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

A share price follows GBM with μ = 12% p.a. and σ = 20% p.a. Under this model, what is the expected value of ln(S(t)/S(0)) per year, i.e. the drift of the log price?

The log price drifts at 10% a year. Itô's lemma gives a drift of μ minus half the variance, which is 0.12 minus 0.02. The 12% figure is the drift of the price itself, not of its logarithm.

  1. A12%
  2. B14%
  3. C10%Correct
  4. D8%
  5. 2%

Explanation

By Itô's lemma, d ln S = (μ − σ²/2)dt + σ dW. Here μ − σ²/2 = 0.12 − 0.02 = 0.10. Choosing 12% ignores the Itô correction, and 14% adds it with the wrong sign.

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