Murrey Math Study Notes -- Part 2

Written by Tim Kruzel

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Murrey Math Reversal Percentage Moves

The following notes are observations regarding the Murrey Math Price Percentage Moves (MMRPM). The MMRPM statistics are a key Murrey Math factor to consider when evaluating a trade. The MMRPM statistics are also key in understanding the importance and function of the Square in Time.
Recall the definition of the MMRPM. The MMRPM statistics specify the probability that a price movement, of some magnitude (X), occurring during some time interval (t), will reverse itself. For example, in Reference Sheet U of the Murrey Math Book, a listing is given for:
```
Price Percentage Moves for Indexes over 500 but under 1000.
(Intraday Basis) (Slow Day).

One of the entries is this listing is:

6/8 ths 85% of the time 1.4648
```
This entry is specifying the following. The Murrey Math Square in Time that is being considered is based upon the perfect square of 1000. The height of the square in time consists of 8 Murrey Math Intervals with each Murrey Math Interval (MMI) being given by:
((((1000/8) /8) /8) /8) = 1000/4096 = 0.244140625
Since each 1/8'th = 0.244 then 6/8'ths = (6 x 0.244) = 1.4648. So, if price moves either up or down by 1.4648 then the probability that the price movement will reverse direction is 0.85 or 85%. This statement of probability assumes that the price movement of 6/8'ths has occurred on an intraday basis in a slow market.
Not being a Murrey-like genius I found the descriptions of time in the MMRPM tables of the Murrey Math Book to be somewhat subjective. I personally have difficulty deciding when a market is long term, short term, fast, slow etc. (just my own personal weakness).
Since the MMRPM statistic is an important part of Murrey Math and we have the Square in Time at our disposal one may wish to generalize the MMRPM tables for any Square in Time. Having one MMRPM table for any given Square in Time has a certain appeal. First of all, the analysis of the price movement of any traded entity is simplified and made more objective. Secondly, having one MMRPM table for all squares has a certain aesthetic appeal. After all, the Square in Time is a fractal that acts as an adjustable reference frame. In the purest sense of Murrey Math only one MMRPM table should be necessary for any Square in Time.

Fractals

To understand the approach that will be used here, certain concepts must be explained. First one must review the definition of a fractal.
The sizes (scale) of basic geometric shapes are characterized by one or two parameters. The scale of a circle is specified by its diameter, the scale of a square is given by the length of one of its sides, and the scale of a triangle is specified by the length of its three sides. In contrast, a fractal is a self similar shape that is independent of scale or scaling. Fractals are constructed by repeating a process over and over. Consider the fractal shown in FIGURE 1.

A rectangle, may be subdivided into four equal sub-rectangles as shown in FIGURE 1. Each sub-rectangle can be divided, likewise, into a set of four smaller sub-rectangles. This process may be carried out ad infinitum (ad nauseum). Each resulting rectangle, no matter how large or small it may be has the exact same ratio of height to width. This property is called self similarity.

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                                  FIGURE 1

The zig-zagging pattern on a chart of price vs. time for a market or traded equity may also be regarded as a fractal. The definition of this type of zig-zagging fractal is not as simple as the definition given above for the rectangle. The price-time behavior of a market or traded equity may be regarded as a STATISTICALLY self similar fractal (if price and time are scaled correctly).

