This chapter introduces basic definitions and results in term structure modeling. It lays the theoretical foundations for interest rate models. A few simple models are presented at the end of the chapter.

Terminology

A period denotes a unit of elapsed time throughout this chapter. Hence, viewed at time t, the next time instant refers to time t + dt in the continuous-time model and time t + 1 in the discrete-time case. If the discrete-time model results from dividing the time interval [s, t] into n periods, then each period takes (t – s)/n time. Here bonds are assumed to have a par value of one unless stated otherwise. We use the same notation for discrete-time and continuous-time models as the context is always clear. The time unit for continuous-time models is usually measured by the year. We standardize the following notation:

t: a point in time.

r(t): the one-period riskless rate prevailing at time t for repayment one period later (the instantaneous spot rate, or short rate, at time t).

P(t, T): the PV at time t of $1 at time T.

r(t, T): the (T – t)-period interest rate prevailing at time t stated on a per-period basis and compounded once per period – in other words, the (T – t)-period spot rate at time t. (This definition dictates that continuous-time models use continuous compounding and discrete-time models use periodic compounding.)