The underlying stock for a European exchange option has S = $27.15,div = 2.0%,and ?σ = 0.18.The strike stock has S = $30.00,div = 0.0%,and σ = 0.22.The two stocks have a correlation coefficient of 0.73.If the exchange option expires in 2 years,what is the price of the call using a Black-Scholes approach?

Question

Quizplus · Accepted Answer

Using the Black-Scholes formula for a European exchange call option, we have:
d1 = [ln(S1/S2) + (r2 - r1 + 0.5(σ1^2 - σ2^2)t)]/[(σ1^2 - 2ρσ1σ2 + σ2^2)t]^(1/2)
= [ln(27.15/30) + (0.02 - 0 + 0.5(0.18^2 - 0.22^2) x 2)]/[(0.18^2 - 2 x 0.73 x 0.18 x 0.22 + 0.22^2) x 2]^(1/2)
= -0.0425
d2 = d1 - (σ1 - σ2) x (t)^(1/2)
= -0.4399
N(d1) = 0.4852
N(d2) = 0.3294
The price of the option is:
C = S1 x N(d1) - S2 x N(d2) x e^(-r1t)
= 27.15 x 0.4852 - 30 x 0.3294 x e^(-0.02 x 2)
= $1.09

The Underlying Stock for a European Exchange Option Has S