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

In the Black-Scholes framework for a non-dividend-paying share, the share price S_t is assumed to follow geometric Brownian motion. Which statement about this model is correct?

The continuously compounded return over any period is normally distributed. Under geometric Brownian motion the share price is lognormal, always positive, with constant volatility and continuous paths, so only the log of the price ratio is normal.

  1. AThe share price can become negative with positive probability
  2. BThe continuously compounded return over any period is normally distributedCorrect
  3. CThe share price at a future time is normally distributed
  4. DThe volatility is assumed to change randomly over time
  5. The price process has jumps at random times

Explanation

Under geometric Brownian motion, ln(S_t/S_0) is normal, so the price is lognormal and always positive. Volatility is constant and paths are continuous. The claim that the price itself is normal confuses price with log return.

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