To value derivatives, traders rely on LIBOR (London Interbank Offered Rate) as the risk-free interest rate. The problem is that these rates are only available for up to 12 months (and note that between 2021–2023, LIBOR was replaced by alternative rates in various countries).
Why do we need to know the implied rates of zero-coupon bonds?
These rates are necessary to value cash flows of different maturities—they represent the discount rates for cash flows of the corresponding periods. In other words, a zero-coupon rate is the interest rate at which a single future payment is valued today.
When we talk about a bond that pays coupons periodically, we need the discount rate for each corresponding period.
If the period of the cash flows being valued exceeds 12 months, the curve must be extended—meaning, we need to derive the correct interest rates for additional periods. For periods up to 2 years, Eurodollar Futures are used, and for longer periods, swap rates are applied. The resulting extended curve is called the Swap Zero Curve.
Eurodollar Futures provide the market’s expectation of future 3-month LIBOR, allowing the curve to be extended up to 1–2 years because the logic is the same. Below, we will discuss the use of swap rates.
The curve extension is done step by step. For example, if we know the interest rates for three 6-month periods, we derive the rate for the fourth period. This is facilitated by knowing the swap rate of the corresponding maturity. From arbitrage considerations, discounting the four-period cash flows separately using zero rates must give the same value as implied by the swap rate for that period:
Excel – Zero Coupon SWAP Rates
P.S.
Since an interest rate swap involves exchanging the future cash flows of bonds with fixed and floating rates, it is assumed at issuance that the value of these bonds is equal. Moreover, the nominal and market value of the bond with a floating rate is the same because the discounting rate and the coupon rate match. Therefore, the swap rate comes out as the rate at which the fixed leg of the swap has a value exactly equal to par—following the same principle as a par-yield bond.
This logic allows confirmation of the equation above. As a result, the resulting curve shows the corresponding zero-coupon rates for all future dates and is used to discount all derivatives and cash flows.
Source: Options, Futures & Other Derivatives, John C. Hull