OVERVIEW
In the original Currency Trader’s Companion: A Visual Approach to Technical Analy-sis of Forex Markets(or CTCfor short) and its sequels, we focused heavily on whichcurrency pairs to trade and when to trade them. Once traders have determined whichand when, they are then confronted with yet another major decision: which position toinitiate. For that purpose, Chapters 16 through 19 in CTC provided a brief introductionto some classical methods in technical analysis for novice traders. In the current chap-ter, we intend to delve more deeply into science of forecasting price movements, contin-uations, and reversals. Many famous traders, notably W. D. Gann, Charles L. Lindsay,and Ralph N. Elliott, have based their trading systems on a curious mathematical rela-tionship that exists in nature, the golden mean.
We have therefore decided to open this section with a look at so-called magic num-bers, their history and calculation. Our goal is to demystify this esoteric realm in techni-cal analysis. In later chapters, we will refer to possible Fibonacci relationships but, tobe quite honest, it is not the motivating force behind the study of swing analysis. Famil-iarity with Fibonacci numbers is, however, a beneficial and interesting branch of fore-casting to be aware of.
We have therefore decided to open this section with a look at so-called magic num-bers, their history and calculation. Our goal is to demystify this esoteric realm in techni-cal analysis. In later chapters, we will refer to possible Fibonacci relationships but, tobe quite honest, it is not the motivating force behind the study of swing analysis. Famil-iarity with Fibonacci numbers is, however, a beneficial and interesting branch of fore-casting to be aware of.
FIBONACCI THE MAN
Leonardo Fibonacci was born in Pisa, Italy, around 1170, the son of Guilielmo Bonacci, asecretary of the Republic of Pisa who was responsible for directing the Pisan trading
colony in Algeria. Sometime after 1192, Bonacci brought his son with him to Algeria. Thefather intended for Leonardo to become a merchant and so arranged for his instructionin calculational techniques, especially those involving the Hindu-Arabic numerals,which had not yet been introduced into Europe. Eventually, Bonacci enlisted his son’shelp in carrying out business for the Pisan republic and sent him on trips to Egypt,Syria, Greece, Sicily, and Provence. Leonardo took the opportunity offered by histravel abroad to study and learn the mathematical techniques employed in these vari-ous regions.
colony in Algeria. Sometime after 1192, Bonacci brought his son with him to Algeria. Thefather intended for Leonardo to become a merchant and so arranged for his instructionin calculational techniques, especially those involving the Hindu-Arabic numerals,which had not yet been introduced into Europe. Eventually, Bonacci enlisted his son’shelp in carrying out business for the Pisan republic and sent him on trips to Egypt,Syria, Greece, Sicily, and Provence. Leonardo took the opportunity offered by histravel abroad to study and learn the mathematical techniques employed in these vari-ous regions.
Around 1200, Leonardo Fibonacci returned to Pisa, where, for at least the next 25years, he worked on his own mathematical compositions. The five works from this pe-riod that have come down to us are: the Liber abbaci (1202, 1228); the Practica geome-
triae (1220/1221); an undated letter to Theodorus, the imperial philosopher to the court
of the Hohenstaufen Emperor Frederick II; Flos(1225), a collection of solutions toproblems posed in the presence of Frederick II; and the Liber quadratorum(1225), anumber-theoretic book concerned with the simultaneous solution of equations qua-dratic in two or more variables.
So great was Leonardo Fibonacci’s reputation as a mathematician as a result of hisearly works that Frederick summoned him for an audience when he was in Pisa around1225. Fibonacci died sometime after 1240, presumably in Pisa. He is recognized as themathematician who introduced the decimal system and the Arabic numeral system toEuropeans. He has also been acclaimed as the greatest European mathematician ofthe Middle Ages. More of Fibonacci’s accomplishments are documented in his biograph-ical entry at the web page www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html.
FIBONACCI THE SERIES
Leonardo Fibonacci is best remembered for the numerical series that he discovered.Each Fibonacci number is the sum of the previous two Fibonacci numbers. (See Fig-ure 21.1.)
The first two Fibonacci numbers are defined as Fib1 = 1 and Fib2 = 1. Table 21.1 lists
the first 50 numbers in the series.
