In earlier chapters of Wilmott’s book, we modeled interest rates by picking one short-term rate and deriving the entire yield curve from it. Works fine for simple stuff. But Heath, Jarrow, and Morton said: why model just the short end when you can model the whole forward rate curve at once?
We have spent several chapters building interest rate models. Vasicek, CIR, Hull and White, Ho and Lee, Black-Derman-Toy. Each one chosen for its nice mathematical properties, clean closed-form solutions, and easy calibration. But here is the uncomfortable question Wilmott asks in Chapter 36: do any of these models actually match what interest rates do in the real world?
One-factor interest rate models have a fundamental problem. They assume that a single number, the spot interest rate, drives the entire yield curve. That means all rates of all maturities move together in lockstep. If the spot rate goes up by 1%, every other rate adjusts accordingly. The yield curve can shift up and down, but it cannot twist or tilt independently at different maturities.
Swaps are one of the biggest markets in finance. The total notional principal is comfortably in the hundreds of trillions of dollars. Chapter 14 of Wilmott’s book explains how they work, why they exist, and how they connect to the bond pricing we covered in the previous post.