An investor currently holds a long European call with a strike of $47 and a short put with a strike of $41. Both the options are on the same stock and they expire in three weeks. Ignoring the premium, which of the following is (are) true?

I. The investor's portfolio is worthless if the expiration stock price is between $41 and $47.
II. The investor gains if the stock price falls below $41.
III. The investor gains if the stock price rises above $47.

Explanation: This problem is best solved by drawing a payoff diagram.

If the expiration stock price is below $47, the stock price is lower than the strike price of the call, rendering the option worthless. If the expiration stock price is above $41, the stock price exceeds the strike price of the put, which then expires out-of-the money. Therefore, I is true.

Now, since the investor is long the call, he gains when the expiration stock price exceeds the strike price of the call.

On the other hand, since he is short the put option, he loses when the put ends up in-the-money at expiration. This happens when the expiration stock price is below the strike price of the put.