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

Under the Black-Scholes model, a European call on a non-dividend-paying share is valued by the risk-neutral method. Which statement about the process followed by the share price under the risk-neutral measure Q is correct?

Under the risk-neutral measure the share price has drift equal to the risk-free rate r, while its volatility sigma is unchanged. The change of measure alters only the drift, so the option price does not depend on investors' risk preferences or the real-world expected return.

  1. AThe share price has drift equal to its expected real-world return, and volatility sigma
  2. BThe share price has drift equal to the continuously compounded risk-free rate r, and the same volatility sigmaCorrect
  3. CThe share price has drift equal to r minus sigma, and volatility sigma
  4. DThe share price has zero drift and volatility sigma
  5. The share price has drift r and volatility equal to sigma multiplied by the risk-free rate

Explanation

Under Q all traded assets earn the risk-free rate in expectation, so dS = rS dt + sigma S dW~. Volatility is unchanged by the change of measure (Girsanov only changes the drift). A real-world drift would make the price depend on risk preferences, which the replication argument removes.

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