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Discovering Fibonacci World

The sequence of the Fibonacci numbers is considered to have been discovered by Leonardo Fibonacci de Piza, a mathematician who lived in the 13^th century. He was one of the most famous scientists in his time. In early 1200s, Leonardo Fibonacci from Piza published his well-known book “Liber Abaci - Book of Abacus, or Book of Calculation”, which represented one of the greatest discoveries of all times, namely decimal numeration system, including the position of zero as the first number in the number sequence. This system, which includes the familiar symbols of 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, became known as Hindu-Arabian system and is currently commonly used instead of Roman numbers.

Leonardo Fibonacci was a greatest mathematician in the Medieval Age. He was honoured with meeting Roman emperor Frederick II who, being a powerful governor of his time, visited Piza to communicate with scientists. Later, in 1228, Fibonacci updated book “Liber Abaci” and devoted it to the emperor. Totally he wrote three essential books on mathematics: “Book of Abacus”, published in 1202 and updated in 1228, “Practical Geometry” (“Practica Geometriae” - a compendium on geometry and trigonometry), published in 1220, and “Liber quadratorum” – “The Book of Squares”. Today you can read about the age of Fibonacci and his works in (Joseph and Frances Gies “Leonard of Pisa and the New Mathematics of the Middle Ages”). In the town of Piza, Italy, the statue after the famous scientist is erected.

The Fibonacci Sequence

In “Liber Abaci” Leonardo presented the following task:

“How many couples of rabbits, placed into a rabbit corral, can be produced for a year by a rabbit couple, if each couple produces one more couple every month since the second month?” A sequence of numbers called the Fibonacci sequence became the task solution.

1,1,2,3,5,8,13,21,34,55,89,144… endlessly.

The mathematic sequence appears when beginning with 1, the next number forms the sum of the two previous ones:

1+1=2; 1+2=3; 2+3=5; 3+5=8, etc.

Why is the sequence so important?

The sequence strives to some constant ratio more and more slowly. However, this ratio is irrational, i.e. a number with endless, unpredictable sequence of decimal numbers in the fractional part. It is impossible to express it exactly. If any number of the Fibonacci sequence is divided by the previous one (for example, 13:8), the result will be the value fluctuating around an irrational value (1.61803398875....). But it will be impossible to find out the ratio exactly, to the last decimal number, even if you spend Eternity for it. We will present it shortly as 1.618.

This ratio was called differently, from Divine proportion in ancient times to the Golden section, the Golden average or the Ratio of rotating squares (or logarithmic spiral) nowadays. Famous mathematician Kepler called the ratio “one of the treasures of geometry”. In mathematics, Greek letter φ is adopted as the symbol of the ratio.

φ equals 1.618.

It is differently called gold coefficient. This abbreviation is also commonly adopted in trading and encountered quite frequently. The Fibonacci sequence, as well as its numbers, can be revealed in plenty of combinations. This is not just a game with numbers, but rather the most important mathematic expression of natural phenomena of all discovered ones. Here are the most well-known and interesting applications of the mathematic sequence:

Egypt pyramids

Pyramids in Mexico

Plants

Fibonacci Ratio in Geometry

Mathematical properties of these ratios are not of great practical interest for us. This is not a lecture on geometry, anyway. But it is worth while paying some attention to the main examples which are of some importance for trading.

Gold Section

Any segment may be divided in such way that ratio between its smaller and bigger parts will be equal to the ratio between the bigger part and the whole segment. This ratio is always equal to 0.618.

Figure 1. Golden section of a segment

1/3 of interval СВ forms the ratio with 2/3 of segment АС.

2/3 of segment АС forms the ratio with segment АВ.

