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IAI Actuarial Core Principles · Economic Modelling · Black-Scholes derivative-pricing model

In deriving the Black-Scholes PDE for a derivative value V(S,t) on a non-dividend-paying share, a portfolio holds one derivative and -Delta shares, with Delta = dV/dS. Which feature of this choice makes the portfolio instantaneously riskless?

Choosing -Delta shares with Delta equal to dV/dS makes the random dW term from Ito's lemma on the derivative cancel the dW term from the shares. The portfolio change is then deterministic, so by no-arbitrage it must earn the risk-free rate.

  1. AThe dW term in dV cancels the dW term in Delta times dS, leaving only dt termsCorrect
  2. BThe drift mu of the share is set equal to r
  3. CThe volatility sigma is set to zero
  4. DThe theta of the derivative is set to zero
  5. The gamma of the derivative is set to zero

Explanation

By Ito's lemma dV contains (dV/dS) sigma S dW. Holding -dV/dS shares removes the Brownian term, so the portfolio change is deterministic and must earn r. Mu drops out of the PDE as a consequence, not by assumption.

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