Question

The 3-month forward price is 50. The put option premium with a strike price 52 is 3 and the put option matures in 3 months. The risk-free interest rate is 4% p.a., compounded quarterly. The stock pays no dividend. What is the price of a call option with a strike of 52 and matures in 3 months?

Answer 1

At first, we begin with calculating the Nominal risk-free interest rate for the given sum. As you can see, the rate is given as an effective interest rate, i.e., 4% p.a. compounded quarterly.

Nominal risk-free interest rate = {[1+ (4%/4)]^4} - 1

Nominal risk-free interest rate = 4.06%

The put-call parity for the relation between the call option, put option, spot price and bond (Strike price discounted at Risk free rate) goes as follows:

Call + Ke^(-rt) = Put + Spot at time 0

The Spot price in the above equation can also be replaced by:

Spot = Forward price/ [(1+r)^t]

Thus, the Put Call parity can also be stated as:

Call + K/[(1+r)^t] = Put + Forward price/ [(1+r)^t]

K (Strike Price) = 52

r (Risk-free rate) = 4.06%/12 = 0.338%

t (time period) = 3/12

Substituting the Values, we get:

Call + 52/[(1+0.00338)^(3/12)] = 3 + 50/ [(1+0.00338)^(3/12)]

Call + 51.956 = 3 + 49.958

Call = 1.002

Thus, the Price of a call option, with a strike of 52 that matures in 3 months is 1.002.