Question

A stock has a price of 100. It is expected to pay a dividend of $3 per share at year-end. An at-the-money European put option with 1 year maturity sells for $8. If the annual interest rate is 4%, what must be the price of an at-the-money European call option on the stock with 1 year maturity.

Answer 1

risk-free rate , rf = 4% = 0.04

present value of dividend , D = expected dividend/(1+rf) = 3/1.04 = 2.884615385

Let the price of call option = c

Exercise price = K = 100 ( since the option is at the money)

stock price = s = 100

let the put option price = p = 8

according to put call parity

c+ D + K/(1+rf) = p + s

c = p + s - D - K/(1+rf) = 8 + 100 - 2.884615385 - 100/1.04 = 105.1153846 - 100/1.04 = 105.1153846- 96.15384615

c = 8.961538446 or $8.96 ( rounding off to 2 decimal places) or $9 ( rounding off to nearest dollar value)

Hence price of call option ,c =  8.961538446 or $8.96 ( rounding off to 2 decimal places) or $9 ( rounding off to nearest dollar value)