Question

Find the price for a call option with the following inputs:

Current stock price = $25 Strike price of option = $30

Time until option expires = 6 months Risk-free rate = 3%

Standard deviation of stock returns = 18%

Using the Black Scholes Option Pricing Model Formula

Given all the information above, what would be the value of a put option on the above stock with a strike price of $30?

Answer 1

As per Black Scholes Model
Value of call option = (S)*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =	25
t = time to expiry =	0.5
K = Strike price =	30
r = Risk free rate =	3.0%
q = Dividend Yield =	0%
σ = Std dev =	18%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(25/30)+(0.03-0+0.18^2/2)*0.5)/(0.18*0.5^(1/2))
d1 = -1.250963
d2 = d1-σ*t^(1/2)
d2 =-1.250963-0.18*0.5^(1/2)
d2 = -1.378242
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.105474
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.084064
Value of call= 25*0.105474-0.084064*30*e^(-0.03*0.5)
Value of call= 0.15

As per Black Scholes Model
Value of put option = N(-d2)*K*e^(-r*t)-(S)*N(-d1)
Where
S = Current price =	25
t = time to expiry =	0.5
K = Strike price =	30
r = Risk free rate =	3.0%
q = Dividend Yield =	0%
σ = Std dev =	18%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(25/30)+(0.03-0+0.18^2/2)*0.5)/(0.18*0.5^(1/2))
d1 = -1.250963
d2 = d1-σ*t^(1/2)
d2 =-1.250963-0.18*0.5^(1/2)
d2 = -1.378242
N(-d1) = Cumulative standard normal dist. of -d1
N(-d1) =0.894526
N(-d2) = Cumulative standard normal dist. of -d2
N(-d2) =0.915936
Value of put= 0.915936*30*e^(-0.03*0.5)-25*0.894526
Value of put= 4.71