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 Introduction
Squares
- As mentioned above, Murrey Math has identified a system of reference
frames (coordinate systems) that can be used to objectively gauge price
movement at all price-time scales. Taken collectively, these reference
frames or "squares in time" constitute a fractal. Each square in time can be
thought of as being a part of (1/4) a larger square in time. Recall the
simple example of the fractal described in the introduction of this paper.
Each set of four squares was created by subdividing a larger square. Unlike
a mathematically ideal fractal, we cannot have infinitely large or small
squares in time since we do not get price data over infinitely large or
small time frames. But for all practical purposes, the Murrey Math squares
in time are a fractal.
- Fractals are created by recursiveley (repeatedly) executing a set of
steps or instructions. This is also true of Murrey Math "squares in time".
- The first step in constructing a square in time for a particular entity
(NOTE: The word "entity" will be used as a shorthand to refer to any traded
equity or derivative such as stocks, commodities, indices, etc.) is
identifying the scale of the smallest square that "controls" the price
movement of that entity. Murrey refers to this as "setting the rhythm".
Murrey defines several scales.
- Let's use the symbol SR to represent the possible values of these scales
(rhythms). SR may take on the values shown below in TABLE 1:
- A larger value of SR could be generated by multiplying the largest value
by 10.
Hence, 10 x 100,000 = 1,000,000 would be the next larger scale
factor.
- The choice of SR for a particular entity is dictated by the maximum
value of that entity during the timeframe in question.
TABLE 1 defines the
possible choices of SR.
TABLE 1:
IF (the max value of AND (the max value of THEN (SR is)
the entity is less the entity is
than or equal to) greater than)
250,000 25,000 100,000
25,000 2,500 10,000
2,500 250 1,000
250 25 100
25 12.5 12.5
12.5 6.25 12.5
6.25 3.125 6.25
3.125 1.5625 3.125
1.5625 0.390625 1.5625
0.390625 0.0 0.1953125
- The value of SR that is chosen is the smallest value of SR that
"controls" the maximum value of the entity being studied.
The word
"controls" in this last statement needs clarification. Consider two
examples.
- EXAMPLE 1)
Suppose the entity being studied is a stock. During the timeframe being
considered the maximum value that this stock traded at was 75.00. In this
case, the value of SR to be used is 100. (Refer to TABLE 1)
- EXAMPLE 2)
Suppose the entity being studied is a stock. During the timeframe being
considered the maximum value that this stock traded at was 240.00. In this
case, the value of SR to be used is also 100. (Refer to TABLE 1)
- In EXAMPLE 2, even though the maximum price of the stock exceeds the
value of SR, the stock will still behave as though it is being "controlled"
by the SR value of 100. This is because an entity does not take on the
characteristics of a larger SR value until the entity's maximum value
exceeds 0.25 x the larger SR value. So, in EXAMPLE 2, the lower SR value is
100 and the larger SR value is 1000. Since the price of the stock is 240 the
"controlling" SR value is 100 because 240 is less than (.25 x 1000) 250. If
the price of the stock was 251 then the value of SR would be 1000. TABLE 1
shows some exceptions to this ".25 rule" for entities priced between 12.5
and 0.0. TABLE 1 takes these exceptions into account.
Murrey Math Lines
- Let us now continue constructing the square in time for our entity. The
selection of the correct scale factor SR "sets the rhythm" (as Murrey would
say) for our entity.
- Remember, Gann believed that after an entity has a price movement, that
price movement will be retraced in multiples of 1/8's (i.e. 1/8, 2/8, 3/8,
4/8, 5/8, 6/8, 7/8, 8/8). So, if a stock moved up 4 points Gann believed the
price of the stock would reverse and decline in 1/2 point (4/8) increments
(i.e. 1/2, 2/2, 3/2, 4/2, 5/2, 6/2, 7/2, 8/2 ...). Since prices move in
1/8's, Murrey Math divides prices into 1/8 intervals. The advantage of
Murrey Math is that a "rhythm" (a scale value SR) for our entity has been
identified. Traditional Gann techniques would have required one to
constantly chase price movements and to try to figure out which movement was
significant. If a significant price movement could be identified then that
price movement would be divided into 1/8's. Murrey Math improves upon
traditional Gann analysis by providing a constant (non-changing) price range
to divide into 1/8's. This constant price range is the value of SR (the
"rhythm") that is chosen for each entity.
- So, having selected a value for SR, Murrey Math instructs us to divide
the value of SR into 1/8's. For the sake of consistency, let's introduce
some notation. Murrey refers to major, minor, and baby Murrey Math lines.
Murrey abbreviates the term "Murrey Math Lines" using MML. Using the MML
abbreviation let;
The symbol: MML be defined as: Any Murrey Math Line
The symbol: MMML be defined as: Major Murrey Math Line
The symbol: mMML be defined as: Minor Murrey Math Line
The symbol: bMML be defined as: Baby Murrey Math Line
and, using the abbreviation MMI to mean "Murrey Math Interval", let;
The symbol: MMI be defined as: Any Murrey Math Interval
The symbol: MMMI be defined as: Major Murrey Math Interval = SR/8
The symbol: mMMI be defined as: Minor Murrey Math Interval = SR/8/8
The symbol: bMMI be defined as: Baby Murrey Math Interval = SR/8/8/8
where the symbol /8/8/8 means that SR is to be divided by 8 three times.
For example, if SR = 100 then the Baby Murrey Math Interval bMMI is:
100/8/8/8 = 12.5/8/8 = 1.5625/8 = 0.1953125
- Let's also introduce the term octave. An octave consists of a set of 9
Murrey Math Lines (MML's) and the 8 Murrey Math Intervals (MMI's) associated
with the 9 MML's. Major, minor, and baby octaves may be constructed. For
example, if SR = 100 then the major octave is shown in FIGURE 2. The octave
is constructed by first calculating the MMMI. MMMI = SR/8 = 100/8 = 12.5.
The major octave is then simply 8 MMMI's added together starting at 0. In
this case 0 is the base.
