What Is Nonlinearity?
Nonlinearity is a statistical term used to describe a situation where there is not a straight-line or direct relationship between an independent variable and a dependent variable. In a nonlinear relationship, changes in the output do not change in direct proportion to changes in any of the inputs.
A linear relationship creates a straight line when plotted on a graph. A nonlinear relationship does not create a straight line but instead creates a curve. Some investments, such as options, exhibit high levels of nonlinearity and require investors to pay special attention to the numerous variables that could impact their return on investment (ROI).
Key Takeaways
- Nonlinearity is a mathematical term describing a relationship between an independent variable and a dependent variable that is not predictable from a straight line.
- Certain investment classes, such as options, show a high degree of nonlinearity, which can make these investments seem more chaotic.
- Investors of asset classes that exhibit a high level of nonlinearity will often use sophisticated modeling techniques to estimate their potential loss or gain over a specific period of time.
Understanding Nonlinearity
Nonlinearity is a common issue when examining cause and effect relationships. These relationships require complex modeling and hypothesis testing to fully explain nonlinear events. Nonlinearity without explanation can seem to lead to random, erratic outcomes.
In investing, we can see examples of nonlinearity in certain investment classes. Options, for example, are nonlinearderivativesbecause changes in the input variables associated with options do not result in proportional changes in output. Investments with high nonlinearity may appear more chaotic or unpredictable.
Investors who include nonlinear derivatives in their portfolio need to use different pricing simulations to estimate the risk profile of their investments than they would for linear assets.
Linear investment assets include shares of stock and futures contracts.
For instance, options traders will use "Greeks," such as the delta, gamma, and theta values for their investments. These assessments can help investors manage their risk and help time the entry and exit points of their trades.
Nonlinearity vs. Linearity
In contrast to a nonlinear relationship, a linear relationship refers to a direct correlation between an independent variable and a dependent variable. A change affecting an independent variable will produce a corresponding change in the dependent variable. When plotted on a graph, this linear relationship between independent and dependent variables will create a straight line.
For example, suppose management at a shoe factory decides to increase its workforce (the independent variable) by 10%. If the company's workforce and production (the dependent variable) have a linear relationship, then management should expect to see a corresponding 10% increase in the production of shoes.
Nonlinearity and Investing
The multiple factors that can impact an option investment's return make options an example of an asset class with high nonlinearity. When trading options, investors have many variables to consider, including:
- The underlying asset price
- Implied volatility
- Time to maturity
- Current interest rate.
For investments with a high degree of linearity, investors generally use a standard value-at-risk technique to estimate the potential loss the investment might incur. However, using a value-at-risk technique is generally not sufficient for options because of their higher degree of nonlinearity.
Instead, options investors might use a more advanced technique, such as a Monte Carlo simulation. This models for a wide variety of variables with different parameters to assess possible investment returns and risks.
Special Considerations
Nonlinear regression is a common form of regression analysis used in the financial industry to model nonlinear data against independent variables in an attempt to explain their relationship. Although the model's parameters are nonlinear, nonlinear regression can fit data using methods of successive approximations to offer explanatory outputs.
Nonlinear regression models are more complicated to create than linear models because they often take considerable trial-and-error to define the outputs. However, they can be valuable tools for investors who are attempting to determine the potential risks associated with their investments based on different variables.
What Is a Nonlinear Example?
A nonlinear relationship is anything that can't be represented by an equation in the form f(x) = ax+b. An example of an equation for a nonlinear relationship is f(x) = x2.
How Can You Tell If a Relationship Is Linear or Nonlinear?
A linear relationship has a constant rate of change, which means it can be plotted with a straight line as the dependent variable changes in response to the independent variable. A nonlinear relationship doesn't have a constant rate of change. When a nonlinear relationship is plotted, it will form a shape other than a straight line.
What Are the Greeks in Investing?
The Greeks are variables that are used by investors and analysts to assess risk in the options market. The name comes from the Greek letter that is used to represent each variable, such as delta, gamma, theta, and vega. These are the first partial derivatives of an options pricing model. Each Greek tells investors something about how that option moves or the risk associated with it.
The Bottom Line
While linear relationships between variables can be plotted with a straight line, nonlinear relationships are not predictable from a straight line. In a nonlinear relationship, changes in the dependent variable are caused by a variety of inputs, so the value doesn't change in direct proportion to the independent variable.
Some investment classes, such as options, are highly nonlinear, which can make it more difficult for investors to predict their losses or gains in response to certain market changes. To understand these investments, investors will use more complex modeling techniques to estimate potential gains or losses over time.
