A call option has a premium of $.50, a strike price of $26, and 3 months to expiration. The current stock price is $25.50. The stock will pay a $.40 dividend two months from now. The risk-free rate is 1%. What is the premium on a 3-month put with a strike price of $40? Assume the options are European style.
A)$.25
B)$.51
C)$1.23
D)$1.33
E)$1.50

Question

Quizplus · Accepted Answer

Using Black-Scholes formula to calculate the value of the call option:
d1 = [ln(S/X) + (r + σ^2/2)t] / [σ√(t)]
d2 = d1 - σ√(t)
where S = 25.50, X = 26, r = 1%, t = 3/12, σ = unknown. 
d1 = [ln(25.50/26) + (0.01 + σ^2/2)(3/12)] / [σ√(3/12)] = -0.343σ
d2 = d1 - σ√(3/12) = -0.619σ
N(d1) = 0.3658, N(d2) = 0.2687
Value of call option = S*N(d1) - X*e^(-rt)*N(d2) + PV(dividend)
PV(dividend) = 0.40*e^(-0.01*(2/12)) = 0.398
Value of call option = 25.50(0.3658) - 26*e^(-0.01*(3/12))*0.2687 + 0.398 = 0.615
Premium on 3-month put with $40 strike price can be calculated using put-call parity:
Premium on put = Premium on call - S + X*e^(-rt)
Premium on put = 0.615 - 25.50 + 40*e^(-0.01*(3/12)) = 1.335 ≈ $1.33

Therefore, the answer is D) $1.33.

A Call Option Has a Premium of