100 -------------------------------------------- 8/8 MMML
87.5 -------------------------------------------- 7/8 MMML
75 -------------------------------------------- 6/8 MMML
62.5 -------------------------------------------- 5/8 MMML
50 -------------------------------------------- 4/8 MMML
37.5 -------------------------------------------- 3/8 MMML
25 -------------------------------------------- 2/8 MMML
12.5 -------------------------------------------- 1/8 MMML
0 -------------------------------------------- 0/8 MMML
FIGURE 2
- A minor octave is constructed in a manner similar to the method shown
for the major octave. Again, let SR = 100. First calculate the mMMI. mMMI =
SR/8/8 = MMMI/8 = 12.5/8 = 1.5625. The minor octave is then simply 8 mMMI's
added together starting at the desired base. The base must be a MMML. In
this case let the base be the 62.5 MMML. The result is shown in FIGURE 3.
75 -------------------------------------------- 8/8 mMML
73.4375 -------------------------------------------- 7/8 mMML
71.875 -------------------------------------------- 6/8 mMML
70.3125 -------------------------------------------- 5/8 mMML
68.75 -------------------------------------------- 4/8 mMML
67.1875 -------------------------------------------- 3/8 mMML
65.625 -------------------------------------------- 2/8 mMML
64.0625 -------------------------------------------- 1/8 mMML
62.5 -------------------------------------------- 0/8 mMML
FIGURE 3
- Naturally, a baby octave would be constructed using the same method used
to construct a minor octave. First calculate bMMI (bMMI = mMMI/8). Then add
bMMI to the desired mMML 8 times to complete the octave.
Characteristics of MMLs
- Since, according to Gann, prices move in 1/8's, these 1/8's act as
points of price support and resistance as an entity's price changes in time.
Given this 1/8 characteristic of price action, Murrey assigns properties to
each of the MML's in an a given octave. These properties are listed here for
convenience.
- 8/8 th's and 0/8 th's Lines (Ultimate Resistance)
These lines are the hardest to penetrate on the way up, and give the
greatest support on the way down. (Prices may never make it thru these
lines).
- 7/8 th's Line (Weak, Stall and Reverse)
This line is weak. If prices run up too far too fast, and if they stall at
this line they will reverse down fast. If prices do not stall at this line
they will move up to the 8/8 th's line.
- 6/8 th's and 2/8 th's Lines (Pivot, Reverse)
These two lines are second only to the 4/8 th's line in their ability to
force prices to reverse. This is true whether prices are moving up or down.
- 5/8 th's Line (Top of Trading Range)
The prices of all entities will spend 40% of the time moving between the 5/8
th's and 3/8 th's lines. If prices move above the 5/8 th's line and stay
above it for 10 to 12 days, the entity is said to be selling at a premium to
what one wants to pay for it and prices will tend to stay above this line in
the "premium area". If, however, prices fall below the 5/8 th's line then
they will tend to fall further looking for support at a lower level.
- 4/8 th's Line (Major Support/Resistance)
This line provides the greatest amount of support and resistance. This line
has the greatest support when prices are above it and the greatest
resistance when prices are below it. This price level is the best level to
sell and buy against.
- 3/8 th's Line (Bottom of Trading Range)
If prices are below this line and moving upwards, this line is difficult to
penetrate. If prices penetrate above this line and stay above this line for
10 to 12 days then prices will stay above this line and spend 40% of the
time moving between this line and the 5/8 th's line.
- 1/8 th Line (Weak, Stall and Reverse)
This line is weak. If prices run down too far too fast, and if they stall at
this line they will reverse up fast. If prices do not stall at this line
they will move down to the 0/8 th's line.
- Completing the square in time requires the identification of the upper
and lower price boundaries of the square. These boundaries must be MML's.
The set of all possible MML's that can be used as boundaries for the square
were specified with the selection of the scale factor (rhythm) SR. Given SR,
all of the possible MMMI's, mMMI's, bMMI's and MMML's, mMML's, and bMML's
can be calculated as shown above. The following rules dictate what the lower
and upper boundaries of the square in time will be.
Rules and Exceptions
- Rule 1:
The lower boundary of the square in time must be an even MML (i.e. 0/8 th's,
2/8 th's, 4/8 th's, 6/8 th's, or 8/8 th's). It may be a MMML, a mMMl, or a
bMML. Generally, the lower boundary will be a mMML.
- Rule 2:
The MML selected for the bottom of the square in time should be close to the
low value of the entity's trading range. The word "close" means that the
distance between the square's bottom MML and the low value of the entity
should be less than or equal to 4/8 of the next smaller octave.
For example, suppose a stock is trading within a range of 28 1/4 to 34
1/2. In this case the value of SR is 100. The MMMI is 12.5 (i.e. 100/8). The
next smaller MMI is a mMMI = 12.5/8 = 1.5625. The MMML closest to 28 1/4 is
the 2/8 th's (i.e. 2 x 12.5 = 25). The closest mMML (measured from 25) is
also a 2/8 th's MML (i.e. 2 x 1.5625 = 3.125). So, the bottom of the square
is 25 + 3.125 = 28.125 (i.e. 28 1/8).
The 28 1/8 MML is the base of the square in time. This MML satisfies rule
1 (it is an even numbered line, 2/8 th's) and it is close to 28 1/4 (28 1/4
- 28 1/8 = 1/8 = .125). The result of .125 is less than 4/8 th's of the next
smaller octave which is a "baby" octave (bMMI = 1.5625/8 = .1953125).
Specifically .125 is less than .78125 (4 x .1953125 = .781254).
- Rule 3:
The height of the square in time must be 2, 4, or 8 MMI's. The type of MMI
(major, minor, or baby) must be the same as the type of MML being used for
the lower boundary. Generally this will be a mMMI.
NOTE: If the bottom MML of the square in time is an even MML, and the top
MML of the square in time is 2, 4, or 8 MMI's above the bottom MML, then the
top MML is also an even numbered MML.
- Rule 4:
The MML selected for the top of the square in time should be close to the
high value of the entity's trading range. The word "close" means that the
distance between the square's top MML and the high value of the entity
should be less than or equal to 4/8 of the next smaller octave. This is
simply rule (2) being applied to the top of the square.
