FRM Exam Part II · The Vasicek and Gauss+ Models

Vasicek Model Basics: How the Short Rate Mean-Reverts

Q: What do kappa and theta mean in the Vasicek model?

Theta (θ) is the long-run level the short rate is pulled toward. Kappa (k) is the speed of that pull. The expected change each instant is k(θ − r)dt, so a bigger k closes the gap faster.

Q: Why can the Vasicek model produce negative interest rates?

The random shock is normally distributed and volatility is constant. A normal distribution has no lower bound, so the rate has a positive probability of falling below zero, especially when it starts low and σ is large.

Q: What happens to the Vasicek model if kappa is zero?

With k = 0 the drift term disappears and dr = σdW. There is no mean reversion, and you have the normal model with zero drift. Rates can wander without limit.

Q: How is Vasicek different from the CIR model?

Both have mean reversion. Vasicek uses constant volatility, so rates can go negative. CIR makes volatility depend on the rate level, which shrinks near zero and keeps rates from going negative under standard conditions.

Updated 11 October 2026 · Fact-checked

The Vasicek model describes the short rate as dr = k(θ − r)dt + σdW. The rate is pulled toward the long-run level θ at speed k, with constant volatility σ. Solve questions by finding the drift k(θ − r), then adding the volatility shock. Because shocks are normal, rates can go negative.

Understand Vasicek Model Basics: Mean-Reverting Short Rate

Start with the simplest rate model: the short rate moves randomly with constant volatility and no pull back to any level. Over time, such a rate can wander far from where it started. Real rates do not behave that way. They tend to drift back toward a long-run level.

The Vasicek model adds that pull. The short rate r follows dr = k(θ − r)dt + σdW. Here θ (theta) is the long-run rate, k (kappa) is the speed of mean reversion, and σ is the constant annual volatility. dW is a normal random shock with mean zero and variance dt.

Read the drift term k(θ − r) as a spring. If r is above θ, the drift is negative and pulls the rate down. If r is below θ, the drift is positive and pulls it up. A larger k means a stronger, faster pull. If k = 0, the pull vanishes and you are back to a plain normal model with zero drift.

The expected rate moves a fraction of the gap each period. For a short step dt, the expected change is k(θ − r)dt. Over longer horizons, the expected rate is E[r(t)] = θ + (r0 − θ)e^(−kt). The gap to θ shrinks exponentially. The half-life of a gap is ln(2) ÷ k.

The weakness is the constant σ and the normal shocks. A normal distribution has no floor, so the model gives a positive probability of negative rates. This was once seen as a flaw. After negative policy rates in the euro area and Japan, it is seen as less damaging, but it still matters for low-rate and option pricing questions. Note also that volatility does not depend on the rate level, unlike in the CIR model.

Key formulas to remember

Vasicek short-rate dynamics: dr = k(θ − r)dt + σdW; k = speed of reversion, θ = long-run rate, σ = constant volatility. The drift is k(θ − r), not kθ.
Expected change over a short interval: E[dr] = k(θ − r)dt; Positive when r < θ, negative when r > θ.
Expected future short rate: E[r(t)] = θ + (r0 − θ)e^(−kt); Gap to θ decays at rate k. As t grows, the expected rate tends to θ.
Half-life of mean reversion: Half-life = ln(2) ÷ k ≈ 0.693 ÷ k; Time for the expected gap to θ to halve. k is per year, so the result is in years.
Variance of short rate over time t: Var[r(t)] = σ² (1 − e^(−2kt)) ÷ (2k); Tends to σ² ÷ (2k) as t grows, so long-run standard deviation is σ ÷ √(2k).
Special case: k = 0 gives dr = σdW; No mean reversion: normal model with zero drift.

How to solve Vasicek Model Basics: Mean-Reverting Short Rate questions

Use this method for any Vasicek question on dynamics, interpretation or negative rates.

1Identify the parameters: k, θ and σ, and the current rate r. Check units are annual and in the same form (decimal or percent).
2Compute the gap θ − r. Its sign tells you the direction of the pull.
3Multiply by k and dt to get the expected change: k(θ − r)dt.
4For a longer horizon, use E[r(t)] = θ + (r0 − θ)e^(−kt) rather than adding drifts repeatedly.
5Add the random part: the shock has standard deviation σ√dt, so a one-period change is normal with mean k(θ − r)dt and sd σ√dt.
6For probability questions, standardise: z = (threshold − mean) ÷ sd, then use the normal table. Negative-rate probability uses threshold 0.
7Interpret: state whether the rate is expected to rise or fall, how fast, and what the constant σ and normality imply.

Quickest way: Gap, speed, shock

When to use it: Use for MCQs asking for expected rate change, direction of drift, effect of changing k, or whether negative rates are possible.

