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Splining over Two Time Intervals

To draw a comparison to Table 2.9, one would need to fit a cubic polynomial in each of the time intervals (0, 0.3333) and (0.3333, 0.4167) using the overnight rate, the 1-month, and 2-month Eurodollar futures contract prices.

As aconsequence, one is interested in fitting a function in the time interval () and the functionin the interval () where t0 = 0, ,. As before, one has to simultaneously guess at the zero rates for and and then apply the cubically splined zero curve to ensure the reproduction of the market rates for the overnight, 1-month, and 2-month instruments.

Letting the appropriate zero rates at times be given by respectively and making the substitution into the cubic function, one gets

(2.12a)

(2.12b)

(2.12c)

(2.12d)

where both and are guessed zero rates that are used so their interpolated cubic spline values can reproduce the given market rates.

Since equations (2.12a) to (2.12d) have eight unknowns and only four equations, one needs another four equations to solve for the unknown variables. To do this, I will assume21 that the cubic functions are continuous in their first and second derivatives at the time node so as to arrive at the equations

(2.12e)

(2.12f)

To get the remaining two equations, I will assume, as before, that the zero-rate curve is instantaneously linear at times 0 and 0.4167, which would imply that the second derivatives at times 0 and 0.4167 are both 0. Doing this yields

(2.12g)

(2.12h)

To solve equations (2.12a) to (2.12h), one can rewrite these equations using matrix notation to obtain MX = Y, where

and, M is an 8 X 8 matrix that takes the form as shown in Figure 2.2.

21Doing this allows one to ensure that at the connecting note (i.e., at time) the 2 cubic functionsand are continuous. As a consequence, the kinks which were present when simply connecting the linear pieces (in the linear interpolation method) will no longer exist.

The Form of the 8×8 Matrix

FIGURE 2.2 The Form of the 8×8 Matrix

The system of equations can now be solved for when the zero rates and are known (and in this case they are guessed). To get the exact values of these rates (and hence unique solutions to the system of equations), one would need two additional equations in and . These equations can be obtained by realizing that the zero rates in the intervals (0, 0.3333) can be interpolated using the function while those in the interval (0.3333, 0.4167) can be interpolated using the function . As consequence, it readily follows that

(2.12i)

(2.12j)

Since equations (2.12i) and (2.12j) are not a function of and one needs two more equations (one connectingwith and the other connecting with ). Drawing on the relation ship between zero rates and forward rates in equation (2.7), one can easily arrive at

(2.13a)

(2.13b)

where , and the corresponding zero rates at these times are given by and respectively.

Putting all the pieces together allows one to arrive at cubic polynomial coefficients as given in Table 2.13.

Using the values in Table 2.13, it is easy to arrive at the zero rates as shown in Table 2.14.

TABLE 2.13 Splining Coefficients When Fitting over Two Time Intervals

TABLE 2.14 Zero Rates Obtained Using First Two Eurodollar Futures Contract and Splining

By comparing the results of Table 2.9 with Table 2.14, the reader will see a slight difference in the zero rates that is caused by the use of different fitting functions.

 
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