The Mathematics of Speculation

5/4/2019 - Monocle Journal

In 1955, Paul Samuelson – then a Professor of Economics at MIT – received a strange postcard from his colleague, Leonard Jimmie Savage – a mathematician and statistician at the University of Chicago. Savage wanted to know whether Samuelson had ever heard of a Frenchman called Louis Jean-Baptiste Alphonse Bachelier. Although well-known among mathematicians for his pioneering work in probability theory and stochastic analysis, Bachelier was largely unheard of in economic circles. By chance, Savage had discovered a book in the Chicago library written by Bachelier and published in 1914 entitled Jeu, la Chance, et le Hasard (Games, Chance, and Randomness), and – having found something that he suspected might be of great importance in economic circles – he had sent out postcards to several of his colleagues from other universities asking if anyone had heard of Bachelier. Intrigued by the postcard, Samuelson went searching for other works by the mysterious Frenchman in the MIT library, and it was there that he located Bachelier’s PhD thesis from 1900, entitled Théorie de la Spéculation, or The Theory of Speculation. In it was contained the first recorded attempt in history to apply advanced mathematics to finance, and so successful was the endeavour that Bachelier appeared to have developed a viable solution to a problem that had been plaguing the financial world for decades – the problem of pricing options.


Little is known about Bachelier, who was born in 1870 in Le Havre, in north-west France, to a wine merchant and the daughter of a prominent banker. Although he had reportedly been a keen scholar, his parents died shortly after he completed high-school and he was forced to take up his father’s business for a short time so that he could look after his siblings. It was there that the young maths prodigy was first exposed to the world of trade and commerce. Undeterred by the detour in his academic career, in 1892, Bachelier travelled to Paris and began studying mathematics at the Sorbonne, where eight years later he successfully defended his PhD thesis to a panel of academics that included Henri Poincare – regarded as one of the most influential mathematicians in the world at the time.

Bachelier’s thesis contained an analysis of options markets, culminating in the development of an options pricing model. Impressed, but not wholly convinced, the marking panel remarked that the subject matter of the study strayed a bit too far out of the realm of pure mathematics to earn Bachelier a perfect grade. Nonetheless, he was awarded a mark of honorable and his thesis was published in the highly-respected mathematics journal Annales Scientifiques de l’École Normale Supérieure. Bachelier’s ideas were well-received in the mathematics community, and the work he had presented in Théorie de la Spéculation set the course for developments in stochastic analysis and probability theory for the next century. However, he remained unknown in the world of finance up until Savage’s postcard brought the long-forgotten PhD to the fore, setting off a wave of research in financial economics that would lead to one of the most important developments in modern finance: the Black-Scholes formula for the pricing of stock options.

The use of options – contracts that give the holder the right, but not the obligation, to either buy or sell an asset at a set price in the future – grew throughout Europe, the UK and the US during the 17th century, although they remained largely unstandardised and unregulated. This meant that the terms of each option contract, including the price of the contract, had to be negotiated between the parties involved in each case. From around the middle of the 19th century, however, attempts at devising a model for option pricing began to emerge, with theorists focusing on the relationship between the price of the option and the price of the underlying asset. This was a relationship made complicated by the inherent difficulty of predicting the future price of the asset involved.

Bachelier based his thesis on an analysis of an exchange-traded futures contract, where the underlying asset was French Government rentes – perpetual coupon paying bonds – listed on the Paris Bourse. At the time, rentes were considered to be relatively secure investments with a nominal value that hovered around 100 francs and fixed returns, typically between 3% and 5%. Bachelier asserted that, at any given moment, there was an infinite amount of new information that influenced the price of the stock and that the movement of the stock prices was therefore random. “The determination of these fluctuations depends on an infinite number of factors,” he notes, “it is, therefore, impossible to aspire to mathematical prediction of it. Contradictory opinions concerning these changes diverge so much that at the same instant buyers believe in a price increase and sellers in a price decrease.” How investors made decisions in a world of such uncertainty, thus, became the subject of the mathematician’s inquiry.