For example, consider the same stock trading within the range 28 1/4 to
34 1/2. The base of the square in time was identified as the 2/8 th's mMML
28.125. In this case the top of the square is the mMML that is 4 mMMI's
above the base: 28.125 + (4 x 1.5625) = 34.375. This MML can also be shown
to be "close" to the high end of the trading range, since, 34.5 - 34.375 =
.125 and .125 is less than .781254 (4 x .1953125 = .781254). Recall that
.1953125 is the bMMI (i.e. the next smaller octave).
- Exception to Rule 1:
The rule, "The lower boundary of the square in time must be an even MML...",
appears to have exceptions. Murrey states, "When a stock is trading in a
narrow range rotating near a MMML you may use only 1 line above and below.
Since a MMML is always an even MML (a 0 or 8 line for the next smaller
octave) then one line above or below would be an odd MML (1 or 7).
An example of this can be seen on Chart #91 in Murrey's book. This is a
chart of Chase Manhatten. In this case the bottom and top MML's of the
square in time are the 5/8 th's and 7/8 th's MML's respectively. These are
obviously odd MML's. Another example of an exception is Chart #83 in
Murrey's book. In this case the bottom of the square in time is 37.5 (an odd
3/8 th's line) and the top of the square in time is 62.5 (an odd 5/8 th's
line).
- Exception to Rules 2, 4:
Rules 2 and 4 address how close the boundaries of the square in time are to
the actual trading range of the entity in question. Murrey states;
"Then you simply count up 2, 4, or 8 lines, and include the top of its
trading range, as long as it's no higher than a) 19, b) 39, c) 78 cents
above the 100% line. (there are exceptions where it will run up a full 12.5,
or 25 or 50% line above the 100% line and come back down..."
- At this point Murrey leaves us on our own to review the charts. The book
is replete with examples in which the bottom and top MML's of the square in
time are far from the actual trading ranges (by as much as 2 mMMI's).
- Consider the two charts (both are labeled Chart #85) of McDonalds. The
lower chart espcially shows McDonalds trading in a range from 28 to 34.
Clearly, the set of mMML's that would best fit this trading range are the
lines 28.125 (2/8 th's) and 34.375 (6/8 th's). Murrey, however, draws the
square from 25 (0/8 th's) to 31.25 (4/8 th's).
- Given the above rules and exceptions I have developed a set of "rules of
thumb" to assist in the construction of squares in time. Using these "rules
of thumb" I have written a simple C program that calculates the top and
bottom MMLs for squares in time. This offers a fairly mechanical approach
that may prove beneficial to a new Murrey Math practitioner. Once a Murrey
Math neophyte becomes experienced using this mechanical system he/she may go
on to using intuition and methods that are a little (a lot) less tedious.
- I have tested this program against all of the charts in Murrey's book
and it seems to work fairly well. There are some exceptions/weaknesses that
are discussed below. First, to illustrate the methodology, a few detailed
examples are included here.
Calculating the MMLs --
Example 1
- Refer to Chart #85B of First American in the Murrey Math book. During
the time frame in question, First American traded in a range with a low of
about 28.0 and a high of about 35.25 (the wicks on the candlesticks are
ignored).
- Let's define a parameter called PriceRange. PriceRange is simply the
difference between the high and low prices of the trading range.
- STEP 1:
Calculate PriceRange.
PriceRange = 35.25 - 28.0 = 7.25
- STEP 2:
Identify the value of SR (the scale factor).
Murrey refers to this as "setting the rhythm" or identifying the "perfect
square". Refer to TABLE 1 in this paper. Reading from TABLE 1 SR = 100 (This
is because the high price for First American was 35.25. Since 35.25 is less
than 250 but greater than 25, SR = 100).
- STEP 3:
Determine the MMI that the square in time will be built from.
Let's define two new parameters. The first parameter is RangeMMI.
RangeMMI = PriceRange/MMI. RangeMMI measures the price range of First
American (or any entity) in units of Murrey Math Intervals (MMI's).
The second parameter is OctaveCount. The purpose of OctaveCount will
become evident shortly. The question to answer is, "What MMI should be used
for creating the square in time?" This question will be answered by dividing
the SR value by 8 until the "appropriate MMI" is found. So:
MMI = MMMI = SR/8 = 100/8 = 12.5
This is a MMMI. Is this the "appropriate MMI"? To answer that question
divide PriceRange by this MMI.
RangeMMI = PriceRange/MMI = 7.25/12.5 = 0.58
Now compare RangeMMI to 1.25. If RangeMMI is less than 1.25 then a
smaller MMI is needed. This is indeed the case because 0.58 is less than
1.25. Since the first MMI calculated was a MMMI, then the next MMI will be a
mMMI. Simply divide the prior MMI by 8 to get the new MMI.
MMI = mMMI = MMMI/8 = 1.5625
This is a mMMI. Is this the "appropriate MMI"? To answer that question
divide PriceRange by this latest MMI.
RangeMMI = PriceRange/MMI = 7.25/1.5625 = 4.64
Now compare RangeMMI to 1.25. If RangeMMI is less than 1.25 then a
smaller MMI is needed. Since RangeMMI is 4.64 and 4.64 is greater than 1.25
we're done. The correct MMI to use is the mMMI which is 1.5625. (Naturally,
in other cases, this process may be repeated further, continuing division by
8, until RangeMMI is greater than 1.25.)
Since we had to divide the perfect square (SR) by 8 two times to arrive
at the appropriate MMI (SR/8/8 = 100/8/8 = 12.5/8 = 1.5625) we'll set the
value of OctaveCount to be 2. The value of OctaveCount will act as a
reminder as we proceed through this example.
Now the question of 1.25. Where did this number come from? Partly trial
and error and partly reasoning. Remember that the parameter RangeMMI
describes the trading range of First American in units of Murrey Math
Intervals. Remember also that the rules for the square in time require that
the square be at least 2 MMI's high, and that the square be close to the
high and low values of the trading range.
