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. (Do not round intermediate calculations. Round your answer to 2 decimal places.)
b. A put option written on the same stock at the same time, with the same exercise price and expiration date. (Do not round intermediate calculations. Round your answer to 2 decimal places.)
(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, Call option premium = $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 | |
=$7.82 |
Hence, Put option premium = $7.82
Problem 21-12 Black–Scholes model Use the Black–Scholes formula to value the following options: a. A call...
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