The Black-Scholes formula, in fact, is elicited from a partial differential equation demonstrating that the fair price for an option is the one that would bring a risk-free return within such a hedging portfolio. Variations on the hedging strategy outlined by Black, Scholes and Merton have proved invaluable to financial- center banks and a range of other institutions that can use them to protect portfolios against market vagaries—ensuring against a steep decline in stocks, for instance.
The basic options-pricing methodology can also be extended to create other instruments, some of which bear bizarre names like “cliquets” or “shouts.” These colorful financial creatures provide the flexibility to shape the payoffs from the option to a customer’s particular risk profile, placing a floor, ceiling or averaging function on interest or exchange rates, for example.
With the right option, investors can bet or hedge on any kind of uncertainty, from the volatility (up-and-down movement) of the market to the odds of catastrophic weather. An exporter can buy a “look-back” currency option to receive the most favorable dollar-yen exchange rate during a six-month period, rather than being exposed to a sudden change in rates on the date of the contract’s expiration.
In the early 1970s Black and Scholes’s original paper had difficulty finding a publisher. When it did reach the Journal of Political Economy in 1973, its impact on the financial markets was immediate. Within months, their formula was being programmed into calculators. Wall Street loved it, because a trader could solve the equation easily just by punching in a few variables, including stock price, interest rate on treasury bills and the option’s expiration date. The only variable that was not readily obtainable was that for “market volatility”— the standard deviation of stock prices from their mean values. This number, however, could be estimated from the ups and downs of past prices. Similarly, if the current option price was known in the markets, a trader could enter that number into a workstation and “back out” a number for volatility, which can be used to judge whether an option is overpriced or underpriced relative to the current price of the stock in the market.
Investors who buy options are basically purchasing volatility—either to speculate on or to protect against market turbulence. The more ups and downs in the market, the more the option is worth. An investor who speculates with a call—an option to buy a stock—can lose only the cost of purchase, called a premium, if the stock fails to reach the price at which the buyer can exercise the right to purchase it. In contrast, if the stock shoots above the exercise price, the potential for profit is unlimited. Similarly, the investor who hedges with options also anticipates rough times ahead and so may buy protection against a drop in the market.
Physicists on Wall Street
Although it can be reduced to operations on a pocket calculator, the mathematics behind the Black-Scholes equation is stochastic calculus, a descendant from the work of Bachelier and Einstein. These equations were by no means the standard fare in most business administration programs. Enter the Wall Street rocket scientists: the former physicists, mathematicians, computer scientists and econometricians who now play an important role at the Wall Street financial behemoths.
Moving from synchrotrons to trading rooms does not always result in such a seamless transition. “Whenever you hire a physicist, you’re always hoping that he or she doesn’t think of markets as if they were governed by immutable physical laws,” notes Charles Smithson, a managing director at CIBC World Markets, an investment bank. “Uranium 238 always decays to uranium 234. But a physicist must remember that markets can go up as well as go down.”