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

A share price follows a mean-reverting Ornstein-Uhlenbeck process dX_t = a(m - X_t)dt + s dW_t with a = 0.5 per year, m = 100 and s > 0. Given X_0 = 120, what is E[X_2]?

The expected value is 107.36. For a mean-reverting process the expected deviation from the long-run mean decays as e^(-at). With a = 0.5 and t = 2 the exponent is -1, so the 20 deviation shrinks to 7.36 above 100.

  1. A100 + 20 × e^(-0.5) = 112.13
  2. B100 + 20 × e^(-1) = 107.36Correct
  3. C100 + 20 × e^(-2) = 102.71
  4. D120 × e^(-1) = 44.15
  5. 100 + 20 × (1 - e^(-1)) = 112.64

Explanation

For the OU process E[X_t] = m + (X_0 - m)e^(-at). Here a t = 0.5 × 2 = 1, so E[X_2] = 100 + 20e^(-1) = 100 + 7.36 = 107.36. Option 0 uses t = 1 and option 2 uses a = 1, so both apply the wrong exponent.

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