If we used the MMMI to build the square in time for First American the
result would have been a square with a height of (2 x 12.5) 25. Because
First American has only traded within a range of 7.25 points, this square
would not represent First American's' behavior very well. The trading range
of First American should approximately fill the square. By choosing a
smaller MMI (i.e. mMMI = 1.5625) the result is a square in time that will be
4 MMI's high (RangeMMI = 4.64 which is rounded to 4. The actual height
selected for the square in time will be determined in STEP 4). Again, recall
the rule that the square must be 2, 4, or 8 MMI's high. (Is the number 1.25
perfect? NO! But, tests conducted on the charts in the Murrey Math book
indicate that 1.25 works in nearly all cases).
- STEP 4:
Determine the height of the square in time.
In STEP 3 above, we selected the appropriate value for the MMI and
calculated the final value of RangeMMI. Given the value of RangeMMI, TABLE 2
may be used to select the actual height of the square in time.
TABLE 2
ALLOWED SQUARES IN TIME:
RangeMMI Square in Time is Bounded by These MML's
1.25 < RangeMMI < 3.0 (0,2) (1,3) (2,4) (3,5) (4,6) (5,7) (6,8) (7,1)
3.0 <= RangeMMI < 5.0 (0,4) (2,6) (4,8) (6,2)
5.0 <= RangeMMI < ... (0,8) (4,4)
TABLE 2 was arrived at using trial and error. The results of the C
program I had written were compared to the charts in the back of the Murrey
Math book. Is TABLE 2 perfect? NO! But it works fairly well. TABLE 2
specifies the allowed upper and lower MML numbers that may be used to create
the square in time. Note that once the upper and lower MML's are specified
so is the height of the square. TABLE 2 attempts to accomodate Murrey's
rules for creating the square in time as well as the exceptions to those
rules.
The first row of TABLE 2 addresses squares that are two MMI's high. Note
that the exception of having squares in time with odd top and bottom MML's
is included.
The second row of TABLE 2 addresses squares that are four MMI's high.
Note that these squares are required to lie on even MML's only.
The third row of TABLE 2 addresses squares that are eight MMI's high.
Note that these squares are required to lie on (0,8) or (4,4) MML's only.
The notation (0,8) means that the bottom of the square will be a 0/8 th's
MML and the top of the square will be an 8/8 th's MML.
Continuing with First American, recall that RangeMMI = 4.64. Reading from
TABLE 2 we see that the square in time will be 4 MMI's high and will lie on
one of the MML combinations (0,4), (2,6), (4,8), or (6,2).
- STEP 5:
Find the bottom of the square in time.
The objective of this step is to find the MML that is closest to the low
value of First American's trading range (i.e. 28.0). This MML must be a mMML
since the MMI we are using is a mMMI (i.e. 1.5625). Actually, the MML we
will find in this step is the mMML that is closest to but is less than or
equal to First American's low value.
This is fairly simple. To repeat, the MML type must correspond to the MMI
type that was selected. We chose an MMI that is a mMMI (i.e. 1.5625), hence,
the MML must be a mMML. We now make use of the parameter OctaveCount. In
this example, OctaveCount = 2. Since OctaveCount = 2 we will perform 2
divisions by 8 to arrive at the desired MML.
MMI = MMMI = SR/8 = 100/8 = 12.5
The base of the perfect square is 0.0, so subtract the base from the low
value of First American's trading range (28.0 - 0.0 = 28.0). Now we find the
MMML that is less than or equal to 28.0. In other words, how many MMMI's
could we stack up from the base (i.e. 0.0) to get close to (but less than
28.0).
28.0/MMMI = 28.0/12.5 = 2.24 ==> 2 (Since there are no partial MMI's)
0.0 + (2 x 12.5) = 25.0
25.0 is the 2/8 th's MMML that is closest to but less than 28.0
Since OctaveCount = 2, this process will be repeated a second time for
the mMMI. The only difference is that the base line is the MMML from the
prior step. So, once again, subtract the base (i.e. 25) from the low value
of First American's trading range (28 - 25 = 3.0). Now find the mMML that is
less than or equal to 28.0. In other words, how many mMMI's could we stack
up from the base (i.e. 25) to get close to (but less than 28.0).
3.0/mMMI = 3.0/1.5625 = 1.92 ==> 1 (Since there are no partial MMI's)
25 + (1 x 1.5625) = 26.5625
26.5625 is the 1/8 th mMML that is closest to but less than 28.0
So, mMML = 26.5625
This mMML is the "best first guess" for the bottom of the square in time.
But there is a problem...
- STEP 6:
Find the "Best Square"
By the end of STEP 5, a square in time has been defined that will be 4
mMMI's in height and have a base on the 1/8 th mMML = 26.5625. Recall,
however, that the rules in TABLE 2 state that a square that is 4 MMI's in
height must lie on an even numbered MML. A 1/8 th line is odd. So, two
choices are available. Referring to TABLE 2 we can choose either a (0,4)
square or a (2,6) square. Which do we choose?
Let's define an error function and choose the square that minimizes this
error. The error function is:
Error = abs(HighPrice - TopMML) + abs(LowPrice - BottomMML)
Where:
- HighPrice is the high price of the entity in question
(in this case the high price of First American 35.25)
- LowPrice is the low price of the entity in question
(in this case the low price of First American 28.0)
- TopMML is the top MML of the square in time
- BottomMML is the bottom MML of the square in time
- abs() means take the absolute value of the quantity in parentheses
(i.e. If the quantity in parentheses is negative, ignore the minus sign
and make the number positive. For example, abs(-2.12) = abs(2.12) =
2.12.
Having now defined an error function it can now be applied to the problem
at hand. The square in time that was determined in STEP 5 has a bottom MML
of 26.5625 and a height of 4 mMMI's. The top MML is therefore 26.5625 + (4 x
1.5625) = (26.5625 + 6.25) = 32.8125. Recall, however, this is still the
square lying upon the 1/8 mMML (a (1,5) square on odd MML's). We want to use
the error function to distinguish between the (0,4) square and the (2,6)
square.
