advanced
10 min read
Finance & economics · Topic
Interest Rate Models
Interest rate models describe the stochastic evolution of the term structure of interest rates. Short-rate models (Vasicek, CIR), HJM, and LIBOR market models form the main families — each balancing tractability against realism for pricing and hedging fixed-income derivatives.
Vasicek model
The short rate $r_t$ follows an Ornstein-Uhlenbeck process:
\[dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t\]- $\kappa$: mean-reversion speed
- $\theta$: long-run mean
- $\sigma$: volatility
Bond price: $P(t,T) = e^{A(t,T) - B(t,T)r_t}$ — affine in $r_t$.
Allows negative rates (a flaw). Analytical bond and option prices exist.
Cox-Ingersoll-Ross (CIR)
\[dr_t = \kappa(\theta - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t\]The $\sqrt{r_t}$ diffusion term ensures $r_t \geq 0$ (Feller condition: $2\kappa\theta > \sigma^2$). Bond prices remain affine. The distribution of $r_t$ is non-central chi-squared.
Heath-Jarrow-Morton (HJM) framework
Models the entire forward rate curve $f(t,T)$ directly:
\[df(t,T) = \alpha(t,T)\,dt + \sigma(t,T)^\top dW_t\]No-arbitrage imposes the HJM drift condition: $\alpha(t,T) = \sigma(t,T)^\top\int_t^T\sigma(t,s)\,ds$.
The Vasicek and CIR models are special cases of HJM.