Nonlinearity in Finance

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 on the 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:

• Nonlinearity modeling
• Examples of nonlinear...

Nonlinearity modeling

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.

Examples of nonlinear models

Many nonlinear models have been proposed for academic and applied research to explain certain aspects of economic and financial data that are left unexplained by linear models. The literature on nonlinearity in finance is simply too broad and deep to be adequately explained...

Root-finding algorithms

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, f, such that f(x)=0, 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, in The implied volatility model section. An option trader would be interested in deriving implied prices based on the Black-Scholes model and comparing them with market prices. In the...

SciPy implementations in root-finding

Before starting to write your root-finding algorithm to solve nonlinear or even linear problems, take a look at the documentation of the scipy.optimize methods. SciPy contains a collection of scientific computing functions as an extension of Python. Chances are that these open source algorithms might fit into your applications off the shelf.

Root-finding scalar functions

Some root-finding functions that can be found in the scipy.optimize modules include bisect, newton, brentq, and ridder. Let's set up the examples that we have discussed in the Incremental search section using the implementations by SciPy:

In [ ]:
    """
    Documentation at
    http://docs.scipy.org...

Summary

In this chapter, we briefly discussed the persistence of nonlinearity in economics and finance. We looked at some nonlinear models that are commonly used in finance to explain certain aspects of data left unexplained by linear models: the Black-Scholes implied volatility model, Markov switching model, threshold model, and smooth transition models.

In Black-Scholes implied-volatility modeling, we discussed the volatility smile, which was made up of implied volatilities derived via the Black-Scholes model from the market prices of call or put options for a particular maturity. You may be interested enough to seek the lowest implied-volatility value possible, which can be useful for inferring theoretical prices and comparing against market prices for potential opportunities. However, since the curve is nonlinear, linear algebra cannot adequately solve for the optimal point...