Question

XYZ Corp. will pay a $2 per share dividend in two months. Its stock price currently is $74 per share. A call option on XYZ has an exercise price of $66 and 3-month time to expiration. The risk-free interest rate is 0.5% per month, and the stock’s volatility (standard deviation) = 13% per month. Find the Black-Scholes value of the American call 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

Risk free Rate = r = 0.5% per month

Dividend Received in 2 months = D2 = $2

Present Value of Dividend = D0 =  D2/(1+r)² = 2/(1+0.005)² = $1.98

S = Dividend Adjusted Current Stock Price = 74 - 1.98 =	72.02
t = time until option expiration(month) =	3
K = Option Strike Price =	66
r = risk free rate(monthly) =	0.005
s = standard deviation(annual) =	0.13
N = cumulative standard normal distribution
d1	= {ln (S/K) + (r +s^2/2)t}/s√t
	= {ln (72.02/66) + (0.005 + 0.13^2/2)*3}/0.13*√3
	0.5669
d2	= d1 - s√t
	= 0.5669 - 0.13√3
	0.3417
Using z tables,
N(d1) =	0.7146
N(d2) =	0.6337
C = Call Premium =	=SN(d1) - N(d2)Ke^(-rt)
	= 72.02*0.7146 - 0.6337*66e^(-0.005*3)
	10.26

Hence, value of call option = $10.26