In a one-period binomial model with up factor u, down factor d, and risk-free rate r over a period h, the replicating portfolio that reproduces payoffs V_u and V_d has initial value Delta*S0 + B, where Delta = (V_u - V_d)/(S_u - S_d) and B = e^{-r h}*(u V_d - d V_u)/(u - d). Which option correctly specifies Delta and B?

• A. Delta = (V_u - V_d)/(S_u - S_d); B = e^{-r h}*(u V_d - d V_u)/(u - d)
• B. Delta = (V_d - V_u)/(S_u - S_d); B = e^{r h}*(u V_d - d V_u)/(u - d)
• C. Delta = (V_u - V_d)/(S_u + S_d); B = e^{-r h}*(u V_u - d V_d)/(u - d)
• D. Delta = (V_u + V_d)/(S_u - S_d); B = e^{-r h}*(u V_d + d V_u)/(u - d)

Answer: (A)

In a one-period binomial model, the replicating portfolio uses Delta shares of the stock and a position B in the risk-free asset so that it matches the option payoffs in both the up and down states. Let S_u and S_d be the stock prices in the up and down states, and V_u and V_d be the corresponding payoffs. The portfolio must satisfy two equations: DeltaS_u + B e^{r h} = V_u and DeltaS_d + B e^{r h} = V_d.

Subtracting these gives Delta*(S_u - S_d) = V_u - V_d, so Delta = (V_u - V_d)/(S_u - S_d). This is the sensitivity of the payoff to the move in the stock price—the amount you hold in the stock to reproduce the payoff difference between the states.

To find B, use one of the equations, say DeltaS_u + B e^{r h} = V_u. Solve for B e^{r h} = V_u - DeltaS_u. Substituting Delta and the relations S_u = S0 u and S_d = S0 d leads to B e^{r h} = (u V_d - d V_u)/(u - d). Therefore B = e^{-r h}*(u V_d - d V_u)/(u - d).

These expressions match the given form, confirming the correct Delta is (V_u - V_d)/(S_u - S_d) and the correct B is e^{-r h}*(u V_d - d V_u)/(u - d).