Short-Rate Interest Rate Models › The Vasicek Model

Oldřich Vašíček's 1977 model established the foundation for modern interest rate modeling. Before this, there was no standard framework for relating interest rate dynamics to bond prices. The Vasicek model assumes the short rate follows an Ornstein-Uhlenbeck process, a continuous-time stochastic process originally developed by physicists to model the velocity of particles subject to friction:

$$dr(t) = \kappa(\theta - r(t))\,dt + \sigma\,dW(t)$$

This equation captures key economic behaviors through terms that govern rate evolution:

• $\kappa$ speed of mean reversion, controlling how quickly rates return to the long-run mean
• $\theta$: long-run mean level toward which the process gravitates
• $r(t)$: instantaneous short rate at time $t$
• $\sigma$: volatility of the short rate
• $W(t)$: standard Brownian motion under the risk-neutral measure

The drift term $\kappa(\theta - r(t))$ provides mean reversion, the model's defining characteristic. To understand this mechanism, consider what happens in different scenarios. When $r(t) > \theta$, the current rate exceeds the long-run mean, making the term $(\theta - r(t))$ negative. Multiplied by the positive constant $\kappa$, this produces a negative drift, pushing rates down toward $\theta$. Conversely, when $r(t) < \theta$, the drift becomes positive, pulling rates up. This self-correcting behavior ensures that rates oscillate around $\theta$ rather than wandering off to arbitrarily high or low values.

The parameter $\kappa$ controls the strength of this gravitational pull. A large $\kappa$ means rates snap back quickly after any displacement from $\theta$, while a small $\kappa$ allows rates to deviate from the long-run mean for extended periods. Empirically, interest rates do exhibit mean reversion: extremely high rates tend to fall as central banks loosen policy to stimulate growth, while very low rates eventually rise as economies recover. The Vasicek model captures this economic reality through its mathematical structure.

The Vasicek SDE is linear and has an explicit solution, making it useful for practical applications. We solve this using the integrating factor method, a technique familiar from ordinary differential equations. The key insight is to transform the equation into one that can be integrated directly.

Consider the transformation $f(t) = r(t)e^{\kappa t}$. This choice is motivated by the observation that if we can eliminate the $r(t)$ term from the right-hand side, integration becomes straightforward. Applying Itô's Lemma to this transformation, we obtain:

$$\begin{aligned} d(r(t)e^{\kappa t}) &= e^{\kappa t}dr(t) + \kappa e^{\kappa t}r(t)dt && \text{(product rule)} \\ &= e^{\kappa t}[\kappa(\theta - r(t))dt + \sigma dW(t)] + \kappa e^{\kappa t}r(t)dt && \text{(substitute } dr(t)\text{)} \\ &= (\kappa\theta e^{\kappa t} - \kappa r(t)e^{\kappa t})dt + \sigma e^{\kappa t}dW(t) + \kappa r(t)e^{\kappa t}dt && \text{(expand drift)} \\ &= \kappa\theta e^{\kappa t}dt + \sigma e^{\kappa t}dW(t) && \text{(simplify)} \end{aligned}$$

The $-\kappa r(t)e^{\kappa t}$ term from the original drift cancels with the term arising from differentiating the exponential, which is the purpose of this transformation. The resulting expression contains only the deterministic term $\kappa\theta e^{\kappa t}dt$ and the stochastic term $\sigma e^{\kappa t}dW(t)$, both of which can be integrated directly.

Integrating from $t$ to $T$:

$$r(T)e^{\kappa T} - r(t)e^{\kappa t} = \int_t^T \kappa\theta e^{\kappa s}ds + \int_t^T \sigma e^{\kappa s}dW(s)$$

The first integral on the right-hand side is a standard calculus integral that evaluates to $\theta(e^{\kappa T} - e^{\kappa t})$. The second integral is a stochastic integral against Brownian motion. Multiplying both sides by $e^{-\kappa T}$ and rearranging yields the explicit solution:

$$r(T) = r(t)e^{-\kappa(T-t)} + \theta(1 - e^{-\kappa(T-t)}) + \sigma\int_t^T e^{-\kappa(T-s)}\,dW(s)$$

Each component of this solution has an intuitive interpretation:

• $r(T)$ short rate at future time $T$
• $r(t)$: current short rate at time $t$
• $\kappa$: speed of mean reversion
• $\theta$: long-run mean level
• $\sigma$: volatility of the short rate
• $W(s)$: standard Brownian motion

The first two terms are deterministic, showing how the initial rate $r(t)$ decays exponentially toward $\theta$. The term $r(t)e^{-\kappa(T-t)}$ represents the "memory" of the initial condition, which fades exponentially over time. The term $\theta(1 - e^{-\kappa(T-t)})$ is the complementary contribution from the long-run mean, which grows as the initial condition fades. As $T - t$ becomes large, the first term approaches zero and the second approaches $\theta$, so the expected future rate converges to the long-run mean regardless of where it started.

The stochastic integral $\sigma\int_t^T e^{-\kappa(T-s)}\,dW(s)$ represents accumulated random shocks, with exponential weighting that gives more influence to recent shocks. Shocks that occurred long ago have been "forgotten" due to the mean-reverting drift, while recent shocks still influence the current rate. This weighting function $e^{-\kappa(T-s)}$ is largest when $s = T$ (equal to 1) and decays for earlier times.

