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

For a European put on a non-dividend-paying share in the Black-Scholes model, the put price is P = K·e^(−rT)·N(−d2) − S·N(−d1). An analyst holds this put and the share price rises slightly, all else unchanged. Which statement about the put's delta and the effect is correct?

The put's delta is N(d1) − 1, which equals −N(−d1) and is negative. So a small rise in the share price lowers the put's value. A positive delta belongs to the call.

  1. ADelta is N(d1) − 1, which is negative, so the put value fallsCorrect
  2. BDelta is N(d1), which is positive, so the put value rises
  3. CDelta is −N(−d2), so the put value rises
  4. DDelta is 1 − N(d1), which is positive, so the put value rises
  5. Delta is zero because the put is European

Explanation

Differentiating the put formula gives delta = −N(−d1) = N(d1) − 1, lying between −1 and 0. A rise in S therefore reduces the put value. Statements with positive delta confuse the put with the call.

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