Question

You buy a share of stock, write a one-year call option with strike X, and buy a one-year put option with same strike X. You know that stock shares are currently traded at the same price as the strike price of the options, i.e. options are at the money and S=X. Your net outlay to establish the entire portfolio is $30. Risk-free interest rate is 3%

-What must the strike price X be? The stock pays no dividends.

- Find the one-year stock futures price.

- Which of the above two same strike options should be more expensive

- Given that one-year put option with same strike X is traded at $3. Find the value of call option with the same strike and maturity.

Answer 1

Put Call Parity

Put-call parity states that a fiduciary call should be equal to protective put. Fiduciary call means buying a call and investing in the PV of the strike price which ensures we have cash to buy the asset. Protective put is a position where we buy a put and we also buy a stock so that it limits the downside of a stock.

1) i.e Call + PV (Strike Price) = Put + Stock

i.e Call - Put = Stock - PV (Strike Price)

i.e 30 = X - X / (1+ 3%)

i.e X = 1030

Hence Strike Price and Stock Price = 1030

2) Future Price = Stock Price (1+ R)^N

Future Price = 1030 (1+3%)^1

Future Price = 1060.90

3) Which of the above option should be more expensive?

As we have solved in part 1) , we assumed, call option to be more expensive than put option such that we got a strike price of 1030. However if we would have assumed put to be expensive than call, then the strike price would have been -1030. Negative strike price / stock price cannot be the case. Hence in the given case, call options are more expensive than put.

4) Call - Put = 30

Hence Call - 3 = 30

Hence value of call = 33