Visualizing Risk and Volatility: A Catastrophic Approach

"I like this concept of “low volatility, interrupted by occasional periods of high volatility”. I think I will call it "volatility". - Daniel Davies

Risk and volatility are rather abstract concepts. They certainly exist in the world, but don't readily manifest themselves in tangible ways. If you get in the car after several beers, you have a mental representation of the risk you are exposing yourself to (e.g. getting pulled over or into an accident) but these possible future outcomes only exist in an abstract space.

I equate this abstract space with Euclidean space. Euclidean space contains dimensions in a similar fashion to the Cartesian space we use in Newtonian physics. Newtonian physics models the behavior of physical systems with dimensions x, y, z, t (representative of left-right, up-down, towards-away, time). However, in the Euclidean system, these dimensions are generalized to n dimensions (the x-axis would be generalized to dimension 1, y-axis to 2, etc.) allowing one to model abstract concepts like risk, volatility, and even subatomic particles. Taking this space, we can create some interesting models of market behavior.

 

Rene Thom's catastrophe theory applied to market behavior.

Above is an interesting visualization of greed and fear in the markets using Rene Thom's catastrophe theory. Rene Thom was a French mathematician who created these landscapes to graphically model catastrophic events. This model would exist in a theoretical Euclidean space, with two dimensions representing "fear" and "greed" as shown on the control surface and the third representing market behavior. To utilize this graph, one would pick a point on the control surface (the square base) and move upwards to the spot directly above it on the "equilibrium surface".

For instance, let's look at that spot on the upper left corner of the control surface indicated by the letter "c", which happens to be on the "fear" half of the surface. This half represents all the possible degrees of fear in the markets (e.g. slight to extreme). Go upwards and see that point "c" represents the amount of fear that corresponds to declining equity markets.

The fascinating (and most important) part of this model however are the points closer to the triangular object on the control surface. It might not be immediately obvious, but that curved triangular object is actually the "shadow" of the folded section on the equilibrium surface above it. Let's take a closer look at this section.

For any point on or inside this triangle, notice that there are two corresponding points on the above equilibrium surface, one that is indicative of bullish behavior and the other bearish. (Also notice how the fold can be described with a polynomial, which often have 2 or more solutions). Within this area, we arrive at catastrophe.

Thom refers to this area as the "cusp", as in, the cusp of catastrophe. At any moment within the cusp, market behavior could spontaneously change from bullish to bearish and from greed to fear. We observe this in the markets. A buying stampede on an earnings beat can change to a selling panic from news of geopolitical instability almost instantaneously.

Specifically, the tip of the triangle is the cusp. When directly on that point, the market is in a delicate balance between greed and fear. They are on the cusp of falling in either direction. Some of you may be familiar with the Minsky Moment, which oddly, can be described as a cusp catastrophe.

Why are these sudden changes in behavior important to the derivatives trader? Cusp catastrophes cause extreme volatility in financial instruments.

 

The financial disaster of 2008-9 can be seen as a cusp catastrophe, where market behavior bifurcated on concerns of economic collapse. A seven year bull market lost all its gains in a matter of months and the VIX blew up to unprecedented levels. A derivatives trader long on volatility would have profited enormously if adequately posed for this cusp of catastrophe. The trick, obviously, would have been to known that this was coming.

Is it possible to predict when a market will hit a turning point? Furthermore, would it have been possible to predict the sub-prime crisis and the collapse of Lehman? Absolutely not. It is, however, possible to protect yourself from such volatility and to mitigate exposure to such risk.

The astute derivatives trader not only hedges him or herself to these events, but can seek to profit off of them.