IAI Actuarial Core Principles · Economic Modelling · Stochastic models for security prices
A stock follows dS = μS dt + σS dW with μ = 0.12 and σ = 0.20. Using Ito's lemma, what is the drift per year of ln S(t)?
The drift of ln S is μ minus half the variance: 0.12 − 0.5 × 0.04 = 0.10 per year. The subtraction arises from the second-order Ito correction term, which is ignored in ordinary calculus.
- A0.12
- B0.10Correct
- C0.14
- D0.08
- 0.11
Explanation
For f = ln S, df = (μ − σ²/2)dt + σ dW. Here σ²/2 = 0.04/2 = 0.02, so drift = 0.12 − 0.02 = 0.10. Choosing 0.14 adds the correction with the wrong sign, and 0.12 ignores the Ito term.
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