# PUT-CALL PARITY

Put-call parity helps to define the relative values of an option to buy a security and the option to sell the same security. For example, suppose Microsoft stock is selling for \$93 per share. Investors will pay something for the right, or option, to buy a share of Microsoft stock for \$95 during the next 90 days. Suppose this amount is \$8. Hence, purchasers of this "call" option expect the price to rise significantly above \$95 by the end of this period. Likewise, other investors may pay \$5 for the option to sell a share of Microsoft for \$95 during the next 90 days. Purchasers of this "put" option expect the price to fall below \$95.

Now consider two portfolios. Portfolio I consists of one put option and one share of Microsoft stock. This portfolio would cost \$98 to create (\$5 + \$93). Portfolio 2 consists of one call option and one bank deposit that will grow to \$95 in 90 days. If interest rates on such bank deposits are 6 percent per year, then an investor would earn about 1.5 percent in 90 days and must deposit \$93.60 today. The total cost of Portfolio 2 is \$101.60 (\$8 + \$93.60).

Since an option will be exercised only if it is beneficial to the holder, there are only two possible outcomes. Either Microsoft sells for \$95 or more at the end of 90 days or it doesn't. Let's examine the payoffs for each portfolio at the end of 90 days. If Microsoft stock is selling for more than \$95, say \$110, then the put is worthless and the call is worth \$15. Portfolio I will pay \$110 (\$0 + \$110) and Portfolio 2 will pay \$110 (\$15 + \$95). The payoffs for the two portfolios will be equal for any given price of Microsoft in excess of \$95. If Microsoft is selling for less than \$95, say \$88, then the put is worth \$7 and the call is worthless. Portfolio I will pay \$95 (\$7 + \$88) and Portfolio 2 will pay \$95 (\$0 + \$95) as well. Again, the payoffs for the two portfolios will be equal for any given price of Microsoft below \$95.

What does this mean? Since the two portfolios always produce the same cash flows at the end of 90 days, they should have the same value today. In this example, however, Portfolio 1 is less expensive than Portfolio 2. This suggests an arbitrage opportunity. If an investor can buy Portfolio 1 and sell Portfolio 2, then an immediate profit of \$3.60 can be achieved. An investor could sell Portfolio 2 by selling a call option and borrowing \$93.60. Since each portfolio produces the same cash flow in 90 days, buying one and selling the other will result in a net cash flow of zero. But all of this pricing information is known today! Hence, many investors will be buying puts and selling calls to capture the arbitrage profit. This will eventually increase the price of the put and decrease the price of the call to a point where the arbitrage profit will vanish and the two portfolios will be priced identically.

The put-call parity relationship is a powerful economic result. It holds true in any reasonably efficient market and keeps investor optimism or pessimism from spiraling out of control.

[ Paul Bolster ]

Brigham, Eugene F., Louis C. Gapenski, and Michael C. Ehrhardt. Financial Management: Theory and Practice. 9th ed. Fort Worth, TX: Dryden Press, 1999.

Hull, John C. Options, Futures, and Other Derivatives. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1997.

Kolb, Robert W. Understanding Options. New York: John Wiley & Sons, Inc., 1995.