Question

Let S = $52, s = 20%, and r = 7% (continuously compounded). The stock is set to pay a single dividend of $1.10 nine months from today, with no further dividends expected this year. Use the Black-Scholes model (adjusted for the dividend) to compute the value of a one-year $50-strike European call option on the stock.

answer= $6.43

Please show all the steps. Thanks

Answer 1

Interest Rate = r = 7% or 0.07/12 monthly

Dividends received in none months = D9 = 1.10

Present Value of Dividend = D0 = D9/(1+r)ⁿ = 1.10/(1+0.07/12)⁹ = 1.044

S = Dividend adjusted Stock Price = 52 - 1.044	50.956
t = time until option expiration(years) =	1
K = Option Strike Price =	50
r = risk free rate(annual) = 7% =	0.07
s = standard deviation(annual) =	0.20
N = cumulative standard normal distribution
d1	= {ln (S/K) + (r +s^2/2)t}/s√t
	= {ln (50.956/50) + (0.07 + 0.2^2/2)*1}/0.2*√1
	0.5447
d2	= d1 - s√t
	= 0.5447 - 0.2√1
	0.3447
Using z tables,
N(d1) =	0.7070
N(d2) =	0.6348
C = Call Premium =	=SN(d1) - N(d2)Ke^(-rt)
	= 50.956*0.707 - 0.6348*50e^(-0.07*1)
	6.4317

Hence, Value of call option = C = $6.43