Chapter 79 is about what happens when your problem has two random factors. Convertible bonds with stochastic interest rates. Exotic options with stochastic volatility. Barrier options where you model both the stock and the rates. These are real problems that real desks face, and they need two-factor numerical methods.
In the previous chapter we built the explicit finite-difference method. It works, it is easy to code, but it has that annoying stability constraint: your time step must be tiny relative to your asset step. Chapter 78 shows how to remove that constraint and get better accuracy at the same time. The price? The code gets a bit more complicated. But as Wilmott says, the extra complexity is worth it.
Chapter 77 is where we stop talking about pricing theory and start actually building a pricer. This is the chapter where the Black-Scholes equation stops being a formula on paper and becomes code on a screen. If you have ever used the binomial model, congratulations, you already know the basic idea. Finite differences are just a more powerful, more flexible version of the same concept.
We have reached the final part of Wilmott’s massive book. Part Six: Numerical Methods and Programs. This is where all the theory meets reality. Because in practice, you almost never get a nice closed-form formula. You get a PDE and a computer.