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Using the Vasicek Model, calculate the change in rate and its standard deviation, along with the rate expected T years out and the half-life.

FRM® Part II · Market Risk Measurement and Management · The Art of Term Structure Models: Drift · FRM2.MR.14.g

The idea

In the Vasicek model, over a step dt the expected change in the rate is written Drift = k × (θ − r) × dt. The standard deviation of the change is the same as in the zero-drift model, written SD = σ × √dt. Over a longer horizon T the gap to the long-run mean shrinks by the factor e raised to the power −kT, so the expected rate approaches θ without overshooting it. The expected rate T years from now, with r0 today's rate, is written Rate = θ + (r0 − θ) × e^(−kT). For any k, the time for the expected gap to halve is the half-life, written Half = ln(2) ÷ k. With k = 0.35 it is about 2 years. The classic mistakes are using 1 ÷ k as the half-life, using linear instead of exponential decay, and scaling the drift by the square root of dt. Exam point: after one half-life half of the gap to θ remains, after two a quarter.

Formulas

Expected change in the short rate over one Vasicek step

Drift = k × (θ − r) × dt

k
the speed of mean reversion
θ
the long-run rate
r
the current rate
dt
the step length in years

Scale by dt, not √dt. The sign follows θ − r; the standard deviation of the change is still σ × √dt.

Expected Vasicek short rate T years ahead

Rate = θ + (r0 − θ) × e^(−k × T)

θ
the long-run rate
r0
today's short rate
k
the speed of mean reversion
T
the horizon in years

The gap decays exponentially, never linearly, and never overshoots θ.

Half-life of mean reversion

Half = ln(2) ÷ k

k
the speed of mean reversion, per year

The time for the expected gap to the long-run level to halve. Do not use 1 ÷ k.

A practice question

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

A Vasicek model has a mean-reversion speed of k = 0.35 per year. The half-life of a deviation of the expected short rate from its long-run mean is closest to:

  1. 2.86 yearsWhy not: Uses 1 / k, the mean-reversion time constant, as the half-life.
  2. 1.43 yearsWhy not: Divides one half by k instead of using ln(2) / k.
  3. 0.50 yearsWhy not: Inverts the formula to k / ln(2).
  4. 1.98 years - correct

The gap decays as exp(-k x t). Setting exp(-k x t) = 0.5 gives t = ln(2) / k = 0.6931 / 0.35 = 1.98 years.

A harder one

In a Vasicek model with k = 0.40, theta = 4.50% and a current short rate of 2.00%, the expected short rate in three years is closest to:

  1. 3.75% - correct
  2. 5.00%Why not: Applies linear instead of exponential adjustment, overshooting theta.
  3. 5.25%Why not: Flips the sign of the exponential term, moving the rate away from the starting side of theta.
  4. 2.82%Why not: Uses exp(-k) for a single year rather than exp(-k x T).

Expected rate = theta + (r0 - theta) x exp(-kT) = 4.50 - 2.50 x exp(-1.2) = 4.50 - 2.50 x 0.3012 = 3.75%.

The hardest tier

A Vasicek model is calibrated with a half-life of 1.5 years, theta = 3.00% and annual volatility 100 bp. The short rate is 6.00% and the tree uses quarterly steps. The up node after one step is closest to:

  1. 6.00%Why not: Sets k = 1 / half-life instead of ln(2) / half-life.
  2. 5.72%Why not: Sets k = ln(2) x half-life, multiplying instead of dividing.
  3. 6.15% - correct
  4. 5.15%Why not: Computes the down node instead of the up node.

k = ln(2) / 1.5 = 0.4621. Drift = 0.4621 x (3.00 - 6.00) x 0.25 = -0.3466%. Shock = 1.00 x sqrt(0.25) = 0.5000%. Up node = 6.15%.

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