Federal Reserve Bank of San Francisco

FRBSF Economic Letter

1998-05 | February 13, 1998

More Economic Letters

- What is an option?
- A simple example of option pricing
- Applications of option pricing theory
- The danger of derivatives
- References

The 1997 Nobel Prize in economics was awarded to Robert C. Merton and Myron S. Scholes. Merton and Scholes and the late Fischer Black are widely credited with developing the tools necessary to price options. This achievement not only has opened new doors for academic research, but also has been widely embraced by practitioners in the financial industry.

The seminal papers by Black and Scholes and Merton were both published in 1973. This year also marked the beginning of organized trading in options contracts on the Chicago Board Options Exchange (CBOE). The first options were written on liquid stocks such as Kodak and IBM. By the late 1970s, it was not unusual for the daily volume of shares traded via options contracts on the CBOE to exceed the volume of these shares traded on the New York Stock Exchange. As trading on options markets increased, new options contracts were developed. Today, the majority of options are written on stock indices, interest rates, and currencies.

The proliferation of options can be explained by the fact that many investors and firms hold assets that expose them to risks that they do not want to bear. The work of Merton, Black, and Scholes revealed how the payoffs of assets could be decomposed in a way such that different sources of risk could be isolated and sold to those most willing to bear them. In this *Letter* I will outline the basic problem that Merton, Black, and Scholes solved and discuss the far-ranging implications of their option pricing formula.

A call option on a stock is a contract that gives the holder (the long position) the right, but not the obligation, to buy the stock at a specified price called the exercise price. The person who sells the option is said to have a short position in the call option. Since the holder of the call option need not exercise the option if the stock price is lower than the exercise price, we see that a call option can never have negative value. This limit in the downside risk of the option implies that an increase in the risk of the underlying stock translates into an increase in the option’s upside potential. Thus, options are more valuable when the risk in the underlying asset is great.

A put option gives the holder the right to sell the stock at a specified price. A European option specifies a date on which the holder can exercise the option. An American option allows the holder to exercise the option at any time prior to a terminal date. Investors may want to buy or sell options in order to hedge or to speculate on future price changes.

**A simple example of option pricing**

The fundamental technique for pricing any asset is to identify the future payoffs of the asset and discount these payoffs to the present time. If future payoffs are uncertain, however, most investors will discount these payoffs to reflect risk. The chief insight of Merton, Black, and Scholes was to point out that if some combination of shares of the asset and an option can produce a risk-free payoff, then the return on such a portfolio must be equal to the risk-free rate. This result will hold regardless of investors’ tastes for risk. This pricing technique is often referred to as risk neutral pricing or pricing by arbitrage.

A simple example illustrates the elegance of their solution. Suppose that a stock currently trades at $20. One period hence the stock price will be either $25 or $15. We will refer to these two possibilities as the “up” state and the “down” state respectively. Also assume that there exists a risk-free investment opportunity with interest rate equal to 5% per period. The objective is to price a put option that allows the holder to sell the stock next period for $18.

Figure 1 shows the payoffs to holding either the stock or the put option in isolation. It is simple to show that if an investor were to buy 3 shares of the stock and buy ten put options, then the payoff on the portfolio is always equal to $75 (see Figure 2). If the stock price increases, the investor earns $75 (3 shares x $25/share) from the stock portion of his portfolio. In this case, the put options expire worthless. If the stock price decreases, then the investor earns only $45 on his shares. However, the investor can exercise the put options to earn $30 [10 options x ($18-$15)].

The payoff on the portfolio is risk-free. Therefore, the return on the portfolio must be equal to the 5% (the risk-free rate). By discounting $75 by the risk-free interest rate we see that the value of the portfolio in the beginning period is $71.43. Since we already know that each share of stock sells for $20, it must be the case that the price of a single put option is $1.14 [($71.43 – $60) / 10 options = $1.14/option].

Black and Scholes price options in a very general setting where investors are able to trade not just once but continuously over a period of time. Merton generalizes the framework by allowing the interest rate to change over the lifetime of the option. Naturally, the pricing formula is more complicated in these more general cases. However, the economic reasoning behind the pricing solution is identical to the argument presented in the example above.

