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The Math Behind the Meltdown

The history of the renowned Black-Scholes formula on options pricing weaves through several centuries and many countries. That history, were it better known, would have inspired a little of the humility that was in such short supply in the world of finance before September 2008.

Published on December 12, 2012
Pricing the Future: Finance, Physics, and the 300-Year Journey to the Black-Scholes Equation
by George G. Szpiro
Basic Books, 2011, 320 pp., $28

The financial product known as an “option” matters. Many people never hear of one on an average day, but they are everywhere. Ask a modern participant in or scholar of finance about options, and he will readily tell you that the world he lives in is inconceivable without them. Ask anyone investigating what went wrong with the U.S. economy in autumn 2008, and you’ll probably hear that the buying and selling of options contributed to the meltdown. But what is an option, and, assuming that it was not among the original flora or fauna in the Garden of Eden, where did it come from?

An option is a financial contract in which a buyer purchases the right, but not the obligation, to buy or sell something at an agreed upon time in the future at an agreed upon price. Instead of buying or selling an equity share, foreign currency or commodity outright at its present market price, you buy the option of buying or selling it in the future at a certain stipulated price. As esoteric a financial concept as this may seem, options contracts have existed for centuries. How and why they arose, and how options trading evolved into a mathematical science in the form of the Black-Scholes equation, is the subject of an illuminating and entertaining new book by George Szpiro.

Szpiro, an Israeli Ph.D. in finance and mathematical economics, has worked in business (McKinsey & Co.), academia (the Wharton School, Hebrew University and the University of Zurich) and now journalism (Neue Zürcher Zeitung). He aspires in Pricing the Future to tell the history of the Black-Scholes equation, focusing on the history of markets and mathematical discovery, along with the biographical elements of the story, rather than on its recent adventures within the world economy. Those looking to further fix the global financial system will find little to help them here, but those in search of the essential background to the problem will be rewarded.

Szpiro begins by introducing the concept of options for those who are not familiar with it. Imagine, he advises his readers, a farmer who needs to buy fertilizer every year for his fields. Then imagine that the price of fertilizer is volatile, exposing the farmer to fluctuating profit margins based on fertilizer costs. If the fluctuations in fertilizer prices are higher than the farmer can tolerate, he could for a relatively modest transactional fee buy an option for fertilizer at a set price for six months in the future. When six months has elapsed, if the open-market price is below that of the option, the farmer can simply let the option expire and buy on the open market. If the open-market price of fertilizer is higher than the price contracted in the option, the farmer can exercise the option and buy below that price. Through the use of an option contract, the farmer has thus established a maximum price he will pay for fertilizer. An option is therefore a kind of insurance policy against input price increases, helping the farmer lower his risk.

For examples in modern business news, a reader might think of Southwest Airlines and its jet fuel strategy. Southwest has upended the air travel industry in recent decades, helping to guarantee its profitability by famously “hedging” (insuring) against rising fuel prices in a more aggressive and skillful way than other airlines. Southwest used a combination of simple and complex options and other contracts to lock in lower fuel prices. On November 28, 2007, the New York Times reported that Southwest’s hedging gains were $455 million in 2004, $892 million in 2005, $675 million in 2006 and $439 million for the first nine months of 2007. This compares to Southwest’s net income (as filed with the Security and Exchange Commission in 2008) of $215 million in 2004, $484 million in 2005, $499 million in 2006, and $645 million in 2007. If Southwest had not hedged skillfully, it would have been exposed to volatile returns and potentially high losses. Southwest might have gone belly up despite its well-focused business strategy, simplified fleet and more productive workforce.

Other examples of options in the modern economy can be found in international trade. Imagine a U.S.-based investor who wants to buy shares of a company listed on the stock exchange of Thailand, but he is afraid of currency risk—namely, the fluctuation of exchange rates that can affect the terms of investment. The investor can buy currency insurance in the form of a foreign exchange option.

The use of options is not always so simple or benign, however. We know all too well about large financial institutions that have made investments in or proffered loans to a company and, at the same time, bought options in the opposite direction. This is standard procedure and is not necessarily problematic; in this case a financial institution is simply controlling its downside risk in case the investment or loan takes a turn for the worse. If it becomes clear that the financial institution won’t need to exercise its options, it can trade them to someone else who might want them. An option contract has a market price just like any other financial product, and in general, these practices lower lending costs and allow more precise control of risk. Things start to flow into ethical gray areas, however, when one branch of an investment firm advises a client while another branch buys options that bring in money when the client does poorly. Is information truly separate? Can conflicts of interest be managed?