Something very similar to the random movement in stock prices that Bachelier described had already been observed in the world of science in 1827, by British botanist Robert Brown. He had noticed that, when viewed under a microscope, pollen particles suspended in water appeared to move randomly, although there did not appear to be any external force moving them and they had no way of propelling themselves. Brown was unable to explain the movement of the particles – which became known as Brownian motion. However, in 1905, Einstein famously proved mathematically, in Investigations on the theory of Brownian Movement, that the movement of the particles was the result of their collision with tiny molecules that were moving in the water, thus confirming the existence of atoms. Einstein described how the constant, random collision of atoms with the pollen particle from all sides would create the spontaneous movement of the particle that Brown had observed. Although the trajectory of the particle was random, Einstein reasoned that it would be possible to calculate the probability of the particle moving a particular distance and found that – like many other random phenomena in the world – the frequency distribution of the particle’s probabilistic movement was normally distributed. This made it possible to determine statistically the most likely paths that the particle would follow. Five years before Einstein, Bachelier had achieved the same insight with regard to stock prices, leading him to derive a mathematical model for option pricing that was based not on the predictability of returns, but rather on the probability distribution of future stock prices.

In developing his model, Bachelier laid the academic groundwork for several pivotal concepts in both mathematics and finance, including the Markov property, the theory of Martingales, the Chapman- Kolmogorov equation, and the connection between Brownian motion and the heat equation. By the time Samuelson discovered Bachelier’s work in 1955, the mathematics used by the Frenchman had advanced greatly, providing Samuelson with a more refined set of “the tools” – as he described them – that Bachelier had established in his thesis. This led to a wave of intensive development in financial modelling, with Bachelier’s option pricing formula forming the core of the Nobel prize-winning Black-Scholes formula developed in 1973.

Although commonly referred to as the Black-Scholes formula, the most widely used version of the formula is actually the result of a collaboration between three men: Fischer Black, Myron Scholes and Robert Merton. In the late 1960s, Merton was completing his PhD at MIT under the supervision of Samuelson, who was researching techniques for warrant pricing at the time. Together, Merton and Samuelson published A Complete Model of Warrant Pricing that Maximizes Utility in 1969, in which they linked the price of a warrant to the price of the underlying stock, in much the same way that Bachelier had demonstrated a relationship between the rentes and the option contract. After receiving his PhD, Merton joined the MIT Sloan School of Management, working alongside Scholes who had received his PhD from the University of Chicago in 1969, under the guidance of Eugene Fama – known for developing the Efficient Market Theory and the Fama-French model – and Merton Miller – known for the Modigliani-Miller theorem. Through Sloan, the two men were introduced to Black who was working at a consultancy firm, and who had obtained his PhD from Harvard in 1964 and had also published several important papers during his career, including one detailing the development of the Capital Asset Pricing Model.

In 1973, Black and Scholes published The Pricing of Options and Corporate Liabilities, describing a partial differential equation known as the Black-Scholes equation, used to calculate the value of options and other derivatives. Using stochastic calculus, Merton proposed certain important modifications to the formula that same year, in his Theory of Rational Option Pricing. The formula makes use of five inputs to determine the option price: the current price of the stock; the period over which the option can be exercised; the interest rate in the market; the strike price; and volatility (determined using the past year’s data). Resting on Bachelier’s assertion that stock price movements are probabilistically normally distributed, the model essentially implies that very large movements in stock prices are unlikely. And empirical tests performed by the three economists proved it to be remarkably accurate for pricing options – at least when the markets behaved rationally.

