; (time in Julian centuries)newmoon{i] = 2451550.09765 + 29.530588853*k+ 0.0001337*@power(T.2)5-0,000000150*@power(T,3) + 0.00000000073*8) 3. : power (T.4);k= next .5; {full moons relative to thT= k / 1236.85; @ year Zone!fullmoon(i] = 2451550.09765 + 29.530588853*k+ 0.0001337*@power(T,2)=0000000150*@power(T,3) + 0.00000000073*@power(T,4);next = next + 1;end;———fone te pce
 
378
BEHAVIORAL TECHNIQUES
4, The moon's mean-anomaly:M’ = 201.5643 + 385.81693528k + 0.01074387?+ 0,000012397? — 0.0000000587*5. The moon’s argument of latitude:F = 160.7108 + 390.67050274k ~ 0.00163417*~ 0,000002277? + 0.0000000117*
6. The ascending node of the lunar orbit:
Q (Omega) = 124.7746 ~ 1.56375580k + 0.0020691T? + 0.000002157?
7. The eccentricity of the Earth’s orbit around the Sun:
E = 1 —0.002516T — 0.00000747*
8. The following calculation, based on F and Q, gives the basis for the solar or lunareclipse; and if F differs from 0°, 180°, or 360° by less than 13°.9 then an eclipse iscertain, If F is more than 21° from these phases, there is no eclipse. If F falls be-tween these values then there is no eclipse if |sin F| > 36.
The next components needed for corrections are:F, =F =0°.02665 sin 2‘A, = 299°.77 + 0°.107408k — 0.009173T*
9. Then to find the exact time of the full solar (or lunar) eclipse, the following correc-tions (in days) should be added to the time of the mean conjunction given by JDEin the first formula above (smaller quantities have been omitted):
‘Time of maximum eclipse= JDE + (lunar or solar component below) .4£0,0161 sin 2M’ — 0.0097 sin 2F; + 0.0073 x Ex sin QM’ ~My~ 0.0050 x £ x sin (M’ + M) ~ 0.0023 sin (M’ ~ 2F,) + 0.0021 x E x sin 2M40.0012 sin (M’ + 2F,) + 0.0006 x E x sin (2M’ + M) — 0.0004 sin 3M’— 0.0003 x Ex sin (M + 2F,) + 0.0003 sin Ay ~ 0.0002 x E x sin (M~ 2F)— 0.0002 x Ex sin (2M’ — M) — 0.00022
solar component = ~0.4075 sin M’ + 0.1721 x ExsinMJunar component = —0.4065 sin M’ + 0.1727 X Ex sin M
10, P= 0.2070 x EX sin M + 0.0024 x E x sin 2M — 0.0392 sin. M” + 0.0116 sin 2’» 0.0073 x Ex sin (M’ +M) - 0.0067 x E x sin (&’ ~ Mf) + 0.0118 sin 21%
11. Q-=+5.2207 — 0.0048 x Ex cos M + 0.0020 x E x cos 2M = 0.3399 cos M’70.0060 x EX cos (A’ + M) + 0.0041 x £ x cos (M’ ~ My
12. W=| cosF; | . .
13, y (gamma) = @ cos F; + Q sin Fy) x (1 ~ 0.004817)
14, w= 0.0059 + 0.0046 cos M — 0.0182 cos A’ + 0.0004 cos 21” — 0.0005 cos (Mf +A’)
 
FINANCIAL ASTROLOGY 379
 
Solar Eclipses
For a solar eclipse, y represents the shortest distance from the axis of the moon’s shadowto the center of the Earth in units of the equatorial radius of the Earth (the distance fromthe center to the surface of the Earth at the equator). its value is positive if the axis of theshadow is passing north of the Earth’s center, and negative if it is passing south. When yisess than +.9972 and greater than -.9972, the solar eclipse is central; that is, there is a lineof central eclipse on the surface of the Earth (see Figure 14-24). ,
“The value u gives the radius of the moon's umbral cone on the fundamental planewhich passes through the centet of the Earth perpendicular to the axis of the moon's
adow (in units of Earth's equ: rt in i i
shadow (in units of Bard ‘sea atorial radius). The radius of the penumbral cone in the
Based on these values, the following situations exist:
   
| 7 | is between 0.9972 and 1.5432 +u The eclipse is not central.
| 7 | is between 0.9972 and 1.0260 Part of the eclipse may touch the polarregions.0.9972 < | y| < 0.9972 + |x| ‘The combination of a noncentral total or
annular eclipse.
ly | > 1.54324 , No eclipse is visible from the Earth’s sur-
face.For a central eclipse,u<0 The eclipse is total.u > 0.0047 The eclipse is annular.0 <u <0.0047 The eclipse is either.annular or annular-total.
‘To remove the ambiguity of this last situation, calculate© = 0.00464 cos Ws where sin W=7
If u <@ then the eclipse is annular-total; otherwise, it is annular.
—_—_
FIGURE 14-24 Geometry of a solar eclipse.
 
