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

Under the Black-Scholes model for a non-dividend-paying share, the price of a European call is given by C = S·N(d1) − K·e^(−rT)·N(d2). Which statement about N(d2) is correct?

N(d2) is the risk-neutral probability that the European call expires in the money, i.e. that S_T exceeds K. The delta is N(d1), and real-world probabilities would use the share's actual drift rather than the risk-free rate.

  1. AIt is the risk-neutral probability that the call finishes in the moneyCorrect
  2. BIt is the delta of the call
  3. CIt is the real-world probability that the share price rises
  4. DIt is the probability that the put finishes in the money
  5. It is the vega of the call divided by the share price

Explanation

Under the risk-neutral measure, ln(S_T) is normal and N(d2) equals P(S_T > K). The delta of the call is N(d1), not N(d2). The put finishing in the money has probability N(−d2), and the real-world probability would use the true drift rather than r.

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