Fitting the Yield Curve

This chapter considers some of the techniques used to fit the model-derived term structure to the observed one. The Vasicek, Brennan-Schwartz, Cox-Ingersoll-Ross, and other models discussed in chapter 4 made various assumptions about the nature of the stochastic process that drives interest rates in defining the term structure. The zero-coupon curves derived by those models differ from those constructed from observed market rates or the spot rates implied by market yields. In general, market yield curves have more-variable shapes than those derived by term-structure models. The interest rate models described in chapter 4 must thus be calibrated to market yield curves. This is done in two ways either the model is calibrated to market instruments, such as money market products and interest rate swaps, which are used to construct a yield curve, or it is calibrated to a curve constructed from market-instrument rates. The latter approach may be implemented through a number of non-parametric methods. 

There has been a good deal of research on the empirical estimation of the term structure, the object of which is to construct a zero-coupon or spot curve (or, equivalently, a forward-rate curve or discount function) that represents both a reasonably accurate fit to market prices and a smooth function—that is, one with a continuous first derivative. Though every approach must make some trade offbetween these two criteria, both are equally important in deriving a curve that makes economic sense. 

This chapter presents an overview of some of the methods used to fit the yield curve. A selection of useful sources for further study is given, as usual, in the References section. 

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In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

THE CUBIC SPLINE METHOD FOR ESTIMATING AND FITTING THE YIELD CURVE... 

Figure 7.3. Dislocation densities required to fit the precursor curves as a function of the initial quasi-static yield stress.

Fitting the swelling curves of Fig. 7a to the form Q(t) — kt yields values of a greater than or equal to 0.8. Thus the swelling must be considered anomalous, or non-Fickian. In the absence of ionic interactions, this would not be expected since BMA/DMA 70/30 is initially not far below its Tg at 25 °C. Indeed, swelling measurements of this copolymer in hexane show kinetics that are nearly Fickian (a 0.55), as shown in Fig. 7b. Therefore, the anomalous swelling observed in Fig. 7a must be attributed to ion transport and binding rates in the gel. We will return to this point later. 

Figure 20 shows the result for a flow curve, where a small positive separation parameter was necessary to fit the flow curve and the linear viscoelastic moduli simultaneously. The data are compatible with the (ideal) concept of a yield stress, but fall below the fit curves for very small shear rates. This indicates the existence of an additional decay mechanism neglected in the present approach [32, 33]. Again, the A-formula describes the experimental data correctly for approximately four decades. For higher shear rates, an effective Herschel-Bulkley law... 

Upon titration of 10 mM enPd(OH2)2 with standard base an endpoint is reached at pH 7.5 after addition of one equiv base. We discovered, however, that the reversible titration curve flattens on the pH axis and cannot be fitted with an equilibrium expression for a simple deprotonation (11). We succeeded in fitting the titration curve to a combination deprotonation and dimerization process that yields a binuclear, dihydroxo bridged dimer. For this reaction the overall equilibrium constant was found to be = 10 3 M. 

In this chapter we consider multi-factor and whole yield curve models. As we noted in the previous chapter, short-rate models have certain drawbacks, which, though not necessarily limiting their usefulness, do leave room for further development. The drawback is that as the single short-rate is used to derive the complete term structure, in practice, this can be unsuitable for the calculation of bond yields. When this happens, it becomes difficult to visualise the actual dynamics of the yield curve, and the model no longer fits observed changes in the curve. This means that the accuracy of the model cannot be observed. Another drawback is that in certain equilibrium model cases, the model cannot be fitted precisely to the observed yield curve, as they have constant parameters. In these cases, calibration of the model is on a goodness of fit or best fit approach. 

The two previous chapters introduced and described a fractiOTi of the most important research into interest-rate models that has been carried out since the first model, presented by Oldrich Vasicek, appeared in 1977. These models can be used to price derivative seciuities, and equitibrium models can be used to assess fair value in the bond market. Before this can take place however, a model must be fitted to the yield curve, or calibrated In practice, this is carried out in two ways the most popular approach involves calibrating the model against market interest rates given by instruments such as cash Libor deposits, futures, swaps and bonds. The alternative method is to model the yield curve from the market rates and then calibrate the model to this fitted yield curve. The first approach is common when using, for example extended Vasicek... 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

The technique for curve fitting presented by Nelson and Siegel and variants on it described by Svensson (1994), Wiseman (1994) and Bjork and Christensen (1997) have a small number of parameters, and generally one obtains a relatively close approximation to the yield curve with them. As we saw above, the Nelson and Siegel curve contains four parameters while the Svensson curve has six parameters. The curve presented by Wiseman contains 2x( +l) parameters, given by f)j, kj]j=o.n- The curve is f2(1 ) ... 

FIGURE 5.5 Comparison of fitted spot yield curve to observed spot yield curve. (Reproduced with permission from the Bank of England Quarterly Bulletin, November 1999.)... 

For the purposes of conducting monetary policy and for central government requirements, little use is made of the short end of the yield curve. This is for two reasons one is that monetary and government policy is primarily concerned with medium-term views, for which a short-term curve has no practical input, the second is that there is often a shortage of data that can be used to fit the short-term curve accurately. In the same way that the long-term term stmcture... 

Use of polynomial functions that pass through the observed market data points create a fitted smooth yield curve that does not oscillate wildly between observations. It is possible to either use a single, high order... 

Recently, because of the difficulty of accurately determining the yield stress, a new model has been developed (Amanullah et al., 1998) based on assuming that a power law model with a low n value fits the flow curve. It also defines the cavern size by a minimum speed at its edge as the motion/no-motion boundary. As yet, an independent report has not been published confirming the effectiveness of this new approach. Finally, it should be noted that although there has been significant work dedicated to defining the size of the zones of motion in yield stress fluids, work has not been done to determine the blend time of the fluid inside the cavern. 

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