  
380 BEHAVIORAL TECHNIQUES
‘The greatest magnitude for a partial solar eclipse is reached at the point on the Earth’ssurface that comes closest to the axis of the shadow at 1433 +u- trl0.5461 + 2u
   
Greatest magnitude of solar eclipse =
Lunar Eclipses
For a lunar eclipse, 7 is the least distance for the center of the moon to the Earth's shadow(see Figure 14-25). The value 7 is positive if the moon’s center is passing north of the axisof the shadow, and negative if it is passing south. At the distance of the moon, the penum-bra radius p = 1.2847 + and the umbra radius 6 = 0.7404 — u. The magnitude of the lunareclipse is
penumbral eclipses: (1.5572 +u— | ¥ |) /0.5450umbral eclipses: (1.0129 ~u- | |) /0.5450
If the magnitude is less than zero, there is no eclipse. The semidurations of the partial andtotal phases in the umbra are calculated as .
P=1.0129-uT= 04679 -unn = 0.5458 + 0.0400 cos M’and the semidurations in minutes are60partial phase = —~ VP?*= 7
total phase = so T=
Example: Solar eclipse of May 21, 1993:‘The following values can be used to verify your calculations: .
May 21 is the 141st day of the year, therefore
k= 1993 + 141/365 = 1993.38T= 1993.38/1236.85 = 1.6116.JDE = 2449128.5894
 
 
 
FIGURE 14-25 Geometry of a lunar eclipse. -
 
 
 
 
FINANCIAL ASTROLOGY
 
 
381M= 135°.9142 P=0.1842
M = 244°.5757 Q=5.3589
F=165°.7296 Y= 1.1348
Q = 253°.0026 u = 0.0097
Fy = 165°.7551
Because 180° —F is between 13°.9 and 12°.0 the eclipse is uncertain, andis between 0.9972 and 1.5433 + u = 1.553, the eclipse is partial. By cee ane Pecause |magnitude of a solar eclipse, we get 0.740. Because F is near 180°, the eclipse occurs nearthe moon's descending node, and because 7 is positive, the eclipse is visible in the northernhemisphere of the Earth. By adding the corrections to JDE, the final time of maximumeclipse is 2449129.0979, which corresponds to May 21, 1993, at 14h21m0s TD. This differsfrom the exact value of 14h20m14s TD by less than 1 minute. . 
 
 
 
380 BEHAVIORAL TECHNIQUES
The greatest magnitute for a partial solar eclipse is reached at the point on the Earth'ssurface that comes closest to the axis of the shadow at :1.5433 +u— 71in rt li =,Greatest magnitude of solar eclipse = “5 S567 4 buLunar EclipsesFor a lunar eclipse, is the least distance for the center of the moon to the Earth's shadow(sce Figure 14-25). The value 7 is positive if the moon's center is passing north of the axisof the shadow, and negative if it is passing south. At the distance of the moon, the penum-bra radius p = 1.2847 + u and the umbra radius 6 = 0.7404 — u. The magnitude of the lunar
eclipse is(1.5572 + w- | |) (0.5450(1.0129 ~ w= | ¥ |) /0.5450
Ifthe magnitude is.tess than zero, there is no eclipse. The semidurations of the partial andtotal phases in the umbra are calculated as
penumbral eclipses:
umbral eclipses:
P=1.0129-uT= 0.4679 —un= 0.5458 +:0.0400 cos M’
and the semidurations in minutes are
2
 
60.partial phase = ~~ (P?
vo phase = 2 VEEP
Example: Solar eclipse of May 21, 1993: -‘The following values can be used to verify your calculations:May 21 is the 141st day of the year, therefore
k = 1993 + 141/365 = 1993.38T = 1993.38/1236.85 = 1.6116.JDE = 2449128.5894
FIGURE 14-25 Geometry of a lunar eclipse.
 