Since the stochastic integral of a deterministic function against Brownian motion is normally distributed (a fundamental property of Itô integrals), $r(T)$ conditional on $r(t)$ follows a normal distribution:

$$r(T) | r(t) \sim \mathcal{N}\left(\mu(t,T), v^2(t,T)\right)$$

The conditional mean and variance can be computed explicitly:

$$\begin{aligned} \mu(t,T) &= r(t)e^{-\kappa(T-t)} + \theta(1 - e^{-\kappa(T-t)}) \\ v^2(t,T) &= \frac{\sigma^2}{2\kappa}\left(1 - e^{-2\kappa(T-t)}\right) \end{aligned}$$

These expressions reveal the long-run behavior of the process:

• $\mu(t,T)$ conditional mean of $r(T)$ given $r(t)$
• $v^2(t,T)$: conditional variance of $r(T)$ given $r(t)$
• $\frac{\sigma^2}{2\kappa}$: long-run variance of the short rate
• $r(t)$: current short rate
• $\kappa$: mean reversion speed
• $\theta$: long-run mean
• $\sigma$: volatility
• $T-t$: time horizon

As the horizon $T - t$ grows large, the conditional mean approaches $\theta$ regardless of the starting point $r(t)$. Similarly, the conditional variance approaches the limiting value $\frac{\sigma^2}{2\kappa}$, which represents the steady-state variance of the rate process. This limiting variance decreases with stronger mean reversion (larger $\kappa$) because the process is pulled back toward $\theta$ more frequently, reducing dispersion.

The Vasicek model yields analytical formulas for bond prices. The Vasicek model belongs to the class of affine term structure models. This means bond prices have the exponential-affine form

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

This structure is powerful because the bond price depends on the current short rate $r(t)$ in a very specific way: exponentially, with a coefficient $B(t,T)$ that depends only on the time to maturity and model parameters. The components of this expression are:

• $P(t,T)$ price of a zero-coupon bond maturing at $T$
• $A(t,T), B(t,T)$: deterministic functions of time depending on model parameters
• $r(t)$: current short rate

To derive the specific forms of $A$ and $B$, we substitute the assumed form into the fundamental partial differential equation governing bond prices and match coefficients. For the Vasicek model, this procedure yields:

$$\begin{aligned} B(t,T) &= \frac{1 - e^{-\kappa(T-t)}}{\kappa} \\ A(t,T) &= \exp\left[\left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(t,T) - (T-t)) - \frac{\sigma^2}{4\kappa}B(t,T)^2\right] \end{aligned}$$

Understanding what these functions represent helps build intuition for bond pricing under mean reversion:

• $A(t,T), B(t,T)$ deterministic functions determining the bond price structure
• $\kappa, \theta, \sigma$: Vasicek model parameters
• $T-t$: time to maturity

The function $B(t,T)$ measures the sensitivity of the bond price to the short rate, which is closely related to the bond's duration. When $\kappa$ is small, $B(t,T)$ approaches $T - t$, and bond price sensitivity grows linearly with maturity. When $\kappa$ is large, the mean reversion prevents long-term rate changes from persisting, so sensitivity saturates at $1/\kappa$ for very long maturities.

The function $A(t,T)$ captures two effects. First, it incorporates the effect of the mean-reverting drift toward $\theta$. Second, it includes a convexity adjustment arising from volatility. This adjustment reflects the fact that bondholders benefit from interest rate volatility due to the convex relationship between rates and prices: when rates fall, prices rise more than they fall when rates rise by the same amount.

These closed-form expressions make the Vasicek model computationally tractable. Given parameters $(\kappa, \theta, \sigma)$ and the current short rate $r(t)$, you can instantly compute the price of any zero-coupon bond and therefore construct the entire yield curve. This stands in contrast to models without analytical solutions, which require time-consuming numerical methods.

The continuously compounded yield for maturity $\tau = T - t$ follows directly from the bond price:

$$y(t,T) = -\frac{\ln P(t,T)}{\tau} = \frac{B(t,T)r(t) - \ln A(t,T)}{\tau}$$

This expression shows that the yield is an affine (linear plus constant) function of the short rate:

• $y(t,T)$ continuously compounded yield for maturity $T$
• $P(t,T)$: zero-coupon bond price
• $A(t,T), B(t,T)$: Vasicek model functions defined above
• $\tau = T-t$: time to maturity
• $r(t)$: current short rate

The affine structure extends to yields, which is why this class of models is called "affine term structure models." The linearity of yields in the short rate has practical implications: parallel shifts in the short rate produce parallel shifts in the yield curve, scaled by the maturity-dependent factor $B(t,T)/\tau$.

A critical feature of the Vasicek model is that $r(t)$ is normally distributed, which means it can take negative values with positive probability. For decades, this was considered a significant drawback of the model. How could interest rates be negative? The reasoning went that if deposit rates were negative, banks would pay you to borrow money, which seemed economically absurd. Rational agents would simply hold cash rather than accept negative returns.

Then came the post-2008 financial crisis era. Central banks in Europe, Japan, and Switzerland pushed policy rates below zero in an effort to stimulate lending and economic activity. Suddenly, negative rates moved from theoretical possibility to empirical reality. Banks charged depositors for holding their money, and some government bonds traded at negative yields.

The Vasicek model's ability to generate negative rates transformed from a bug to a feature for modeling modern interest rate environments. While the model was not designed with negative rates in mind, its mathematical structure accommodates this reality naturally. This historical reversal illustrates an important principle: model properties that seem problematic under one economic regime may become valuable under different circumstances.