Fractional Brownian Motion

Statistical self similarity implies that if we look at the zig-zagging price-time pattern under different time scales (e.g. intraday, daily, weekly, etc.) the statistics that characterize the zig-zagging pattern are the same. Fortunately, a relatively simple statistical model exists for describing the zig-zagging price-time behavior of markets. That model is known as fractional brownian motion (FBM) and is specified quite simply in EQUATION 1 (EQ 1).
EQ 1: < (X(t2) - X(t1)) ^ 2 > = k*((t2 - t1) ^ (2*H))
While EQ 1 may appear complicated it really is not. Let's break it down.
- X(t1) is the price of an entity at some initial time t1 (e.g. the price of gold at 2:21 PM on an intraday chart). Let X(2:21) = $320 an ounce.
- X(t2) is the price of an entity at some later time t2 (e.g. the price of gold at 3:09 PM on the same intraday chart). Let X(3:09) = $323 an ounce.
- ^ 2 symbolizes that the preceeding number enclosed in parentheses is raised to the power of 2 (i.e. square the difference of X(t2) - X(t1)). So, $323 - $320 = $3 ($3 ^ 2) = ($3 * $3) = $9 Where * is used to symbolize multiplication.
- < > These brackets symbolize the average of the enclosed number over many samples. So the number < (X(t2) - X(t1)) ^ 2 > is the result of looking at many sampled pairs of gold prices at 48 minute intervals. One could imagine a spread sheet with the following information:
```
               A         B              C           D

       1    X(9:33)   X(10:21)   COL_B - COL_A   COL_C ^2
       2    X(9:34)   X(10:22)   COL_B - COL_A   COL_C ^2
       3    X(9:35)   X(10:23)   COL_B - COL_A   COL_C ^2
       .
       .
       .
       R    X(2:21)   X(3:09)    COL_B - COL_A   COL_C ^2
```
  So < (X(t2) - X(t1)) ^ 2 > would be the sum of all the numbers in Column D divided by the number of samples (R). Where COL_A, COL_B, and COL_C denote the numbers in Column A, Column B, and Column C respectively.
- k is simply an undefined proportionality constant (i.e. just some number we don't know yet). The character * is used to symbolize multiplication.
- t2-t1 is simply the time interval. In this case 48 minutes.
- ^(2*H) symbolizes that the preceeding number enclosed in parentheses is raised to the power of 2*H. The character * is used to symbolize multiplication. The exact value of H is also unknown, however, the FBM model states that H will have a value between 0 and 1.
What does EQ 1 tell us? For simplicity, let H = 1.0. In this case, EQ1 is saying that on average, the price range of some entity over any given time interval is proportional to that time interval. The key phrase here is "on average". One would look at the spread sheet of gold prices and find that the value in each row of Column D is different. But, when averaged together they will be proportional to the time interval (in this case 48 minutes).
If, in fact, gold prices behaved according to the FBM model (with H set equal to 1.0) then one would observe this same relationship for all time intervals. So, if one built a second spreadsheet looking at the range of gold prices over many 96 minute time intervals (96 = 2 x 48) one would find that the range of gold prices would be twice as large as the range of gold prices observed over 48 minute time intervals.
For example, if the average range of gold prices observed over many 48 minute time intervals was $3, then the average range of gold prices observed over many 96 (2 x 48) minute time intervals would be $6 (i.e. $6 = 2 x $3).

Statistical Nature of Price Changes

The next part of the FBM model to understand is the statistical nature of price changes. Let's define a price change that occurs over some time interval as:
| X(t2) - X(t1) |
Where the | | symbol means to take the absolute value of the number inside the vertical brackets. This just means that if X(t2) - X(t1) happens to be a negative number, then ignore the minus sign. Treat the number as if it was positive.
Let's define the symbol X21, where X21 = | X(t2) - X(t1) |.
This next statement is abhorrent and anathema to anyone wanting to trade the markets (forgive me my sin). Are you ready?
Assume that X21 is a random number that is normally distributed. Being "normally distributed" simply means that the probability distribution that describes a collection of X21 values is the good old bell shaped curve that our teachers used to grade us in school.

Here is a quick refresher for those who do not remember the properties of the bell curve (formally known as the Gaussian distribution). Refer to FIGURES 2A and 2B.

       | P(X12)                        *
       |                            *|||||*
       |                           *|||||||*
       |                          *|||||||||*
       |                         *|||||||||||*
       |                        *|||||||||||||*
       |                       *|||||||||||||||*
       |                      *|||||||||||||||||*
       |                     *|||||||||||||||||||*
       |                    *|||||||||||||||||||||*
       |                   *|||||||||||||||||||||||*
       |                 *|||||||||||||||||||||||||||*
       |              *  |||||||||||||||||||||||||||||  *
       |          *      |||||||||||||||||||||||||||||      *
       |      *          |||||||||||||||||||||||||||||           *
       |*                |||||||||||||||||||||||||||||                 *
        -----------------------------------------------------------------
                      -z * S                      +z * S              X12
                                   FIGURE 2A


       | P(X12)                        *
       |                            *     *
       |                           *       *
       |                          *         *
       |                         *           *
       |                        *             *
       |                       *               *
       |                      *                 *
       |                     *                   *
       |                    *                     *
       |                   *                       *
       |                 *                           *
       |              *                                 *
       |          *                                         *
       |      *                                                  *
       |*||||||                                                  ||||||*
        -----------------------------------------------------------------
           -z * S                                             +z * S  X12
                                   FIGURE 2B

