IAI Actuarial Core Principles · Economic Modelling · Stochastic models for security prices
A share price follows GBM with μ = 12% p.a. and σ = 20% p.a. What is the expected value of ln(S(t)/S(0)) over one year, i.e. the mean of the continuously compounded return?
The mean log return over one year is μ − σ²/2 = 12% − 2% = 10%. The volatility adjustment of half the variance is subtracted because the log of the price is normal with that reduced drift.
- A12%
- B10%Correct
- C14%
- D8%
- 2%
Explanation
ln(S(t)/S(0)) is normal with mean (μ − σ²/2)t. Here σ²/2 = 0.04/2 = 0.02, so the mean is 0.12 − 0.02 = 0.10, or 10%. Using 12% ignores the volatility drag; 14% has the wrong sign.
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