The opportunity to invest in a project can be thought of as a three-year real option that is worth $500 million with an exercise price of $800 million. Calculate the value of the option given that, N(d1) = 0. 3 and N(d2) = 0.15. Assume that the interest is 6% per year.
A) $150 million
B) $49.25 million
C) Zero
D) None of the above.

Question

Quizplus · Accepted Answer

First, we need to calculate the d1 and d2 values using the Black-Scholes formula:

d1 = [ln(V/P) + (r + σ^2/2)t]/(σsqrt(t))
d2 = d1 - σsqrt(t)

where:
V = value of the real option = $500 million
P = exercise price = $800 million
r = interest rate = 6%
σ = volatility = not given
t = time to expiration = 3 years

Using the given N(d1) and N(d2) values, we can solve for d1 and d2:

N(d1) = 0.3 => d1 = 0.52
N(d2) = 0.15 => d2 = 0.31

Now we can use the Black-Scholes formula to solve for the volatility, σ:

V = SN(d1) - Pe^(-rt)N(d2)
500 = S(0.3) - 800e^(-0.06*3)(0.15)
500 = S(0.3) - 86.09
S(0.3) = 586.09

586.09 = S*e^(0.5σ^2t)
Solving for σ, we get σ = 37.69%

Now we can use the Black-Scholes formula to solve for the value of the option:

V = SN(d1) - Pe^(-rt)N(d2)
V = Se^(-0.06*3)N(d1) - 800e^(-0.06*3)N(d2)
V = Se^(-0.18)(0.3) - 800e^(-0.18)(0.15)
V = 0.3S - 97.87

Substituting the value of S from the previous equation, we get:

V = 0.3(586.09) - 97.87 = $49.25 million

Therefore, the option is worth $49.25 million and the correct answer is B.

The Opportunity to Invest in a Project Can Be Thought