In the preceding section, we discussed some nonlinear models commonly used for studying economics and financial time series. From the model data given in continuous time, the intention is therefore to search for the extrema that could possibly infer valuable information. The use of numerical methods, such as root-finding algorithms, can help us find the roots of a continuous function such that , which can either be the maxima or the minima of the function. In general, an equation may either contain a number of roots or none at all.

One example of the use of root-finding methods on nonlinear models is the Black-Scholes implied volatility modeling discussed earlier. An option trader would be interested in deriving implied prices based on the Black-Scholes model and comparing them with the market prices. In the next chapter, we will see how we can combine a root-finding method with a numerical option pricing procedure to create an implied volatility model based...