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

A security price follows geometric Brownian motion dS = μS dt + σS dW under the real-world measure. Which statement about the distribution of S(t) given S(0) is correct?

Under geometric Brownian motion the price at time t is lognormally distributed because its logarithm is normal. A lognormal variable is always positive, so the model cannot generate negative prices, unlike an arithmetic Brownian motion for the price itself.

  1. AS(t) is normally distributed with mean S(0) + μt
  2. BS(t) is lognormally distributed, so the price can never become negativeCorrect
  3. Cln S(t) is lognormal with variance σ²t
  4. DS(t) has independent increments in absolute price changes
  5. S(t) is lognormal with a variance that does not grow with t

Explanation

Under GBM, ln S(t) is normal, so S(t) is lognormal and strictly positive. Absolute price changes are not independent or identically distributed because they scale with S. The normal-price option ignores the exponential structure, and variance of S(t) grows with t.

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