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