# CHAPTER 22

# MULTIFACTOR MODELS

In this chapter we introduce multifactor models. This is an important extension of the no arbitrage model discussed so far, as from Chapter 4 we saw that we need at least three factors to “explain” the variation in yields. In other words, the yield curve not only moves up and down, but it also changes slope as well as curvature. The models developed so far did not allow for independent variation of these quantities: For instance, in the Vasicek model, the level, slope, and curvature of the yield curve are all tied to the short-term interest rate *r*_{t}, and thus they are perfectly correlated.

The good news is that the methodology extensively covered in the previous chapters readily extends to many factors, including the risk neutral pricing methodology. This implies that (almost) all of the pricing technique learned in previous chapters can be applied to the multifactor case. We now illustrate the changes to be made.

## 22.1 MULTIFACTOR ITO’S LEMMA WITH INDEPENDENT FACTORS

As in the single factor case, the most important tool is the extension of Ito’s Lemma, discussed in Chapter 14, to the multifactor case. We start with the simple case in which there are only two independent factors affecting the term structure of interest rates. The general multifactor model is dealt with below.

Consider two independent factors, generically denoted by *φ*_{1, t} and *φ*_{2, t}, and assume they move according to the processes

where *X*_{1, t} and *X*_{2, t} are two independent Brownian motions. ...