Catastrophe Theory and Financial Markets
The boom of the 1990s has led to investors having unrealistic expectations of financial markets. In particular, there seems to be a widely held perception that what goes down must go up, and that the current state of the financial markets is merely a temporary aberration and that the markets will shortly resume their upward march rewarding all those who have "kept the faith" in a New Era of a permanent bull market.
Unfortunately, human history has many examples of catastrophic events. Such calamities happen suddenly with very little warning and can change lives and, perhaps, even the world forever. Sudden changes occur very commonly in nature, and are known as "discontinuities," it is only when such a sudden event impacts human lives in a tragic way is a "discontinuity" referred to as a "catastrophe."
In quantum mechanics an electron orbiting around a nucleus of an atom can change its orbit instantaneously in a quantum leap and appear in the new orbit as soon as it leaves the old orbit without ever being physically in between the two orbits. Imagine if this occurred with the planets in our solar system that Mercury could suddenly appear in the same orbit as the earth and then just as mysteriously suddenly be back in its own orbit without ever having physically traveled the intervening distance!
If you bend a piece of wood it will bend and then suddenly break. The failure of the wood is a discontinuity in the distortion of the wood as it bends. The eruption of volcanoes and earthquakes are other examples of sudden discontinuous events.
It is strange that with many discontinuities occurring in nature that humans always predict the future events in their environment as a linear extrapolation of events in the present or past. This is stranger still if one considers that the central organ to life itself, the heart, pumps blood through a sudden muscle spasm which is in itself a discontinuous event.
An "event" occurs essentially because something changes. The change may occur slowly and gradually or it may occur abruptly. A slow and gradual change would lead to what might be considered a normal, everyday event while an abrupt change could be considered catastrophic.
Imagine you pour water into a glass to fill it up. This occurs gradually without any discontinuity. If you pour water out of the glass to empty it, and in so doing reversing the first event, the water flows out gradually and so is also performed with no discontinuity. This is not always the case. If you have ever lived in hot, humid climates like the southern United States you will know that when you walk out of an air-conditioned building your glasses will suddenly fog up. If you go back into the building they will take several minutes to clear.
If you take ice out of the refrigerator and leave it at room temperature it may take many hours to melt into liquid water. Even after several hours there may still be some solid ice left. If, however, you put water in the refrigerator and check it frequently it will remain liquid for some time and then suddenly and abruptly it will become ice.
Why is it that such events can not be reversed in a gradual way? Why is it that you may approach your neighbor's dog on many different days and stroke him but one day he growls and bites you? Why is it that markets can move up and down smoothly but one day they can crash? The answer lies in Catastrophe Theory. Catastrophe Theory was invented in the 1960's by a Frenchman called René Thom. Thom described catastrophes by a three dimensional surface. As shown in the figure below. The surface is a sheet that is folded to create an upper level, a lower level and a cusp or fold.
An event is defined as a transition from either the lower state to the upper state or vice versa. If the transition occurs on the back edge of the surface following the green line the event is normal and continuous. If , however, the transition occurs along the front edge following the red line then the event is a sudden and abrupt change which is a catastrophe.
Let's use this theory to explain the fogging-up of our glasses in a humid climate. When you are inside the building you are in the upper state and as you exit the building the event follows the red line. The sudden condensation on the glasses causes them to fog instantaneously. When you return inside the building the event follows the green line because the condensation slowly evaporates.
We can also explain the behavior of your neighbor's dog. Before you enter your neighbor's yard the dog is peaceful and his behavior is in the lower state. You now approach him slowly and his behavior gradually follows the green line to the upper state as he becomes wary of you and starts barking. You stroke him and he realizes you are no threat so his behavior follows the greenline back to his lower state of peacefulness. There is no catastrophe. Now you do the same thing but while the dog is barking instead of stroking him you move toward him shouting for him to be quiet. His behavior follows the red line. There is a sudden attack and he bites you and as you run from the yard he stops barking and growling and returns to his peaceful state.
How the surface is traveled depends on whether the event is a normal event or a discontinuity.
So now let us consider how catastrophe theory relates to financial markets.
Trading in financial instruments will generally follow the greenline in that in that the price change will move gradually and continuously and will reverse direction from time to time such that prices change up and down in a continuous fashion during a trading session. There may even be discontinuities in price movements that do not lead to what we would term catastrophes. Let us say that crude oil closes a trading day at $25 per barrel. After the close the API inventory report shows that crude oil inventories rose sharply. This means that the price of oil (the upper level) and the actual true value (based on supply and demand) of oil (the lower level) are significantly different. When the market opens the following day the price of oil will not follow the green line but instead follow the red line and make a discontinuous price change. It may instantaneously trade at $24 per barrel. While this is a discontinuity it does not create a market crash in crude oil prices. That is because there is a different three dimensional surface that describes the trading behavior of traders. This behavior surface has a lower level which is a calm behavior and an upper level which is an agitated behavior. If traders move up to the agitated state by way of the green line route the selling is brisk but not uncontrolled. If the API report were to be released during the trading session in this example then the crude oil price could decline gradually to $24 (the green line) or drop suddenly to $24 (the red line).
So what causes a market crash? A market crash is caused by a sudden discontinuity in price which then leads to a trading behavior which takes the red line route from calm to agitated. The sudden, discontinuous change of state from calm to agitated is what we know as "panic." This would lead to excessive selling and result in a catastrophic market drop. If, on the other hand, the price correction takes the green line and occurs over many months slowly and gradually this will never trigger the change of state in trading behavior to a state of panic. In this case there may be a very large market correction but over a long period of time. This means that despite the investors who buy and hold may lose just as much money as in a market crash it can not be called a catastrophic event because any investor could have liquidated his position at any time during the slide if he so desired.
Based on catastrophe theory when are market crashes likely?
As we have seen if there exists a significant difference between the true value of a financial asset and the market price there exists the possibility of either a gradual adjustment of market price or a sudden adjustment. If the difference between market price and true value is extremely large then if the market price follows the red line this could easily trigger a catastrophic change in state of trading behavior to one of panic which would lead to a market crash.
Currently, by historical p/e measures the DJIA, the Nasdaq and the S&P are all very significantly over-valued. This leads to a very real possibility of a catastrophic price change which would trigger a catastrophic change in trading behavior which would result in a catastrophic market crash. Recently, the markets have seen large price declines but, to the astonishment of many observers, still return to or close to their overvalued state. Nothing yet has triggered the catastrophic change in trading behavior. Each day that the markets trade the prices move around the three dimensional surface. Unless average company earnings rise to reduce the p/e ratios, which would in turn reduce the height between the upper and lower levels of the three dimensional catastrophe model, there is the clear and present danger of a sudden price change due to the market following the red line that would then trigger a similar sudden change in trading behavior. The rest would then be history.
In the gold market many observers have remarked that the supply and demand fundamentals indicate that the price of gold is very significantly below its true value. Many market participants who are very "short" gold appear to be trying to make the gold price follow the green line on any given day such that no sudden change in trading behavior is triggered. With the huge gap between the market price of gold and its true value (based on supply and demand) there exists a real possibility that at any time the price could follow the red line creating a very large price change which would then trigger a sudden and abrupt change in trading behavior... a buying frenzy. Catastrophe theory hence explains the hypothesis often proposed by Bill Murphy that one day "gold will gap up at the open and run from there."
So by this theory would this be a catastrophe?... only if you are short gold!!!
Adrian Douglas
June 13, 2002
adriandouglas@earthlink.net
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