 
FINANCIAL ASTRe OGY
 
381M = 135°.9142 P=0.1842M! = 244°.5757 Q=5.3589F=165°.7296 y= 1.1348Q = 253°.0026 u=0.0097
F,=165°.7551
Because 180° — F is between 13°.9 and 12°.0 the eclipse is uncertain, and alis between 0.9972 and 1.5433 + u = 1.553, the eclipse is partial. By calcolaune ‘he meetmagnitude of a solar eclipse, we get 0.740. Because F is near 180°, the eclipse occurs nearthe moon’s descending node, and because is positive, the eclipse is visible in the northernhemisphere of the Earth. By adding the corrections to JDE, the final time of maximumeclipse is 2449129.0979, which corresponds to May 21, 1993, at 14h21m0s TD. This differsfrom the exact value of 14h20m14s TD by less than 1 minute. . 
 
15.
SePattern Recognition
attern recognition forms the basis for most trading systems. It is most obvious in tra-Pe nal charting, which is entirely the identification ‘of common formations; even
moving averages attempt to isolate, using mathematical methods, what has been visu-ally determined to be a trend. Traders have always looked for patterns in price movement.Because they were not equipped with computers, their conclusions are considered marketlore rather than fact and are handed down from generation to generation s6 proverbs, suchas “Up on Monday, down on Tuesday,” “Locals even up on Fridays,” and “Watch for keyseversals.” Because these three sayings have endured, they are candidates for analysis laterin this chapter.’
The earliest technical systems based on patterns were of the form: “If after a sharp rise
jhe market fails to advance for 3 days, then sell.” As compute’s became more powerful,more complex approaches could be taken, For example, by ‘observing the closing pricesstarting on an arbitrary day, all patterns of higher ‘and lower closes can be recorded to findtheir tendency to repeat. A computer is well-equipped to perform this task. First, the 2- and3-day patterns are eliminated because the recurrence of up-down or down-up and theequivalent 3-day patterns would be too frequent to be meaningful. Then, the closing pricesare scanned for occurrences of predefined patterns. For example, if an up-up-down-up-down-up is to be matched, every six consecutive prices must be tested. From a table ofoccurrences we can conclude that these patterns can be predicted in advance or that theyare leading indicators of other price moves. This approach, as well as the combination ofevents used to forecast profitable situations, are discussed later in this chapter.
Pattern recognition may appear to be more of a game than a business, but it is a sourceof many valuable ideas. Figure 15-1, a graph of the New York Stock Exchange (1854-1959),shows the simplification of patterns to the point where it is difficult not to count the recur-rences of the more obvious patterns and look for the formations that precede them to seewhether they could be predictive. For example, in 1922, 1924, and 1927, there were sharpadvances in the market; the years preceding those showed an identical U pattern. It wouldbe interesting to see whether another occurrence of a symmetric U was followed by a sim-ilar rise. Another pattern that stands out is that of two consecutive years of sharp rise,
_ 1862-1863, 1908-1909, 1918-1919, and 1927-1928; in no case was there a third consec-utive year, but neither the preceding nor following years seem consistent.
Patterns frequently provide the foundation for a trading method or the justification forbeginning work in the development of a method. They have been applied in many ways to.price analysis, from the time of day to place an order to the compound relationship ofprice, voltime, and open interest. Daily trading-opportunities may be a function of patternsbased on the strength or weakness of the daily opening price. ‘These are discussed in thenext section, Weekly and weekend traits are studied, as well as types of reversals and theirbffects. These techniques can be considered complementary when used sequentially, orthey can confirm the results of another test when ‘used together. The end of this chapterdiscusses more general issues in pattern recognition.
«phe only other known work that concems similar patterns, although for the stock marker, is by Arthur A. Mer-till Bobavior of Prices on Wall Street (Analysis Press, Chappaqua, NY, 1966)
PROJECTING DAILY HIGHS AND LOWS383
 
FIGURE 15-1 Graph of the New York Stock market.
pbib
RM le ae
 
 
 
 
 
 
 
 
 
wt
SA
 
 
 
 
 
 
 
 
 
 
 
 
 
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i
 
 
 
 
te
Pia a Ge aml it
 
 
 
 
 
 
    
 
 
BWA SS SPSS pL Sa
 
 
 
 
 
