A single branch on a Binomial tree is represented by the following diagram. The riskfree rate is 6% and the branch length (delta t) = 3 months Calculate how much money is required to invest in the bank ("B") to replicate the derivative. Enter your answer to two decimal places. Do not enter the dollar sign "$". A positive (negative) number means invest (borrow).数值题
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Question

A single branch on a Binomial tree is represented by the following diagram. The riskfree rate is 6% and the branch length (delta t) = 3 months Calculate how much money is required to invest in the bank ("B") to replicate the derivative. Enter your answer to two decimal places. Do not enter the dollar sign "$". A positive (negative) number means invest (borrow).数值题

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BlackTom AI · Accepted Answer

Delta = (payoff_up − payoff_down) / (S_up − S_down). Here S0 = 40, u = 1.33 → S_up = 40×1.33 = 53.20, d = 0.75 → S_down = 40×0.75 = 30.00. The derivative payoffs shown are P_u = 1.80 and P_d = 25.00 (these correspond to a put with strike 55).

1) Relevant concepts:
- Replicating portfolio in a one-period binomial: hold Δ shares and invest/borrow B now so terminal value Δ·S_T + B·(1+r)^{Δt} equals the derivative payoff in each state.
- Solve for Δ from the two terminal equations, then find B by matching one state and discounting back to present.

2) Step-by-step reasoning and calculations:
- Compute Δ:
  Δ = (P_u − P_d) / (S_up − S_down)
    = (1.80 − 25.00) / (53.20 − 30.00)
    = (−23.20) / 23.20
    = −1.00.
  So replicate by shorting 1 share.

- Use the up-state equation to find B·(1+r)^{Δt}:
  Δ·S_up + B·(1+r)^{Δt} = P_u
  ⇒ (−1)·53.20 + B·(1.06)^{0.25} = 1.80
  ⇒ B·(1.06)^{0.25} = 1.80 + 53.20 = 55.00.

- Discount to find B (the amount to invest now):
  (1.06)^{0.25} ≈ 1.014716 (quarter-year compounding).
  B = 55.00 / 1.014716 ≈ 54.19.

3) Why this is correct:
- The Δ found ensures the difference in terminal portfolio values matches the payoff gap between states; the B found makes the portfolio match the absolute payoff level. Discounting uses the risk-free growth over 3 months.

4) Common incorrect approaches:
- Forgetting to discount (reporting $55.00) or using an incorrect compounding convention (e.g., continuous compounding) would give a different value. Using wrong Δ (sign or magnitude) would also fail to replicate payoffs.