FRM Exam Part II · The Art of Term Structure Models: Drift

Vasicek Model and Mean Reversion Explained for FRM Part 2

Updated 11 October 2026 · Fact-checked

The Vasicek model is a short-rate model: dr = k(θ − r)dt + σdw. The drift pulls the rate toward the long-run mean θ at speed k. Solve questions by finding the expected rate, then the volatility. Mean reversion makes long-rate volatility lower than short-rate volatility. Ho-Lee has no mean reversion.

Understand Vasicek Model and Mean Reversion

A term structure model describes how the short-term interest rate moves over time. The Vasicek model says the change in the short rate has two parts: a drift and a random shock. The drift is k(θ − r). The shock is σ times a normal random move.

Mean reversion means the drift depends on where the rate is now. 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. The parameter k is the speed of reversion. A larger k means a faster pull back to θ. The parameter θ is the long-run mean (in pricing, the risk-neutral long-run rate).

This changes volatility along the curve. A shock to the short rate fades as the rate reverts. So the rate far in the future, and long-maturity yields, move less than the short rate. In Vasicek, the T-year yield has a sensitivity to the short rate of B(T) ÷ T, where B(T) = (1 − e^(−kT)) ÷ k. This falls as T rises. In Ho-Lee, which has no mean reversion, a shock to the short rate moves all rates equally, so the volatility term structure is flat at σ.

Because rates are normally distributed in Vasicek, rates can turn negative. That is a known weakness. The model also has a closed-form bond price, P(T) = A(T) e^(−B(T) r). This is why exam questions on it can be solved by hand.

The Vasicek model with constant θ cannot fit an arbitrary market curve exactly. Ho-Lee can, because its drift is a function of time chosen to match the curve. Keep that contrast in mind.

Key formulas to remember

Vasicek dynamics: dr = k(θ − r)dt + σdw; k is the speed of reversion, θ the long-run mean, σ the constant normal (basis-point) volatility, dw a normal shock.
Expected future short rate: E[r(T)] = θ + (r₀ − θ)e^(−kT); The gap between r₀ and θ shrinks by the factor e^(−kT). As T gets large, E[r(T)] approaches θ.
Variance of future short rate: Var[r(T)] = σ²(1 − e^(−2kT)) ÷ (2k); Rises with T but levels off. Long-run standard deviation is σ ÷ √(2k). Compare with Ho-Lee, where variance is σ²T and keeps growing.
Half-life of a shock: Half-life = ln(2) ÷ k; Time for the gap between the rate and θ to halve in expectation.
Bond price: P(T) = A(T) e^(−B(T) r), with B(T) = (1 − e^(−kT)) ÷ k; B(T) is the sensitivity of ln P to the short rate. A(T) depends on k, θ and σ and is usually given.
Yield volatility by maturity: Volatility of T-year yield = σ × B(T) ÷ T; This is below σ for any k > 0 and falls as T rises. As k → 0, B(T) ÷ T → 1 and the volatility goes to σ, as in Ho-Lee.
Bond price volatility: Volatility of dP ÷ P = σ × B(T); Use B(T) as a duration-like measure for the short-rate factor.

How to solve Vasicek Model and Mean Reversion questions

Use this order for any Vasicek or mean-reversion question. It keeps you from mixing the short rate with longer yields.

1Read the parameters: k, θ, σ, current short rate r₀ and the horizon T. Check units are in years and the rates are in the same form (all % or all decimals).
2Decide what is asked: an expected rate, a standard deviation, a yield volatility, a bond price or a conceptual comparison.
3For expected rates, use E[r(T)] = θ + (r₀ − θ)e^(−kT). Compute e^(−kT) first.
4For rate uncertainty, use Var[r(T)] = σ²(1 − e^(−2kT)) ÷ (2k) and take the square root.
5For yields or bond prices, compute B(T) = (1 − e^(−kT)) ÷ k. Use B(T) ÷ T for yield volatility and B(T) for price volatility.
6For model comparison, check whether the drift depends on r. If yes it is Vasicek (mean reverting). If the drift is a function of time only, it is Ho-Lee.
7Interpret the answer: say if the rate is pulled up or down, and whether volatility is lower than the short-rate σ because of mean reversion.

Quickest way: Direction, then size, then compare

When to use it: Use this when the question is conceptual or when the options are far apart and you only need to rule out wrong ones.

Direction: compare r₀ with θ. Below θ means the expected rate rises. Above means it falls.
Size: the remaining gap is (r₀ − θ)e^(−kT). A high k × T makes the rate close to θ.
Volatility: any k > 0 means long-rate volatility is below short-rate volatility. Only k = 0 gives equal volatility.
Eliminate: drop options where the expected rate overshoots θ, or where variance grows without limit in Vasicek.
If the question says time-dependent drift to fit the curve, think Ho-Lee.

