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

A share price follows GBM with μ = 8% and σ = 25% p.a. For a 1-year horizon, which pair gives the variance of ln(S_1/S_0) and the probability that S_1 > S_0 expressed in terms of the standard normal CDF Φ?

The variance of the log return is σ²t = 0.0625, and its mean is μ − σ²/2 = 0.04875. The probability of a rise is Φ(0.04875/0.25), about Φ(0.195). Using μ directly or flipping the sign would be incorrect.

  1. A0.0625; Φ(0.0488/0.25)Correct
  2. B0.0625; Φ(0.08/0.25)
  3. C0.25; Φ(0.0488/0.25)
  4. D0.0625; Φ(−0.0488/0.25)
  5. 0.0625; Φ(0.1113/0.25)

Explanation

Var = σ²t = 0.0625. Mean of log return = 0.08 − 0.03125 = 0.04875. P(S_1 > S_0) = P(Z > −0.04875/0.25) = Φ(0.04875/0.25) ≈ Φ(0.195). Using μ alone ignores the drift correction; using the negative sign gives the probability of a fall.

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