A single branch on a Binomial tree is represented by the following diagram. Calculate how many shares of the underlying asset (delta) are required to delta hedge.Enter your answer to two decimal places. If your answer is negative, then enter the negative sign.数值题
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Question

A single branch on a Binomial tree is represented by the following diagram. Calculate how many shares of the underlying asset (delta) are required to delta hedge.Enter your answer to two decimal places. If your answer is negative, then enter the negative sign.数值题

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BlackTom AI · Accepted Answer

Delta in a one-step binomial model is the slope of the option payoff with respect to the underlying: delta = (V_up − V_down) / (S_up − S_down).

1) Knowledge points and concepts
- Delta for a replicating portfolio in a one-step binomial tree is found by taking the difference in option values between the up and down states divided by the difference in stock prices between those states.
- For a put option, the option value falls when the stock price rises, so delta is negative.

2) Step-by-step reasoning and intermediate calculations
- From the diagram: S0 = $12, S_up = $13.80, S_down = $10.44. Put option values: P_up = $0, P_down = $2.56.
- Compute differences:
  - V_up − V_down = 0 − 2.56 = −2.56
  - S_up − S_down = 13.80 − 10.44 = 3.36
- Delta = (V_up − V_down) / (S_up − S_down) = −2.56 / 3.36.

3) Arithmetic shown
- −2.56 / 3.36 = −256/336. Reduce fraction: divide numerator and denominator by 16 → −16/21 ≈ −0.76190476.
- Round to two decimal places → −0.76.

4) Why this is correct
- This delta is the number of shares of the underlying to hold (short or long) to replicate the option’s payoff for the one-step tree. The negative sign indicates a short position in the underlying is required to hedge a long put (put value falls when stock rises).

5) Why other approaches are incorrect
- Using current stock price S0 or mixing up numerator sign (e.g., V_down − V_up) would give wrong sign or magnitude. Solving the two linear equations for a replicating portfolio yields the same delta: subtracting the equations eliminates the cash account and gives the same slope calculation.