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130 18 Price Distribution Systems Price movement is usually viewed as a chart on which each new time period is seen as a new bar or point recorded to the right of the previous prices. There are many applications that need to look at the way prices cluster, or dishibute, rather than sequences of pattems. In options, it is important to evaluate the current maitet volatility to decide the chances of prices remaining in a specific range for a specific amount of time. To get that value, we use the standard deviation calculation first introduced in Chapter 2. The standard deviation gives the most basic measure of price disUibution. From the value of 1 standard deviation we can estimate the chances of a price remaining within a range over time. The key values to remember are that 1 standard deviation defines 68°o of the price movement (both up and down), 2 standard deviations contain 95"o. and 3 standard deviations contain 99°o of all price movement based on the sample of data used to calculate the standard deviation value. USING THE STANDARD DEVIATION The data used to determine the standard deviation is very important. Because it is a statistical measure, it is most accurate when a large amount of data is applied. For example, you might find that 1 standard deviation of the crude oil daily price move is only S0.25 per barrel when measured over the past 10 years, but during the 6 months of the Gulf War the same measurement yielded $ JO, twice as large. Most trading applications using the standard deviation tend to apply short data intervals to ite calculation, such as 20 dajs. This short period is not likely to represent the same price disUibution as a 10year calculation; therefore, the probabilities given by the resulting standard deviation value must be interpreted differently. While it is less likely that the price will make a move of 3 standard deviations compared with 1 standard deviation, the probabilities can be misleading. Statistics tell us that there is only a Po chance that prices will move a distance of 3 standard deviations higher or lower; however, that value is reliable only when measured over a long data period. If you selected 20 dajs of unusually low volatility, the chance of a 3standard deviation move Would be very high. The frequency disUibution is another very practical approach to measuring price distributions. This was also described in Chapter 2. it has the advantage of having a much clearer visual interprdation. While the standard deviation gives us what appears to be a highly mathematical probability, the large error factor that is caused by small amounts of data may make its usefulness about the same as the frequency disUibution In the following sections, both techniques will he used Standard Deviation Bands Bollinger bands, discussed in Chapter 5 ("Trend Sjstems"), are a very popular application of price dishibutions. They do not detrend price, but calculate the stantbrd des iation of prices over a period of 20 dajs and form a band of 2 standard deviations around the trendline. It is common for traders to varj both the period and the number of standard devia lions used to conshnct the band. Once calculated, Bollinger bands can be displayed on any price chart and used to generate buy and sell signals, much the same as any other channel breakout sjstem. Using a smaller Bollinger band, for example, 1 standard deviation, will give many more lignals than using one of 3 standard deviations. At the same time, a band of 3 standard deviations translates into rid; that is 3 times greater than 1 standard deviation. Signals produced with a larger band tend to be more reliable, but have greater rid;. Bollinger bands also describe maitet volatility. A relatively narrow band franslates into low volatility. By comparing a 2 leriod Bollinger band with a 65eriod band, you can see the relative difference between shorter4erm and longer4erm maitet volatility in Figure 181. The thicker lines, representing the 65period calculation, cross the short4erm band at points that diow relative overbought and ovesBpld situations. If this was a daily rather than a 154ninute chart, the 21eriod band would give monthly volatility and the 65period band would be quarterly volatility, useful values for options fraders. Problems in Using Moving Standard Deviations Appljing any technique to a rolling time interval of the most recent N bars is a common method of keeping in tune with current market conditions, in the case of a simple moving average, we should be very familiar with the lag that is infroduced. For trends, when prices are moving steadily higher, the lag causes the trendline to be much lower.
