Question

Assume an initial underlying stock price of $20, an exercise price of $20, a time to expiration of 3 months, a risk free rate of 12% and a underlying stock return variance of 16%. If the risk free rate decreased to 6% and assuming other variables are held constant, the call option value would

A) increase

B) remain the same

C) decrease

D) indeterminate from the information given

Answer 1

Std dev = variance^(1/2) = 0.16^0.5 = 0.4 = 40%

As per Black Scholes Model
Value of call option = S*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =		20
t = time to expiry =		0.25
K = Strike price =		20
r = Risk free rate =		12.0%
q = Dividend Yield =		0.00%
σ = Std dev =		40%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(20/20)+(0.12-0+0.4^2/2)*0.25)/(0.4*0.25^(1/2))
d1 = 0.25
d2 = d1-σ*t^(1/2)
d2 =0.25-0.4*0.25^(1/2)
d2 = 0.05
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.598706
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.519939
Value of call= 20*0.598706-0.519939*20*e^(-0.12*0.25)
Value of call= 1.88

As per Black Scholes Model
Value of call option = S*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =		20
t = time to expiry =		0.25
K = Strike price =		20
r = Risk free rate =		6.0%
q = Dividend Yield =		0.00%
σ = Std dev =		40%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(20/20)+(0.06-0+0.4^2/2)*0.25)/(0.4*0.25^(1/2))
d1 = 0.175
d2 = d1-σ*t^(1/2)
d2 =0.175-0.4*0.25^(1/2)
d2 = -0.025
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.56946
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.490027
Value of call= 20*0.56946-0.490027*20*e^(-0.06*0.25)
Value of call= 1.73

Price has decreased with r = 6% compared to r = 12%