It is against that reality that the scale of options trading comes as such a shock. In 2011, the CIA estimated world GDP at approximately $79 trillion. According to the Bank for International Settlements, in that same year over $1 quadrillion worth of derivatives were traded.1 The sum of $1 quadrillion sound like a big number, and it is: It’s almost 13 times as much as the value of total world output of final goods and services. The notional amount outstanding of over-the-counter derivatives (non-standard contracts that are not openly traded on exchanges) in 2011 was more than $645 trillion, up from just below $400 trillion in the first half of 2006, amounting to roughly eight times the value of world output. Finally, the gross market value (cost of replacing existing contracts) was around $27 trillion, roughly a third of world output. Over 24 billion derivative contracts were traded on exchanges in 2011 alone.

What accounts for the very rapid popularity of such financial products, and, some agree, the concomitant risks they pose to global economic order if they are mismanaged? Very simply, confidence. Options proliferate in proportion to the confidence the buyer has in the value of the product, which is by definition something that can only be ratified by the future. The Black-Scholes equation has enabled buyers of options to accurately predict the future with vastly more confidence than had previously been the case.

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efore the 1970s, even though options contracts had existed for centuries, there was no scientific, agreed-upon way to price them. How much should the farmer pay for that fertilizer option? How much should the investor fork out for protection against currency fluctuations? It took hundreds of years of progress in mathematics, physics, statistics and other disciplines for Fischer Black, Myron Scholes and Robert C. Merton to create their formula, which, to put it simply, projected the price of an option by feeding five variables into a partial differential equation: the strike price of the option, the current price of the underlying asset, the risk-free rate, the time to maturity and the volatility of returns of the underlying asset. As it happened, their discovery coincided with the rapid rise of computing power, which enabled traders to apply the formula quickly and inexpensively. The market for options exploded as a result, as contracts could be more confidently priced and then traded as commodities themselves.

Obviously, Black, Scholes and Merton stood on the “shoulders of giants”, as the great sociologist Robert K. Merton, Robert C.’s father, put it in a famous 1985 book. Szpiro takes us back to the origins of modern financial exchanges, and their respective disasters: the Dutch Tulip craze and the Dutch East India Company, the first publicly traded company; and the Paris Bourse, the first French stock exchange, created to boost the disastrous finances of the French monarchy. Both histories richly describe the speculative trading on margin, efforts to get around bans on short-selling, runs, counterparty failures, and market crashes that are now so uncomfortably familiar to us.

But Szpiro soon gets to the meat of the book in his investigation of the lives and research of three French pioneers of financial analysis. Jules Regnault, born in 1834, was the first to attempt a mathematical understanding of bourse activity, and the first to make foundational assertions about efficient markets and the potential for abuse of information. Henri Lefèvre, a secretary to the banker James de Rothschild, created the first known graphical representations of option payouts. And Louis Bachelier, a professor ignored and spurned, laid the foundations for much of modern options analysis. Bachelier was the first to apply Gauss’s bell curve and heat diffusion equations from physics to stock price movements, the first to assert that an option’s value would depend on the volatility of a stock, and one of the first to attempt mathematical and statistical explanations of economic theory.

Szpiro then leaves the world of early financial analysis to describe important innovations elsewhere in science and mathematics that eventually became foundational for the options pricing formula. He shows how the British army doctor and botanist Robert Brown applied the Gaussian error distribution to the irregular, random movement of particles suspended in fluid—Brownian motion. He also dives into the work of Albert Einsten, Marian Smoluchowski, Paul Langevin, Theodor Svedberg, and the politics of Nobel Prize selection. All eventually influenced the emergence of Black-Scholes. He spends a bit more time on Andrei Kolmogorov, who, in a life of prolific mathematical discovery, wrote the axioms that eventually defined the foundations of probability theory, and on the Japanese mathematician Kiyoshi It?, who made breakthroughs in stochastic differential equations. We also learn of the tragic life of Wolfgang Doblin, who died an early death as a soldier fighting the Nazis in World War II. Doblin independently created prototypes of It?’s work that he had hidden for decades in a pli cacheté (a sealed envelope) in the Academie des Sciences in Paris to prevent his work from falling into the hands of the Nazis.

Having surveyed the relevant scientific and mathematical inputs that eventually came together to produce Black-Scholes, Szpiro returns to financial theory in the modern era and specifically to the discovery of the options pricing model. He describes the career of Paul A. Samuelson, the modern origin of mathematical rigor in economics, the politics of Harvard University economics department appointments, and the eventual revival of Bachelier’s work. Then we are introduced to the academic, personal and career histories of Fischer Black, Myron Scholes and Robert C. Merton themselves. We follow them from various departments at Harvard and MIT to the private sector and other destinations beyond. We witness serendipity in science: Their discovery coincided with the opening of the Chicago Board Options Exchange and the explosion of options markets.

