A one‑factor short‑rate model in which the instantaneous short rate r_t follows an Ornstein–Uhlenbeck (mean‑reverting Gaussian) diffusion dr_t = a(b − r_t) dt + σ dW_t; the model is affine, admits closed‑form expressions for zero‑coupon bond prices and yields a Gaussian distribution for future short rates, and therefore can produce negative rates.

Principle

Mean reversion and affine (linear in r_t) dynamics imply analytic bond pricing: bond prices are exponential affine functions of r_t, and the short rate has Gaussian transitions determined by a, b and σ.

Demonstration

Illustrative scenario → At time t an analyst observes r_t and uses the Vasicek closed‑form zero‑coupon bond price P(t,T)=exp(A(t,T) − B(t,T) r_t), where A and B are functions of (a,b,σ); consequence → the bond price and its sensitivities are computed without numerical simulation.

Misapplication

Treating the Vasicek short rate as guaranteed nonnegative. Why plausible → Gaussian shocks may appear to model small fluctuations; semantic error → the Gaussian diffusion admits negative rates; correct interpretation → Vasicek can produce negative rates and is not appropriate where positivity is essential.

Consequence

Analytic tractability: closed‑form bond and bond‑option formulas enable fast pricing and calibration under constant parameters; modeling consequence → possible negative short rates and limited ability to fit an arbitrary initial term structure with constant parameters.

Reversal

When parameters are made time‑dependent or multiple factors are added the Gaussian affine structure may be preserved but the original stationary Gaussian distribution and simple constant‑parameter calibration properties change; enforcing positivity requires replacing the diffusion with non‑Gaussian (e.g., square‑root) terms.

Boundary

Clearly within → one‑factor Ornstein–Uhlenbeck short‑rate model with constant parameters. Boundary case → σ→0 yields deterministic exponential approach to b. Clearly outside → square‑root short‑rate models (CIR) or forward‑rate frameworks (HJM) that either enforce positivity or model the full forward curve.

Semantic Tension

Tractability (closed‑form prices, analytic Greeks) ↔ Realism (rate positivity and flexible fit to market term structure).

Synthesis

Vasicek cleanly exhibits how an affine Gaussian short‑rate produces analytic prices: it is a useful, tractable benchmark but trades off realism (positivity and flexible curve fit) for closed‑form simplicity.