The Vasicek Model¶
The Vasicek model (1977) is the foundational mean-reverting short-rate model in the affine class. Under the risk-neutral measure \(\mathbb{Q}\), the short rate follows an Ornstein-Uhlenbeck process:
This short overview places the model in the general short-rate framework. All detailed derivations live in the dedicated Vasicek model folder.
Where to find each topic¶
Recall (see § General Short-Rate Framework): the bond pricing PDE and the expectation \(P(t,T) = \mathbb{E}^{\mathbb{Q}}_t[\exp(-\int_t^T r_s\,ds)]\) apply to any Markov short-rate model, including Vasicek.
Recall (see § Affine Term Structure): Vasicek is the canonical Gaussian affine model, with bond price \(P(t,T) = \exp(A(\tau) - B(\tau)r_t)\) arising from Riccati ODEs (linear in this case since \(\sigma\) is constant in \(r\)).
| Topic | Canonical location |
|---|---|
| SDE and OU representation | § Vasicek SDE and OU Process |
| Explicit solution and Gaussian distribution of \(r_t\) | § Explicit Solution and Distribution |
| Zero-coupon bond pricing (PDE, ansatz, \(A\) and \(B\) formulas) | § Zero-Coupon Bond Pricing |
| Yield curve shapes and inversions | § Yield Curve Shapes and Inversions |
| Change of measure and market price of risk | § Change of Measure |
| Bond options via Jamshidian | § Bond Options (Jamshidian) |
| Caplets and swaptions | § Caplet and Swaption Formulas |
| Negative rates | § Negative Rate Problem |
| Calibration | § Calibration |
| Monte Carlo simulation | § Monte Carlo Simulation |
For Hull-White's time-dependent drift extension that fits the initial curve exactly, see § Hull-White Model. For side-by-side comparisons with CIR and Hull-White, see § Vasicek vs CIR vs Hull-White.
Key takeaways¶
- Vasicek: \(dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t\), Ornstein-Uhlenbeck dynamics.
- Short rate is Gaussian: closed forms but negative rates possible.
- Affine bond prices: \(P(t,T) = \exp(A(\tau) - B(\tau)r_t)\) with \(B(\tau) = (1 - e^{-\kappa\tau})/\kappa\).
- Hull-White extends Vasicek with \(\theta(t)\) for exact curve fit.
Further reading¶
- Vasicek, O. (1977), "An Equilibrium Characterization of the Term Structure".
- Hull & White (1990), "Pricing Interest-Rate-Derivative Securities".
- Brigo & Mercurio, Interest Rate Models, Chapter 3.