The (0,4) square is simply the (1,5) square shifted down by one mMMI and
the (2,6) square is the (1,5) square shifted up by one mMMI.
0/8 th mMML = 26.5625 - 1.5625 = 25.0
4/8 th's mMML = 32.8125 - 1.5625 = 31.25
So, the bottom of the (0,4) square is 25.0 and the top of the (0,4)
square is 31.25.
Likewise for the (2,6) square:
2/8 th's mMML = 26.5625 + 1.5625 = 28.125
6/8 th's mMML = 32.8125 + 1.5625 = 34.375
So, the bottom of the (2,6) square is 28.125 and the top of the (2,6)
square is 34.375.
Now apply the error function to each square to determine "the best square
in time".
Error(0,4) = abs(35.25 - 31.25) + abs(28.0 - 25.0) = 7.0
Error(2,6) = abs(35.25 - 34.375) + abs(28.0 - 28.125) = 1.0
Clearly the (2,6) square is the better fit (has less error). Finally, we
have arrived at a square in time that satisfies all of the rules. We can now
divide the height of the square by 8 to arrive at the 1/8 lines for the
square in time.
(34.375 - 28.125)/8 = 6.25/8 = .78125
So the final square is:
100.0% 34.375
87.5% 33.59375
75.0% 32.8125
62.5% 32.03125
50.0% 31.25
37.5% 30.46875
25.0% 29.6875
12.5% 28.90625
0.0% 28.125
Exactly as seen on Chart #85B of the Murrey Math book.
Calculating the MMLs --
Example 2
- Refer to Chart #294, the OEX 100 Cash Index in the Murrey Math book.
During the time frame in question (intraday), the OEX traded in a range with
a low of about 433.5 and a high of about 437.5 (the wicks on the
candlesticks are ignored). EXAMPLE 1 above contains all of the detailed
explanations regarding the mechanics of setting up the MML's. The following
examples will just show the basic steps.
- STEP 1:
Calculate PriceRange.
PriceRange = 437.5 - 433.5 = 4.0
- STEP 2:
Identify the value of SR (the scale factor).
Refer to TABLE 1: SR = 1000
- STEP 3:
Determine the MMI that the square in time will be built from.
Octave 1:
- MMI = MMMI = SR/8 = 1000/8 = 125
- RangeMMI = PriceRange/MMI = 4.0/125 = .032
- (RangeMMI is less than 1.25 so divide by 8 again)
Octave 2:
- MMI = mMMI = MMMI/8 = 125/8 = 15.625
- RangeMMI = PriceRange/MMI = 4.0/15.625 = .256
- (RangeMMI is less than 1.25 so divide by 8 again)
Octave 3:
- MMI = bMMI = mMMI/8 = 15.625/8 = 1.953125
- RangeMMI = PriceRange/MMI = 4.0/1.953125 = 2.048
- (RangeMMI is greater than 1.25 so 1.953125 is the desired MMI)
Since the scale factor SR was divided by 8 three times, OctaveCount = 3.
- STEP 4:
Determine the height of the square in time.
Refer to TABLE 2: RangeMMI = 2.048 so the height of the square is 2.
- STEP 5:
Find the bottom of the square in time.
First Octave:
- 433.5 - 0.0 = 433.5
- 433.5/MMMI = 433.5/125 = 3.468 ==> 3.0
- 0.0 + (3.0 x 125) = 375 (3/8 th's MMML)
Second Octave:
- 433.5 - 375 = 58.5
- 58.5/mMMI = 58.5/15.625 = 3.744 ==> 3.0
- 375 + (3.0 x 15.625) = 421.875 (3/8 th's mMML)
Third Octave:
- 433.5 - 421.875 = 11.625
- 11.625/bMMI = 11.625/1.953125 = 5.952 ==> 5.0
- 421.875 + (5.0 x 1.953125) = 431.640625 (5/8 th's bMML)
This results in a square with a height of 2 bMMI's and a base on the 5/8
th's bMML 431.64.
- STEP 6:
Find the "Best Square"
The result of STEP 5 is a square with a height of 2 bMMI's and a base on
the 5/8 th's bMML 431.64. Refer to TABLE 2: The likely "best square" is
either the (5,7) or the (6,8).
The bottom and top of the (5,7) square are:
Bottom: 431.64
Top: 431.64 + (2 x 1.953125) = 435.55
The bottom and top of the (6,8) square are:
Bottom: 431.64 + 1.953125 = 433.59
Top: 435.55 + 1.953125 = 437.50
Calculate the fit errors:
- Error(5,7) = abs(437.5 - 435.55) + abs(433.5 - 431.64) = 3.81
- Error(6,8) = abs(437.5 - 437.50) + abs(433.5 - 433.59) = 0.09
The "best square" is the (6,8) square since the (6,8) square has the
smallest error.
So the final square is:
100.0% 437.5
87.5% 437.01
75.0% 436.52
62.5% 436.03
50.0% 435.54
37.5% 435.05
25.0% 434.57
12.5% 434.08
0.0% 433.59
Calculating the MMLs --
Example 3
- Refer to Chart #300, the Deutsche Mark, in the Murrey Math book. During
the time frame in question (intraday), the Mark traded in a range with a low
of about .7110 and a high of about .7170 (the wicks on the candlesticks are
ignored). The Deutsche Mark is an example of an entity that trades on a
scale that is different from the literal choice on TABLE 1. The price values
for the Deutsche Mark must be re-scaled so that the appropriate SR value is
selected. All of the Deutsche Mark prices are multiplied by 10,000. So, the
trading range to be used to calculate the square in time is 7110 to 7170.
After the square in time is determined, the resulting MML values may be
divided by 10,000 to produce a square that can be directly compared to the
quoted prices of the Deutsche Mark.
- STEP 1:
Calculate PriceRange.
PriceRange = 7170 - 7110 = 60.0
- STEP 2:
Identify the value of SR (the scale factor).
Refer to TABLE 1: SR = 10000
- STEP 3:
Determine the MMI that the square in time will be built from.
