Question

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.

Answer 1

(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