… the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of generality is, in essence, the same as a small and concrete special case.

Paul Halmos

This appendix serves a number of purposes. First, it elaborates on the topics in Chapter 19 by focusing on a specific problem (computing implied volatility) using the various methods discussed in that chapter. Second, it uses some of the concepts, models and methods that we discussed in various chapters of this book, for example:

- Computing implied volatility.
- Numerical methods for root finding and univariate optimisation.
- Homotopy methods and the solution of nonlinear equations.
- Creating a software framework for function optimisation based on the design principles that we introduced in Chapter 9 (see also Figure 19.1).
- Doing mathematics in C++; higher-order functions. This is a small research topic.
- Unconstrained multivariate optimisation.

Finally, we shall discuss a number of new topics (such as higher-order mathematical functions, Differential Evolution and multivariate optimisation) in more detail in a later work.

In the Black–Scholes model, the theoretical value of a vanilla option is a monotonic increasing function of the volatility of the underlying asset. This means that it is usually possible to compute a unique ...

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