Derivatives: Markets, Valuation, and Risk Management

CHAPTER 20

Valuing Interest Rate Products Numerically

Valuing interest rate derivatives written on short-term bonds is trickier than valuing derivatives on other types of assets for two reasons. First, for an asset such as a stock, a currency or a commodity, price can roam freely through time without constraint. For a fixed income security, however, price is often forced to take a particular level when the security matures. A T-bill, for example, has a value of 100 when it matures, and a T-note has a terminal payment equal to its final coupon interest payment plus the par value. Second, in the fixed income markets, there is often a wide range of securities available on the same underlying source of uncertainty. The U.S. Treasury, for example, has T-bills, T-notes and T-bonds with a wide range of maturities. In modeling interest rate dynamics, care must be taken to ensure that all of these securities are simultaneously valued at levels consistent with observed market prices.

The purpose of this chapter is modest—to develop a binomial procedure for valuing interest rate derivative contracts where the short-term interest rate (“short rate”) is the single underlying source of interest rate uncertainty. To begin, we discuss a number of constant-parameter short rate processes to lay a foundation for interest rate behavior. While these models are often useful in developing economic intuition regarding interest rate behavior, they produce zero-coupon bond values that are different from ...