**Applications of option pricing theory**

To a large degree, the success of the Black-Scholes-Merton pricing formula can be attributed to two features. First, the formula is very easy to use. The inputs for the formula are either directly observable or relatively easy to estimate. Option prices can be computed with a pocket calculator or a computer, making the formula practical for real time trading.

Second, the theory is wonderfully flexible. Researchers and practitioners quickly realized that many securities and economic decisions possess option-like features. For example, if a firm’s capital structure consists solely of debt and equity, the equity claim is valuable only if the firm is able to pay off the bondholders. We can view corporate stock as a call option on the firm’s assets with exercise price equal to the face value of the corporate debt. This observation helps to clarify why stockholders and bondholders so often differ in opinion about the appropriate way to run the firm. As holders of a call option, stockholders have more to gain than bondholders do if the firm adopts a risky strategy.

A convertible bond gives the bondholder the option to exchange the bond for a predetermined number of shares. If the stock price rises enough, bondholders will wish to make this exchange. If the stock price does not rise by a suitable amount, the bondholders will simply let the conversion option expire. Clearly, convertible bondholders possess a call option on the firm’s stock. The convertible bond, therefore, can be priced as a package of a straight bond and a call option.

For another example, the Federal Deposit Insurance Corporation provides a guarantee of bank deposits. We may view this insurance as a hedge for banks if the value of their loans declines. If the bank insures all of its deposits, then the price of this insurance can be obtained from the price of a put option on the bank’s assets. The exercise price of the put option is equal to the value of the bank’s deposits. In effect, when a bank’s asset value falls below the value of its deposits, the bank fails and puts its assets to the FDIC.

Recently, one of the more active areas of research in corporate finance has been the application of option pricing theory to project analysis. Consider a firm that is contemplating whether or not to build a new factory. The factory may be deemed unprofitable if built today. However, if future demand for the firm’s product improves, it might be advantageous to build the factory later. The firm essentially holds a call option that they will exercise if demand is favorable. The value of the project today, and hence part of the overall value of the firm, can be derived from option pricing theory.

Option pricing techniques have been used to value research and development, to value the ability to abandon a project, to determine the optimal timing of oil drilling projects, and to price land when there is a development option.

Options are referred to as derivatives because they “derive” their value from some underlying asset. There are numerous other kinds of derivative instruments such as futures, swaps, and collateralized mortgage obligations. Derivatives all share the same economic function in that they allow investors to shift exposure to risk.

There is no doubt that the use of options and other derivative instruments is widespread. There is not unanimous agreement, however, about whether this development is a good thing. Derivatives-based hedging is not always foolproof. Several well-publicized derivatives losses have involved investors who believed they were hedged when, in fact, they were not. Some derivatives positions can be extremely sensitive to changes in the underlying asset value. If the price of the underlying asset declines, this decline will be magnified in the derivatives position. Some of the more spectacular financial disasters in recent memory have involved speculation with the aid of derivatives.

These problems related to derivatives must be acknowledged. However, it is not necessary, as some commentators have suggested, to restrict the use of derivatives. These problems are best resolved by reminding buyers to beware and by encouraging users to adopt more thorough controls over their derivatives trading.

John Krainer* Economist*

Black, F., and Scholes, M. 1973. “The Pricing of Options and Corporate Liabilities.” *Journal of Political Economy* 81 (May-June) pp. 637-659.

Merton, R., 1973. “Theory of Rational Option Pricing.” *Bell Journal of Economics and Management Science *4 (Spring) pp. 141-183.

______. 1989. “Options.” *The New Palgrave: Finance.* New York, W. W. Norton & Company.

Opinions expressed in FRBSF Economic Letter do not necessarily reflect the views of the management of the Federal Reserve Bank of San Francisco or of the Board of Governors of the Federal Reserve System. This publication is edited by Sam Zuckerman and Anita Todd. Permission to reprint must be obtained in writing.

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