Use the Black-Scholes formula to price a call option for a stock whose share price today is $16 when the interest rate is 4%, the maturity date is 6 month, the strike price is $17.5 and the volatility is 20%. Find the price of the same option half way to maturity if the share price at that time is $17.
(a) Time to Maturity = 6 months
S = Current Stock Price = | 16 |
t = time until option expiration(years) = 6/12 = | 0.5000 |
X = Option Strike Price = | 17.5 |
r = risk free rate(annual) = | 0.04 |
s = standard deviation(annual) = | 0.20 |
N = cumulative standard normal distribution | |
d1 | = {ln (S/K) + (r +s^2/2)t}/s√t |
= {ln (16/17.5) + (0.04 + 0.2^2/2)*0.5}/0.2*√0.5 | |
= -0.4215 | |
d2 | = d1 - s√t |
= -0.4215 - 0.2√0.5 | |
= -0.5629 | |
Using z tables, | |
N(d1) = | 0.3367 |
N(d2) = | 0.2868 |
C = Call Premium = | =SN(d1) - N(d2)Ke^(-rt) |
= 16*0.3367 - 0.2868*17.5e^(-0.04*0.5) | |
= 0.4676 = $0.47 |
(b) Time to Maturity = 3 months (half way to maturity)
S = Current Stock Price = | 17 |
t = time until option expiration(years) = 3/12 = | 0.2500 |
X = Option Strike Price = | 17.5 |
r = risk free rate(annual) = | 0.04 |
s = standard deviation(annual) = | 0.20 |
N = cumulative standard normal distribution | |
d1 | = {ln (S/K) + (r +s^2/2)t}/s√t |
= {ln (17/17.5) + (0.04 + 0.2^2/2)*0.25}/0.2*√0.25 | |
= -0.1399 | |
d2 | = d1 - s√t |
= -0.1399 - 0.2√0.25 | |
= -0.2399 | |
Using z tables, | |
N(d1) = | 0.4444 |
N(d2) = | 0.4052 |
C = Call Premium = | =SN(d1) - N(d2)Ke^(-rt) |
= 17*0.4444 - 0.4052*17.5e^(-0.04*0.25) | |
= 0.5344 = $0.53 |
Use the Black-Scholes formula to price a call option for a stock whose share price today...
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