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congestion for the look-back period and lowest mobility. For the analysis here, the mode is easy to observe and to calculate directly from the distribution function. It is determined as

PDF. =PDF =Max{PDF) for / = 1 to M

im max r

where im is the interval number containing the maximum PDF value. The Price Mode (PM) is the price value at the center of the interval containing PDFnax. Since the PDF is defined over intervals and not at points, the interval center is chosen as the best value for the interval:

PM = BC.

The Mobility Oscillator (MO) is now constructed from the PDF. The intent here is to gauge mobility by determining the amount of congestion at the current price and then comparing it with the extremes in congestion for the look-back period. In addition, the MO is constructed to provide a sense of direction, large and positive when there is an upward bias and large and negative when there is a downward bias.

To accomplish this, first form the ratio of the value of the PDF in the interval containing the current price with the maximum PDF value. When this ratio is one, then the current price is sitting in the interval with the mode and there is maximum congestion. When the ratio is equal to zero, there is minimum congestion. To convert from congestion to mobility, this ratio is subtracted from one since maximum congestion represents minimum mobility and conversely. This gives a direct relationship rather than an inverse relationship. The MO is assigned a positive sign when the current close is above the mode and a negative sign when below the mode. If the current price is below the mode, then there is downward mobility, and if it is above the mode, there is upward mobility. Convert the entire expression to a percentage by multiplying by 100.

MO = 100 (] - PDFJPDFJ for > PM MO = 0 for = PM

MO = - 100 (1- PDFJ PDF for Cn<PM

where Cn is the current price and PDFC is the PDF value for the interval containing the current price. Normally Cn is the current close; however, other options are possible such as the current high, current low, average of current high and low, or the average of the current high, low, and close. One of these alternatives can sometimes provide confirmation or reduction in noise.

The MO ranges from -100 to +100. If the price is above the mode, it is supported and is "upwardly mobile" and if below the mode it is resisted and "downwardly mobile." It is not to say that the price will move up or down, but rather it may



be easier to do so based on what it did previously. Price momentum should also be considered as well as the potential for surprises, such as the release of economic data.

MO values near zero correspond to large congestion and are important for forecasting. When the MO is moving toward zero, from either above or below, price action is contracting and the market is pausing, waiting, and building something. Congestion building often occurs before a large move, and the MO is a useful indicator under these circumstances. This behavior appears to be analogous to the narrowing of the Bollinger Bands that is frequently observed prior to a large move. When the MO moves away from zero, the price action is expanding and there is mobility. Sometimes the MO will have momentum and move rapidly through zero. This confirms the move as a strong move that will likely continue. Congestion building also frequently occurs after a large move and is a good indicator of when the move is over.

It is also useful to define the Slow Mobility Oscillator (MOS) as

MOS=EMA{MO)

where EMA represents the Exponential Moving Average. Typical smoothing periods for the moving average are a portion of the look-back period; for example, for N = 14 bars use about 7 bars for smoothing. This slows and smooths the MO and can be used to identify direction as well as crossing signals that indicate change.

Application and Interpretation of Mobility Oscillators

Consider several examples of mobility analysis applied to recent data. The first example, Figure 4.9, is a recent OEX history. Here a top with a large cycle below the top characterizes the price history. The MO (dark lower line) and the MOS (lighter lower line) are plotted on the same time scale as the OEX high-low-close data. Crossings of the MO and MOS have significance, although in some cases the noise level is high and whipsawing is possible.

The MO moving toward zero precedes principal moves. This represents congestion building. The MO builds congestion prior to major moves and gives some warning of what is to come. However, price action can be erratic just prior to the move so be careful of direction. Also note that congestion building following a move often signals the end of the move.

Figure 4.10 shows mobility analysis for a long and steady uptrend of the OEX in early 1995. Even this relatively steady trend consists of many small steps where the market flattens out and pauses for a few days before continuing higher. The MO reflects this behavior by approaching zero during these steps signaling congestion building. This does signal a move, only here it is up to the next rung on the ladder. Note that the MO remains positive, with the price above the mode of the PDF reflecting the upward bias for the price trend.



Figure 4.9 Mobility analysis for recent OEX data. Here a top with

a large cycle characterizes the price history. the MO (dark lower line) and the MOS (lighter lower line) are plotted on the same time scale as the OEX high-low-close data.

tCMOCMOJMOt-CNJCO

OOOOOOOOOOO °i4 7 40 1

Figure 4.11 shows a significant move to the upside in early 1996. There is evidence of congestion building prior to moves, although the action can again be erratic just prior to a large move. This is seen with the very brief, but significant, dip prior to the up move. The MO picks up the dip with an abrupt move to negative territory and crossing of the MOS in time for damage control. The next congestion building phase signals the up move and there is follow-thro ugh with the MO remaining positive for a long period. Once again, the congestion slowly builds signaling the move is over. Slowly building congestion is a more reliable signal with less likelihood of an abrupt reversal prior to the major move.

Other examples of mobility analysis are shown in Figures 4.12 and 4.13. These examples show several features including the fast and slow Mobility Oscillators. The Mobility Oscillators use a look-back period of 14 bars, a smoothing period of 10 bars, and 40 intervals for the PDF. The PDF for the entire plot period, 59 bars, is shown at the right, tilted on its side, for reference. These plots were produced using Visual Basic macros in EXCEL.

Figure 4.12 shows the rise, sell-off, and bounce of the NASDAQ Composite in the summer and fall of 1995. The rising periods generally correspond to positive



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