The timing of the introduction of the Black-Scholes- Merton model in 1973 was uncanny, seeming to bring scientific order to a volatile financial environment in the wake of the failure of the Bretton-Woods system, the OPEC oil crisis, and a prolonged recession in the US. In addition, that same year, the Chicago Board of Trade became the first exchange to list standardised, exchange-traded stock options when it opened the Chicago Board Options Exchange (CBOE), which became the official exchange for options trading in the US. With investors emboldened by the mathematical support that the Black- Scholes-Merton model provided for important decision-making, options trading quickly swelled into a market worth billions of dollars. When the CBOE opened, it was restricted to call options for just 16 stocks, but by 1975 it had introduced computerised trading and by 1977 it had grown to include put options. In time, the CBOE further expanded to accommodate a wide range of financial products, with annual exchange volumes reaching 100 million option contracts in 1984 and over one billion option contracts by 2008.

However, the Black-Scholes-Merton model itself was limited in its capabilities. For one thing, market participants have pointed out that whereas the assumption of a normal distribution in the delta value, or in the value of stock price movements, is useful in a rational market, empirical data demonstrates that there have been exceptionally large movements, especially negative movements, that are probabilistically extremely unlikely. Secondly, the model did not have the capability to embed the value of, and the effect of, dividends issued by the underlying stocks, and to consider how these would affect the option price. The model’s use was also limited to European options, where the right to exercise the option is limited to a single point in time, as opposed to American options, which can be exercised at any time up to a specified point in time.

But the true underlying danger of systematic overreliance on financial models would not be realised until 25 years later. In 1998, Long Term Capital Management – a hedge fund led by former Salomon Brothers trader John Meriwether, together with Scholes and Merton – lost extraordinarily large sums of money as a result of the inability of their models to accurately predict unusual and extreme market movements. The assumption generally embedded in financial probability models is that extreme movements in asset prices are exceptionally unlikely – and, thus, so is the likelihood of an extreme market event occurring that would cause such movements. And yet, the history of banking is littered with these types of events.

To illustrate just how frequently statistically-unlikely stock price movements actually occur, Sam Stovall – chief investment strategist at the Centre for Financial Research and Analysis – conducted an historical analysis of the S&P 500 index in 2017. He estimates that since World War II, there have been 56 instances where the S&P 500 index has experienced declines of between 5% and 9.9% over a period of about one month – a move of between five and ten standard deviations. To put this in perspective, a movement of five standard deviations on the S&P 500 should only occur once every 6 900 years. A movement of ten standard deviations should only occur once in 2.6 x 1020 years. And yet, on average, significant declines have occurred almost annually. During the period under investigation, Stovall also notes that the S&P 500 has had 21 corrections – that is, the index has shown a decline of between 10.0% and 19.9%, typically over a period of five months – which is a movement of between 10 and 20 standard deviations. These movements are statistically highly unlikely to occur, and yet on average there has been a correction every 2.8 years. There have also been twelve bear markets – a decline of between 20% and 39.9% usually over more than a year, indicating a move of 20 standard deviations or more – on the S&P 500. That is an average of one bear market every 4.8 years, when statistically these events should almost never occur.

Myron Scholes correctly stated in The New York Times in 1986 that “all models have faults – that doesn’t mean you can’t use them as tools for making decisions.” However, a problem arises when those who use these tools refuse to acknowledge the threat of model risk – the risk that arises when scientific theories are presented as undeniable facts and deemed the only reliable way of interpreting how the world operates. Though theories may be useful devices for gaining insight into complex phenomena, they are always limited in their ability to fully encapsulate the true nature of what they are trying to measure or explain. An over-reliance on mathematical models is therefore problematic because it limits peoples’ understandings of the world, rather than enhancing it – leading, in the case of economics, to financial crises. Fischer Black himself was well-aware of the limits of financial mathematics, claiming in a 1986 article for The Journal of Finance: “In the end, a theory is accepted not because it is confirmed by conventional empirical tests, but because researchers persuade one another that the theory is correct and relevant.” And over the years, the failure to remember this has led many individuals and institutions alike to over-estimate the accuracy of financial mathematics. As a result, there was a time when financial models were applied almost universally, despite the fact that they were essentially based on a set of simple assumptions made by Bachelier, many years ago, on a very limited market of French bonds.


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