hae
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
PROJECTING DAILY HIGHS AND LOWS
Statisticians daim that the best forecast of tomorrow's price is today’s price; they state that
no on . has b en able to reliably account for all the factors that would be needed to accu-¢ changes that result in the high and low pri
a prices for tomorrow. Neverthe-
less, many traders project these values and seem to use them with success. “
Pivot Technique
The simplest approach to projecting tomorrow's highs and lows is to base those figures on
the average price of today plus or minus a value that somehow relies on the current trad-, of volatility. One technique, based on a pi int,? proj
ing range, of volt pivot point,? projects two levels of sup-
Pivot point P=(H+L+Cy2 -Resistance 1 RI=2xP-L
Support 1 S1=2xP-H
Resistance 2 R2=(P-SI)+RI
Support 2 $2 =P -(R1-SI)
DeMark’s Projected Ranges
Another author and trader, Tom DeMark,? has based his projections on the positioning ofy's opening price relative to yesterday’s closing price, which gives him-added timely
? Mark e " -s Etzkorn, “All in a day's work,” Futures (January 95). This technique is attributed to William Greenspan.fom DeMark, The New Sctence of Technical Analysis (John Wiley & Sons, New York, 1994). 
384 PATTERN RECOGNITION
information. If today's optn is higher than the previous close, the projections are biasedupward; if lower than the close, they are biased downward. '
1. If today’s close is below today’s open, thenTomorrow’s projected high = (H +C+2xD/2-L‘Tomorrow’s projected low = (H + C+2xL/2-H
2. If today’s close is above today’s open, thenTomorrow's projected high = (2x H+L+C)/2-LTomorrow’s projected low = (2 x H+ L+O)/2-H
3, If today’s close is the same as today's open, thenTomorrow's projected high = (H+L+2xC)/2-LTomorrow's projected low = (H+L+2xC)/2-H
 
Once the basic formula is determined, biasing tomorrow's projection in the directionof today’s close relative to today’s open, then the projected high is found by removing two.nite of the low price, and the projected low is found by removing two units of the highprice from the formula. This effectively shifts the projection in the direction of the newhigh by one-half the difference of today’s close and today’s low, (C ~ L)/2, and the new lowby one-half the difference of today’s close and today’s high, (H - €)/2.
Comparing the Two Ranges
Because the two methods, the pivot point technique and projected ranges, use the samebasic prices, combining them with simple multiples, it is not clear ‘whether the results givevalues that are the same or different-For example, if the Deutschemark had a high, low,wad close of 6600, 6450 and 6500, the first pivot method would give a projected first resis-tance level (high) of 2 x (6600 + 6450 + 6500)/3 ~ 6450 = 6583. DeMark’s projected high isdifferent based on how today's opening price relates to the previous close. If the open islower than the previous close, we get (6600 + 6500 + 2 x 6450)/2 — 6450 = 6550; if theopen is higher, we have (2 x 6600 + 6450 + 6500)/2 ~ 6450 = 6625; and if the open is aboutcee eame as the previous close, then the projected high is (6600 + 6450 + 2 x 6500)/2 -6450 = 6575. It seems reasonable that the pivot point method returns a value close toDeMark’s neutral case, when the market opens unchanged from the previous close.\TIME OF DAY
Market participants, especially floor traders, are the cause of periodic movement during theday. Angas called these the “tides of the daily prices.” Over the years, the great increase inparticipants has added liquidity to each pit but has not altered the intraday time patterns.
‘There are a number of reasons for the regular movement of prices. Because mow ofthe daily volume is the result of day trades, those positions entered in the morning will beclosed out by the afternoon to avoid the need for the margin required of positions heldovernight, Orders that originate off the floor are the result of overnight analysis and areexecuted at the open. Scalpers and floor traders who hold trades for only a few minutesfrequently have a midmorning coffee break together; this natural phenomenon causes liq-uidity to decline and may result in a temporary price ceversal. All traders develop habits oftrading at particular times. Some prefer the opening, others 10 minutes after the open.Large funds and managed accounts will have a specific procedure for entering the market,such as using close-only orders.
A day trader must watch certain key times. The opening moments of trading are nor-mally used to assess the situation. A floor trader will sell a strong Open and buy a weak one;this seans the trade must be evened-up later and thus reinforces the opening direction.Ona strong open without a downward reaction, all local selling is absorbed by the marketand later attempts of the locals to liquidate will hold prices up. In any event, floor trades
 
