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

A share price is assumed to follow geometric Brownian motion. Which one of the following is a correct consequence under that model, and hence a limitation when compared with observed equity returns?

Under geometric Brownian motion log returns are normally distributed, so the price is lognormal and positive. Normal tails are thinner than the fat tails and crashes observed in real markets, so the model understates the probability of extreme moves.

  1. ALog returns are normally distributed, so extreme moves are rarer than the fat tails seen in practiceCorrect
  2. BShare prices can become negative with positive probability
  3. CLog returns are skewed to the left with fat tails
  4. DVolatility is a random process correlated with the share price
  5. The share price over any interval is normally distributed

Explanation

Under geometric Brownian motion the log of the price is normal and the price itself is lognormal, so it is never negative. Normal log returns have thin tails relative to empirical returns, which show excess kurtosis and crashes. Hence the model underestimates extreme moves.

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