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 on the market prices of a particular option.

Root-finding methods use an iterative routine that requires a start point or the estimation of the root. The estimation of the root can either converge toward a solution, converge to a root that is not sought, or may not even find a solution at all. Thus, it is crucial to find a good approximation to the root.

Not every nonlinear function can be solved using root-finding methods. The following figure shows an example of a continuous function where root-finding methods may fail to arrive at a solution. There are discontinuities at and for the y values in the range of -20 to 20:

There is no fixed rule as to how a good approximation can be defined. It is recommended that you bracket or define the lower and upper search bounds before starting the root-finding iterative procedure. We certainly do not want to keep searching in the wrong direction endlessly for our root.