# CHAPTER 46

# INSTANTANEOUS FORWARD RATE MODELS

The short rate models presented in the previous chapter are not flexible for calibration. As a consequence, Heath et al. (1992) introduced a new uniform framework, called the HJM framework, for modeling the interest rates. In the Heath-Jarrow-Morton (HJM) framework, the entire forward curve is modeled directly. These interest rate models are called *HJM models* or *instantaneous forward rate* models. In this chapter, we introduce some concepts related to the HJM framework.

# 46.1 Basic Concepts and Facts

**Definition 46.1** (Forward Rate). Let 0 ≤ *t* < *T* < *S.* The simple (or simply compounded) forward rate for [*T, S*] prevailing at *t* is defined to be

where *P*(*t, T*) is the value at time *t* of a dollar at time *T.* The continuously compounded forward rate for [*T, S*] prevailing at *t* is defined to be

The instantaneous forward rate with maturity *T* prevailing at *t* is defined as

The function *T* → *f*(*t, T*) is called the forward curve at time *t.*

**Definition 46.2** (Forward Contract). Let *t* < *T.* A forward contract on an underlying , entered at time *t*, with maturity *T* is ...

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