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

A stock price S(t) follows geometric Brownian motion dS = μS dt + σS dW. Which statement about the process is correct?

Under geometric Brownian motion the logarithm of the price is normally distributed, so the price itself is lognormal and always positive. Ito's lemma gives ln S a constant drift of μ − σ²/2 and volatility σ.

  1. AS(t) is normally distributed for every t
  2. Bln S(t) is normally distributed, so S(t) is lognormalCorrect
  3. CS(t) can become negative with positive probability
  4. DThe proportional increments dS/S have variance σ dt
  5. The drift of ln S(t) equals μ

Explanation

Applying Ito's lemma to ln S gives d(ln S) = (μ − σ²/2)dt + σ dW, so ln S(t) is normal and S(t) is lognormal. S stays strictly positive. The drift of ln S is μ − σ²/2, not μ, and dS/S has variance σ² dt.

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