Since the first interest rate swap transaction between World Bank and IBM in 1991, interest rate swaps have grown to be one of the most widely traded derivatives contracts on the over-the-counter (OTC) markets. In its simplest form, a swap basically is a financial instrument that allows two parties to exchange floating interest rate payments for fixed rate ones over a fixed

TABLE 2.3 Calculating Continuously Compounded Forward Rates from a Eurodollar Futures Price

period of time based on a predefined notional amount.^{[1]} This type of swap is also called a simple, fixed-floating swap. See Ravindran (1997) for further details on this type of a swap and variations on this theme.

Unlike the instruments discussed earlier, pricing an interest rate swap requires a zero-rate curve as an input. Suppose that a discount factor associated with a given time is denoted by (where the current time is t, the zero-rate is assumed to be given for time and ) and (since the present value of a dollar today is simply a dollar), and that one is interested in pricing^{[2]} an n-period swap where the floating rates are reset n times during the life of the swap at times and Assume further that the cash flows arising from the settlement of the floating and fixed interest rates are done so at times where

TABLE 2.4 Calculating Swap Rates from a Zero Curve

t2 < ... < tn+1. Then the fixed rate of the swap can be shown to be given by the following formula.^{[3]}

(2.6)

An application of equation (2.6) is given in Table 2.4.

[1] To illustrate, consider an example of a two-year, semiannually reset interest rate swap on a notional of USD100MM. In this instance, one party pays interest payments using the floating rate of б-month LIBOR that are reset once every 6 months (at 0- month, 6-month, 12-month, and 18-month time) and receives interest payments once every 6 months (on the dates when the floating-rate payments are settled at the 6- month, 12-month, 18-month, and 24-month periods) that are based on a fixed rate (called the swap rate). As a consequence, there will be four cash-flow exchange dates at 6 months, 12 months, 18 months, and 24 months. On each cash-flow date the interests arising from both the floating and fixed interest rates would be netted off. The party resulting in a negative cash flow position would end up writing a check to its counterparty. Thus, a swap rate can be intuitively thought of as a blending of the market's expectation on how the future 6-month floating rates will unfold. Furthermore, it may be of interest to the reader to note that a single-period swap (i.e., n = 1) is equivalent to a forward rate agreement.

[2] Pricing a fixed-floating interest rate swap is analogous to finding a fixed rate associated with the swap, so the practice of exchanging interest-rate payments based on a floating-rate LIBOR for those based on a fixed swap rate would require no upfront exchange of cash flows.

[3] This is discussed in detail in the Appendix of this chapter.

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