In our case the quantity of interest is the price range (X12) that our entity will trade in during the next time interval (t2 - t1). The Gaussian distribution has the nice property that it considers all possible values of X12 (i.e. X12 can take on any value ranging from minus infinity to plus infinity).
The vertical axis in Figures 2A and 2B represents P(X12). P(X12) is the probability that X12 (shown on the horizontal axis) will take on some specific value X (inside an infinitely narrow range).
FIGURE 2A may be interpreted as follows. The shaded area specifies the probability that X12 will lie in a range between (-z * S) and (z * S) (i.e. (-z * S) <= X12 <= (z * S)). The total area under the Gaussian distribution curve (from minus infinity to plus infinity) is 1.0. So, in the extreme case that (-z * S) = minus infinity and (z * S) = plus infinity then the entire area under the Gaussian curve would be shaded and the probability would be 1.0 that X12 will have some value at the end of the next time interval (t2 - t1). We wouldn't know what that value is, but we are guaranteed with 100% certainty that it would be something. In practical terms, one would feel 100% confident making the prediction that the price of gold will change by some amount in the next 48 minutes (where some amount is any number from minus infinity to plus infinity).
Consider a practical example. One would find credible the prediction that in the next 48 minutes the price of gold would increase by $1 per ounce or less, or that the price of gold would decrease by $1 per ounce or less. This scenario is depicted in FIGURE 2A with (-z * S) = -$1 and (z * S) = +$1. In this case more than half of the area under the Gaussian distribution is shaded. Hence, based upon history, the prediction of a $1 per ounce (or less) swing in the price of gold over the next 48 minutes has a better than 50% chance of being correct.
Consider another example. If someone came up to you and told you that in the next 48 minutes the price of gold would go up $2000 or more per ounce, or that in the next 48 minutes gold would become so devalued that people would pay you $2000 or more per ounce just to take it off their hands, you would not be likely to make that trade. This is because history has shown that the probability of either of those events occurring is so small that you would be better off buying a lottery ticket. This scenario is depicted in FIGURE 2B. In this case (-z * S) = -$2000 and (z * S) = +$2000. Notice that the shaded area under the Gaussian distribution is at the tails of the distribution. Most of the area under the Gaussian is at the center. Very little area lies under the right and left tails of the distribution. Since the shaded area is very small when compared to 1.0 then we can see that the chances (probability) of gold making a $2000 per ounce price swing are very small.
The shaded area in FIGURE 2A can also be thought of in another way. The shaded area is the probability that prices will reverse after moving out to (z * S) or (-z * S). This is because the probability of moving further out into the tails of the Gaussian distribution is given by the unshaded area under the tails (FIGURE 2A). So, if the the price of gold happened to move far enough in the next 48 minutes so that 90% of the area under the Gaussian was shaded then only 10% of the Gaussian would be unshaded. Thus gold would only have a 10% chance of moving further. Therefore, the chance of reversal is 90%.
Let's repeat the prior point more symbolically. Refer again to FIGURE 2A. Let the current time be t1 and the price of the traded entity (e.g. gold) be specified by X(t1). Let the future time be t2 and the price of the traded entity be specified by X(t2).
X12 = X(t2) - X(t1)
The shaded area in FIGURE 2A specifies the probability that gold will increase in price by an amount of X12 or less or decrease in price by an amount of X12 or less during the future time interval t2 - t1. The probability that gold will increase in price by an amount greater than X12 or decrease by an amount greater than X12 is specified by the unshaded area in FIGURE 2A. Recall that the total area under the Gaussian distribution is 1.0
1.0 - Shaded Area = Unshaded Area
The shaded area is specifying the probability that a price swing of X12 (occurring during the future time interval t2 - t1) will be reversed. This is exactly the definition of the Murrey Math MMRPM's.
The above examples illustrate the fact that the behavior of the Gaussian distribution is consistent with the expected price behavior of traded markets. That is to say, within a given future time interval (t2 - t1), small to moderate price swings around the current price are more likely (more probable) than very large price swings. All of this discussion assumes that one is using the correct Gaussian distribution.
The shape of the Gaussian distribution is controlled by the parameter S. The parameter S is called the standard deviation. The parameter z is just some number that allows X12 to be expressed in units of standard deviations (i.e. X12 = (z * S)). The larger the value of S, the shorter and wider (more spread out) the bell shaped curve becomes. As S becomes smaller the bell shaped curve becomes more narrow and tends to look more like a spike than a bell. The larger the value of S the greater the price volatility over the time interval of interest.
