Question

You would like to practice your knowledge of investment and quantitative methods for
finance that you learnt at Oxford, so you found two call options A and B, both options sell in the market for $5, for an underlying stock which currently sells at $30. Both options mature in one year, and the risk free rate is 6% and a variance of 30%. Upon maturity, you can exercise
option A for $25, and option B for $35.
Which option you choose to buy? Explain your answer with details

Answer 1

Std dev = variance^(1/2) = 30^(1/2) = 5.477

Call A

As per Black Scholes Model
Value of call option = (S)*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =	30
t = time to expiry =	1
K = Strike price =	25
r = Risk free rate =	6.0%
q = Dividend Yield =	0%
σ = Std dev =	5.477%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(30/25)+(0.06-0+0.05477^2/2)*1)/(0.05477*1^(1/2))
d1 = 4.451733
d2 = d1-σ*t^(1/2)
d2 =4.451733-0.05477*1^(1/2)
d2 = 4.396963
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.999996
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.999995
Value of call= 30*0.999996-0.999995*25*e^(-0.06*1)
Value of call= 6.46

Call B

As per Black Scholes Model
Value of call option = (S)*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =	30
t = time to expiry =	1
K = Strike price =	35
r = Risk free rate =	6.0%
q = Dividend Yield =	0%
σ = Std dev =	5.477%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(30/35)+(0.06-0+0.05477^2/2)*1)/(0.05477*1^(1/2))
d1 = -1.691634
d2 = d1-σ*t^(1/2)
d2 =-1.691634-0.05477*1^(1/2)
d2 = -1.746404
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.045358
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.04037
Value of call= 30*0.045358-0.04037*35*e^(-0.06*1)
Value of call= 0.03

Buy Call A as it has higher intrinsic value than B