Mathematical Techniques in Finance

CHAPTER 7Interest Rate Derivatives

In the previous chapters, we made the simplifying assumption that interest rates are deterministic and can be represented by a single constant quantity, . In reality, interest rates for different maturities (terms) are different, giving rise to the term structure of interest rates. For example, a 3‐month deposit earns a different interest rate than a 6‐month deposit, with the 6‐month interest rate typically higher.

7.1 TERM STRUCTURE OF INTEREST RATES

There are a variety of equivalent ways to represent the term structure, among them the discount factor curve, the zero curve, and the forward rate curve (see Figure 7.1). Starting with the market prices of actively traded instruments, we can use the bootstrap method introduced in Section 2.7 to extract the discount factor curve.

7.1.1 Zero Curve

Once we have extracted the discount factor curve, , we can price any fixed income instrument and extract any spot or forward rates. For example, we can compute the zero‐coupon curve, which is the yield of zero‐coupon bonds versus their maturity. We can express the yields using any quote convention, for example, with semiannual compounding, we have