Chapter 31: Equilibrium Models of the Short Rate

Background

Equilibrium models start with assumptions about economic variables and derive a process for the short rate $r$. The initial term structure is an output of the model, not an input. These are also called "endogenous" term structure models.

The short rate process takes the general form:

$dr=m(r)dt+s(r)dz$

One-Factor Models

In one-factor models, the entire term structure is driven by a single source of uncertainty — the short rate $r$. All rates move together (perfectly correlated changes).

The Rendleman and Bartter Model

Assumes the short rate follows geometric Brownian motion:

$dr=\mu rdt+\sigma rdz$

This means interest rates can grow without bound, and rates are always positive. However, it does not incorporate mean reversion — a key feature of interest rates.

The Vasicek Model (1977)

$dr=a(b-r)dt+\sigma dz$

• $a$ = speed of mean reversion
• $b$ = long-run mean short rate
• $\sigma$ = volatility

Key features:

• Mean reversion: The short rate is pulled toward $b$ at speed $a$
• Normal distribution: $r$ can become negative (a flaw, but analytically convenient)
• Closed-form solutions exist for zero-coupon bond prices and European options on bonds

The zero-coupon bond price:

$P(t,T)=A(t,T)\cdot e^{-B(t,T)\cdot r(t)}$

where $A$ and $B$ are deterministic functions.

The Cox–Ingersoll–Ross (CIR) Model (1985)

$dr=a(b-r)dt+\sigma \sqrt{r}dz$

• Same mean reversion as Vasicek
• Volatility is proportional to $\sqrt{r}$ (prevents negative rates)
• If $2ab\ge {\sigma }^2$, the short rate never reaches zero

The CIR model also has closed-form solutions for bond prices, with $A$ and $B$ functions being modified versions of Vasicek's.

Real-World vs. Risk-Neutral Processes

The real-world process for the short rate differs from the risk-neutral process:

$\text{Real-world drift}=\text{Risk-neutral drift}+\lambda \cdot s(r)$

where $\lambda$ is the market price of interest rate risk. The models described above are risk-neutral processes — they directly give prices. To calibrate to historical data, the market price of risk must be estimated.

Estimating Parameters

Parameters can be estimated from:

• Historical time series of short-term interest rates (real-world parameters)
• Cross-section of current bond prices (risk-neutral parameters)
• Maximum likelihood or generalized method of moments (GMM)

More Sophisticated Models

Two-Factor Models

Add a second stochastic variable (e.g., the long-run mean itself follows a stochastic process, or add a stochastic mean-reversion level). Example: two-factor Vasicek adds a stochastic $b(t)$.

Multi-Factor Models

More factors allow richer term structure dynamics (non-parallel shifts). In practice, two to three factors capture most yield curve variation.

The Hull–White Two-Factor Model

Adds a mean-reverting process for the reversion level itself. This allows the short rate to mean-revert to a level that itself changes through time.

Limitations of Equilibrium Models

• The model output rarely matches today's market bond prices exactly
• Single-factor models assume perfect correlation across maturities (not true in reality)
• Cannot exactly fit the observed volatility structure

These limitations motivated the development of no-arbitrage models (Chapter 32), which take today's term structure as an input.