One-Factor Short Rate Models: Vasicek, CIR and Hull-White

Understand One-Factor Short Rate Models

Key rules to remember

Vasicek SDE

dr = α(μ − r)dt + σ dW

α > 0 is the speed of mean reversion, μ the long-run mean, σ constant volatility. Treat the parameters as risk-neutral when pricing bonds.

Vasicek distribution of r(t)

r(t) ~ Normal with mean μ + (r(0) − μ)e^(−αt) and variance σ²(1 − e^(−2αt)) ÷ (2α)

As t → ∞, the mean tends to μ and the variance tends to σ² ÷ (2α). Negative rates have positive probability.

Bond price (affine form)

P(t,T) = exp(A(τ) − B(τ)r(t)), τ = T − t

Vasicek and Hull-White take this form. CIR is written as A(τ)exp(−B(τ)r) with a different A and B.

Vasicek B(τ)

B(τ) = (1 − e^(−ατ)) ÷ α

The same B is used in Hull-White. It measures the sensitivity of ln P to r.

Vasicek A(τ)

A(τ) = (μ − σ²/(2α²))(B(τ) − τ) − σ²B(τ)² ÷ (4α)

Here μ is the risk-neutral long-run mean. If a market price of risk is used, adjust μ accordingly.

Vasicek long-run yield

R(∞) = μ − σ² ÷ (2α²)

Yield is R(τ) = −ln P ÷ τ. The long-run yield is below μ because of the convexity term.

CIR SDE and Feller condition

dr = α(μ − r)dt + σ√r dW; r stays above 0 if 2αμ ≥ σ²

If the condition fails, r can touch zero but is reflected, so it remains non-negative.

CIR bond price

P = A(τ)e^(−B(τ)r), γ = √(α² + 2σ²), B = 2(e^(γτ) − 1) ÷ [(γ + α)(e^(γτ) − 1) + 2γ], A = [2γe^((α+γ)τ/2) ÷ ((γ + α)(e^(γτ) − 1) + 2γ)]^(2αμ/σ²)

These are the risk-neutral forms with no extra market price of risk term. Check which version your question gives.

Hull-White SDE

dr = (θ(t) − αr)dt + σ dW

θ(t) is chosen to match the initial forward curve f(0,t): θ(t) = ∂f(0,t)/∂t + αf(0,t) + σ²(1 − e^(−2αt)) ÷ (2α).

How to solve One-Factor Short Rate Models questions

1. 1Identify the model from the SDE. Constant σ means Vasicek or Hull-White. A √r term means CIR. A time-dependent θ(t) means Hull-White.
2. 2Write down the parameters and check they are risk-neutral. State any assumption about the market price of risk.
3. 3For distribution questions, use the known distribution. Vasicek and Hull-White are normal. Compute the mean and variance at the required time, then standardise.
4. 4For bond prices, compute τ = T − t, then B(τ), then A(τ), then P = exp(A − Br) in Vasicek or P = A·e^(−Br) in CIR. Keep six decimal places in the exponent.
5. 5For yields, use R = −ln P ÷ τ. Quote it as a continuously compounded rate unless told otherwise.
6. 6For discussion parts, link each point to a model feature: negative rates, mean reversion, volatility level dependence, fit to the initial curve, and tractability.
7. 7For calibration, say what is fitted. Vasicek and CIR parameters are fitted by least squares to market prices or yields. Hull-White fits θ(t) to the initial curve exactly, and α and σ to volatility data such as option prices.
8. 8Finish by checking reasonableness. Bond prices must be below 1 when yields are positive. For positive yields, prices fall as τ rises. Prices always fall as r rises.

Quickest way: Compare-the-three table in your head

Common mistakes in One-Factor Short Rate Models

• Saying Vasicek rates can never be negative, or that CIR rates can be negative.

  Students remember that mean reversion pulls rates up, and confuse that with a floor at zero.

  Fix: Vasicek r(t) is normal, so P(r < 0) > 0. CIR has σ√r, which removes noise at zero, so r ≥ 0. Mean reversion alone does not stop negative values.

• Forgetting the convexity term σ²/(2α²) in the Vasicek A(τ) formula.

  Students use μ directly in place of μ − σ²/(2α²).

  Fix: Write the formula down in full before substituting. Remember that the long-run yield is μ − σ²/(2α²), not μ.

• Using B(τ) = τ or leaving out division by α in B(τ).

  B looks like the integral of e^(−αs) and the 1/α gets dropped.

  Fix: B(τ) = (1 − e^(−ατ)) ÷ α. Check that B tends to τ as α → 0 and to 1/α as τ → ∞.

• Claiming Hull-White fits the whole yield curve with no parameters.

  Students stop at 'fits the initial curve' and ignore α and σ.

  Fix: θ(t) is fitted to the initial forward curve. α and σ still need calibrating, usually to the volatility of bond or option prices.

• Mixing up real-world and risk-neutral parameters.

  Students plug the observed long-run mean into bond price formulae without checking.

  Fix: Bond prices use risk-neutral parameters. If a market price of risk λ is given, adjust the drift before using the formulae, and state this assumption.

• Applying the Feller condition the wrong way round.

  The inequality is memorised without the meaning.

  Fix: 2αμ ≥ σ² means the drift is strong enough to keep r from hitting zero. If it fails, r can reach zero but not go negative.

Worked examples