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. Your net outlay to establish the entire portfolio is $50. Risk-free interest rate is 4%

-What must the strike price 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

According to the put-call parity on options

C + X*e^(-rT) = P + S

X is the strike price of the options

S is the current stock price = X

C and P are the call and put option price with the same strike price and same maturity

We have, S=X, r=0.04, T=1

C+X*e^(-0.04*1) = P +X

0.9608X + C = P+X

C-P = 0.03921X -------------------(equation 1)

Now, our portfolio = S-C+P = 50

=> X+P-C = 50 -----------------------(equation 2)

from the above equation 1 and equation 2, we get

X-0.03921X =50

X = 52.0405

Hence, the strike price must be = $52.0405

One-year stock futures price = S(0)*e^(rT)

Here S(0)=X

r=0.04 and T =1

One-year stock futures price = 52.0405*e^(0.04*1) =$54.1643

Since, both options have the same strike price, the options are at-the-money and we have a risk-free rate of interest greater than 0, hence the call option will always be more expensive than the put option in this case.

The logic behind this is that since the stock has more probability to go up since the risk-free interest rate is greater than 0.

a one-year put option with same strike X is traded at $3

Using equation 2, and the value of X, we get

52.0405+P-C = 50

P = 3

C = 52.0405-50+3

The call option price C =$5.0405