Question

Consider an asset that trades at $100 today. Suppose that the European call and put options on this asset are available both with a strike price of $100. The options expire in 275 days, and the volatility is 45%. The continuously compounded risk-free rate is 3%. Determine the value of the European call and put options using the Black-Scholes-Merton model. Assume that the continuously compounded yield on the asset is 1,5% and there are 365 days in the year.

Answer 1

As per Black Scholes Model
Value of call option = (S*e^(q*t))*N(d1)-N(d2)*K*r^(-r*t)
Where
S = Current price =		100
t = time to expiry =		0.753424
K = Strike price =		100
r = Risk free rate =		3.000%
q = Dividend Yield =		1.50%
σ = Std dev =		45%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(100/100)+(0.03-0.015+0.45^2/2)*0.753424)/(0.45*0.753424^(1/2))
d1 = 0.224233
d2 = d1-σ*t^(1/2)
d2 =0.224233-0.45*0.753424^(1/2)
d2 = -0.166367
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.588712
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.433934
Value of call= 100*e^(-0.015*0.753424)*0.588712-0.433934*100*e^(-0.03*0.753424)
Value of call= 15.79

As per Black Scholes Model
Value of put option = N(-d2)*K*e^(-r*t)-(S*e^(q*t))*N(-d1)
Where
S = Current price =		100
t = time to expiry =		0.753424
K = Strike price =		100
r = Risk free rate =		3.000%
q = Dividend Yield =		1.50%
σ = Std dev =		45%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(100/100)+(0.03-0.015+0.45^2/2)*0.753424)/(0.45*0.753424^(1/2))
d1 = 0.224233
d2 = d1-σ*t^(1/2)
d2 =0.224233-0.45*0.753424^(1/2)
d2 = -0.166367
N(-d1) = Cumulative standard normal dist. of -d1
N(-d1) =0.411288
N(-d2) = Cumulative standard normal dist. of -d2
N(-d2) =0.566066
Value of put= 0.566066*100*e^(-0.03*0.753424)-100*e^(-0.015*0.753424)*0.411288
Value of put= 14.67