In the above examples of gold, price swings were considered over the future time interval (t2 - t1) of 48 minutes. If one wished to consider a different time interval (e.g. 96 minutes) then one would need to have a new value of S to describe a new Gaussian distribution. One would need a Gaussian distribution for each future time interval (i.e. for our purposes, the standard deviation S is a mathematical function of time S = S(t)).
If one knows the value of S for all desired time intervals (i.e. if one knows the function S(t)) then one can refer to tables to determine the probability that price swings will reverse after reaching some particular value X12.
Fortunately, based upon how the Gaussian distribution is defined, the following relationship is true:
(S ^ 2) = < (X(t2) - X(t1)) ^ 2 > = k*((t2 - t1) ^ (2*H))
Hence we now know S as a function of time. A new problem arises in that the values of k and H are not known for gold or any other market. We do, however, have Murrey Math and the Square in Time. Given the assumptions made by Murrey Math, and by making some additional assumptions, one can arrive at the final goal of specifying the MMRPM's for all markets.
Let's stop for a moment and consider the key assumptions that must be made to achieve the desired result.
- 1) The zig-zagging price-time behavior of markets is described by the model known as fractional brownian motion (FBM) (Eq 1).
  EQ 1: < (X(t2) - X(t1)) ^ 2 > = k*((t2 - t1) ^ (2*H))
- 2) The values of X(t2) - X(t1) (i.e. X12) are random numbers that are normally distributed (the Gaussian distribution). This imples that < (X(t2) - X(t1)) ^ 2 > = (S ^ 2) where S is the standard deviation of the Gaussian distribution.
  Assumptions 1 and 2 are pretty good assumptions. Together, these two assumptions make up the random walk model of markets (When H = 1/2). Some have questioned whether or not (X(t2) - X(t1)) is normally distributed. In general, however, the normal distribution is considered to be a good approximation.
- 3) All markets exhibit the same statistical behavior specified in assumptions (1) and (2).
  This assumption is the basis of Murrey Math. Rejecting this assumption would require the rejection of Murrey Math.
- 4) The Square in Time scales the price-time action of markets so that the parameter H from EQ1 is equal to 1.0 (i.e. H = 1.0).
  This is a big assumption, but an argument may be made in favor of it. The Square in Time is a fractal. The rules for changing the scale of this fractal are to simply multiply the height and width of the square by 2 or by 1/2. This is a linear scaling. This can only be valid if H = 1.0. H relates the typical change in price < (X(t2) - X(t1)) > to the time interval (t2 - t1) i.e.
  < (X(t2) - X(t1)) > is proportional to ((t2 - t1) ^ H)
  The same statistical properties should be observed in a larger Square in Time as well as in a smaller Square in Time. This is the statistical self similarity property of price-time behavior. If we wished to consider price action over a longer time frame then we would multiply the time interval by 2.0 (this is how we scale the fractal). Lets do that:
  ((2 * (t2 - t1)) ^ H) = (2 ^ H) * ((t2 - t1) ^ H)
  Note the term (2 ^ H). This term shows that if the time interval is doubled, then one would have to multiply the price range by (2 ^ H). If the scaling rule of the Square in Time is valid then H must be 1.0. Otherwise, we could not simply double price and double time when scaling the Square in Time.
- 5) The proportionality constant (from Eq 1) k = 1.0.
  I have no argument for this assumption other than convenience and wishful thinking. One has to start somewhere. This assumption may be valid based upon the way the Square in Time is defined. There may be a theoretical observation that could be used to prove k = 1.0 as was done for assumption (4) showing that H = 1.0. Algorithms are available for identifying the value of k. This would, however, require some computer programming that I do not have the time to perform currently. So, for now, k = 1.0.
Recall that when the price-time behavior of a market has been scaled inside a Square in Time the actual price-time units of dollars vs. days or points vs. minutes are replaced by 1/8'ths of price vs. 1/8'ths of time. Each Square in Time extends 8/8'ths in height and 8/8'ths in time.
Once the price-time behavior of a market has been scaled inside a Square in Time the following formula may be applied:
(S ^ 2) = < (X(t2) - X(t1)) ^ 2 > = k*((t2 - t1) ^ (2*H))
Setting H = 1.0 and k = 1.0 yields:
(S ^ 2) = < (X(t2) - X(t1)) ^ 2 > = ((t2 - t1) ^ 2)
or
S = t2 - t1
with changes in X and t (price and time) expressed in units of 1/8'ths. Let's represent a change in X (price) using M/8 and let's represent a change in t (time) using N/8, where
M = 1, 2, 3, 4, 5, 6, 7, or 8
N = 1, 2, 3, 4, 5, 6, 7, or 8
Refer back to FIGURE 2A and the discussion about the Gaussian distribution. Recall the statement that X12 = (z * S).
Solving for z yields z = X12/S = |X(t2) - X(t1)|/(t2 - t1) where the | | brackets symbolize the absolute value of (X(t2) - X(t1)). If changes in price and time are expressed in 1/8'ths then
z = (M/8)/(N/8) = M/N