The first two Fibonacci numbers are defined as Fib1 = 1 and Fib2 = 1. Table 21.1 lists
the first 50 numbers in the series.
The golden mean (or ratio) expresses itself throughout nature, such as in the spiralcompartments in a nautilus shell, the population growth of rabbits, the location of col-ored tiles on a turtle shell, seed patterns in sunflowers, the arrangement of bumps on apineapple, the ratio of bone lengths in human anatomy, and numerous other phenom-ena. The Fibonacci series has also been observed in phyllotaxis (the study of the or-dered position of leaves on a stem) and extensively studied in three different spiralarrangements.
FIBONACCI THE EQUATIONS
The golden mean can be solved by applying the quadratic formula to the followingequation:
x2 – x – 1 = 0
which yields:
The two roots of the quadratic are conventionally designated as:
Phi = +1.61803 39887 49894 84820 45868 34366
And
phi = –0.61803 39887 49894 84820 45868 34366
where
Please note the use of upper- and lower-case letters to identify the different roots.The upper- and lower-case Greek equivalents are also used to denote the golden ratio: Ffor Phi and ffor phi.
52 23606
=.
79774 99789 69640 91736 68731
x =±
x2 – x – 1 = 0
which yields:
The two roots of the quadratic are conventionally designated as:
Phi = +1.61803 39887 49894 84820 45868 34366
And
phi = –0.61803 39887 49894 84820 45868 34366
where
Please note the use of upper- and lower-case letters to identify the different roots.The upper- and lower-case Greek equivalents are also used to denote the golden ratio: Ffor Phi and ffor phi.
52 23606
=.
79774 99789 69640 91736 68731
x =±
BINET’S FORMULA
In 1843, the French mathematician Philippe Marie Binet discovered a method to calcu-late the nth term in the Fibonacci series. (See Figure 21.2.)
Despite the floating-point operations involved, Binet’s formula always produces aninteger result when n is also a positive integer.
Despite the floating-point operations involved, Binet’s formula always produces aninteger result when n is also a positive integer.
INTERPRETATION
Many technical analysts use Fibonacci numbers when trying to determine support andresistance, and frequently use 38.2 percent, 50 percent, 61.8 percent retracements.
•38.2 percent retracement. It is commonly thought that a 38.2 percent retrace-
ment from a trend move will tend to imply a continuation of the original trend.
(See Figure 21.3.)
•38.2 percent retracement. It is commonly thought that a 38.2 percent retrace-
ment from a trend move will tend to imply a continuation of the original trend.
(See Figure 21.3.)
50% Retracement. A 50 percent retracement shows indecision and may be followedby a lateral movement of prices. (See Figure 21.5.)Many such rules have been adopted by technicians. The steps involved when look-
ing for a continuation of a trend after a retracement are:
1.Calculate the total value of a significant price move (high to low or vice versa).2.Calculate a Fibonacci retracement (in this case 38.2 percent) of the previous move.3.Look for price to confirm by resistance (or support in an up move) near that pre-
dicted retracement area.
Other technicians have devised very complex systems involving Fibonacci fans,arcs, spirals, squares, zones, and other geometric motifs. Traders are advised to reviewthe works of W. D. Gann and his followers for more information. As usual, we recommend thorough testing via paper trading before traders incorporate such techniquesinto their overall trading systems. We admit that the preceding examples are somewhatsimplistic in nature. In later chapters, we examine the retracement phenomena in muchgreater detail.
RESOURCES
Traders who are interested in more detailed information on Fibonacci numbers will bepleased to know that there is even an official Fibonacci Association, incorporated in1963, which focuses on numbers and related mathematics, research proposals, chal-lenging problems, and new proofs of old ideas (http://mscs.dal.ca/Fibonacci). Someother interesting Fibonacci web sites and web pages are:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.htmlhttp://ulcar.uml.edu/~iag/CS/Fibonacci.htmlhttp://math.holycross.edu/~davids/fibonacci/course.htmlhttp://ccins.camosun.bc.ca/~jbritton/fibslide/jbfibslide.htmhttp://www.thinkquest.org/library/site_sum.html?tname=27890&url=27890/mainIndex.html
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