In recent years, there has been a growing interest in research on nonlinear phenomena in economic and financial theory. With nonlinear serial dependence playing a significant role in the returns of many financial time series, this makes security valuation and pricing very important, leading to an increase in studies of nonlinear modeling of financial products.
Practitioners in the financial industry use nonlinear models to forecast volatility, price derivatives, and compute Value at Risk (VAR). Unlike linear models, where linear algebra is used to find a solution, nonlinear models do not necessarily infer a global optimal solution. Numerical root-finding methods are usually employed to converge toward the nearest local optimal solution, which is a root.
In this chapter, we will discuss the following topics to explore some methods that will help us extract information from nonlinear models:
- Examining the definition of nonlinearity
- Discussing the volatility smile in implied volatility modeling
- Discussing Markov switching models, threshold models, and smooth transition models as nonlinear models
- An overview of root-finding to find the optimal point of nonlinear models
- Examining the incremental search algorithm, bisection algorithm, Newton's algorithm, and secant method in root-finding
- Combining root-finding methods with Brent's method
- SciPy's implementation of root-finding methods as scalar functions
- SciPy's general nonlinear solvers in root-finding
While linear relationships aim to explain observed phenomena in the simplest way possible, many complex physical phenomena cannot be explained using such models. A nonlinear relationship is defined as follows:
Even though nonlinear relationships may be complex, to fully understand and model them we will take a look at the examples that are applied in the context of finance and in time series models.