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Calculate the movements in the Gauss+ model's three rate factors, namely the short, medium and long ones.

FRM® Part II · Market Risk Measurement and Management · The Vasicek and Gauss+ Models · FRM2.MR.16.c

The idea

To move each Gauss+ factor over a short interval, apply the stochastic differential equation in discrete form. Let z be a standard normal draw. The change in each factor, written Move, is Move = k × (target − level) × dt + σ × √dt × z. For the short factor the target is the medium rate, for the medium factor it is the long rate, and for the long factor it is the constant long-run level. The first term is the drift, which is proportional to dt. The second is the random shock, which scales with the square root of dt. All three changes use the levels at the start of the step, not the freshly updated ones. Typical errors are using dt in place of its square root for the shock, flipping the sign of the gap, and updating factors sequentially.

Formulas

One-step move in a Gauss+ factor

Move = k × (target − level) × dt + σ × √dt × z

k
the factor's speed of mean reversion
target
the medium rate for r, the long rate for m, the constant level for l
level
the factor's current value
dt
the step in years
σ
the factor's annual volatility
z
a standard normal draw

All three factors use their start-of-step levels. Drift scales with dt, the shock with √dt.

A practice question

From the question bank, with the reason each wrong option is wrong.

In a Gauss+ model dr = k_r(m - r)dt + sigma_r dW. Given r = 3.00%, m = 3.60%, k_r = 0.80 per year and dt = 1/12 (one month), the expected change in the short rate, with zero shock, is closest to:

  1. +48 bpWhy not: Omits dt, so the annual drift of 48 bp is reported as a monthly change.
  2. -4 bpWhy not: Reverses the sign of the gap, using (r - m) instead of (m - r).
  3. +8 bpWhy not: Uses the long gap of 120 bp (as if the target were 4.20%) instead of the medium rate m.
  4. +4 bp - correct

Drift = k_r x (m - r) x dt = 0.80 x 60 bp x (1/12) = +4 bp. The short rate is pulled up toward the higher medium rate.

A harder one

In a Gauss+ model dm = k_m(l - m)dt + sigma_m dW. Given m = 3.60%, l = 4.20%, k_m = 0.25, sigma_m = 70 bp per year, dt = 1/12 and a draw z = +0.50, the one-month change in the medium factor is closest to:

  1. +10.10 bpWhy not: Includes only the random part and leaves out the drift of 1.25 bp.
  2. +25.10 bpWhy not: Leaves out dt on the drift, so it adds the annual drift of 15 bp instead of the monthly 1.25 bp.
  3. +11.4 bp - correct
  4. +4.2 bpWhy not: Scales the shock by dt rather than the square root of dt.

Drift = 0.25 x 60 x (1/12) = 1.25 bp. Shock = 70 x sqrt(1/12) x 0.5 = 10.10 bp. Total = 11.35 bp, about +11.4 bp.

The hardest tier

A Gauss+ model with quarterly steps (dt = 0.25) has r = 3.00%, m = 3.60%, l = 4.30% and long-run level theta = 4.90%. Speeds are k_r = 1.00, k_m = 0.30, k_l = 0.06 and volatilities are 100, 70 and 40 bp per year. The draws are z_r = -0.5, z_m = +1.0, z_l = +0.5. All three factors are updated from their starting levels. The long-minus-short spread l - r after the step is closest to:

  1. 165 bpWhy not: Leaves out all three drift terms and applies only the random shocks.
  2. 141 bpWhy not: Updates the factors one after another, using the new l in the m drift and the new m in the r drift, instead of the starting levels.
  3. 133 bpWhy not: Scales the random parts by dt instead of the square root of dt.
  4. 151 bp - correct

dr = 1.0 x 60 x 0.25 - 100 x 0.5 x 0.5 = 15 - 25 = -10 bp. dl = 0.06 x 60 x 0.25 + 40 x 0.5 x 0.5 = 0.9 + 10 = +10.9 bp. New spread = (430 + 10.9) - (300 - 10) = 440.9 - 290 = 150.9 bp.

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