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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 is assumed to follow geometric Brownian motion dS = μS dt + σS dz. Which statement about the implied distribution of the stock price at a future time T is correct?

The stock price at time T is lognormally distributed, which means it cannot fall below zero. The log of the price, and hence the continuously compounded return, is normal with variance σ²T that increases with the horizon.

  1. AThe stock price is normally distributed, so it can become negative
  2. BThe stock price is lognormally distributed, so it cannot become negativeCorrect
  3. CThe continuously compounded return is lognormally distributed
  4. DThe stock price is lognormally distributed with variance that decreases as T increases

Explanation

Under geometric Brownian motion, ln S_T is normally distributed, so S_T is lognormal and bounded below by zero. Option A confuses price with log price. Option C reverses the roles: the continuously compounded return is normal. Option D is wrong because variance of ln S_T equals σ²T, which grows with T.

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