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

A stock has S_0 = 80, μ = 10% and σ = 40% per year. Under the lognormal model, what is the median of the stock price after 1 year?

The median price is S_0 × exp[(μ − σ²/2)T] = 80 × e^0.02 ≈ 81.62. The mean is higher at about 88.41 because the lognormal distribution is right-skewed. Subtracting the full variance instead of half gives the wrong figure.

  1. A80 × e^(0.10) ≈ 88.41
  2. B80 × e^(0.10 − 0.08) ≈ 81.62Correct
  3. C80 × e^(0.10 − 0.16) ≈ 75.34
  4. D80 × e^(0.10 + 0.08) ≈ 95.78

Explanation

The median of S_T is S_0 exp[(μ − σ²/2)T]. σ²/2 = 0.16/2 = 0.08, so exponent is 0.02 and the median is 80 × e^0.02 ≈ 81.62. The first option is the mean; 75.34 subtracts the full variance 0.16 instead of half.

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