Devlin's Angle

November 1997

A Nobel Formula

The discovery of a single mathematical formula in 1970 led to this year's Nobel Prize for economics, shared between Stanford University professor of finance (emeritus) Myron Scholes and economist Robert C. Merton of Harvard University. The prize would undoubtedly have been shared with a third person, Fischer Black, but for the latter's untimely death in 1995.

Discovered by Scholes and Black, and developed by Merton, the Black-Scholes formula tells investors what value to put on a financial derivative, such as a stock option. By turning what would otherwise be a guessing game into a mathematical science, the Black-Scholes formula made the derivatives market into the hugely lucrative industry it is today.

So revolutionary was the very idea that you could use mathematics to price derivatives that initially Black and Scholes had difficulty publishing their work. When they first tried in 1970, Chicago University's Journal of Political Economy and Harvard's Review of Economics and Statistics both rejected the paper without even bothering to have it refereed. It was only in 1973, after some influential members of the Chicago faculty put pressure on the journal editors, that the Journal of Political Economy published the paper.

Industry was far less shortsighted than the ivory-towered editors at the University of Chicago and Harvard. Within six months of the publication of the Black-Scholes article, Texas Instruments had incorporated the new formula into their latest calculator, announcing the new feature with a half-page ad in The Wall Street Journal.

Modern risk management, including insurance, stock trading, and investment, rests upon the fact that you can use mathematics to predict the future. Not with 100% accuracy, of course. But well enough so that you can make a wise decision as to where to put your money. In essence, when you take out insurance or purchase stock, the real commodity you are dealing with is risk. The underlying ethos in the financial markets is that the more risk you are prepared to take, the greater the potential rewards. Using mathematics can never remove the risk. But it can tell you just how much of a risk you are taking, and help you decide on a fair price.

The idea of using mathematics to predict the future goes back to two seventeenth century French mathematicians, Blaise Pascal and Pierre De Fermat (he of Fermat's Last Theorem fame). In a series of letters exchanged in 1654, the two mathematicians worked out the probabilities of the various outcomes in a game where two dice are thrown a fixed number of times. For example, suppose Mary and Bill are playing a best-of- five series, and after three throws, Mary is ahead two to one. What would be a fair stake for you to wager on Bill winning the series, if I offer to pay out $100 if he wins? Pascal and Fermat showed how to find the correct answer. According to their mathematics, the probability that Bill will go on to win the series is 25%. So, if I allow you to take the bet for exactly $25, then I am making you a totally fair offer. A stake of less than $25 would be more attractive to you than to me; a stake of more than $25 would be unfair to you. Of course, we would both still be gambling. The mathematics does not eliminate the risk. It simply tells you what the fair price is.

What Black and Scholes did was find a way to determine the fair price to charge for a derivative such as a stock option. The idea with stock options is that you purchase an option to buy stock at an agreed price prior to some fixed later date. If the value of the stock rises above the agreed price before the option runs out, you buy the stock at the agreed lower price and thereby make a profit. If you want, you can simply sell the stock immediately and realize your profit. If the stock does not rise above the agreed price, then you don't have to buy it, but you lose the money you paid out to purchase the option in the first place.

What makes stock options attractive is that the purchaser knows in advance what the maximum loss is: the cost of the option. The potential profit is theoretically limitless: if the stock value rises dramatically before the option runs out, you stand to make a killing. Stock options are particularly attractive when they are for stock in a market which sees large, rapid fluctuations, such as the computer and software industries. Most of the many thousands of Silicon Valley millionaires became rich because they elected to take a portion of their initial salary in the form of stock options in their new company.

The question is, how do you decide a fair price to charge for an option on a particular stock? This is precisely the question that Scholes, Black, and Merton investigated back in the late 1960s. Black was a mathematical physicist with a recent doctorate from Harvard, who had left physics and was working for Arthur D. Little, the Boston-based management consulting firm. Scholes had just obtained a Ph.D. in finance from the University of Chicago. Merton had obtained a bachelor of science degree in mathematical engineering at New York's Columbia University, and had found a job as a teaching assistant in economics at MIT.

The three young researchers -- all were still in their twenties -- set about trying to find an answer using mathematics, exactly the way a physicist or an engineer approaches a problem. After all, Pascal and Fermat had shown that you can use mathematics to determine the fair price on a bet on some future event, and gamblers ever since had used mathematics to figure the best odds in card games and roulette. Similarly, actuaries used mathematics to determine the right premium to charge on an insurance policy, which is also a bet on what will or will not happen in the future.

But would a mathematical approach work in the highly volatile, new world of options trading which was just being developed at the time. (The Chicago Board Options Exchange opened in April 1973, just one month before the Black-Scholes paper appeared in print.) Many senior market traders thought such an approach could not possibly work, and that options trading was beyond mathematics. If that were the case, then options trading was an entirely wild gamble, strictly for the foolhardy.

The old guard were wrong. Mathematics could be applied. It was heavy duty mathematics at that, involving an obscure technique known as stochastic differential equations. The formula takes four input variables -- duration of the option, prices, interest rates, and market volatility -- and produces a price that should be charged for the option.

Not only did the new formula work, it transformed the market. When the Chicago Options Exchange first opened in 1973, less than 1,000 options were traded on the first day. By 1995, over a million options were changing hands each day.

So great was the role played by the Black-Scholes formula (and extensions due to Merton) in the growth of the new options market that, when the American stock market crashed in 1978, the influential business magazine Forbes put the blame squarely onto that one formula. Scholes himself has said that it was not so much the formula that was to blame, rather that market traders had not grown sufficiently sophisticated in how to use it.

The award of a Nobel Prize to Scholes and Merton shows that the entire world now recognizes the significant effect on our lives that has been wrought by the discovery of that one mathematical formula.

- Keith Devlin


Devlin's Angle is updated at the beginning of each month.


Keith Devlin ( devlin@stmarys-ca.edu) is the editor of FOCUS, the news magazine of the MAA, and the author of Mathematics: The Science of Patterns, a Scientific American Library paperback published by W. H. Freeman in 1996.