Octave 1:
- MMI = MMMI = SR/8 = 10000/8 = 1250
- RangeMMI = PriceRange/MMI = 60/1250 = .048
- (RangeMMI is less than 1.25 so divide by 8 again)
Octave 2:
- MMI = mMMI = MMMI/8 = 1250/8 = 156.25
- RangeMMI = PriceRange/MMI = 60/156.25 = .384
- (RangeMMI is less than 1.25 so divide by 8 again)
Octave 3:
- MMI = bMMI = mMMI/8 = 156.25/8 = 19.53125
- RangeMMI = PriceRange/MMI = 60/19.53125 = 3.072
- (RangeMMI is greater than 1.25 so 19.53125 is the desired MMI)
Since the scale factor SR was divided by 8 three times, OctaveCount = 3.
- STEP 4:
Determine the height of the square in time.
Refer to TABLE 2: RangeMMI = 3.072 so the height of the square is 4.
- STEP 5:
Find the bottom of the square in time.
First Octave:
- 7110 - 0.0 = 7110
- 7110/MMMI = 7110/1250 = 5.688 ==> 5.0
- 0.0 + (5.0 x 1250) = 6250 (5/8 th's MMML)
Second Octave:
- 7110 - 6250 = 860
- 860/mMMI = 860/156.25 = 5.504 ==> 5.0
- 6250 + (5.0 x 156.25) = 7031.25 (5/8 th's mMML)
Third Octave:
- 7110 - 7031.25 = 78.75
- 78.75/bMMI = 78.75/19.53125 = 4.032 ==> 4.0
- 7031.25 + (4.0 x 19.53125) = 7109.375 (4/8 th's bMML)
This results in a square with a height of 4 bMMI's and a base on the 4/8
th's bMML 7109.375.
- STEP 6:
Find the "Best Square"
The result of STEP 5 is a square with a height of 4 bMMI's and a base on
the 4/8 th's bMML 7109.375. Refer to TABLE 2: The likely "best square" is
the (4,8). One could, of course, perform a test using the error function and
check other squares as was done in the prior examples. A quick visual check
of Chart #300, however, shows that the (2,6) or (6,2) squares will result in
errors that are greater than the error associated with the (4,8) square.
The bottom and top of the (4,8) square are:
Bottom: 7109.375
Top: 7109.375 + (4 x 19.53125) = 7187.5
Since the original price values were multiplied by 10000, the reverse
operation is performed to arrive at MML values that match the quoted prices
of the Deutsche Mark.
The "corrected" bottom and top of the (4,8) square are:
Bottom: .7109
Top: .7187
So the final square is:
100.0% .7187
87.5% .7177
75.0% .7168
62.5% .7158
50.0% .7148
37.5% .7138
25.0% .7129
12.5% .7119
0.0% .7109
Calculating the MMLs --
Example 4
- Refer to Chart #298, the 30 Year Bond, in the Murrey Math book. During
the time frame in question (intraday), the 30 Yr Bond traded in a range with
a low of about 102.05 and a high of about 102.75 (the wicks on the
candlesticks are ignored). The 30 Yr Bond is another example of an entity
that trades on a scale that is different from the literal choice on TABLE 1.
The price values for the 30 Yr Bond must be re-scaled so that the
appropriate SR value is selected. All of the 30 Yr Bond prices are
multiplied by 100. So, the trading range to be used to calculate the square
in time is 10205 to 10275. After the square in time is determined, the
resulting MML values may be divided by 100 to produce a square that can be
directly compared to the quoted prices of the 30 Yr Bond.
- STEP 1:
Calculate PriceRange.
PriceRange = 10275 - 10205 = 70.0
- STEP 2:
Identify the value of SR (the scale factor).
Refer to TABLE 1: SR = 10000
- STEP 3:
Determine the MMI that the square in time will be built from.
Octave 1:
- MMI = MMMI = SR/8 = 10000/8 = 1250
- RangeMMI = PriceRange/MMI = 70/1250 = .056
- (RangeMMI is less than 1.25 so divide by 8 again)
Octave 2:
- MMI = mMMI = MMMI/8 = 1250/8 = 156.25
- RangeMMI = PriceRange/MMI = 70/156.25 = .448
- (RangeMMI is less than 1.25 so divide by 8 again)
Octave 3:
- MMI = bMMI = mMMI/8 = 156.25/8 = 19.53125
- RangeMMI = PriceRange/MMI = 70/19.53125 = 3.584
- (RangeMMI is greater than 1.25 so 19.53125 is the desired MMI)
Since the scale factor SR was divided by 8 three times, OctaveCount = 3.
- STEP 4:
Determine the height of the square in time.
Refer to TABLE 2: RangeMMI = 3.584 so the height of the square is 4.
- STEP 5:
Find the bottom of the square in time.
First Octave:
- 10205 - 0.0 = 10205
- 10205/MMMI = 10205/1250 = 8.164 ==> 8.0
- 0.0 + (8.0 x 1250) = 10000 (8/8 th's MMML)
Second Octave:
- 10205 - 10000 = 205
- 205/mMMI = 205/156.25 = 1.312 ==> 1.0
- 10000 + (1.0 x 156.25) = 10156.25 (1/8 th's mMML)
Third Octave:
- 10205 - 10156.25 = 48.75
- 48.75/bMMI = 48.75/19.53125 = 2.496 ==> 2.0
- 10156.25 + (2.0 x 19.53125) = 10195.3125 (2/8 th's bMML)
This results in a square with a height of 4 bMMI's and a base on the 2/8
th's bMML 10195.3125.
- STEP 6:
Find the "Best Square"
The result of STEP 5 is a square with a height of 4 bMMI's and a base on
the 2/8 th's bMML 10195.3125. Refer to TABLE 2: The likely "best square" is
the (2,6). One could, of course, perform a test using the error function and
check other squares as was done in the prior examples. A quick visual check
of Chart #298, however, shows that the (0,4) or (4,8) squares will result in
errors that are greater than the error associated with the (2,6) square.
