attempt to fit a particular form. A bar graph representation of the PDF not only is useful for interpretation, but also accurately reflects the method that was used to determine the distribution. Each bar is defined for one interval or bin and the bar height is a measure of congestion within that bin.

Up to this point, we have considered a single time period and a single PDF for that period. When a long history of data is analyzed, then each bar will have a lookback period and PDF for each bar. This series of PDFs can be plotted bar by bar to view the time dependence. This plot is a complicated 3-D surface. While rich with information, is it not easy to interpret. There are too many trees and not enough forest. To have a more practical tool, it is necessary to distill the PDF to a simpler, oscillator form that represents the level of congestion as well as one that captures the momentum, direction, and extreme conditions. And it must have a single numerical value for each bar so that it can be used to make trading decisions. This oscillator has been constructed and is called the Mobility Oscillator (MO). This oscillator and its smoothed companion, the Slow Mobility Oscillator (MOS), provide insight into price behavior and have utility as a forecasting tool.

In the discussion to follow, mobility analysis is described in detail, step by step. The six steps are:

1. Measuring congestion-Price Distribution Functions (PDF).

2. PDF examples and interpretation.

3. Reference points (support, resistance, and moving averages).

4. Mobility Oscillators.

5. Applications and interpretation of the Mobility Oscillators.

6. Steps for assessing price mobility.

One final point. Besides providing a useful trading tool, this chapter also provides an example of how an oscillator is constructed. To develop an oscillator from a hypothesis, determine parameters that characterize the hypothesis, determine reference points for these parameters, and combine the parameters to produce the oscillator. Once accomplished, then test the method by back-testing and by adjusting variables that make up the parameters. Sometimes refinement of the method is needed. Keep the analysis as simple as possible and avoid curve fitting. This was the process followed for developing the Mobility Oscillators.

Measuring Congestion-Price Distribution Functions (PDF)

The PDF is a distribution of price action for a fixed period of time. The computing procedure is straightforward. Select a particular time. Tabulate the high, low, and close data from that time over a look-back period of N bars. The range of the PDF is just the range of prices from the minimum low Z, = Mxa(L.) to the maximum high

= Max(H.) over the look-back period from j = 1 to N bars. This is the same range as is used for stochastics and may implicidy capture some of the same features.

Next divide the range into M equal intervals or bins separated by boundaries. The range is bounded by the lowermost boundary Z, and the uppermost boundary H . The boundary values B. are

max J i

Bi ~ Anin + -tj- X ( Anin )

for i = 1 to M+ 1, where M- number of intervals.

The number of intervals M should be chosen no larger than necessary, but large enough to delineate any important structure. The interval size should be smaller than the smallest bar height by some factor, say two or so.

Figure 4.2 shows an example of a short S&P 100 history (N= 14 bars) with the price range divided into 10 intervals (M = 10). The interval boundaries are extended through the data history.

Figure 4.2 price Distribution Function (PDF) constructed from 14 days of data.

the price range for the 14-day period is divided into 10 intervals and the price duration within each interval is projected to the right. the PDF is a bar graph tilted on its side. the bar length is proportional to the amount of congestion.

ft. 647.15 ™ 645.27

639.63 637.75

\Q \Q

\

9\ ri t»5

JJ5

1 VI Ifl 1 «

e e e e e

9 48 4S 4S 40~

t- ® n ?5 f*3

n ri ri ri

PDF N = 14 M =10

Weightings and spreads for each bar are chosen as follows. Each bar, within the look-back period, is equally weighted. It is also assumed that it is equally likely that prices will occur anywhere between the low and high for that bar. In other words, the price is uniformly distributed between the high and low of each bar. These choices allow the contribution of each bar to be easily divided within each interval and tallied. This is simple and minimizes the number of adjustable parameters. Some variations on the assumptions have been tried by the author, but found to add little to the discussion presented here.

The next step is to calculate the fraction of time or frequency of occurrence of the price within each of the intervals (the PDF values). One method for doing this would be to go bar by bar, compute the fraction of the total bar that falls within each interval, and then tally these fractions for each interval over all bars to get the PDF. Although this is perfectly valid, it is not the most numerically efficient.

A more efficient method is to first compute a Cumulative Price Distribution Function (CPDF) for the same intervals. The PDF is then the difference between adjacent values of the CPDF. The CPDF values represent the fraction of all the price action that is below the interval boundary values. Since we assumed that the price for any bar is uniformly distributed between the high and low values H. and L., then the term T.j represents the fraction of the bar j that is below boundary £>.: 1

1 for . > H.

• j

(B-L)I(H-L) fortf. > B>L.

fir 1 i

0 fori. >

i <

The CPDF. collects all the individual fractions .. below boundary B. and normalizes by the number of bars ./V. This gives the fraction of the price action during the lookback period that is below boundary .:

CPDFi = Sum( T.j) IN for j= 1 to N

Note that CPDF = 0 as there is no price action below B} and that CPDFM+1 = 1 as all the price action is below BM+].

Normally the terms T. and the CPDF are only a means to an end (i.e., parametric variables) and are not viewed or printed, unless to familiarize the reader with the process. By using Visual Basic macros to perform the calculations, all the intermediate steps can be done within the macro. It is not necessary to store these variables in a worksheet. Should the reader choose to use standard spreadsheet methods for the preceding calculations, however, it would be necessary to insert these terms in a spreadsheet. This latter choice is possible, but very cumbersome, and I do not recommend it based on my experience.

The PDF is given as

PDF = CPDF , - CPDF. for i = ltoM

i / +J I