Step 1: Convert Eurodollar Futures Prices to Forward Rates

From the discussion on Eurodollar futures, one can convert the 1-, 2-, 3-, 6-, and 12-month quoted Eurodollar futures prices into implied 3-month futures rates (as in Table 2.6), which can be converted to implied 3-month forward rates.

TABLE 2.6 Extracting Futures Rates from Eurodollar Prices

By using equation (2.5) and assuming a short-rate standard deviation of 1.2 percent, one gets the continuously compounded 3-month forward rates as shown in Table 2.7.

Step 2: Calibrate Zero Rates for First Year

From Table 2.7, the reader should observe that since the continuously compounded overnight rate is 0.1 percent, I have trivially assumed that the zero rate is 0.1 percent at time 0 (i.e., ).

To obtain the other zero rates, it is first important to understand the relationship between zero rates and continuously compounded forward rates. More precisely, if and represent the respective zero rates for maturities и and w (obtained from the zero curve at current time t) respectively where , then the continuously compounded forward rate applied from time и to w is given by the expression

(2.7)

Since the continuously compounded 3-month forward rate at time 0.0833 years is 0.15 percent, if and represent the continuously compounded zero rates at times 0.0833 years and 0.3333 years respectively, then using equation (2.7) one gets

(2.8a)

Additionally because of the linear interpolation assumption, it follows that

(2.8b)

Solving equations (2.8a) and (2.8b), one has and . Putting all these together, one can arrive at the zero rates given in Table 2.8.

Continuing to build the zero curve for longer maturities, one now needs to get (zero rate at time 0.1667 years) and (zero rate at time

TABLE 2.7 Extracting Continuously Compounded Forward Rates from Futures Rates

TABLE 2.8 Zero Rates Obtained Using the First Eurodollar Futures Contract

TABLE 2.9 Zero Rates Obtained Using the First Two Eurodollar Futures Contracts

0.4167 years) so as to match up with the 3-month forward rate starting 2 months from now. Using equation (2.7) one gets

(2.8c)

Since r0,0.1667 can be linearly interpolated using the zero rates r0,0.0833 and r0,0.3333, it readily follows that r0,0.1667 = 0.00 12. Using this value, in equation (2.8c), it is easy to see that r0,0.4167 = 0.00228.

Putting this together with Table 2.8, one can arrive at Table 2.9.

Repeating this for all the subsequent times, one can arrive at Table 2.10.^{[1]}

Using Table 2.10, the zero rates in the time interval (0, 0.0833) or (0.0833, 0.1677) or (0.1617, 0.2500) or (0.2500, 0.3333) or (0.3333, 0.4167) or (0.4167, 0.5000) or (0.5000, 0.7500) or (0.7500, 1.0000) or

TABLE 2.10 Zero Rates Obtained Using All Eurodollar Futures Contracts

(1.0000,1.2500) can be easily obtained using linear interpolation. For example, to compute r0,0.3, the zero rate at 0.3 years, one has to use the rates at times 0.2500 and 0.3333 to linearly interpolate and obtain

[1] Throughout this example I have assumed that the tenor underlying the forward rates is the same as that for the Eurodollar futures contract and is 0.25 years.

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