This chapter begins with an introduction to the goals of constructing curves of discount factors or rates and then recommends and presents in detail two popular methodologies, namely, flat forwards and a smoothing of those forwards based on piecewise quadratic interpolation. To present ideas and techniques, the focus here is on building a single London Interbank Offered Rate (LIBOR)-based curve. The techniques for implementing the two-curve methodology of Chapter 17 are essentially the same.
In some very special cases a security can be priced by arbitrage relative to a set of other securities, e.g., a 5% 2-year swap can be priced relative to par swaps with maturities of six months, 1 year, 1.5 years, and 2 years. More frequently, however, arbitrage pricing is not possible because a security to be priced makes cash flows on one set of dates while benchmark securities make cash flows on another set of dates. Continuing with another swap example, it might be necessary to value a 5% 13.4-year swap relative to a set of more frequently traded swaps, none of which makes payments on exactly the same set of dates as the 5% swap. Or, to take another example, it might be necessary to calculate the spread to swaps of a 9% 7.3-year corporate bond with annual coupon payments. Problems of this sort could, in theory, be solved with the models of Part Three: a term structure model could be calibrated based on both historical data and selected current ...