Question

A price on a non-dividend paying stock is currently £50. Over each of the next two six-month periods the stock is expected to go up by 5% or down by 10%. The risk- free interest rate is 3% per annum with continuous compounding.
(a) What is the value of a one-year European call option with a strike price of £48? [10 marks]

(b) What is the value of a one-year American call option with a strike price of £48? [4 marks]
(c) Discuss how your answer to (b) would change if the stock instead actually did pay cash dividend [6 marks]

Answer 1

Ans a.) We have to compute the one year European call option.Which is calculated as

Step 1 : Compute the option values on expiry date . If it goes up by 5 % or go down by 10 %.

Hence 50* 0.05 + 50 = 52.5 when up

And 50 - 5 = 45 . when down

Hence if the stock finishes at 45 it will lapse value =0. If the stock finishes at 52.5 then the call has value = 52.5-48= 4.5 (as 48 is yhe strike price )

Then we have to compute the present value of lower price which = 45 *e-^(0.03*0.5) = 44.3300.

Step 2 we need to calculate the number of calls to be brought , Calls to be brought = spread in stock price / Spread in call option value

= 52.5-45/4.5 -0 = 7.5/4.5 = 1.66.

Step 3 : We apply the formula

Current market price = Pv of lower price +call bought *value of call . Putting in the formula below.

50 = 44.330+1.66 * value of call

50 -44.330 = 1.66* value of call

5.67/ 1.66 = 3.41 = value of call.

Ans b ) The American call option can be exercised aby time before the expiry date.In the absence of dividend the value of call option increases with time to maturity hence it would not be fruitfull to excercise an American call early.Hence the value of European call is equal to American call as in the above 3.41 is value of call.

Ans c) When the stock do pay dividend the only time to exercise the call option is before the ex-dividend date , as after the dividend declared the price of the stock falls , hence the value of American option is different from European option.