Question

Problem 21-12 Black–Scholes model

Use the Black–Scholes formula to value the following options:

a. A call option written on a stock selling for $68 per share with a $68 exercise price. The stock's standard deviation is 6% per month. The option matures in three months. The risk-free interest rate is 1.75% per month.

b. A put option written on the same stock at the same time, with the same exercise price and expiration date.

Answer 1

(a)

S = Current Stock Price =	68
t = time until option expiration(years) = 3/12 =	0.25
X = Option Strike Price =	68
r = risk free rate(annual) = 1.75*12/100 =	0.21
s = standard deviation(annual) = 6*12/100 =	0.72
N = cumulative standard normal distribution
d1	= {ln (S/K) + (r +s^2/2)t}/s√t
	= {ln (68/68) + (0.21 + 0.72^2/2)*0.25}/0.72*√0.25
	= 0.3258
d2	= d1 - s√t
	= 0.3258 - 0.72√0.25
	= -0.0342
Using z tables,
N(d1) =	0.6277
N(d2) =	0.4864
C = Call Premium =	=SN(d1) - N(d2)Ke^(-rt)
	= 68*0.6277 - 0.4864*68e^(-0.21*0.25)
	= 11.30

Hence, Value of call option = $11.30

(b)

S = Current Stock Price =	68
t = time until option expiration(years) = 3/12 =	0.2500
X = Option Strike Price =	68
r = risk free rate(annual) = 1.75*12/100 =	0.21
s = standard deviation(annual) = 6*12/100 =	0.72
N = cumulative standard normal distribution
d1	= {ln (S/K) + (r +s^2/2)t}/s√t
	= {ln (68/68) + (0.21 + 0.72^2/2)*0.25}/0.72*√0.25
	= 0.3258
d2	= d1 - s√t
	= 0.3258 - 0.72√0.25
	= -0.0342
Using z tables,
N(-d1) =	0.3723
N(-d2) =	0.5136
P = Put Premium =	=N(-d2)Ke^(-rt) - SN(-d1)
	= 0.5136*68e^(-0.21*0.25) - 68*0.3723
	=7.8221

Hence, Value of put option = $7.82