Risk Latte - What are “Linear” and “Non-Linear” systems?

What are “Linear” and “Non-Linear” systems?

Team latte
September 9, 2009

Are financial and economic systems linear or non-linear?

To tell you the truth, guys, we would have a difficult time answering this question. If we have to shoot from the hip, then of course the answer is: linear. Take the example of the celebrated Black-Scholes model. The classical Black-Scholes partial differential equation to value financial derivatives is a linear (parabolic) equation as is the generalized diffusion equation which models the asset price.

But as traders and risk managers working in the financial markets would know that while valuing financial derivatives and analyzing their risks, we cannot ignore the transaction costs, large investor's preferences, lack of liquidity and its impact on volatility, and many other important factors that do not figure in a classical Black-Scholes model. Add all these and the Black-Scholes model (or the system that the model will describe) will start to become non-linear, and perhaps, far more complex.* Non-linear modifications to the classical linear Black-Scholes PDE by modifying volatility have recently been studied in some detail**.

In fact, both financial systems and economic systems are strongly non-linear, even though at times they may appear linear. But more on that later.

First, let's try to understand what is meant by “linear” and "non-linear"?

In high school, while doing differential calculus, we learnt that a straight line is a linear equation because the slope never changes, it is constant. On the other hand, if we take any curve then it will represent a non-linear equation of some kind; the slope of the curve changes every time we move up or down the curve (changing slope is called “curvature” or “convexity”). This changing slope gives rise to non-linearity.

In a linear equation the power of the variable is one. Therefore, the second mathematical derivative does not exist. In a non-linear equation, the power of the variable is 2 or more than 2. Therefore, higher order mathematical derivatives exist.

Systems, such as physical systems, financial systems, biological and weather systems, etc, also display linearity and non-linearity. Linear systems use "linear operators" and are much easy to understand and study. They exhibit simple and well defined properties. Signal processing and telecommunication systems are linear systems. Quantum mechanics, the branch of physics that concerns itself with the study of atomic and sub-atomic particles, is essentially, at the most fundamental level, also a linear theory.

An essential property of linear systems is the superposition principle. This simply means that if we have an input to the system that produces A and another input that produces B then if we add both the inputs the output should be equal to A + B. Simply speaking, two plus two should be equal to four.

A non-linear system is one where the output is not proportional to the input. Also, a non-linear system fails to satisfy the superposition principle.

In a non-linear system, when external forces are applied - inputs - these forces interact both with themselves as well as with the initial conditions creating a coupling between linear and non-linear dynamics of the system.***

In a non-linear system, the sum of the parts does not add up to the whole. It's a bit like stating 2 + 2 = 5. Take a plane sheet of paper and place it on a table top. This is a linear system. Now crumple this sheet of paper and you get a non-linear system. (A crumpled sheet of paper is an example of non-Euclidean space and explains the curvature of space and how that gives rise to gravity. Incidentally, Einstein's General Theory of Relativity is a non-linear theory.) Once the sheet of paper is crumpled it cannot be torn into parts that will add up to give back the crumpled sheet.

Weather systems are non-linear. Small changes in one part of the system can produce very complex output in other parts of the system. The weather systems are deterministic but extremely complex; one moment you are cruising smoothly at 39,000 ft above the sea level on an airplane with clear air around you and the next instant your plane is rocked by severe turbulence that neither the pilot nor his radar could anticipate.

In fact, turbulence, a common occurrence on airplanes (and in fluids) remains one of the most fascinating and least understood problems of physics. It is strongly non-linear in the way it manifests itself in air/atmosphere or other fluids and some believe that Navier-Stokes' non-linear partial differential equations can describe turbulence well.

Most biological systems are non-linear in nature. One of the famous biological system is the predator-prey system (mathematically described as the Lotka-Volterra equations). This system describes the dynamics of interaction between two species, one predator and the other prey.

And now the million dollar question: are financial and economic systems linear or non-linear?

*On the Numerical Solution of Non-linear Black-Scholes Equations by Julia Ankudinova and Matthias Ehrhardt

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