The three most common statements that people will make about financial markets are that they are like a roller coaster, or up and down like a Yoyo, or that they bounce back and they are a cyclic.
Then in the next breath they will tell you it is totally random and cannot be predicted.
Well it can’t possibly be both random and a cycle, cycles have a wave lengths and therefore are not random and are discoverable.
What can we deduce from these three anecdotal observations?
The one thing that all these things have in common is that they can be described by differential equations and if the market tends to bounce back then this is due to the elastic properties of markets, economics 1.01 tells us that markets are elastic and will therefore display properties that can be described by Hooke’s law of springs.
If I pluck an elastic medium it will oscillate with the equation
y = a * Cosine ( t ) . e^-st
If many people pluck it with respect to time it will form an interference pattern that will display complex harmonic motion where we will observe the superposition of the harmonics of the system forming complex interference patterns.
This opens a Pandora’s box of possibilities of calculations to choose from and as such there will be a lot of trial an error involved in determining just what the equation in question will look like, however the first principles of calculus are not as complex as they first seem provided we keep in mind that we may already be looking for an equation that is well understood in another field that has not be fully applied to finance.
If you think about it money is an imaginary quantity that circulates in an imaginary field know as the market, I use the word field in its strictest scientific sense in the same way a radio engineer refers to the field that surrounds a radio antenna. In the case of a market this takes the form of information that is not observable in the chart unless we measure it and add it.
The simplest form of field equation we use is the trend line or an indicator, however all modern indicators that I have looked do no qualify as they only accept one input in the form of price with limited or inadequate treatment of the passage of time, and as such are not true differential equations and are all just averaging technics and all have phase delays which renders them only partially useful as pricing or trend modelling methods.
What is required is a more definitive understanding of the basic equations that we already know are central to monetary systems, I am referring to the compound interest rate function, which can be expanded on by adding an integration that when it is all said and done makes it identical to the Laplace transform in every way.
By making this conversion we are translating the market to the complex plane so the entire field of the calculus of unstable systems can be applied, or to put it more simply we are looking for equivalent of a mass damper for portfolio management.
It is no great step to consider market prices on the complex plane, after all money circulates and markets are by definition elastic and prices are inherently exponential with respect to time.
The problem arises when we start measuring markets using the current methods of looking at rectangular charts with the time axis measured in terms of the Roman calendar and base ten number in the price axis which obscures more than it revels.
The reality of markets and crowd behaviour in the market place is very simple, the market is going up because people are buying it and people are buying it because it is going up, the same applies on the down side, this is where the wave like behaviour orginates
These are the ingredients required for self referential exponential growth and decay function, which is exactly what we see in both buying and selling panics, as the margin calls go out on both sides.
Markets also have propagation delay so movements that accumulate over time form support and resistance levels which when broken lead to runaway behaviour in both directions, this is the instability I referred to earlier.
It is this instability that we need to understand in detail before any automated purely equation based portfolio management system can be developed into reliable trading algorithms. With all of this in mind any candidate solution will have the following properties and will contain .
- A Real Component.
- An Imaginary Component.
- It will be a Differential Equation.
- It will explain the observations.
- It will reliably predict or at least price future observations.