TIME OF DAY385
can be expected to take the opposite positio ing directin to thecan be expected to take the P opening direction, usually causing aTubbs’ Stock Market Correspondence Lessons, Chapter 13, explains the dominant pat-terns in the stock market (given a 10:00 a.M.-to-3:00 PM. session). p
1. Ifa rally after the open has returned to the opening price by 1:00, the day =enipening :00, the day is ex-2. If the market is strong from 11:00 to 12:00, it continue from 12:00 to 1will continue fro! 00 to 1:00.3. Ifa reversal from 1:00 to 1:30 finds support at 1:30, it will close strong.4. If the market has been bullish until 2:00, it will probably continue until the close, Pp ue until cl
5. A rally that continues for 2 or 3 days as in (4) will most likely end on an 11:00 re-sal.
. In general, a late afternoon reacti, ction down a .versal. fter a strong day shows a pending re-
Putting these together, the following patterns (among others) can be expected:
1. Astrong open with a reversal at 11:00 not reachii i i4 ching the opening price, then strengthfrom 11:00 to 1:00, a short reversal until 1:30, and then a strong close; accordiny to; (5), another strong open the following day. , 8. A strong open that reverses by 11:00, continuing 1 i200, lowe: E iuntil 1:30, and then closes weak. * until 1:00, reverses again
Merrill's work shows the hourly pattern of the stock market in - iclearly indicates a bullish bias with 1963 the most obvious. The Sattern ts wniforin and anilar to what would be expected: early trending, an adjustment, then a wave of down-up-down. Contrary to commodities, these 4 years show extremely consistent strength on theopening with a sell-off on the close. Commodity prices may have regular patterns, butthese patterns will occur with downward moves as well as upward ones. It is just as likelto see an opening sell-off with a rally on the close. *
1 commodities, the patterns are similar but may be compacte i i-cultural products, due to shorter, earlier hours 630 AM. enrate” ere ee ieeegrains and livestock), or shifted as with currencies that open earlier. Knowing the Gailtime patterns would not only help a day trader but would also aid any speculator to enteran order at a better place. Figure 15-2 shows the time pattern for the Chicago MercantileExchange live cattle contract (open from 9:05 a.m. to 12:45 RM.). Seventy-six consecutivetrading days were tabulated for the June 75 contract from February 1 through May 311975, an active period for cattle. The left scale of the chart shows the frequency of occur,renrces and the bottom shows the time of day. A line appearing in the top half of the chartindicates price movement in the direction of the opening price (from the prior close); thelower half shows a reversal trend, Measurements were first taken every 5 minutes and laterevery 15 minutes, due to a more variable price direction near the open.
———
TABLE 15-1 Merrill’s Hourly Stock Market Patterns
Time during Trading Session
 
10:00 £1:00 42:00 1:00 00 3:001962 - + - - + -1963 + + + + 4+ -1964 + + - = + -—1965 + + - - - =
————386 PATTERN RECOGNITION
Figure 15-2 shows that follow-through in the direction of the opening price slowsdown immediately and by 9:30, 25 minutes after the open, the price direction has usuallyreversed. There is a subsequent steady change in direction back and forth throughout theday. This pattern is remarkably similar to Merrill's stock market observations.
By showing two consecutive days of cattle price movement, Table 15-2 demonstrateshow the data was accumulated. Although the table actually used 5-minute intervals from9:00 to 10:15 and then 15-minute intervals from 10:15 until the close at 12:45, Table 15-2js summarized in half-hour increments. If Friday's closing price was 45.00, Monday's openof 45.15 was indicated by a “+” for absolute direction and an O for opening direction. Anyhalfhour interval showing a price rise was marked with the relational symbol O and areversal with an X. Tuesday opened lower, but the opening direction is still given as an oO.Every interval that followed with lower prices is in the opening direction and is marked O.Adding the 76 test days together resulted in the scores shown in Table 15-3.
Knowing the direction of the price movement does not mean that trading these inter”vals will be profitable. To evaluate the potential, it is necessary to know the approximatesize of the price move; the interval can then be selected with the greatest potential fortnovement in the direction of the time-of-day pattern. By observing Figure 15-2, the inter-vals to study can be picked from the tops and bottoms of the cycles:
——_—_—_—Selected Time Intervals
a. Opening to 9:20b. 9:20 to 10:00
€. 10:00 to 11:00
d, 9:20 t0 11:00
e. 9:20 to 12:00
f. 12:00 to closing——————
FIGURE 15-2 Intraday time patterns, June 75 cattle.
 
 
 
TIME OF DAY -387
 
TABLE 15-2 Accumulating Time-of-Day Data
 
 
 
 
Direction DirectionTime Price from Last from OpenMonday
() 45.15 + °
Q) 45.25 + °
@) 45.10 ~ x
«) 45.05 ~ x
6) 45.20 + °
©) 45.30 + °
Q) 45.40 + °
8) 45.30 - x
Tuesday
0°) 45.25 - °(19) 45.35 + xaly 45.45 + x(12) 45.40 ~ °(13) 45.50 + x(4) 45.40 - °(is) 45.20 - °(16) 45.05 °
 
 
_ For each interval, the average range from high to low (volatility) and the net
this span (additive bias) were measured without conaideng the celavive «ection thatwas used to create the curve of time patterns. The results, shown in Table 15-4, indicatethat the narrowest ranges, (b) and (c), were early in the day moving only a total of 14points. The combined period (d) was 21.7 points, 50% greater than the individual periods(b) and (c). Figure 15-3 describes the likely combination of the intervals.
The results show a midmorning dr