Use a binomial tree to value to following option.Assume rf = 0.045,r_p = 0.14,σ = 0.20,E(CF₁)= $62 million,g = 0.03,time horizon = 2 years,binomial period = 1 year,and cost = $500 million.What is the value of this project option? A) $47 million B) $57 million C) $67 million D) $77 million

Question

Quizplus · Accepted Answer

First, we need to calculate the value of the underlying asset at each node of the binomial tree:

At t=0:
S0 = $500 million

At t=1:
S1U = S0(1+u) = $630 million
S1D = S0(1+d) = $380 million

At t=2:
S2UU = S1U(1+u) = $793.8 million
S2UD = S1U(1+d) = $476.3 million
S2DU = S1D(1+u) = $476.3 million
S2DD = S1D(1+d) = $287.8 million

where u and d are calculated as follows:

u = e^(σ√(binomial period)) = e^(0.20√1) = 1.22
d = 1/u = 0.82

Next, we calculate the cash flows at each node:

At t=2:
CF₂UU = max(E(CF₁) - S2UU, 0) = $0
CF₂UD = max(E(CF₁) - S2UD, 0) = $62 million - $476.3 million = $0
CF₂DU = max(E(CF₁) - S2DU, 0) = $0
CF₂DD = max(E(CF₁) - S2DD, 0) = $62 million - $287.8 million = $34.2 million

where E(CF₁) is given as $62 million.

Then, we work backwards from t=2 to t=0 to calculate the option value at each node:

At t=2:
V2UU = CF₂UU = $0
V2UD = CF₂UD = $0
V2DU = CF₂DU = $0
V2DD = CF₂DD = $34.2 million

At t=1:
V1U = e^(-rf√(binomial period)) * (pV2UU + (1-p)V2UD) = $17.8 million
V1D = e^(-rf√(binomial period)) * (pV2DU + (1-p)V2DD) = $22.1 million

where p is calculated as follows:

p = (e^(rf√(binomial period)) - d) / (u - d) = (e^(0.045√1) - 0.82) / (1.22 - 0.82) = 0.5782

Finally, we calculate the option value at t=0:

V0 = e^(-rf√(binomial period)) * (pV1U + (1-p)V1D) = $66.8 million

Therefore, the value of the project option is $66.8 million, which is closest to choice C ($67 million).

Use a Binomial Tree to Value to Following Option