CHAPTER 5 Fitting the Yield Curve

In this chapter we present some of the techniques used to actually fit the term structure. The term-structure models described in the previous chapter defined the interest-rate process under various assumptions about the nature of the stochastic process that drives these rates. However, the zero-coupon curve derived by models such as those described by Vasicek (1977), Brennan, and Schwartz (1979), and Cox, Ingersoll, and Ross (1985) do not fit the observed market rates or spot rates implied by market yields, and generally market yield curves are found to contain more variable shapes than those derived using term-structure models. Hence, the interest-rate models described in Chapter 4 are required to be calibrated to the market and, in practice, they are calibrated to the market yield curve. This is carried out in two ways; the model is either calibrated to market instruments such as money-market products and interest-rate swaps, which are used to construct the yield curve, or the yield curve is constructed from market-instrument rates, and the model is calibrated to this constructed curve. If the latter approach is preferred, there are a number of non-parametric methods that may be used. We will consider these later.

The academic literature contains a good deal of research into the empirical estimation of the term structure, the object of which is to fit a zero-coupon curve 1that is a reasonably accurate fit to the market prices and is a smooth ...

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