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 ...

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