We need to choose a quantity to serve as a state variable that determines the interest rate term structure and its evolution in time. The first generation of stochastic interest rate models use the instantaneous short rate as the state variable. The two key advantages of short-rate models are their general simplicity and the fact that they often lead to analytic formulae for bonds and associated vanilla options. The tractability of short-rate models means that the price of a given derivative can often be computed quickly, important in situations where a large number of securities need to be valued. Indeed, throughout this chapter, we focus on short-rate models that allow discount bonds to be priced in closed form.

One-factor models assume that the entire interest rate term structure is driven by a one-dimensional Wiener process. Such models are usually suitable when pricing securities that depend on a single rate only, but for more complex products which depend on two or more different rates we may need to move to a multi-factor model driven by multi-dimensional Brownian motion. In the final section we present one of the most popular multi-factor short-rate models, the two-factor Hull–White model.

A weakness of the short-rate approach is that the instantaneous short rate is a mathematical idealisation rather than something that can be observed directly in the market. In the past decade, short-rate models have, to some extent, been superseded by the LIBOR market model (covered in Chapter 5), in which the stochastic state variable is a set of benchmark forward LIBOR rates. Nonetheless, short-rate models are particularly useful and remain popular due to their analytic tractability.

We continue to work under Assumption 2.1, which stipulates the existence of a probability measure Q (the risk-neutral measure) equivalent to P such that the value of any security discounted by the money market account is a martingale under Q.