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

A share follows GBM with S_0 = 200, mu = 8% and sigma = 25% p.a. Using the normal distribution of ln(S_1/S_0), which expression gives P(S_1 < 200)?

The log return is normal with mean 0.08 minus 0.03125, which is 0.04875, and standard deviation 0.25. The probability the price falls below its starting value is therefore the standard normal cdf at minus 0.04875 divided by 0.25.

  1. APhi(-0.08/0.25)
  2. BPhi(-0.04875/0.25)Correct
  3. CPhi(-0.08875/0.25)
  4. DPhi(-0.11125/0.25)
  5. Phi(0.04875/0.25)

Explanation

ln(S_1/S_0) ~ N(mu - sigma^2/2, sigma^2) = N(0.08 - 0.03125, 0.0625) = N(0.04875, 0.0625^... sd 0.25). P(S_1<200) = P(ln ratio < 0) = Phi((0 - 0.04875)/0.25). Option 1 omits the Ito correction; others use wrong arithmetic.

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