The Black-Scholes formula quickly became the industry standard, so much so that it was directly programmed into Texas Instruments calculators for traders. Fame, new professorships, and work on Wall Street quickly followed. Fischer Black died of cancer in 1995 at the age of 57, two years before the Nobel Prize was awarded to the creators of the options pricing formula. Thus, Scholes and Merton, both still very much alive, won the award. (The formula is most frequently called Black-Scholes because these two originally teamed up to work and publish on the subject together, while Merton’s work basically confirmed and expanded upon theirs.)

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ricing the Future is essentially a math and finance history book—or, put a bit more generously, a treatise on the history and sociology of science. It is as much a demonstration of how scientific progress is made as it is specifically about the Black-Scholes equation. In the process, Dr. Szpiro shows a fondness for sometimes superfluous anecdote, and he is clearly enraptured by the mystical beauties of mathematics.

Yet Szpiro, journalist that he is nowadays, doesn’t ignore the present. He discusses the failure of Long-Term Capital Management, the hedge fund that Scholes and Merton joined that eventually required a federally coordinated private sector bailout. Nor is he mute about the pitfalls of the model, particularly when those using it generate large market distortions of scale the theory was never meant to account for. For the model to function as intended, Szpiro points out, the equation assumes risk-free borrowing and lending at a fixed rate, no transaction costs, constant forward-looking volatility, liquid markets and normal distributions of price movement. This last point in particular has caused trouble in many financial models, as “rare” events happen with greater frequency than the models assume: the 1997 Asian financial crises, the 1998 Russian financial crisis, the popping of the internet/tech bubble and, of course, the current financial mess following from the housing bubble collapse. The model’s assumptions are clearly stated, but most users of the formula are not as mindful of these limitations as they should be. Put a bit differently, the world seems to behave in ways a bit less normal than the risks accounted for in the Black-Scholes model.

Clearly, mistakes happen. Let’s go back to Southwest Airlines for a moment: According to Forbes, hedging “saved Southwest Airlines over $3.5 billion and made up almost 83 percent of the company’s total profits between 1998–2008.”2 This description, by itself, makes hedging seem like a no-brainer, so why don’t all firms do it? Because hedging isn’t free; it’s a game that costs tens of millions of dollars to play at that scale, meaning that only large-sized players can really take advantage of it. And what if you’re wrong? Firms must account for the value of their options portfolio, and what happens when their value crashes? Fuel hedges that lost value caused the majority of Southwest’s $140 million third-quarter loss in 2011.

In addition to the potential for firm-level failure, risk exists at the system level when interconnectedness is not adequately understood. For example, “over-the-counter” (OTC) options, the customized options contracted directly between two parties, open up both sides to counterparty risk. There is no intermediary in such arrangements to help guarantee the settlement of the trade. When a single firm enters into a large number of OTC options contracts, it quickly becomes difficult to understand who is exposed to risk from what firm and what event or class of events. If a firm at the center of many OTC options contracts were to fail and default on its contracts, default could spread when other firms no longer get the money they expected from Peter to pay Paul. This is the “contagion” risk that sparked government intervention to prevent an uncontrolled default by Long-Term Capital Management and, later, AIG. The failure of a large and very active firm at the nexus of complex webs of lightly disclosed options transactions can and most likely still will cause a systemic crisis.

On the whole, greater market efficiencies enabled by a more accurate pricing of risk is a good thing. So is the ability of a firm to insure against risk, especially if this insurance allows a firm to take on more projects and grow. Black-Scholes only turns into a formula from hell when its users either glibly or unknowingly violate the parameters of its assumptions, or when the internal controls at firms fail to contain trading that was intended to reduce risk rather than increase it. When that happens, hedging can produce great losses, as last summer’s events at J.P. Morgan have illustrated.

With the enormous volume of derivatives being traded and the gargantuan value they represent, the stability of the financial system depends on participants understanding the details and pricing of the options contracts in which they engage. These contracts, the risk they mitigate and the risks they create, must be properly monitored by all participants, including but not restricted to regulators without and corporate risk managers within. The stability of the financial system depends on a proper understanding and monitoring of interconnectedness and the systemic risk raised by large players in these markets. This understanding is aided by a better grasp of the history behind it all, including the history of the Black-Scholes formula. Thanks to George Szpiro, we now have that history.

 

1A derivative designates a broader category of financial instruments in which the value of the payout of the contract is derived from another source, such as forwards, futures, options, credit default and other kinds of swaps, and combinations thereof. For our purposes, derivatives and options fall into a similar category of financial contracts that have exploded in recent decades, enabled by the Black-Scholes equation and other factors.

2“Southwest Airlines Flies to $14 Unless Hedging Losses Eat Profits”, Forbes, June 30, 2011.

 

Chris Taylor is a graduate of Northwestern University and the Harvard Business School.