What is the basic idea behind the delta-hedging approach to valuing a derivative?[Fill in the blank]

Question

BlackTom AI · Accepted Answer

Delta-hedging forms a riskless portfolio by choosing Δ = ∂V/∂S so that the portfolio’s immediate exposure to small moves in the underlying is eliminated.

1) Relevant knowledge points and concepts
- V(S,t) is the derivative price; S is the underlying. Delta (Δ) = ∂V/∂S.
- Construct portfolio Π = Δ·S − V (i.e., long Δ shares and short one derivative).
- Use Ito/total differential to see how Π changes; if the dS term cancels, the portfolio is locally riskless and must earn the risk-free rate r.
- This is the core of delta-hedging and the Black–Scholes derivation.

2) Complete reasoning process step by step
- Write the change in the portfolio: dΠ = Δ dS − dV.
- Expand dV by Ito: dV = V_t dt + V_S dS + (1/2) V_SS σ^2 S^2 dt.
- Substitute into dΠ:
  dΠ = Δ dS − [V_t dt + V_S dS + (1/2)V_SS σ^2 S^2 dt]
      = (Δ − V_S) dS − [V_t + (1/2)V_SS σ^2 S^2] dt.

3) Intermediate steps and choice of Δ
- Choose Δ = V_S so the dS term vanishes:
  dΠ = −[V_t + (1/2)V_SS σ^2 S^2] dt.
- The portfolio is now riskless over the instant dt, so it must grow at rate r: dΠ = r Π dt.
- Equating and rearranging gives the Black–Scholes PDE. The key is the creation of a locally riskless portfolio by shorting the derivative and holding Δ shares.

4) Why the correct answer is correct
- Option b states exactly this construction: buy Δ shares and enter one short derivative position to make the two-piece portfolio certain/riskless (locally). That describes delta-hedging.

5) Why the other options are incorrect
- a and e: buying Δ shares and investing B in the bank describes a static replicating portfolio (Δ and bond) that can match payoffs in some models, but it does not state the delta-hedging idea of neutralizing instantaneous risk by shorting the derivative.
- c: buying Δ shares and shorting the derivative does not “precisely match the payoff to the derivative” — it produces a riskless hedge, not the same payoff as the derivative.
- d: using a long derivative with shares to be riskless reverses the sign; the standard construction shorts the derivative.

6) Conclusion
- Therefore the correct choice is b: buying Δ shares and entering one short derivative position makes the two-piece portfolio certain/riskless, matching the provided answer.

What is the basic idea behind the delta-hedging approach to valuing a derivative?[Fill in the blank]单项选择题

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