Short-Rate Interest Rate Models › The Short Rate and Why We Model It

The short rate is a theoretical construct that is the fundamental building block of interest rate modeling. You cannot observe it directly in markets because no instrument has an instantaneous maturity. The closest observable proxies are overnight rates like the federal funds rate or overnight repo rates, but even these apply to discrete time periods rather than true instants. Despite its abstract nature, the short rate is a useful modeling device because it gives a parsimonious way to describe the entire yield curve with a single state variable.

Short rates are fundamental because of their connection to bond prices. If we know how $r(t)$ evolves from now until some future time $T$, we can compute the discount factor from the fundamental relationship:

$$P(t,T) = \mathbb{E}^Q\left[\exp\left(-\int_t^T r(s)\,ds\right)\right]$$

This equation encapsulates the pricing framework. Let's examine each component:

• $P(t,T)$ price at time $t$ of a zero-coupon bond paying a face value of 1 at time $T$
• $\mathbb{E}^Q$: expectation taken under the risk-neutral measure $Q$
• $r(s)$: instantaneous short rate at time $s$

The integral $\int_t^T r(s)\,ds$ accumulates the short rate over the entire period from $t$ to $T$. This accumulated rate, when exponentiated with a negative sign, produces the discount factor. The expectation operator is necessary because $r(s)$ is random: we do not know today what the short rate will be at future times $s$. By taking the risk-neutral expectation, we obtain a price consistent with no-arbitrage principles.

This formula, which follows from the risk-neutral valuation framework we developed in Chapter 4 of this Part, tells us that the bond price equals the expected value of the discount factor, where discounting occurs at the random path of short rates. This approach generalizes: once we specify a stochastic process for $r(t)$, we can price any interest rate derivative by computing appropriate expectations of payoffs discounted at this random rate.

A single stochastic differential equation (SDE) for $r(t)$ determines the term structure. Different short-rate models make different assumptions about this SDE, leading to different yield curve shapes and dynamics. The choice of SDE reflects your beliefs about how interest rates behave: whether they exhibit mean reversion, how their volatility changes with the rate level, and whether they can become negative.