In the previous section, I discussed the quoting, pricing, and settling of some commonly traded interest-rate instruments. In addition, I also discussed how a zero curve can be used as an input to value a fixed-floating interest rate swap. Given this backdrop, I will now focus on the construction of a zero curve using observed, tradeable prices and rates.

Although a zero rate associated with a specific maturity can be used to determine the present value of a cash flow happening on the maturity date, the purpose of a zero curve is to:

■ Reproduce the prices/rates of the instruments that are used to construct the zero curve.

■ Interpolate and extrapolate relevant zero rates that can be used to determine the present value of cash flows linked to maturities that do not coincide with those used for the zero rate curve calibration.

The process of building a zero curve using a collection of liquid tradeable interest-rate instruments is called zero-curve construction. It starts with the shortest instrument maturity and then gradually builds the curve out (or

TABLE 2.5 Market Information

bootstraps^{[1]}) over longer maturities. In this section, I discuss the process of zero-curve building, assuming that

■ Liquid instruments used to build the zero curve are given.

■ Linear method is used to interpolate between two given zero rates.

■ Zero rates beyond the longest available instrument maturity are extrapolated horizontally.

Using the market data given in Table 2.5 and assuming that the zero rates are extrapolated horizontally beyond the 5-year maturity (as this is the longest maturity associated with the given market information), I will now discuss the construction of a zero curve using a four-step process for terms less than five years.

Assuming that rt,u represents the zero rate associated with a maturity time и when the current time is t (where и > t), one can obtain the appropriate zero rates by going through the following four steps.

[1] The term bootstrap is used to reflect the fact that in putting on a pair of boots and tightening up the laces, one has to start from the bottom-most end of the holes (which the lace has to go through) and then tighten the lace across each hole to eventually arrive at the topmost end of the holes.

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