Replicating portfolio (B) equals which expression?

• A. e^{-rh} * (u V_d - d V_u) / (u - d)
• B. e^{rh} * (u V_d - d V_u) / (u - d)
• C. e^{-rh} * (u V_d - d V_u) / (u - d)
• D. e^{-rh} * (V_u - V_d) / (u - d)

Answer: (C)

In a one-period binomial model, to replicate the claim you choose a number of stock shares Δ and an amount B in the risk-free asset so that the portfolio matches the claim in both the up and down states. The risk-free asset grows by e^{r h} over the period, so the state equations are:

Δ S_u + B e^{r h} = V_u and Δ S_d + B e^{r h} = V_d.

Subtracting eliminates B and gives Δ = (V_u − V_d) / (S_u − S_d) = (V_u − V_d) / ((u − d) S_0). Solve for B using one of the equations, for example the up state:

B e^{r h} = V_u − Δ S_u = V_u − Δ u S_0.

Substitute Δ to get:

B e^{r h} = [u V_d − d V_u] / (u − d).

Thus B = e^{−r h} [u V_d − d V_u] / (u − d). This is the amount in the risk-free asset today in the replicating portfolio, which is the expression shown.