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Outline how a simple, recombining short-rate tree is built under the mean-reverting Vasicek Model.

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

The idea

The Vasicek model lets the drift depend on the current rate. The rate change is dr = k × (θ − r) × dt + σ × dw. Here k is the speed of mean reversion and θ is the long-run rate. In a tree, from a node with rate r the next rates are r + k × (θ − r) × dt plus or minus σ times the square root of dt, each with probability one half. Because the drift now depends on the node, up-then-down no longer lands exactly where down-then-up does. The gap between the two end rates is small for short steps. It is written Gap, where Gap = 2 × k × σ × dt × √dt. So the middle node is set to one value, such as the average, to give a recombining tree. Exam point: the drift is k × (θ − r) × dt, with its sign set by θ minus r.

Formulas

Gap between the up-down and down-up rates in a Vasicek tree

Gap = 2 × k × σ × dt × √dt

k
the speed of mean reversion
σ
the annual volatility
dt
the step length in years

Small for short steps, but not zero, which is why the middle node is set to one value such as the average to make the tree recombine.

A practice question

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

In a tree for the Vasicek model, dr = k x (theta - r) x dt + sigma x dw, from a node with rate r the next rates are:

  1. r + k x (theta - r) x dt plus or minus sigma x sqrt(dt), each with probability one half. - correct
  2. r + k x theta x dt plus or minus sigma x sqrt(dt), each with probability one half.Why not: Uses a constant drift k x theta, omitting the dependence on the current rate.
  3. r + k x (r - theta) x dt plus or minus sigma x sqrt(dt), each with probability one half.Why not: Reverses the sign of the gap, pushing the rate away from theta instead of toward it.
  4. r + k x (theta - r) plus or minus sigma x sqrt(dt), each with probability one half.Why not: Omits dt from the drift term.

The Vasicek drift is k x (theta - r) x dt: proportional to the gap between the long-run mean and the current rate and scaled by the step length. The shock is plus or minus sigma x sqrt(dt).

A harder one

A Vasicek tree uses an initial rate of 4.00%, k = 0.50, theta = 2.50%, annual volatility 0.90% and a quarterly step (dt = 0.25). The two possible rates after one step are closest to:

  1. 4.45% and 3.55%Why not: Ignores mean reversion entirely.
  2. 4.64% and 3.74%Why not: Applies the drift with the wrong sign, pushing the rate away from theta.
  3. 4.04% and 3.59%Why not: Scales volatility by dt instead of the square root of dt.
  4. 4.26% and 3.36% - correct

Drift = 0.50 x (2.50 - 4.00) x 0.25 = -0.1875%. Shock = 0.90 x sqrt(0.25) = 0.45%. Up = 4.00 - 0.1875 + 0.45 = 4.2625%, down = 3.3625%.

The hardest tier

A Vasicek tree has an initial rate of 4.00%, k = 0.80, theta = 2.00%, annual volatility 1.00% and dt = 0.25. Each node's drift is computed from that node's rate, and the middle node at date 2 is defined as the average of the up-down and down-up rates so the tree recombines. The middle rate at date 2 is closest to:

  1. 3.20%Why not: Applies the initial rate's drift again at the second step instead of the drift at the new rates.
  2. 3.18%Why not: Reports only the up-down value, not the average with the down-up value.
  3. 3.38%Why not: Reports only the down-up value, not the average with the up-down value.
  4. 3.28% - correct

Date-1 rates: 4.00 - 0.40 +/- 0.50 = 4.10% and 3.10%. Up-down = 4.10 + 0.8 x (2.00 - 4.10) x 0.25 - 0.50 = 3.18%; down-up = 3.10 + 0.8 x (2.00 - 3.10) x 0.25 + 0.50 = 3.38%. Average = 3.28%.

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