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

A share price S_t follows geometric Brownian motion dS = S(μ dt + σ dW). Which statement about S_t is correct?

Under geometric Brownian motion the logarithm of the price is normally distributed, so the price itself is lognormal and always positive. Prices cannot become negative. It is log returns, not absolute price changes, that are independent and identically distributed.

  1. AS_t is normally distributed, so negative prices are possible
  2. Bln S_t is normally distributed, so S_t cannot become negativeCorrect
  3. CS_t is lognormal only if μ = 0
  4. DSuccessive absolute price changes S_{t+1} − S_t are independent and identically distributed
  5. The variance of ln(S_t/S_0) is σ t²

Explanation

Under GBM, ln(S_t/S_0) is normal with mean (μ − σ²/2)t and variance σ²t. Exponentiating gives a strictly positive lognormal variable. Normality of S_t itself is wrong since it would permit negative prices; it is the log returns that are independent and stationary, not absolute changes.

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