Skip to content

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

A log-price follows an AR(1) process in discrete time: Y_{t+1} - m = φ(Y_t - m) + e_{t+1}, where the e's are independent N(0, σ²) and |φ| < 1. Which statement is correct?

The stationary variance is σ²/(1 - φ²). Setting the variance equal to φ² times itself plus the innovation variance σ² and solving gives this result, which is finite because |φ| < 1. The process reverts to the mean m, not at φ = 1.

  1. AThe variance of Y_t grows without limit as t increases
  2. BThe stationary variance is σ²/(1 - φ²)Correct
  3. CThe stationary variance is σ²/(1 - φ)
  4. DThe process is mean-reverting only when φ = 1
  5. The stationary mean depends on φ and on Y_0

Explanation

Stationarity requires Var(Y) = φ²Var(Y) + σ², giving Var(Y) = σ²/(1 - φ²). Option 2 omits the squaring of φ. With φ = 1 the process is a random walk with no reversion, and the stationary mean is m, independent of Y_0.

Did you get it right without looking?

One question tells you little. A timed set on Stochastic models for security prices shows your real accuracy, how long you take and where you lose marks.

Practise Stochastic models for security pricesFree diagnostic mock

More Stochastic models for security prices questions