Write gap = θ − r. Positive gap means rates drift up.
Expected change = k × gap × dt. Do this first.
If the question asks for the rate after time t, shrink the gap by e^(−kt).
Higher k means faster pull and lower long-run variance σ² ÷ (2k).
Any option saying rates cannot be negative is wrong for Vasicek, since shocks are normal.

Common mistakes in Vasicek Model Basics: Mean-Reverting Short Rate

Using kθ as the drift instead of k(θ − r).
Students remember the parameters but not the structure of the pull.
Fix: Always compute the gap θ − r first, then multiply by k.
Reading θ as the expected rate after one step.
Mean reversion is confused with instant jumps to the mean.
Fix: θ is the level the rate tends to over the long run. Each step covers only a fraction of the gap.
Treating k as a percentage-per-year change in the rate.
k is called speed, so it feels like a rate change.
Fix: k is the fraction of the gap closed per year (in continuous time). The half-life is ln(2) ÷ k.
Saying Vasicek rates cannot go negative.
Confusion with CIR, where volatility shrinks near zero.
Fix: Vasicek has normal shocks and constant σ, so negative rates have positive probability.
Using σ instead of σ√dt for the shock over a period.
The time scaling of volatility is forgotten.
Fix: Standard deviation of a shock over dt is σ√dt. Monthly dt = 1/12.
Thinking higher k raises long-run volatility.
Faster adjustment sounds like more movement.
Fix: Long-run sd is σ ÷ √(2k). Higher k pulls the rate back harder and reduces it.

Worked examples

Example 1

In a Vasicek model, k = 0.40, θ = 4%, σ = 1.2% per year, and the current short rate is 2%. (a) What is the expected change in the rate over one month? (b) What is the expected rate in 2 years? Use e^(−0.8) = 0.4493.

Show the solution

Gap = θ − r = 4% − 2% = 2%.
(a) Expected change = k × gap × dt = 0.40 × 2% × (1/12) = 0.8% ÷ 12 = 0.0667%.
(b) E[r(2)] = θ + (r0 − θ)e^(−kt) = 4% + (2% − 4%) × e^(−0.8).
e^(−0.8) = 0.4493, so (−2%) × 0.4493 = −0.8986%.
E[r(2)] = 4% − 0.8986% = 3.1014%.

Answer: (a) About +0.067% (6.7 bp) over one month. (b) About 3.10%.

Example 2

A Vasicek model has k = 0.50, θ = 3%, σ = 1.5% per year. The current short rate is 0.5%. Over one year approximated as a single step (dt = 1), what is the mean and standard deviation of the change in the rate, and what is the probability the rate ends below zero under this approximation? Use N(−0.78) ≈ 0.2177.

Show the solution

Gap = 3% − 0.5% = 2.5%.
Mean change = k × gap × dt = 0.50 × 2.5% × 1 = 1.25%.
Sd of change = σ√dt = 1.5% × 1 = 1.5%.
Mean ending rate = 0.5% + 1.25% = 1.75%.
Probability rate < 0: z = (0 − 1.75%) ÷ 1.5% = −1.167.
N(−1.167) ≈ 0.1216 (about 12%).
Check the given value: N(−0.78) is not needed here, since z = −1.17, not −0.78.

Answer: Mean change +1.25%, standard deviation 1.5%, ending mean 1.75%. The probability of a negative rate is about 12%, which shows Vasicek allows negative rates.

Exam tips

Questions are often conceptual: direction of drift, effect of raising k, or what the model implies about negative rates. Settle these before computing.
Check units. If σ is annual and the step is monthly, scale by √(1/12).
If an option says volatility falls as rates approach zero, that is CIR, not Vasicek.
Remember Vasicek is the mean-reverting extension of the normal model with constant drift and volatility. Link it to Ho-Lee, where drift is time-dependent but there is no reversion.
Use the half-life ln(2) ÷ k as a fast sense check: k = 0.7 gives about 1 year.

Practice questions from The Vasicek and Gauss+ Models

Vasicek Model Basics: Mean-Reverting Short Rate: frequently asked questions

What do kappa and theta mean in the Vasicek model?

Theta (θ) is the long-run level the short rate is pulled toward. Kappa (k) is the speed of that pull. The expected change each instant is k(θ − r)dt, so a bigger k closes the gap faster.

Why can the Vasicek model produce negative interest rates?

The random shock is normally distributed and volatility is constant. A normal distribution has no lower bound, so the rate has a positive probability of falling below zero, especially when it starts low and σ is large.

What happens to the Vasicek model if kappa is zero?

With k = 0 the drift term disappears and dr = σdW. There is no mean reversion, and you have the normal model with zero drift. Rates can wander without limit.

How is Vasicek different from the CIR model?

Both have mean reversion. Vasicek uses constant volatility, so rates can go negative. CIR makes volatility depend on the rate level, which shrinks near zero and keeps rates from going negative under standard conditions.