# CHAPTER 3

# Short-Rate Models

## 3.1 INTRODUCTION

When pricing equity derivatives, we generally need to model only a single market instrument: the stock price.^{1} The interest rate world, on the other hand, consists of many instruments: futures, swaps, and the like, all of which can move independently. These are generally combined to form the yield curve, commonly expressed in terms of zero coupon bond prices *P*(*t*, *T*) (i.e., the value seen at time *t* of 1 unit of currency paid at time *T*) or the zero coupon rate *R*(*t*, *T*), defined by

Another useful representation is in terms of the forward rate, *f*(*t*, *T*). This is defined as the rate, fixed at time *t*, for instantaneous borrowing at time *T*. If we agree at time *t* that we will invest 1 at time *T* for an infinitesimal period *δ*, the amount we will get back at time *T* + *δ* is 1 + *f*(*t*, *T*)*δ*. We can hedge this by shorting the zero coupon bond with maturity *T* and buying 1 + *f*(*t*, *T*)*δ* units of the zero coupon bond with maturity *T* + *δ*, making

The EUR yield curve is shown in terms of *R*(0, *T*) and *f*(0, *T*) in figure 3.1.

Two different approaches to interest rate modeling are

*Market models*, where we model the market instruments such as LIBOR^{2}or CMS^{3}rates directly. Examples of market models include the well-known BGM model [59].

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