Question

XYZ Corp. will pay a $2 per share dividend in two months. Its stock price currently is $90 per share. A call option on XYZ has an exercise price of $85 and 3-month time to expiration. The risk-free interest rate is 0.6% per month, and the stock’s volatility (standard deviation) = 24% per month. Find the Black-Scholes value of the option. (Hint: Try defining one “period” as a month, rather than as a year, and think about the net-of-dividend value of each share.)

Answer 1

Dividend paid after 2 months = $2

Risk free rate = r = 0.006

Present Value of the dividend = D/(1+r)ⁿ = 2/(1+0.006)² = 1.98

Net of Dividend Value of share = 90 - 1.98 = 88.02

S = Net Value of share =	88.02
t = time until option expiration(years) = 3/12 =	0.2500
K = Option Strike Price =	85
r = risk free rate(annual) = 0.6*12/100 =	0.072
s = standard deviation(annual) = 24% =	0.24
N = cumulative standard normal distribution
d1	= {ln (S/K) + (r +s^2/2)t}/s√t
	= {ln (88.02/85) + (0.072 + 0.24^2/2)*0.25}/0.24*√0.25
	0.500900
d2	= d1 - s√t
	= 0.5009 - 0.24√0.25
	0.3809
Using z tables,
N(d1) =	0.6918
N(d2) =	0.6484
C = Call Premium =	=SN(d1) - N(d2)Ke^(-rt)
	= 88.02*0.6918 - 0.6484*85e^(-0.072*0.25)
	6.7614
N(-d1) =	0.3082
N(-d2) =	0.3516
P = Put Premium =	=N(-d2)Ke^(-rt) - SN(-d1)
	= 0.3516*85e^(-0.072*0.25) - 88.02*0.3082
	2.2251

Hence, Value of call option = $6.76