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

Under the Black-Scholes-Merton model, the stock price S follows dS = μS dt + σS dz. Which statement about the distribution of the stock price at a future time T is correct?

Under geometric Brownian motion the stock price at T is lognormal, meaning its natural logarithm is normally distributed. This keeps prices positive, and the continuously compounded return over the period is normal rather than lognormal.

  1. AThe stock price is normally distributed with mean S0 e^(μT)
  2. BThe stock price is lognormally distributed, so its natural logarithm is normally distributedCorrect
  3. CThe stock price is lognormally distributed, so the stock can take negative values with small probability
  4. DThe continuously compounded return is lognormally distributed

Explanation

Geometric Brownian motion implies ln(S_T) is normally distributed, so S_T is lognormal. Prices are therefore bounded below by zero, which rules out the normal-price and negative-value options. It is the log return that is normal, not lognormal.

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