Vasicek Model and Mean Reversion: Formula and Interpretation

Understand Vasicek Model and Mean Reversion

Key formulas to remember

Vasicek short-rate process

dr = k(θ − r)dt + σ dW

k is the speed of mean reversion, θ the long-run rate, σ the constant volatility. Under the risk-neutral measure, θ is the risk-neutral long-run level.

Expected rate over a short step

E[Δr] = k(θ − r) × dt

Positive if r < θ, negative if r > θ.

Expected rate at horizon T

E[r(T)] = θ + (r(0) − θ) × e^(−kT)

The gap to θ shrinks by the factor e^(−kT).

Variance of the short rate at T

Var[r(T)] = σ² × (1 − e^(−2kT)) ÷ (2k)

For k > 0, this tends to σ² ÷ (2k) as T grows, so long-horizon variance is bounded.

Volatility of the T-year zero-coupon yield

Yield volatility = σ × (1 − e^(−kT)) ÷ (kT)

This is the volatility of the T-year zero-coupon yield (the T-year spot rate). It equals σ × B(T) ÷ T, where B(T) = (1 − e^(−kT)) ÷ k. It does not apply to the short rate itself, whose volatility stays σ. It falls as T increases, so the volatility term structure slopes downward. Under Ho-Lee, B(T) = T, so the zero-coupon yield volatility is σ × T ÷ T = σ at every maturity.

Half-life of a shock

Half-life = ln(2) ÷ k

Time for the gap to θ to halve.

Limit k → 0

k(θ − r)dt → 0, so dr = σ dW

As k goes to zero the drift term vanishes. The model becomes a driftless, constant-volatility normal model. Mean reversion disappears and long-rate volatility equals short-rate volatility.

How to solve Vasicek Model and Mean Reversion questions

1. 1Identify the parameters: k, θ, σ, current r, and the horizon T.
2. 2Decide what is asked: expected rate, variance, volatility of a long rate, or a comparison with Ho-Lee.
3. 3For the expected rate, compute θ + (r − θ) × e^(−kT). For one short step, use k(θ − r)dt.
4. 4For variance or standard deviation, use σ² × (1 − e^(−2kT)) ÷ (2k) and take the square root if needed.
5. 5For the volatility of long rates, apply the damping factor (1 − e^(−kT)) ÷ (kT) to σ.
6. 6Check direction: r below θ means upward drift; r above θ means downward drift.
7. 7Interpret: state that mean reversion makes long rates less volatile than short rates, and note any negative-rate possibility.

Quickest way: Gap, decay, damp

Common mistakes in Vasicek Model and Mean Reversion

• Treating θ as the current rate or as the expected rate at all horizons.

  The name 'long-run level' sounds like a forecast for tomorrow.

  Fix: θ is only the level toward which rates are pulled. The expected rate at T is θ + (r − θ)e^(−kT), which sits between r and θ.

• Getting the drift sign wrong.

  Students compute (r − θ) instead of (θ − r).

  Fix: Always write k(θ − r). If r is below θ, drift is positive.

• Saying Vasicek rates cannot be negative.

  Mean reversion to a positive θ feels like a floor.

  Fix: Shocks are normal with constant volatility, so rates can be negative. Cox-Ingersoll-Ross uses dr = k(θ − r)dt + σ√r dW, so volatility depends on the rate level and rates stay non-negative. In that model the rate cannot reach zero if the Feller condition 2kθ ≥ σ² holds.

• Confusing Vasicek with Ho-Lee on volatility of long rates.

  Both are normal models with constant σ.

  Fix: Ho-Lee has no mean reversion, so all rates have the same volatility. Vasicek's reversion makes long-rate volatility lower than short-rate volatility.

• Using e^(−k) instead of e^(−kT), or mixing units of k and T.

  Rushing and forgetting the horizon.

  Fix: Write kT first and check that k is per year when T is in years.

• Thinking higher k means higher volatility.

  A stronger pull feels like more movement.

  Fix: Higher k pulls shocks away faster, so long-horizon variance σ² ÷ (2k) falls.

Worked examples