Science and the Stock Market

On March 27, 2012, in economic theory, by Jon Lewis

George G. Szpiro, Pricing the Future: Finance, Physics, and the 300-Year Journey to the Black-Scholes Equation. Basic Books, 2011, pp. 298, $28.00 In the wake of the financial crisis of 2008, economists and pundits alike have questioned what went wrong. Some put the blame squarely on Washington, singling out both the Federal Reserve’s near-zero interest [...]

George G. Szpiro, Pricing the Future: Finance, Physics, and the 300-Year Journey to the Black-Scholes Equation. Basic Books, 2011, pp. 298, $28.00

In the wake of the financial crisis of 2008, economists and pundits alike have questioned what went wrong. Some put the blame squarely on Washington, singling out both the Federal Reserve’s near-zero interest rate policy and a deeply flawed federal housing policy. Others look toward Wall Street, with many rethinking if a largely unregulated and non-transparent derivatives market has been serving the public interest. While there is certainly some validity to questioning whether complex financial instruments should have been better regulated, it must not be forgotten that the same advances in economics and mathematics that allowed investment banks to create risk-transferring derivatives also have allowed us to better understand how financial markets work. When it comes to economics and finance, math matters.

In Pricing the Future, a recent work that does not deal specifically with the recent financial crisis, the author takes the reader on a fascinating intellectual journey in exploring the origins, and development of, the Black-Scholes options-pricing equation. Written by George Szpiro, Ph.D., “a mathematician-turned-journalist” and the Israel correspondent for the Neue Zürcher Zeitung, a Swiss newspaper, Pricing the Future is an engaging study in the history of ideas. More importantly, perhaps, it is a study of the personalities behind the mathematical and scientific discoveries that created this financial model.

The problem solved mathematically by the Black-Scholes equation, eponymously named after Fischer Black and Myron Scholes, is how to price options, defined succinctly by the author as “contracts that give the right, but do not entail the obligation, to buy or sell something, usually a good or a security, at a certain date or a certain price.” Options can be used in commodities themselves, such as when a farmer decides to purchase an option to buy a certain amount of grain for a certain price not in the present, but in the future. They are used regularly in the world of finance to hedge risk, such as when an investor buys the right, but not the obligation, to purchase stock in a given publicly-traded corporation at a certain price in the future. In other words, one can pay a premium today for the right, but not the obligation, to buy or to sell a security in the future. The Black-Scholes equation, which won the two men a Nobel Prize (note that, because the Nobel Committee does not give awards posthumously, Robert Merton did not receive the prize and, unfortunately, his name is not on the equation), gives a “mathematically precise value” to options.

One particularly compelling aspect of the Black-Scholes options pricing formula was the discovery “that the value of options does not depend upon the investors’ attitude toward risk.” This, in many ways, is counterintuitive. One would think that an investor’s wariness about the potential risk of a security should factor into how much said investor should be willing to pay for the option either to purchase or to sell a security at a given point in the future. As Black, Scholes, and Merton discovered, however, “the volatility of the underlying stock plays a crucial role and that the investors’ attitude toward risk plays none.” The value of an option, it turns, out, depends upon five variables. They are as follows: “the price of the underlying stock, the option’s exercise price, the time to maturity, the risk-free interest rate, and the variability of the stock’s price movements.”

The Black-Scholes equation, of course, did not materialize out of thin air. Numerous people and their discoveries and theories, at times both flawed and groundbreaking, paved the way for the options-price formula. In Pricing the Future, Szpiro discusses how the lives and works of such diverse personalities as Jules Regnault, Louis Bachelier, Robert Brown, and Kiyoshi Itō, among other notable scholars, all laid the intellectual foundations for what was to become the Black-Scholes equation. The reader learns, for instance, that Regnault, in nineteenth-century France, “was the first person to try to understand the workings of the stock exchange in mathematical terms.” Although his work was flawed, he did presage the intriguing fact that “stock prices do move in proportion to the square root of time.”

The most important character in the narrative, however, may be the English doctor and botanist, Robert Brown, after whom Brownian motion is named. In 1827, Brown conducted a scientific experiment in which he suspended pollen in water and observed it under a microscope. What he saw was nothing short of amazing. The particles were continuously moving, engaging in a “‘rapid oscillatory motion’.“ This rapid, continuous movement became known as Brownian motion. Albert Einstein, as well as Marian Smoluchowski would discover that “when particles hit the particle from random directions, the probability of a certain displacement of the particle would be governed by the normal distribution.” This connection between randomness and the normal distribution was important. Furthermore, the Japanese mathematician, Kiyoshi Itō would, through his work on stochastic calculus, allow for the understanding of Geometric Brownian motion, which would aid economists in understanding how the stock market, which cannot fall in value below zero, works. Probability theory, statistics, and mathematical economics, including the work of Paul Samuelson, plays a significant role in this work of intellectual history.

Szpiro discusses how Merton and Scholes, through their work at the Long-Term Capital Management (LTCM), sought to apply their models to the financial sector. LTCM, of course, held billions in assets, relied highly on borrowed money, and eventually had to be bailed out, albeit by private banks. The reader also learns what is perhaps one of the more fascinating aspects of how the stock market works. “The really interesting things in the stock market, like booms and busts, are not described the figure’s central parts, but by its tails. And this is where the normal distribution leads us astray.”

In other words, be wary of unexpected and largely unpredictable events. They matter more than one might initially expect them to. “The conclusion,” writes the author, “is that the normal distribution is not a good description of stock price movements. More weight must be given to extreme events; the distribution of stock price movements obviously has fatter tails than the normal distribution.” That said, Szpiro does not believe in making Black, Scholes, and Merton out to be villains. In fact, he equates blaming them and their work for “aberrations” such as the LTCM crash, the financial crisis, and Lehman Brothers “would be like accusing Newton and the laws of motion for fatal traffic accidents.” This is something worth consideration, especially since it is not likely that academic economics and finance will become divorced from mathematics anytime soon.

Because Szipro’s work is filled with both personal anecdotes and detailed scientific theory, it is difficult to encapsulate fully the richness of this recent contribution to the growing corpus of literature on the interaction between math, statistics, and finance. For those readers interested in learning about randomness, mathematical modeling, and how those models, when applied by real people in the real world, do not always work exactly as envisioned, Pricing the Future is an excellent choice.

Jon Lewis (c) 2012

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