Common mistakes in Vasicek Model and Mean Reversion

Treating θ as the current short rate or as the expected rate at every horizon.
Students read θ as a target the rate hits straight away.
Fix: θ is only the long-run level. The expected rate moves from r₀ toward θ by the factor e^(−kT), and reaches θ only as T → ∞.
Using e^(−kT) in the variance formula instead of e^(−2kT).
The expected-rate and variance formulas look alike.
Fix: Variance has (1 − e^(−2kT)) ÷ (2k). The variance is a squared quantity, so the exponent doubles.
Saying Vasicek variance grows like σ²T.
Mixing it up with Ho-Lee or a plain normal model.
Fix: With mean reversion the variance levels off at σ² ÷ (2k). Only k = 0 gives σ²T.
Using σ as the volatility of long yields.
σ is the short-rate volatility, and the yield looks like the same thing.
Fix: The T-year yield volatility is σ × B(T) ÷ T, which is smaller than σ for k > 0.
Saying Ho-Lee and Vasicek differ only in the number of parameters.
Both are normal models with a drift term.
Fix: The key difference is that Vasicek's drift depends on the rate level and pulls toward a mean. Ho-Lee's drift is a function of time chosen to fit the market curve, with no reversion.
Forgetting that Vasicek can produce negative rates.
Students assume mean reversion keeps rates positive.
Fix: Rates are normal, so negative values have positive probability. CIR avoids this by making volatility depend on the rate.

Worked examples

Example 1

A Vasicek model has k = 0.20, θ = 5%, σ = 1% per year and a current short rate of 3%. Find the expected short rate in 2 years and its standard deviation.

Show the solution

e^(−kT) = e^(−0.4) = 0.67032.
E[r(2)] = 5% + (3% − 5%) × 0.67032 = 5% − 1.34064% = 3.659%.
Variance factor: (1 − e^(−2kT)) ÷ (2k) = (1 − e^(−0.8)) ÷ 0.4 = (1 − 0.449329) ÷ 0.4 = 1.37668.
Standard deviation = 1% × √1.37668 = 1% × 1.1733 = 1.173%.
Interpretation: the rate is expected to rise from 3% toward 5%. About 1.34% of the 2% gap remains, so about 33% of the gap has closed.

Answer: Expected rate ≈ 3.66%; standard deviation ≈ 1.17%.

Example 2

A Vasicek model has k = 0.25 and short-rate volatility σ = 1.2% per year. Compare the volatility of the 2-year and 10-year yields, and find the half-life of a rate shock.

Show the solution

For T = 2: e^(−0.5) = 0.60653, so B(2) = (1 − 0.60653) ÷ 0.25 = 1.57388. B ÷ T = 0.78694.
2-year yield volatility = 1.2% × 0.78694 = 0.944%.
For T = 10: e^(−2.5) = 0.08208, so B(10) = (1 − 0.08208) ÷ 0.25 = 3.67166. B ÷ T = 0.36717.
10-year yield volatility = 1.2% × 0.36717 = 0.441%.
Half-life = ln(2) ÷ k = 0.6931 ÷ 0.25 = 2.77 years.
Interpretation: both yields move less than the short rate, and the 10-year moves much less, because shocks fade before they affect the average rate over a long horizon.

Answer: 2-year yield volatility ≈ 0.94%; 10-year ≈ 0.44%; half-life ≈ 2.77 years.

Exam tips

Expect a mix of calculation and concept questions. Know both the formula and the story: reversion lowers long-rate volatility.
Always work out e^(−kT) first and keep four decimals. Most arithmetic errors start there.
Watch for the contrast with Ho-Lee and with the normal model of zero drift. Flat volatility term structure points to no mean reversion.
Check what σ means in the question. It is the short-rate volatility, not the volatility of a long yield or a bond price.
If a question asks about the effect of raising k, answer: faster pull to θ, lower long-rate volatility, lower long-run variance σ² ÷ (2k).

Practice questions from The Art of Term Structure Models: Drift

Vasicek Model and Mean Reversion: frequently asked questions

What does mean reversion mean in the Vasicek model?

It means the drift pushes the short rate back toward a long-run mean θ. The pull is stronger the further the rate is from θ and the larger k is. The rate is still random, but it does not wander off without limit.

How does mean reversion affect the term structure of volatility?

It makes volatility fall with maturity. The T-year yield volatility is σ × B(T) ÷ T, which is below σ and decreases as T rises. With no mean reversion, as in Ho-Lee, volatility is the same across maturities.

What is the difference between the Vasicek and Ho-Lee models?

Vasicek has a drift k(θ − r) that depends on the rate and pulls toward a mean. Ho-Lee has a drift that depends only on time and is chosen to fit the market curve, with no mean reversion. Ho-Lee variance grows with time, while Vasicek variance levels off.

Can the Vasicek model give negative interest rates?

Yes. Rates are normally distributed, so negative values have positive probability. This is a known drawback, and models such as CIR address it by making volatility depend on the rate.