The bottom and top of the (4,8) square are:
Bottom: 10195.3125
Top: 10195.3125 + (4 x 19.53125) = 10273.4375
Since the original price values were multiplied by 100, the reverse
operation is performed to arrive at MML values that match the quoted prices
of the 30 Yr Bond.
The "corrected" bottom and top of the (4,8) square are:
Bottom: 101.95
Top: 102.73
So the final square is:
100.0% 102.73
87.5% 102.63
75.0% 102.54
62.5% 102.44
50.0% 102.34
37.5% 102.24
25.0% 102.15
12.5% 102.05
0.0% 101.95
Calculating the MMLs --
Example 5
- Refer to Chart #85 (the one at the top of the page), McDonalds, in the
Murrey Math book. During the time frame in question McDonalds traded in a
range with a low of about 26.75 and a high of about 32.75 (the wicks on the
candlesticks are ignored). In EXAMPLES 1 through 4 the MML's that were
determined for the square in time matched the examples the the Murrey Math
book. This example will not match the result in the Murrey Math book. This
will lead to a discussion regarding the weaknesses of this calculation
method.
- STEP 1:
Calculate PriceRange.
PriceRange = 32.75 - 26.75 = 6.0
- STEP 2:
Identify the value of SR (the scale factor).
Refer to TABLE 1: SR = 100
- STEP 3:
Determine the MMI that the square in time will be built from.
Octave 1:
- MMI = MMMI = SR/8 = 100/8 = 12.5
- RangeMMI = PriceRange/MMI = 6/12.5 = .48
- (RangeMMI is less than 1.25 so divide by 8 again)
Octave 2:
- MMI = mMMI = MMMI/8 = 12.5/8 = 1.5625
- RangeMMI = PriceRange/MMI = 6/1.5625 = 3.84
- (RangeMMI is greater than 1.25 so 1.5625 is the desired MMI)
Since the scale factor SR was divided by 8 two times, OctaveCount = 2.
- STEP 4:
Determine the height of the square in time.
Refer to TABLE 2: RangeMMI = 3.84 so the height of the square is 4.
- STEP 5:
Find the bottom of the square in time.
First Octave:
- 26.75 - 0.0 = 26.75
- 26.75/MMMI = 26.75/12.5 = 2.14 ==> 2.0
- 0.0 + (2.0 x 12.5) = 25.0 (2/8 th's MMML)
Second Octave:
- 26.75 - 25.0 = 1.75
- 1.75/mMMI = 1.75/1.5625 = 1.12 ==> 1.0
- 25.0 + (1.0 x 1.5625) = 26.5625 (1/8 th's mMML)
This results in a square with a height of 4 mMMI's and a base on the 1/8
th's mMML 26.5625
- STEP 6:
Find the "Best Square"
The result of STEP 5 is a square with a height of 4 mMMI's and a base on
the 1/8 th's mMML 26.5625. Refer to TABLE 2: Two squares are candidates for
the "best square", the (0,4) square and the (2,6) square.
The bottom and top of the (0,4) square are:
Bottom: 26.5625 - 1.5625 = 25.0
Top: 25.0 + (4 x 1.5625) = 31.25
The bottom and top of the (2,6) square are:
Bottom: 26.5625 + 1.5625 = 28.125
Top: 28.125 + (4 x 1.5625) = 34.375
Now apply the error function to each square to determine "the best square
in time".
Error(0,4) = abs(32.75 - 31.25) + abs(26.75 - 25.0) = 3.25
Error(2,6) = abs(32.75 - 34.375) + abs(26.75 - 28.125) = 3.0
The (2,6) square has the smallest error and one would expect it to be the
square of choice. Refer to Chart #85 in the Murrey Math book. The square
selected in the book was the (0,4) square.
Other Considerations When
Selecting the MMLs
Mapping of Murrey Math Lines
Gann Minor 50% Lines, and
19- & 39-cent Reversals
- The prior discussion on the mapping of MML properties provides a nice
lead into this topic (the Gann Minor 50%, 19 cent and 39 cent lines). These
lines are simply the result of the subdividing the MMI currently being used
for the square in time.
- Consider a stock trading between 50 and 62.5. Referring to TABLE 1, the
scale factor, SR = 100. The square in time would be composed of eight
mMMI's. Each mMMI would have a height of 1.5625 (i.e. MMMI=100/8 = 12.5, and
mMMI = MMMI/8 = 12.5/8 = 1.5625). Now suppose one of the mMMI's as
subdivided into its eight bMMI's (bMMI = mMMI/8 = 1.5625/8 = .1953125). One
can now see that the 1/8 th bMML is the 19 cent line (i.e. $ 0.1953125 is
rounded off to 19 cents). Likewise the 39 cent line is just the 2/8 th's
bMML (i.e. 2 x 19 cents = 39 cents). What Murrey refers to as the Gann 50%
line is merely the 4/8 th's (4 x 19 cents = 78 cents) bMML.
- Since the 19 cent, 38 cent, and Gann 50% lines, are simply 1/8 th, 2/8
th's, and 4/8 th's lines, one can assign the appropriate support and
resistance properties to these lines. One may then use these lines to
evaluate price behavior just as one would use any other 1/8 th, 2/8 th's or
4/8 th's line.
- If one were to create a square in time for an entity with a scale factor
(SR) other than 100 (e.g. 1000), one would apply the same logic to the
bMML's. In this case the 1/8th bMML would be 1.953125, the 2/8 th's would be
3.90625 and the 4/8 th's line (Gann minor 50% line) would be 7.8125.
Time
- The term "square in time" has been used liberally throughout the prior
discussions without any specific statements regarding time. All that has
been addressed so far is the vertical price dimension of the square in time.
This is justified since the process of identifying the MML's and MMI's
requires a little more effort than the divisions of time.
- The fact that less discussion has been devoted to the time dimension
should not be interpreted to mean that the time dimension is any less
important than the price dimension. Time and price are equally important.
