Question

Calculate the value of a three-month at-the-money European call option on a stock index when the index is at 250, the risk-free interest rate is 10% per annum, the volatility of the index is 18% per annum, and the dividend yield on the index is 3% per annum.

Answer 1

As per Black Scholes Model
Value of call option = (S*e^(q*t))*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =		250
t = time to expiry =		0.25
K = Strike price =		250
r = Risk free rate =		10.0%
q = Dividend Yield =		3%
σ = Std dev =		18%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(250/250)+(0.1-0.03+0.18^2/2)*0.25)/(0.18*0.25^(1/2))
d1 = 0.239444
d2 = d1-σ*t^(1/2)
d2 =0.239444-0.18*0.25^(1/2)
d2 = 0.149444
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.59462
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.559398
Value of call= 250*e^(-0.03*0.25)*0.59462-0.559398*250*e^(-0.1*0.25)
Value of call= 11.15