But this is not just algebraic division of a segment into three equal parts. The gold section is encountered in nature commonly. The human body is embodiment of the Gold section in everything, from the general dimensions to the face arrangement. Pieces of art are significantly improved through the application of the Golden section. Its value and application were especially attractive in ancient Egypt and Greece and during Renaissance period, i.e. in all most important periods of civilization. Leonardo da Vinci believed that the Golden proportion was of great value. In addition, he found it quite pleasant in its ratios and used to say, “if an object does not have appropriate appearance, it does not work”. Many of his pictures have appropriate appearance, because he would apply the Golden section in order to strengthen its attractiveness. Besides, such expressions as the Gold rectangle, the Golden spiral (logarithmic spiral) exist in geometry. In order to draw these things Fibonacci ratio is employed. You can read more about them in the relevant literature.

In living nature many spiders spin their cobweb as the logarithmic spiral. Pine cones, sea-horses, snail shells, mollusk shells, ocean waves, animal horns, ferns, sunflower and daisy seed arrangement, all they form logarithmic spirals. In addition, the Fibonacci sequence is the mathematic base of Elliott wave theory. In addition, Fibonacci and the Fibonacci numbers are mentioned as a code to unlock a vessel in Dan Brown's best selling novel, The DaVinci Code, and its movie adaptation.

The Fibonacci Sequence at Market

The existence of Fibonacci ratios in geometry is well-known, however, application of these ratios in trading practice is gaining more and more popularity as well. Even a mere glance on a price graph suggests the conclusion that the price movement is alternation of rises and falls. And with ascending trend, for example, each rise and each fall is higher than the previous one. But how can one measure possible length of the correction emerged? One of the most popular methods consists in application of Fibonacci levels.

There are three major correction levels:

38.2%

50%

61.8%

Fig. 2

It is considered that with strong trend, correction amounts to approximately 38.2 %, pullback after middle trend lasts up to 50 %. And in case of deep correction, the trend should be expected to finish near level 61.8%. Out of all the levels, mark 61.8 is the most important one, as according to the general rule, after the level is broken the trend existed before the break is eliminated. With conservative trading with a trend, it is the very level traders direct their attention to as the most important and reliable. In addition, the entrance point from level 61.8 provides the best ratio of potential profit and loss among other correction levels in the following situation. When you wait for correction after ascending trend movement, you always have a guiding point for setting the first target on Take Profit. It is nothing but the last maximum set by the price, since now it is the nearest resistance. It is reasonable to place Stop Loss below the support level which the movement started from. And now look how much profit exceeds your risk in this case, and how worse the ratio between S/L and T/P would have been if you had bought from 38.2% or 50% and transferred S/L below the same support. Of course, it is more profitable to buy from the support level, but do not forget that we act basing on the fact that we are within the trend and not within the corridor.

Fig. 3

Figure 4. Note how clearly the price has worked on correction level 61.8 in the following example.

Fibonacci Expansion

Fibonacci expansion is drawn by three points describing two waves. Then three lines crossing “assumed” wave at Fibonacci levels 61.8%, 100% and 161.8% are drawn. It is believed that essential price changes should be expected near these lines.

Fig. 5

This case is subject to the following logic. According to Dow theory, main trend has three development phases. The brightest phase possessing the largest potential for profit gaining is the second phase. Unlike the first one, which is regarded as a mere bounce from support level, during the second development phase skilled traders, who use technical methods of following the trends join trading. Prices rise vigorously, and economic information becomes more optimistic. And, as a rule, the second phase (in wave analysis it is called the third wave) cannot be smaller than the second phase. According to Elliott wave theory, the phase either equals the first one or frequently exceeds it 1.618 (number φ) times. Note, in Figure 5 the third wave appeared to be longer than the first one by 161.8%. These values can be used to forecast the target. You can always check the practical value of these Fibonacci tools you have got familiar today, on history. I would like to add that Fibonacci correction levels and expansion can be used with any time scale with similar success. Anyway, when used with other tools of technical analysis, they shall become your useful assistants in your work.

Alex Sabodin.
Pro Finance Group Inc.