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IAI Actuarial Core Principles · Economic Modelling · Stochastic models for security prices

A share follows GBM with μ = 10% and σ = 30% p.a. Which of the following is the 1-year median price as a multiple of S_0?

The median multiple is e raised to μ − σ²/2 times time, which is 0.10 − 0.045 = 0.055. So the median price is S_0 × e^0.055. The mean is higher at e^0.10 because of the lognormal skew.

  1. Ae^(0.10)
  2. Be^(0.145)
  3. Ce^(0.055)Correct
  4. De^(0.19)
  5. e^(0.30)

Explanation

ln(S_1/S_0) is normal with mean μ − σ²/2 = 0.10 − 0.045 = 0.055. The median of the lognormal is S_0 e^0.055. e^0.10 is the mean multiple, and e^0.145 wrongly adds σ²/2.

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