. If this seems confusing, it turns out you can do a little better. Bernoulli didn't realize this (in fact, it wasn't fully worked out until the twentieth century), but it is possible to prove that if the chance of getting heads when you flip a coin is 50%, and you flip a coin an infinite number of times, then it is (essentially) certain that half of the times will be heads. Or, for de Méré's strategy, if he played his dice game an infinite number of times, betting on 6 in every game, he would be essentially guaranteed to win 51.7477% of the games. This result is known as the law of large numbers. It underwrites one of the most important interpretations of probability.

Pascal was never much of a gambler himself, and so it is ironic that one of his principal mathematical contributions was in this arena. More ironic still is that one of the things he's most famous for is a bet that bears his name. At the end of 1654, Pascal had a mystical experience that changed his life. He stopped working on mathematics and devoted himself entirely to Jansenism, a controversial Christian movement prominent in France in the seventeenth century. He began to write extensively on theological matters. Pascal's Wager, as it is now called, first appeared in a note among his religious writings. He argued that you could think of the choice of whether to believe in God as a kind of gamble: either the Christian God exists or he doesn't, and a person's beliefs amount to a bet one way or the other. But before taking any bet, you want to know what the odds are and what happens if you win versus what happens if you lose. As Pascal reasoned, if you bet that God exists and you live your life accordingly, and you're right, you spend eternity in paradise. If you're wrong, you just die and nothing happens. So, too, if you bet against God and you win. But if you bet against God and you lose, you are damned to perdition. When he thought about it this way, Pascal decided the decision was an easy one. The downside of atheism was just too scary.