Question

. Use the Black-Scholes model to find the price for a call option with the following inputs: (1) current stock price is $45, (2) exercise price is $50, (3) time to expiration is 3 months, (4) annualized risk-free rate is 3%, and (5) variance of stock return is 0.50.

. Using the information from question above, find the value of a put with a $50 exercise price.

Answer 1

Std dev = variance^(1/2)=0.5^(1/5)=87.06%

As per Black Scholes Model
Value of call option = S*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =		45
t = time to expiry =		0.25
K = Strike price =		50
r = Risk free rate =		3.0%
q = Dividend Yield =		0%
σ = Std dev =		87%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(45/50)+(0.03-0+0.8706^2/2)*0.25)/(0.8706*0.25^(1/2))
d1 = -0.007162
d2 = d1-σ*t^(1/2)
d2 =-0.007162-0.8706*0.25^(1/2)
d2 = -0.442462
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.497143
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.329077
Value of call= 45*0.497143-0.329077*50*e^(-0.03*0.25)
Value of call= 6.04

As per put call parity

Call price + PV of exercise price = Spot price + Put price

6.04+50/(1+0.03)^0.25=45+Put value

Put value = 10.67