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Build a short-rate tree for normally distributed rates, with and without drift, and describe how well it works.

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

The idea

A normal short-rate model says the rate changes by a random amount that does not depend on the rate level, so dr = σ dw. Here σ is the annual basis-point volatility. In a tree with time step dt, each node moves up or down by the amount below, with probability one half each. Move = σ × √dt. With no drift the expected rate never changes, and an up move followed by a down move equals the reverse. The tree therefore recombines and has n + 1 nodes after n steps rather than 2ⁿ. A constant drift λ adds λ × dt to every move. Zero drift implies a convexity effect that makes the spot curve slope down with maturity, lower than r0 by roughly σ² × T² ÷ 6. One constant drift can fit only one point of the curve, so neither version matches a real term structure. Exam point: volatility scales with the square root of time, drift scales with time itself.

Formulas

Spot rate in a normal model with constant drift

Spot = r0 + λ × T ÷ 2 − σ² × T² ÷ 6

r0
today's short rate
λ
the constant drift per year
σ
the annual basis-point volatility
T
the maturity in years

Continuously compounded approximation. With zero drift only the convexity term remains, so the curve slopes down; one λ can match only one maturity.

A practice question

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

In the simplest normally distributed short-rate tree with no drift and time step dt, where sigma is the annual volatility in basis points, what happens at each node?

  1. The rate moves to r + sigma x sqrt(dt) or r - sigma x sqrt(dt) with equal probability, so the expected rate is unchanged. - correct
  2. The rate moves to r + sigma x dt or r - sigma x dt with equal probability.Why not: Scales volatility by dt rather than by the square root of dt.
  3. The rate moves to r x (1 + sigma x sqrt(dt)) or r x (1 - sigma x sqrt(dt)), so the move is proportional to r.Why not: Describes a lognormal-style proportional move, not a normal model whose moves do not depend on r.
  4. The rate moves to r + sigma x sqrt(dt) or r - sigma x sqrt(dt), with probabilities set so the expected rate equals the market forward rate.Why not: That is a fitted-drift model such as Ho-Lee; the zero-drift model has equal probabilities and no fitting.

With zero drift every node moves by plus or minus sigma x sqrt(dt) with probability one half, because the variance of the change over dt must be sigma² x dt. The expected next rate is the current rate.

A harder one

A short-rate tree with normally distributed rates and a constant drift is built with quarterly steps (dt = 0.25). The initial rate is 3.00%, the drift is 0.40% per year and the volatility is 1.00% per year. What are the three possible rates (up-up, middle, down-down) after two steps?

  1. 4.00%, 3.00%, 2.00%Why not: Leaves out the drift entirely, treating the tree as the zero-drift model.
  2. 4.20%, 3.20%, 2.20% - correct
  3. 4.80%, 3.80%, 2.80%Why not: Adds the full annual drift at each step instead of drift x dt.
  4. 3.70%, 3.20%, 2.70%Why not: Scales volatility by dt instead of sqrt(dt), so each move is 0.25% rather than 0.50%.

Drift per step is 0.40 x 0.25 = 0.10%, volatility per step is 1.00 x sqrt(0.25) = 0.50%. Date 2: 3.00 + 2 x 0.10 = 3.20% in the middle, plus or minus 2 x 0.50 = 1.00% for up-up and down-down.

The hardest tier

In a normal-rate model with constant drift lambda and volatility sigma, the continuously compounded T-year spot rate is approximately r0 + lambda x T / 2 - sigma² x T² / 6. The short rate is 3.00%, sigma is 1.00% per year, and the market 10-year spot rate is 4.50%. The drift, in basis points per year, that matches the 10-year rate is closest to:

  1. 30.0Why not: Ignores the convexity term when solving for the drift.
  2. 16.7Why not: Uses lambda x T instead of lambda x T / 2 in the formula.
  3. 33.3 - correct
  4. 26.7Why not: Subtracts the convexity term instead of adding it back when solving for lambda.

Convexity term = 0.0100² x 100 / 6 = 0.167%. Solve 4.50 = 3.00 + lambda x 10 / 2 - 0.167: lambda = 2 x (1.50 + 0.167) / 10 = 0.3333% per year, about 33.3 bp.

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