Options represent the right (but not the obligation) to buy or sell stock or some other asset at a given price on or before a certain date. Another major class of derivatives, called forwards and futures, obligates the buyer to purchase an asset at a set price and time. Swaps, yet another type of derivative, allow companies to exchange cash flows—floating-interest- rate for fixed-rate payments, for instance. Financial engineering uses these building blocks to create custom instruments that might provide a retiree with a guaranteed minimum return on an investment or allow a utility to fill its future power demands through contractual arrangements instead of constructing a new plant.
Creating complicated financial instruments requires accurate pricing methods for the derivatives that make up their constituent parts. It is relatively easy to establish the price of a futures contract. When the cost of wheat rises, the price of the futures contract on the commodity increases by the same relative amount. Thus, the relationship is linear. For options, there is no such simple link between the derivative and the underlying asset. For this reason, the work of Scholes, Merton and their deceased colleague Fischer Black has assumed an importance that prompted one economist to describe their endeavors as “the most successful theory not only in finance but in all of economics.”
Einstein and Options
The proper valuation of options had perplexed economists for most of this century. Beginning in 1900 with his groundbreaking essay “The Theory of Speculation,” Louis Bachelier described a means to price options. Remarkably, one component of the formula that he conceived for this purpose anticipated a model that Albert Einstein later used in his theory of Brownian motion, the random movement of particles through fluids. Bachelier’s formula, however, contained financially unrealistic assumptions, such as the existence of negative values for stock prices.
Other academic thinkers, including Nobelist Paul Samuelson, tried to attack the problem. They foundered in the difficult endeavor of calculating a risk premium: a discount from the option price to compensate for the investor’s aversion to risk and the uncertain movement of the stock in the market.
The insight shared by Black, Scholes and Merton was that an estimate of a risk premium was not needed, because it is contained in the quoted stock price, a critical input in the option formula. The market causes the price of a riskier stock to trade further below its expected future value than a more staid equity, and that difference serves as a discount for inherent riskiness.
Black and Scholes, with Merton’s help, came up with their option-pricing formula by constructing a hypothetical portfolio in which a change of price in a stock was canceled by an offsetting change in the value of options on the stock—a strategy called hedging. Here is a simplified example: A put option would give the owner the right to sell a share of a stock in three months if the stock price is at or below $100. The value of the option might increase by 50 cents when the stock goes down $1 (because the condition under which the option can be used has grown more likely) and decrease by 50 cents when the stock goes up by $1.
To hedge against risks in changes in share price, the investor can buy two options for every share he or she owns; the profit then will counter the loss. Hedging creates a risk-free portfolio, one whose return is the same as that of a treasury bill. As the share price changes over time, the investor must alter the composition of the portfolio—the ratio of the number of shares of stocks to the number of options—to ensure that the holdings remain without risk.