Question

Use the following option prices for options on a stock index that pays no dividends to answer questions. The options have three months to expiration, and the index value is currently 1,000.

STRIKE (K)	CALL PRICE	PUT PRICE
975	77.716	43.015
1000	64.595	X
1025	53.115	67.916

a. Using put-call parity, what is the implied continuously compounded interest rate?

Using put-call parity, what is the correct price for the put option with a strike of 
1,000? (i.e., what is X?) 


Answer 1

Put-call Parity formula :   

C = Call price premium , r = interest rate (cc) , P = Put price premium

X = Strike Price , T = time to maturity (in years) , S = Current Stock Price

As current stock price is same, C₁ + X₁ e^-rT - P₁ = C₂ + X₂ e^-rT - P₂{ Substitute the 1st & 3rd Values }

( C_{1 -} C₂ ) + ( X₁ e^-rT - X₂ e^-rT ) = P₁ - P₂

24.601 + ( 50 )  e^{-r * 0.25} = 24.901

By solving the above equation, we get, r = 4% continuously compounding {Answer}

Using the above r , e^-rT = 0.99005

Put Price, X = C + K e^-rT - S

X = 64.595 + 1000 * 0.99005 - 1000 = $ 54.645 { X : Answer }