Arbitrage: The Key to Pricing Options

Federal Reserve Bank of Cleveland. Economic Commentary, Jan 1, 2004 by Nosal, Ed, Wang, Tan

Arbitrage is the act of simultaneously buying and selling assets or commodities in an attempt to exploit a profitable opportunity. Although the idea behind arbitrage is fairly simple, it is quite powerful because the ability to exploit such opportunities is needed for markets to operate efficiently. Arbitrage ensures, for example, that buyers and sellers of foreign exchange can be assured that they are getting the "correct" rates for the currencies they are buying and selling independent of the national foreign-exchange markets they happen to be using.

When markets are efficient, the prices of the objects being traded reflect their true value. And having prices reflect true values is important in decentralized economies, such as the United States, since it is the relative prices of various goods, services, and assets that determines how many will be produced, how they will be allocated, and how funds will be invested. If prices did not reflect true value, then the resulting allocation of goods, services, and investment would not be, in general, economically efficient.

This Commentary focuses on a particular episode in which the recognition of an arbitrage "opportunity" made financial markets more efficient. It wasn't a chance to make a profit that got noticed, it was the way the principles of arbitrage could be applied to the problem of correctly pricing options. Once financial economists figured it out, the solution enhanced the efficiency of financial markets because it made options useful as hedging instruments. Such instruments can be used to manage cash holdings only if they are correctly priced.

* In Search of the Option Pricing Formula

Options are phenomenally popular these days, but when they were introduced, they didn't take off at first. No one knew how to price them. People were using methods based on familiar financial instruments, like equities and bonds, and these did not work well for pricing options.

An option is called a derivative security because it derives both its payoff structure and its value from some (other) underlying security. A call option on a stock, for example, gives the holder of the call option the right, but not obligation, to purchase a particular stock at some date in the future. The price at which the holder can purchase the stock is called the strike price and the latest possible date at which the holder can purchase the stock-or exercise the option-is called the expiration date. If the holder is only allowed to exercise on the expiration date, the option is called a European; if the holder can exercise any time between now and the expiration date, the option is called an American.

The ability to exercise-or not-gives options different characteristics from the underlying securities on which they're based, and that fact makes them difficult to price. Consider a European call option on ABC stock with a strike price equal to $50 and an expiration date in one year. If you own this option, what will be your possible payoffs one year from now? Clearly, if ABC's stock price is below $50 one year from now, you will not exercise because you would be paying $50 for something that is worth less than that; in these circumstances your payoff will be zero. If the stock price is above $50, you will exercise the option, and for every dollar that ABC's slock is above $50, you will receive that amount.

Determining the payoff structure for an option doesn't seem that difficult. How might one go about determining the value or price of this option? A standard way to think about pricing any financial asset is to first specify the payoffs that the asset is expected to generate in the future; then to appropriately discount these payoffs; and finally add up all of the discounted payoffs. The sum of the discounted payoffs represents the value of the asset, and the price of the asset should be equal to this value.

Future payoffs for financial assets are discounted for at least two reasons. First, a dollar in the future is worth less than a dollar today; so, future dollars must be discounted to make them comparable with dollars today. Second, if one asset's payoff stream is riskier than another's, then, holding all other things equal, the former asset is more valuable than the latter. Hence, riskier payoff streams should be discounted more heavily than less risky payoff streams.

It might seem as if calculating the value of the option should be fairly straightforward. We know when the payoffs to the option will be received and what possible values they may take. All we have to do is to discount these expected future payoffs to determine the option's value.

But here's the rub: What discount factor do we use? A discount factor should reflect the underlying risk of the asset. Since the option's value is related to movements in the stock price of the ABC company, it might seem reasonable to discount the option's future expected payoffs with ABC's discount factor. The problem here is although the movement in the option's payoff perfectly tracks the stock price movements when they are above $50, it does not at all track the movement of the stock price when the price is below $50.