Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 45

SHORT RATE MODELS

Short rate models are the earliest stochastic interest rate models. In this chapter, we present several short rate models and relevant concepts.

45.1 Basic Concepts and Facts

Definition 45.1 (Zero-Coupon Bond). A zero-coupon bond with maturity date T is a contract that guarantees the holder a cash payment of one unit on the date T. Zero-coupon bonds are also referred to as T-bonds. The price at time t of a zero-coupon bond maturing at T is denoted by P(t, T).

Definition 45.2 (Spot Rate and Short Rate). Let 0 ≤ t < T. The simple spot rate for [t, T] is defined to be

The continuously compounded spot rate for [t, T] is defined to be

The function T → R(t, T) is called the zero-coupon yield curve or yield curve. The instantaneous short rate at time t is defined as

Definition 45.3 (Bank Account). A bank account or money-market account represents a risk-free investment. The value β(t) at time t of a bank account is defined as

where r(s) is the short rate at time s.

Definition 45.4 (Discount Factor). Let 0 ≤ t ≤ T. The (stochastic) discount factor (t, T) between ...