Question

Use the Black-Scholes formula for the following stock:

Time to expiration	6 months
Standard deviation	51% per year
Exercise price	$41
Stock price	$41
Annual interest rate	6%
Dividend	0

Recalculate the value of the call with the following changes:

a.	Time to expiration	3 months
b.	Standard deviation	30% per year
c.	Exercise price	$45
d.	Stock price	$45
e.	Interest rate	9%

Calculate each scenario independently. (Round your answers to 2 decimal places.)

Answer 1

a

As per Black Scholes Model
Value of call option = (S)*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =		41
t = time to expiry =		0.25
K = Strike price =		41
r = Risk free rate =		6.0%
q = Dividend Yield =		0%
σ = Std dev =		51%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(41/41)+(0.06-0+0.51^2/2)*0.25)/(0.51*0.25^(1/2))
d1 = 0.186324
d2 = d1-σ*t^(1/2)
d2 =0.186324-0.51*0.25^(1/2)
d2 = -0.068676
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.573904
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.472624
Value of call= 41*0.573904-0.472624*41*e^(-0.06*0.25)
Value of call= 4.44

b

As per Black Scholes Model
Value of call option = (S)*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =		41
t = time to expiry =		0.5
K = Strike price =		41
r = Risk free rate =		6.0%
q = Dividend Yield =		0%
σ = Std dev =		30%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(41/41)+(0.06-0+0.3^2/2)*0.5)/(0.3*0.5^(1/2))
d1 = 0.247487
d2 = d1-σ*t^(1/2)
d2 =0.247487-0.3*0.5^(1/2)
d2 = 0.035355
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.597734
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.514102
Value of call= 41*0.597734-0.514102*41*e^(-0.06*0.5)
Value of call= 4.05

c

As per Black Scholes Model
Value of call option = (S)*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =		41
t = time to expiry =		0.5
K = Strike price =		45
r = Risk free rate =		6.0%
q = Dividend Yield =		0%
σ = Std dev =		51%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(41/45)+(0.06-0+0.51^2/2)*0.5)/(0.51*0.5^(1/2))
d1 = 0.005365
d2 = d1-σ*t^(1/2)
d2 =0.005365-0.51*0.5^(1/2)
d2 = -0.355259
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.50214
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.361198
Value of call= 41*0.50214-0.361198*45*e^(-0.06*0.5)
Value of call= 4.81

d

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 =		0.5
K = Strike price =		41
r = Risk free rate =		6.0%
q = Dividend Yield =		0%
σ = Std dev =		51%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(45/41)+(0.06-0+0.51^2/2)*0.5)/(0.51*0.5^(1/2))
d1 = 0.521638
d2 = d1-σ*t^(1/2)
d2 =0.521638-0.51*0.5^(1/2)
d2 = 0.161014
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.699039
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.563959
Value of call= 45*0.699039-0.563959*41*e^(-0.06*0.5)
Value of call= 9.02