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FRM Part I · FRM Exam Part I · The Black-Scholes-Merton Model

In the Black-Scholes-Merton framework, the stock price S is assumed to follow dS = μS dt + σS dz. Which statement about the distribution of the stock price at a future time T is correct under this model?

Under geometric Brownian motion the log of the stock price is normally distributed, so the future stock price is lognormal and cannot be negative. The return variance grows linearly with time, not with its square, and not constant across horizons.

  1. AS_T is normally distributed, so the price can become negative
  2. BS_T is lognormally distributed, so the price cannot become negativeCorrect
  3. CS_T is lognormally distributed and its continuously compounded return has a variance that grows with the square of time
  4. DS_T is normally distributed and the return over [0,T] has a constant variance independent of T

Explanation

Geometric Brownian motion implies ln S_T is normal, so S_T is lognormal and bounded below by zero. Normal S_T would allow negative prices. The return variance grows linearly with T (σ²T), not with the square of time, so the third option is wrong.

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