The Logarithmic Spiral
You can get wound up with spirals.
-----------------------------------------------------------------------------
------------------------------------------
A spiral is a curve that winds around a center of rotation at an ever-increasing distance from it. In a flat plane, the curve winds around a fixed point in the plane. (In 3-space the curve winds around a fixed line, like a winding staircase, except that it widens as it winds.)
Our concern here is with spirals in a plane -- i.e., spirals in the price-time chart of equities -- to predict price turnarounds. The spirals are to be drawn on a price-time graph whose price dimension is logarithmic. This highlights an important characteristic of prices, because the log scale provides a graphic picture of equal percentage changes. Equal percentage changes in price have the same chart length for high-, medium-, or low-priced stocks.
----------------------------------------------
To better understand the logarithmic spiral, xonsider first the linear spiral, a spiral drawn on a linear chart, a price-time graph in which both dimensions are linear. This is the kind you're most likely to see for stock prices, because prices on semi-log charts can be harder to read. In the linear format, equal differences in price and time, respectively, are represented by equal distances on the graph. But the percentage changes are very different. Similarly, distances on the time scale are equivalent to real time differences.
-------------------------------------------
When dealing with spirals, we are dealing with fairly complex non-linear systems and non-linear functions that represent them. By contrast, linear systems and linear functions are simple. The value of a linear function is proportional to the value(s) of its variable(s).
Perhaps the easiest way to look at a linear function is to think of it in terms of its graph. If the graph is a straight line, the function is linear. Consider, for example, the linear function, y, of a single variable, x, namely y = ax, where a is a constant. The value of y is a fixed multiple of the value of the variable. If y = 3x, for instance, then y is always three times the value of x, no matter what value you choose for x. The function has no memory. If x = 6, for example, then y = 18. The graph is a straight line.
----------------------------------------------
Semi-log graphs are essentially graphs of mixed scales. As used in the stock market, they are logarithmic in price only. In the other dimension -- time -- the values are linear.
Log graphs reflect a special property of logarithmic functions: the value of the function is proportional to the logarithm of the variable rather than the value of the variable itself. In the single-valued logarithmic function, y = alog(z), for instance, if z is doubled, it is not the case that y is doubled. I.e., y isn't linear with respect to z -- it is linear only with respect to log(z).
On a logarithmic scale, it is the logarithms that have equal graphical distances
, not the prices they represent. In fact, the prices can change dramatically, yet the percentage change will still be directly visible. That's the importance of the log scale for stock prices. In particular, low prices don't get crunched to the point where you can't detect price changes, as in linear graphs.In y = log10x, increasing the value of x doesn't increase the value of y proportionately. In this single-variable function -- which depends on the number base 10 -- if x has the value 10, y has the value 1, and if the variable is 100, the value of y is only 2. Similarly, if x is 1000, y is only 3. One step in prices could take you from 1 to 10. The next (equal) step would take you from 10 to 100. The third, from 100 to 1000. And so on.
-------------------------------------------
The trading objective is to superimpose logarithmic spirals on semi-log price-time charts of equities to estimate price turning points, a la Robert Fischer. To express this graph in equation form, let q represent the rotation angle (in degrees), ranging upward from zero. Let L0 be the starting length of the short line (at q = 0), and let L be its length thereafter. At any angle, then,
L = L0 + q /720
The x and y coordinates of the end points of the line are therefore:
x = Lcos(q )
y = Lsin(q )
We might superimpose this graph on the price-time graph, but there would be no rationale to connect it with either prices or time or scale of graph. We need a spiral with a logarithmic component for prices (for percentage changes) that mediates a meaningful principle relating future prices to current prices. Otherwise it would have no predictive value.
Introducing Logarithms to Fibonacci
Details of the pure logarithmic spiral can be found in MathWorld.
Following Fischer, the mediating principle is the Fibonacci number ratio, 1.618. The formula for the logarithmic spiral is:
Cota = 2/p x lnf,
where x is the Fibonacci ratio.
There are two points to note. One is that Fischer uses natural logs (ln) to scale prices. The second point is that the horizontal dimension is time. But Fischer brings in an angle a. Where did it come from? Well, I think it has to be the angle q , as discussed above, but it's now wrapped in a new set of clothes.
As q increases from 0 to 360 degrees (2p radians), a goes from 0 to 90 (p /2), back to 0, again, and so on, repeating every 360 degrees. However, the spiral doesn't begin at the center but rather at a finite distance from the center. The trick in using the spiral is to choose a center point on the price chart and start the curve at another well-defined point. See Fischer for examples. You have to eyeball the spiral against the prices to estimate price turning points.
The key is that each point of the spiral has a price component and a time component, and each component is 1.618 times the distance from the center to the previous point at the given angle in the spiral. So both price and time are related by the Fibonacci ratio. Think of it as a ratio- stretching function in both dimensions. That's where the magic is, I guess.
Seems to me, if the stretching ratio is to make sense, it has to reflect the peculiarities of the trading mechanics as controlled by the trading environment. The specialist carries a book of buy/sell orders -- orders that are conditional and depend on price. Prices are moved up or down by the specialist to pick up our orders, so the ratio reflects his performance (personality) and the state of the price/order list.
Ney has a lot to say about the immediate trading environment. And Prechter has more to say about the broader environment -- its psychology. Our job is to uncover this personality in the context of the broader environment and formulate it so we can place our orders more intelligently. It gets awfully sticky!
------------------------------------------