- Time is divided up in a very reasonable (and practical manner). The year
is broken into quarters of 64 trading days each. Note that 64 is a power of
2 (i.e. (2 x 2 x 2) x (2 x 2 x 2) = 8 x 8 = 64). An interval of 64 can
easily be subdivided into half intervals. Note that 8 (the number of
vertical intervals in the square in time) is also a power of 2 (i.e. (2 x 2
x 2) = 8). Thus, the square in time can easily be scaled in both the price
(vertical) and the time (horizontal) dimensions simply by multiplying or
dividing by 2 (very clever). Consider also that a year consists of four
quarters. Four is also a power of 2. So, a square in time based upon a year
long scale can also easily be subdivided.
- The ability to subdivide the square in time gives the square in time the
ability to evolve as an entity trades through time. The square in time acts
as a reference frame (coordinate system) that can adjust itself as needed.
As an entity reaches new high or low prices, the reference frame can be
expanded by doubling the square in both the price and time dimensions.
Alternatively, if one wishes to look at the price of an entity during some
short time frame one can simply halve the square in both the price and time
dimensions (resulting in a quarter square). This halving and doubling may be
carried out to whatever degree is practical (i.e. Practical within the
limits of how much price and time data may be subdivided. A daily chart
can't be subdivided into intraday prices or time). Refer back to the
description of the rectangular fractal at the beginning of this paper.
- The argument for breaking the year into quarters intuitively makes
sense. The business world (including mutual fund managers) is measured on a
quarterly basis. Each of the four quarters roughly correspond to the four
seasons of the year which drive weather and agriculture (as well as
commodity contracts). Clearly humans are geared to a quarterly cycle.
- Murrey resets the time = 0 point on an annual basis. This is done the
first week of October and corresponds to the day of the U.S. Treasury's
monthly and quarterly bond auctions (This year 10/8/97). Once the time = 0
point is set one may simply count off daily increments of 4, 8, 16, 32, or
64 days relative to the time = 0 point to set the desired square in time (or
256 days if one wants an annual chart).
- At this point one should realize that specifying a time interval is
critical to setting up the square in time. In the above examples that were
used to illustrate the selection of MML's and MMI's the time frame was
implied. All that was specified in the examples was the price range that the
entity traded at. Naturally, one has to ask the question, "The price range
it traded at during what time frame?". One will probably want to set up the
square in time for annual and quarterly time frames. The quarterly square in
time will probably be subdivided into a 16 day time frame for intermediate
term trading.
- One would need intraday data to set up an intraday square in time. The
time coordinate of an intraday chart is simply divided into 4 or 8 uniform
intervals. The intraday MML's and MMI's are then set up using the intraday
trading range. If one is looking at a weekly chart then a quarter should
consist of 13 weeks.
- Another key use of the time dimension is estimating when a trend in
price will reverse itself. The horizontal MML's of a square in time
represent points of support and resistance in the price dimension. The
vertical lines that divide the square in the time dimension represent likely
trend reversal points. My own personal studies, done on the DJIA, showed
that on average the DJIA has a turning point every 2.5 days. Since we know
that the market does not move in a straight line we would expect to see
frequent trend reversals. Murrey uses the vertical time lines (1/8 th lines)
in the square to signal trend reversals.
Circles of Conflict
- The circles of conflict are a by product of the properties of the
horizontal MML's that divide price and vertical time lines (VTL's) that
divide time. MML's represent points of support and resistance. VTL's
represent reversal points. Put it all together and the result is the
"circles of conflict".
- Consider a square in time divided into eight price intervals and eight
time intervals. The five circles of conflict are centered on the 2/8 th's,
4/8 th's, and the 6/8 th's MML's and the 2/8 th's, 4/8 th's, and 6/8 th's
VTL's. Recall that prices spend 40% of their time between the 3/8 th's and
5/8 th's MML's. Recall also that the 2/8 th's, 4/8 th's, and 6/8 th's MML's
represent strong points of support and resistance. If we can assume that the
2/8 th's, 4/8 th's, and 6/8 th's VTL's represent strong points of reversal,
we can expect that in slow trendless markets that prices will be deflected
around the circles of conflict. In a fast up or down market prices will move
through the circles quickly since the price momentum exists to penetrate
support and resistance lines.
- The circles of conflict are an example of the value of a standard
reference frame (square in time) in divining market action. This reference
frame and its associated geometry and rules can be applied to all price-time
scales in all markets.
The Square in Time
No One Ever Went Broke
Taking a Profit
- As we all know, traded markets do not move in a straight line. The
prices zig and zag. A fast large movement in one direction is usually
followed by a reversal as traders take profit from that movement.
- Murrey provides tables that list the probability of certain price
movements for stocks in terms of square in time MMI's. For example, one
table is listed for stocks trading over 50 and less than 100. (This is for
price movements over a short time span (i.e. the MMI for the square in time
is the 1.5625 mMMI). The table is listed here:
1/8 th + .78 cents 50% of the time = 2.34
2/8 ths (3.125) 75% of the time = 3.12
3/8 ths (4.68) 85% of the time = 4.68
4/8 ths (6.25) 90% of the time = 6.25
5/8 ths (7.81) 95% of the time = 7.81
- The way to read an entry in this table is as follows (row 3): If a stock
moves up or down in price (within the square in time) by 4.68 then the
probability that it will reverse direction is 85%.
- Another way to look at it is:
If a stock moves up or down in price (within the square in time) by 4.68
then the probability that it will continue to move in the same direction is
15% (100% - 85%).
- This table could also be re-written in terms of MMI's: (This assumes
that the scale factor (SR) for the square in time is 100)
If Price Moves By: The probability of reversal is:
(1 x mMMI) + (4 x bMMI) 50%
(2 x mMMI) 75%
(3 x mMMI) 85%
(4 x mMMI) 90%
(5 x mMMI) 95%
- The message here is that large fast price movements are short lived.
Take profit and move on to the next trade.
Part 2
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
Fractional Brownian Motion
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.
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-z * S +z * S X12
FIGURE 2A
| P(X12) *
| * *
| * *
| * *
| * *
| * *
| * *
| * *
| * *
| * *
| * *
| * *
| * *
| * *
| * *
|*|||||| ||||||*
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-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
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