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IAI Actuarial Core Principles · Economic Modelling · Black-Scholes derivative-pricing model

A share price S_t follows geometric Brownian motion dS = S(mu dt + sigma dW). Which statement about the distribution of S_t given S_0 is correct?

Under geometric Brownian motion the log of the price is normal with mean ln S_0 plus (mu minus half sigma squared) times t, and variance sigma squared times t. The drift is reduced by the Ito correction, so the price itself is lognormal.

  1. AS_t is normally distributed with mean S_0 e^{mu t}
  2. Bln S_t is normally distributed with mean ln S_0 + (mu - sigma^2/2)t and variance sigma^2 tCorrect
  3. Cln S_t is normally distributed with mean ln S_0 + mu t and variance sigma^2 t
  4. DS_t is lognormal with E[S_t] = S_0 e^{(mu - sigma^2/2)t}
  5. S_t has independent additive increments over disjoint intervals

Explanation

Applying Ito's lemma to ln S gives d(ln S) = (mu - sigma^2/2)dt + sigma dW, so ln S_t is normal with mean ln S_0 + (mu - sigma^2/2)t and variance sigma^2 t. Option 3 omits the Ito correction. Option 4 confuses the mean of ln S with E[S_t], which equals S_0 e^{mu t}.

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