Lawrence D. Cavanagh

Are Puts Cheaper Than Calls?

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In our upcoming example, we price one-year calls and puts using a standard Black-Scholes model with the stock at $100, the interest rate at 6% and the dividend rate at 1%.

At the $100 strike price, the call premium is higher than the put premium. At the $105 strike price (equal to the future delivery price), the call and put premiums are exactly equal.

You will also notice that when you use the $105 future delivery price as your underlying, rather than the $100 current stock price, all the calculated time premiums of puts and calls for each strike price are equal.

An Arbitrage Opportunity

"Fine," you might say "that's the theory, but is it the way options really trade?"

The answer is yes. The reason is that when call or put time premiums get out of line with each other, option market makers can make a risk-free arbitrage profit.

Let us look at the simplest arbitrage. Suppose that the time premium of a particular put is greater than the time premium of the corresponding call (same stock, strike and expiration).

Here the market maker can take advantage of the difference in the time premiums by writing the put and buying the call. What the market maker has done is to create a "synthetic" long stock position with a free credit of premium. The market maker can fully offset his risk by selling the stock. The net credit of time premium from this transaction will be his arbitrage profit.

From the above, you get the first put-call parity rule: If you buy a call and sell a put at the same strike price and expiration, you get the equivalent of a long stock position. Or …

Rule 1: (+) Buy Call and (-) Write Put = (+) Buy Stock

From the aforementioned, it follows that if you write a call and buy a put at the same strike price, you get the equivalent of a short stock position.

Rule 2: (-) Write Call and (+) Buy Put = (-) Short Stock

The other put-call parity rules come from the fact that option time premiums are essentially the same for calls and puts at the same strike price.

Rule 3: (+) Buy Stock & (+) Buy Put = (+) Buy Call

Rule 6: (+) Buy Stock and (-) Write Call = (+) Write Put = Covered Call

Notice that if you move any of these transactions to the other side of the equal sign, you change the strategy from a short to a long (or vice versa) and (remembering your algebra) the "+" sign to a "-" sign (or vice versa).