The value of PDF. represents the fraction of the price action between the boundaries 5. and B{. The PDF. value is assigned at the interval center at 5C, the central price, where

= {B.+B.JI2 ioii = ltoM

The values of the PDF, as defined, represent the fraction of the look-back period that the price has spent in each respective interval. For this definition, the magnitudes of the PDF values depend on the number of intervals M. To remove this dependence, simply normalize PDF. by dividing each value by the interval size AB = (H - L )IM. This normalization is not necessary for the following discussion, maybe confusing, and thus is omitted here. However, if the reader wants to make absolute comparisons between PDFs, then it would be important to include this normalization.

Figure 4.2 includes a plot of the PDF distribution to the right of the High-Low-Close plot. Note that the bar chart has been turned 90 degrees, on its side. This is a convenient way of viewing the distribution function as it combines the time history with the PDF with a shared ordinate axis and range on the same scale. For the example in Figure 4.2, the high H = 656.55 was on 5/28/96 and the low L = 637.75 on 6/7/96. This defines the price range. For M= 10, the interval AB =7656.55- 637.75)/ 10= 1.88.

The value of M should be large enough to resolve all the structure present. That is, M > (H - LmjJ I A, where A is the smallest interval that is to be resolved. Typically A values range from the minimum of the bar height (high-low) to the average of the bar height for the look-back period. These recommended M values, named MRX and MRA, are shown in Table 4.1 for comparison, assuming two bins per A. The reader may also want to adjust M and observe changes to the PDF as a basis for selection.

PDF Examples and Interpretation

Tabulation of the input data for the example given in Figure 4.2 as well as boundary values and the CPDF and the PDF are shown in Table 4.1. These are loaded onto a worksheet for viewing. Next, the T matrix is computed following methods and formulas previously outlined, and the columns are summed and normalized to give the CPDF and PDF.

Table 4.2 gives the same output as the sheet given in Table 4.1, except that the parametric variables are not shown. Only output data is shown. This is the format that is used in my practice for routine analysis and is calculated using Visual Basic macro functions in EXCEL. Accompanying the sheet in Table 4.2 is a bar chart of the PDF as shown in Figure 4.3.

For the example shown in Figure 4.4, the current close was below the peak. The subsequent price moved further lower on the several following days. The mobility

table 4.1 Spreadsheet showing calculation of price Distribution Function

(illustration version)

HMAXandLMIN are computed as are MRA and MRX, the estimates of the required M values based on average and minimum daily price range. The T matrix is shown here for illustration and is a parametric variable that is normally not used for working calculations of the PDF.

Table 4.2 Spreadsheet showing calculation of price Distribution Function

(working version)

A more useful format for PDF calculations with the same input data format but a more streamlined output format than in Table 4.1. The distribution junctions are listed along columns instead of rows, as in Table 4.1.

Figure 4.3 price Distribution Function (PDF) constructed from 14 days of OEX data. The PDF versus is obtained from Table 4.2. The bin containing the last close is highlighted.

PDF CHART 06/14/96 (N = 14 m= 10)

02500

02000

0.1500

0.1000

0.0500

0.0000

analysis would suggest that it is easier to move lower than higher for these conditions and the subsequent price movement was consistent with this. To properly assess the situation, however, requires additional information and an assessment of the momentum to be discussed shortly.

Distribution function shapes can vary. They can be symmetric about the center of the range, peaked to one side or the other, skewed, and even have multiple peaks. Quite often, prices will form bottoms or tops and sometimes steps. This is quite natural before or after a large move, for example. These flat regions correspond to regions of congestion and will produce corresponding peaks in the PDF. The PDF analysis gives a precise means of observing this behavior. Figure 4.5 is an example of a distribution function with multiple peaks. This is not too common for short periods of 14 days, but happens occasionally. For this example, during the look-back period, the OEX moved from a level (flat region) of about 610 to another level of about 640. The move was quick, and little time was spent between these levels. Although this is apparent from simple examination of the time series history, the PDF provides a means of quantifying these conditions for precise comparisons and analysis.

Multiple peaks are more common for longer look-back periods than for shorter ones since it is more likely the price movement will experience cycles or steps within a