There is a similar lag when using the most recent N bars to calculate a standard deviation, even when the data has been detrended. If we are measuring the volatility of the maitet, and prices rally quickly, the volatility rises. This will be seen in the larger value of I standard deviation measured over a fixed number of dajs, or bars. If we are looking for a confirmation of a buy signal based on an increase in volatility, we should get it. However, the volatility represented by the standard deviation will not decline as fast as we expect because of the same lag. Once higher volatility has occurred on a single day. it FIGUREISIComparisonof 21  and 65i3eriod Bollinger bands. h2BMae c72015  0=731 js 73475 L71975 Bands 73716 721Se 2 S2rt6 7 Source: Chart created with TrodeStaiion® by Omega Research, Inc. will remain part of the standard deviation value until it passes completely out of the calculation window. That will prevent a new volatility event from being recognized soon after a short decline in volatility. It will also make il difficult, if not impossible, to get a timely exit signal on reduced volatility, because the lag keeps the volatility appearing high until at least part of this new, more active price movement begins to pass out of the end of the calculation window. This can also be seen in Figure I8I, where the 65eriod bands expand quickly and narrow* slowly. USE OF PRICE DISTRIBUTIONS AND PATTERNS TO ANTIOPATE MOVES Prices often form pattems that can be evaluated using probability methods, or simply viewed in much the same way as a frequency dishibution, or histogram. Because the concepts are sound, but the statistical analjsis is often difficult because of limited amounts of data or changing conditions, analjsis have taken a much more empirical approach toward shidjing price distributions. The following section will look at some innovative ways to look at price dishibutions and how they are interpreted into trading opportunities. Analjsis of Zones Rather than using standard deviations to identify the chance of a price move above or below yesterdays closing level, Bruce Gould observed that historic prices could be divided into five zones, each 20°o of the price range over the previous 3 years. Using this long4erm approach, it is easj to see that selling in zone I (the lowest price levels) would have less opportunity for profit than selling in zone 2, just above it. Similarly, bujing in zone 5. the highest band would both the greatest rid; and the least opportunity of profit. This is likely to remain the case unless prices move to much higher levels and all zones need readjusting. J.T Jackson has used this concept to define five shortterm zones, based only on yesterdays prices, which can
he associated with the sfrength or weakness of todays move. These daily zones, which are popular with floor traders, are calculated as:  Zone  (f Price fs Above Ther?  Higti2 = Average + Hjghl  Low I   Strong up  High 1 = 2*Avere  Low   Moderate up  Average = (Hjgti + Low + Close) / 3   \ up  Lowl = 2*Avere  High   Mildly down  Low2 =Avere + Low)  Highl   Moderately down    Stroneiv down 
where the High, Low, and Close are yesterdays prices. Note that there are five calculations but six zones needed to separate them. A test of how the S&P 500 falls into these relative rankings gives: S&P frequency 20% 44% 83% 79% 42% 20% JT jack.™,Drterttt,gHighlT..fitDayTra.leEbTheFutureEli=kets Vin.torBc.k., 1 1) which shows the slightly upward bias expected of the overall equities maricet These disfributions may varj depending on the interval used in their calculation and whether there is a dominant trend during that period. A longer interval that includes bull, bear, and sideways maitets would be safest: otherwise, there is the chance that the calculations will create a bull maricet profile, while some trading will occur during a bear maitet reaction. Although you can avoid trend bias by using longer intervals for the calculations, the zones tend to get very large. The sfrategies for trading price zones focus on short4erm trends and holding periods. For example, you can sell when prices move into zone 4 (mildly up) with a stop in zone 5. If you consider zones 3,4, and 5 as containing mostlj maitet noise, then selling at the top of zone 4 and closing out that trade at about the average, or bujing near the bottom of zone 3 and closing out at about the average, could capture the majority of price moves that have no direction. Nonrandom Pattems In evaluating the zone approach, the maitets that offer the greatest potential for this stttegj are those that show an atnormal disfribution of prices within the six zones. For example, if the six zones were all equal in size, and the frequency of prices declined by one4ialf as they moved from the center to the exfremes, there would be a perfectly random distribution and no profit potential: Zone of Equal Size 6 5 4 3 2 1 Normal disuibution 7% 14% 28% 28% 14% 7% If the disfribution is normal, but zones 3 and 4 are much wider than the outer zones, then the ride of selling at the top of zone 4 with a stop at the top of zone 5 and a profit target at the average (of zones 3 and 4) would result in an equal number of profits and losses, but the profits would be larger. If the zones are of equal size, the opportunities come when the disttibution is clustered in the center: Zone of Equal Sizt 6 5 4 3 2 1 Norma) disoibution In this case, selling at the top of zone 4 would result in many more profits than losses, although profits and losses would be of equal size. Appljing a Moving Average Disttibution To extend the time frame and include frend bias within the zone values, the same zones may be created by
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