In our last installment, we looked at creating higher dimensional phase space diagrams from a single time series. This technique is useful for inferring the dynamics of complex systems from individual time series. In the phase space, the system at each instant of time is represented by a single point in a 2- or higher-dimensional graph. A sequence of points describes the evolution of the system through time. Traditionally we draw a smooth curve through the sequential points, which is described as the trajectory of the system.
Today I would like to look at reconstructing the phase space of some idealized stock chart patterns.
Let's look at a simple one: the symmetrical triangle. I have digitized an hypothetical stock price series in excel and plotted the series. I made the limbs of the formation approximately linear, but rounded the edges, as the reconstructed phase space plot will look better if the limbs aren't perfectly linear.
Wedge pattern in a hypothetical stock. Breakout is circled.
The figure above represents a wedge--though not a falling one. Like all time series, the data are one-dimensional. We can use excel to create a 2-dimensional phase space portrait by lagging the price data by a time that is roughly half the time taken to move from a short-term low to a short-term high (this may seem vague but it is because I have not presented a time scale on the above graph).
Two-dimensional phase space portrait of the above stock price. The two dimensions are the price, and the lagged price. The flow of the system is in the direction of the arrow. As time advances, the system follows the trajectory around first the outer loop, then the smaller inner loop, and after the breakout, towards the upper right corner of the graph.
In the first part of this essay, we looked at a simple chaotic function to see the relationship between the one-dimensional and two dimensional projections. The Lorenz equations give us a function that exhibits simple chaotic behaviour, which is believed to be characteristic of many natural systems. We saw that complicated looking butterfly function. Now let's look at a smaller portion of it--say 501 points (the ones I have selected are points 100-600) from the excel file generated last time.
Lorenz calculations showing x-values only from point 100 (at left) to 600. Looks a little familiar. . .
The above does look rather suspiciously like the wedge pattern in the stock price shown above. Except for one thing--time is flowing from left to right. So the wedge is actually running backwards when compared to the stock price pattern.
The 2-D reconstructed phase space portrait from the above graph, using a lag of 12 points. The abscissa is price (higher to the right), and the ordinate is lagged price (higher towards the top). At first glance, this looks a little familiar. We see what appears to be the same flow starting from the larger loop, spiralling inwards until Pow! Moonshot!
We wish all investing was this easy. But there is something wrong. Unfortunately, this trajectory starts at the upper right and moves to the region of low prices at the bottom left. Not such a good investment after all!
The characteristic feature of stock price charts is the apparent time reversal--the dynamics of stock prices flows through time backwards in comparison to those of natural systems. The dynamics of the natural system is driven by the flow of energy, with consequences following from causes. We see a rapid evolution towards a new area of phase space, and subsequent fluctuations represent instabilities which grow until the system rapidly shoots over to a new region of phase space.
So how do we explain the reversal in time observed in the stock chart patterns?
As my background is science, you will forgive me if I insist that the natural systems are going the "right" way through time and the economic systems are moving backwards.
Stock prices we normally consider to be driven by greed and fear. But what does this really mean? They are driven by flows of money, which represents information in the economic system (it is not the only information in the system). That the dynamics of the stock prices evolve through time in the opposite direction relative to natural systems suggests that players with foreknowledge of some forthcoming announcement (what we call "insider" information) are making good use of their information.
In the absence of potential profit, that "insider" information would not flow out into the market, but rather the market would learn the information at the time that some announcement is made by the company in question. The profit margin actually causes the information to flow out into the market more rapidly than it would flow otherwise. Furthermore, the very existence of such an opportunity argues against the efficient-market hypothesis.
In conclusion, it appears that stock charting gives us some limited forecasting ability in stock prices because trading on insider information is endemic in the markets.