Question

QUESTION 1 (5 points) You own a European call option and an American Call option, each on one share of Smart `R' Us, and each with an exercise price of $80. The current share price is $120 and it is an instant before Smart `R' Us pays dividends by an amount of $10. An instant after the ex-dividend date, the share price would fall to $110, and the two options would have one period until expiration. By expiration date the end of the period the share price can either increase to $130 or fall to $90. The riskless interest rate over this period (which starts an instant after the ex-dividend date and ends on the expiration day) is 13%.

a) Find the prices, intrinsic values and time values of the European and American call options, an instant AFTER the ex-dividend date (that is, when the stock price is $110). (3 points)

b) Should the American call be exercised an instant before the ex-dividend date? Explain. (2 points)

Answer 1

Here, Option paramters are

u = 130/110 = 1.1818

d =90/110 = 0.8182

So, p = (1.13-0.8182)/(1.1818-0.8182) = 0.8575

a) Value of both European and American call option after ex-dividend date will be same (as there are no more dividends, the price of American and European call are same)

Now, if Stock price reaches $130 at maturity , value of both options = $130-$80 =$50

& if Stock price reaches $90 at maturity , value of both options = $90-$80 =$10

So, the value of stock today = present value of the expected payoffs at maturity

= (50*0.8575+10*0.1425)/1.13 = $39.203

So, price of both options = $39.20

Intrinsic value of both options= $110 -$80 = $30

Time value of both options = $39.20 -$30 = $9.20

b) If the American call option is exercised an instant before the ex-dividend date

value of the option = $120-$80 =$40

As this value is more than the value of the option an instant after the ex-dividend date, it is optimal to exericise the option. This is because on exercising the option just before the ex-dividend date, one also gets the benefit of the dividend .