Given z, one can simply go to any statistics handbook and look up the probability that price will reverse after moving M/8'ths in N/8'ths of time. In other words, a general table of MMRPM values for any square in time (given the fact that the above assumptions are true). Refer to TABLE 1 (A Square of 64).

 PRICE
   M   ^
       |
   8   |  .999    .999    .992   .954    .890    .816    .746    .683
       |
   7   |  .999    .999    .980   .920    .838    .757    .683    .621
       |
   6   |  .999    .997    .954   .866    .770    .683    .610    .547
       |
   5   |  .999    .988    .905   .789    .683    .593    .522    .471
       |
   4   |  .999    .954    .816   .683    .576    .497    .431    .383
       |
   3   |  .997    .866    .683   .547    .451    .383    .332    .296
       |
   2   |  .954    .683    .497   .383    .311    .259    .228    .197
       |
   1   |  .683    .383    .259   .197    .159    .135    .111    .103
        -------------------------------------------------------------->
        N   1       2       3      4       5       6       7       8
                                                                   TIME
                                TABLE 1
                           (A SQUARE OF 64)

TABLE 1 may only be used in the context of the Square in Time. To use TABLE 1, set the price-time action into the appropriate Murrey Math Square in Time. Once the Square in Time has been defined, changes in price are expressed in 1/8'ths of the square's height. Changes in time are expressed in 1/8'ths of the square's time width. One can then look at the most recent price movement within the square as M/8'ths of price over N/8'ths of time (the table is the same for price increases and price decreases). The entry in the M'th row and N'th column specify the probability that the price movement will reverse itself.

General Discussion

The validity of the results shown in TABLE 1 are of course dependent upon the correctness of the assumptions used to derive them. The most questionable assumption is k = 1.0. If the value of k is something other than 1.0, the qualitative nature of the results would still be the same. The term "qualitative nature" meaning that the probabilities of price reversal would still be a function of the ratio M/N. A different value for k would change the magnitude of the probabilities but not their general pattern within the square.
A point worth noting is the fact that M/N is the slope of a line drawn within the Square in Time. The slope of any line is simply rise/run. Within the Square in Time rise/run is:
(change in price)/(change in time) = M/N
This implies that all trendlines within the Square in Time are lines of constant price reversal probabilities. One could think of trendlines as iso-MMRPM lines (just as lines of constant temperature on a weather map are called iso-therms). If prices were to move exactly along a trendline then the probability of price reversal would be constant at any point along the trendline.
Having a Square in Time that is correctly constructed is obviously crucial to using the MMRPM. The current square must be immediately re-drawn when prices move beyond its boundaries.
The reversal probabilities shown in TABLE 1 are in general agreement with the MMRPM numbers presented by Murrey in the Murrey Math Book. Certainly the qualitative behavior of the probabilities in TABLE 1 agree with one's expectations. Large price movements that occur over short time intervals are more likely to reverse than smaller price movements occurring over longer periods of time.
Understanding the source of the MMRPM probabilities helps to put Murrey Math in perspective. Points that Mr. Murrey makes in his book take on a greater clarity (at least for me) after seeing where the MMRPM probabilities seem to come from. While trading cannot be based solely upon MMRPM, they are a valuable part of Murrey Math. Understanding the MMRPM helps to build confidence in the Murrey Math system and confidence in trading.

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