Call Option and Price

Question 1: Consider an option on dividend-paying stock when stock price $30, the exercise price is $29, the risk-free interest rate is 5% p.a., the volatility is 25%p.a. and time to maturity is 4 months. Assume that the stock is due to go ex-dividend in 1.5 months. The expected dividend is 50cents.
a. b. c. what is the price of the option if it is a European call? What is the price of the option if it is a European put? Use the results in the Appendix to this chapter to determine whether there are any circumstances under which the option is exercised early.
(a) The present value of the dividend must be subtracted from the stock price. This gives a new stock price of: and 30−0.5e−0.125x0.05 = 29.5031 1n(29.5031/29)+(0.05+0.252 /2)x0.3333 d1 = 0.25 0.3333 = 0.3068 d2 =1n(29.5031/29)+(0.05−0.252 /2)x0.3333=0.1625
N(d1) = 0.6205; N(d2 ) = 0.5645 The price of the option is therefore
29.5031x0.6205 − 29e−0.05×4 /12 x0.5645 = 2.21 or $2.21. (b) Because
N (−d1 ) = 0.3795, N (−d2 ) = 0.4355 the value of the option when it is a European put is
0.25 0.3333
29e−0.05×(4 /12) x0.4355 − 29.5031x0.3795 = 1.22 (c) If t1 denotes the time when the dividend is paid:
X (1 − e− r (T −t1 ) ) = 29(1 − e−0.05 x 0.2083 ) = 0.3005
This is less than the dividend. Hence the option should be exercised immediately before the ex-dividend date for a sufficiently high value of the stock price. or $1.22.
Question 2: A foreign currency is currently worth $1.50. The domestic and foreign risk-free interest rates are 5% and 9% respectively. Calculate a lower bound for the value of a six-month call option on the currency with a strike price of $1.40 if it is (a) European and (b) American.
Lower bound for European option is
S0e−rf T − Xe−rT =1.5e−0.09x0.05 −1.4e−0.05x0.5 = 0.069 Lower bound for American option is
S0 − X = 0.10
Question 3: Show that if C is the price of an American call with exercise price X and maturity T on a stock paying a dividend yield of q, and P is the price of