Encyclopedia of Quantitative Finance, IV Volume Set

Yield Curve Construction

The objective of an interest rate model is to describe the random movement of a curve of zero-coupon bond prices through time, starting from a known initial condition. In reality, however, only a few short-dated zero-coupon bonds are directly quoted in the market at any given time, a long stretch from the assumption of many models that an initial curve of zero-coupon bond prices is observable for a continuum of maturities. Fortunately, a number of liquid securities depend, in relatively straightforward fashion, on zero-coupon bonds, opening up for the possibility of uncovering zero-coupon bond prices from prices of such securities. Still, as only a finite set of securities are quoted in the market, constructing a continuous curve of zero-coupon bond prices will require us to complement market observation with an interpolation rule, based perhaps on direct assumptions about functional form or perhaps on a regularity norm to be optimized on. A somewhat specialized area of research, discount curve construction relies on techniques from a number of fields, including statistics and computer graphics. We discuss the basic topics in detail and refer the reader to appropriate sources for advanced applications. In the same spirit, we pay scant attention to the subtle intricacies of actual swap and bond market conventions.

Discount Curves

Let P(t, T) be the time ...