## Return to Murrey Math |

- 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.

- 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.
---------------------------------------------------------------- | | | | | | | | | | |-------|-------| | | | | | | | | | | | | |---------------|----------------| | | | | | | | | | | | | | | | | | | | | | |--------------------------------|-------------------------------| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ---------------------------------------------------------------- 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).

- 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.

- 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.
- 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).

- 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.

- 1) The zig-zagging price-time behavior of markets is described by the
model known as fractional brownian motion (FBM) (Eq 1).
- 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 - t1with 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.

- 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.

created by Bonnie Lee Hill,

bonniehill@verizon.net

last modified on June 8, 2009

`URL: http://mysite.verizon.net/bonniehill/pages.aux/murrey/TKnotes.2.html`