Short-rate models are fundamental tools in interest rate quant roles. They describe the evolution of the instantaneous short rate, allowing quants to price bonds, swaps, swaptions, and construct arbitrage-free term structures. Vasicek, CIR, and Hull–White are among the most common models tested in interviews because of their tractability and closed-form solutions.
1. What is a short-rate model?
A short-rate model specifies the stochastic process followed by the instantaneous short rate $r_t$. It defines the drift, volatility, and randomness of interest rate movements.
General form:
\[dr_t = \mu(t,r_t)dt + \sigma(t,r_t)dW_t.\]Given $r_t$, the time-$t$ price of a zero-coupon bond maturing at $T$ is:
\[P(t,T) = \mathbb{E}_t \left[ e^{-\int_t^T r_s ds} \right].\]Short-rate models allow closed-form bond pricing under certain assumptions, making them widely used for fixed-income derivatives.
2. Vasicek Model
SDE: \(dr_t = a(b - r_t) dt + \sigma dW_t\)
Parameters:
- $a$: speed of mean reversion
- $b$: long-term mean
- $\sigma$: constant volatility
Key properties:
- Rates revert to $b$ at speed $a$.
- Gaussian distribution → negative rates possible.
- Closed-form bond price.
Bond price: \(P(t,T) = \exp\left(A(t,T) - B(t,T) r_t\right)\)
Where: \(B(t,T) = \frac{1 - e^{-a(T-t)}}{a}\) \(A(t,T) = \left(b - \frac{\sigma^2}{2a^2}\right)(B(t,T) - (T-t)) - \frac{\sigma^2}{4a}B(t,T)^2\)
Pros: analytically simple, widely used for intuition and pedagogy.
Cons: allows negative rates; cannot match the initial yield curve exactly.
3. CIR Model (Cox–Ingersoll–Ross)
SDE: \(dr_t = a(b - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t\)
Differences from Vasicek:
- Volatility increases with the level of rates $\sqrt{r_t}$.
- Rates remain non-negative if the Feller condition holds: \(2ab \ge \sigma^2\)
Bond pricing is also available in closed form: \(P(t,T)=A(t,T)\exp(-B(t,T)r_t)\)
Pros:
- Ensures positive rates (unlike Vasicek).
- Captures level-dependent volatility.
Cons:
- Less flexible; cannot match the initial yield curve exactly.
- Still a one-factor model.
Common in credit risk and Monte Carlo simulation of short rates.
4. Hull–White One-Factor Model (Extended Vasicek)
SDE: \(dr_t = a(\theta(t) - r_t)\,dt + \sigma\,dW_t\)
Here, $\theta(t)$ is a time-dependent function chosen so that the model fits the current market yield curve exactly.
This is the key distinction:
- Vasicek and CIR do not fit today’s term structure.
- Hull–White does, by construction.
Bond price remains affine: \(P(t,T) = A(t,T)\exp(-B(t,T)r_t),\) but with $A(t,T)$ computed from the observed yield curve and (\theta(t)).
Pros:
- Accurately fits today’s yield curve.
- Closed-form pricing for swaps and vanilla IR derivatives.
- Used widely in practice.
Cons:
- Gaussian → negative rates possible.
- Still one-factor → cannot capture slope/twist of the curve.
5. Model Comparison
| Model | Mean Reversion | Volatility | Rates Always Positive | Fits Initial Curve | Typical Use |
|---|---|---|---|---|---|
| Vasicek | Yes | Constant | No | No | Intro modeling, analytical pricing |
| CIR | Yes | $\sigma\sqrt{r}$ | Yes (Feller) | No | Credit risk, simulation |
| Hull–White | Yes | Constant | No | Yes | Curve calibration, swaption pricing |
6. Calibration Approaches
Vasicek / CIR:
- Estimate $a, b, \sigma$ from historical short rates using MLE or GMM.
- Fit to bond prices using least-squares.
Hull–White:
- Calibrate $\theta(t)$ directly from the yield curve.
- Calibrate $(a, \sigma)$ to swaption or cap/floor vol surfaces.
Common interview question:
“Explain how to calibrate the Hull–White one-factor model to the market.”
Answer:
Fit $\theta(t)$ to match the initial term structure, and fit $a$ and $\sigma$ to match market swaption/cap volatilities.
7. When to use each model
- Use Vasicek when you need tractability and simple closed-form intuition.
- Use CIR when positivity of rates is required or when rate-level-dependent volatility is important.
- Use Hull–White when you need an arbitrage-free model that fits the current yield curve exactly.
For more complex curve dynamics, two-factor extensions such as G2++ or multi-factor Hull–White models are used.
8. Limitations of One-Factor Short-Rate Models
- Cannot capture multi-factor yield curve movements (twist, butterfly).
- Gaussian models can produce negative rates.
- Limited flexibility in matching volatility surfaces.
- Oversimplify correlation structures across maturities.
Modern desks often use HJM models or the Libor Market Model (LMM), but Vasicek, CIR, and Hull–White remain core interview topics.
Summary
Short-rate models describe the stochastic evolution of interest rates and provide analytical pricing tools for fixed-income securities.
Vasicek provides simplicity, CIR enforces positivity, and Hull–White ensures exact calibration to the observed yield curve. A clear understanding of their dynamics, calibration methods, and use cases is essential for fixed-income quantitative roles.