{"id":55333,"date":"2020-08-07T09:50:00","date_gmt":"2020-08-07T13:50:00","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=55333"},"modified":"2024-09-11T13:47:14","modified_gmt":"2024-09-11T17:47:14","slug":"the-magic-of-fibonacci-numbers","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/the-magic-of-fibonacci-numbers\/","title":{"rendered":"The Magic of Fibonacci Numbers"},"content":{"rendered":"\n<p><em>This article was first posted on&nbsp;<a href=\"https:\/\/quantdare.com\/the-magic-of-fibonacci-numbers\/\">QuantDare Blog<\/a>.<\/em><\/p>\n\n\n\n<p>Rabbits, triangles, and triplets. These three things are linked by an important series often&nbsp;characterised&nbsp;by shells and flowers. But what do the Fibonacci numbers (and their subsequent sequence) have to offer the world of finance?&nbsp;<br>&nbsp;<\/p>\n\n\n\n<p>In mathematics, the Fibonacci series refers to the ordered sequence of numbers described by\u202f<a href=\"https:\/\/en.wikipedia.org\/wiki\/Fibonacci\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Leonardo of Pisa<\/strong><\/a>, a 12th-century Italian mathematician.&nbsp;<\/p>\n\n\n\n<p>0,1,1,2,3,5,8,13,21,34,55,89,144,\u20260,1,1,2,3,5,8,13,21,34,55,89,144,\u2026&nbsp;<\/p>\n\n\n\n<p>Each element in the series is known as a Fibonacci number.&nbsp;<\/p>\n\n\n\n<p><strong>The history of the Fibonacci sequence<\/strong>&nbsp;<\/p>\n\n\n\n<p>This sequence was described by Fibonacci as\u202f<strong>the solution to a rabbit breeding problem<\/strong>: \u201ca certain man has a pair of rabbits in a closed space and wants to know how many are created from this pair in a year when, according to nature, each couple requires one month to grow old and each subsequent month procreates another couple.\u201d (Laurence Sigler, Fibonacci\u2019s Liber Abaci, pg. 404).&nbsp;<\/p>\n\n\n\n<p>The answer to this question is as follows:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>1st month: we start from a pair of rabbits.\u00a0<\/li>\n\n\n\n<li>2nd month: the couple grows older but does not procreate.\u00a0<\/li>\n\n\n\n<li>3rd month: the pair procreates another pair (that is, we now have two couples).\u00a0<\/li>\n\n\n\n<li>4th month: the first couple procreate and the second age without procreating (we now have three couples).\u00a0<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>5th month: the two older couples procreate, while the new pair ages (five pairs in total).\u00a0<\/li>\n\n\n\n<li>Etc.\u00a0<\/li>\n<\/ul>\n\n\n\n<p>Schematically, this would be:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"576\" height=\"272\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/08\/image-35.png\" alt=\"Fibonacci is the key - rabbit chart\" class=\"wp-image-55358 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/08\/image-35.png 576w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/08\/image-35-300x142.png 300w\" data-sizes=\"(max-width: 576px) 100vw, 576px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 576px; aspect-ratio: 576\/272;\" \/><\/figure>\n\n\n\n<p>Where:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Black arrow: the pair of rabbits ages.\u00a0<\/li>\n\n\n\n<li>Red arrow:\u202fthe pair of rabbits age for the first time (and therefore don\u2019t procreate).\u00a0<\/li>\n\n\n\n<li>Green arrow:\u202fthe pair of rabbits procreates.\u00a0<\/li>\n<\/ul>\n\n\n\n<p><strong>How are the Fibonacci numbers calculated?<\/strong>&nbsp;<\/p>\n\n\n\n<p>There are different ways to calculate Fibonacci numbers:&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>From the numbers\u202f00\u202fand\u202f11, the Fibonacci numbers are\u202f<strong>defined by the function:<\/strong>\u00a0<\/li>\n<\/ol>\n\n\n\n<p>fnf0f1f2f3\u2026=fn\u22121+fn\u22122=0=1=f1+f0=1=f2+f1=2fn=fn\u22121+fn\u22122f0=0f1=1f2=f1+f0=1f3=f2+f1=2\u2026&nbsp;<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>A generating function\u202f<\/strong>for any sequence\u202fa0,a1,a2,\u2026a0,a1,a2,\u2026\u202fis the function\u202ff(x)=ao+a1x+a2x2+\u2026f(x)=ao+a1x+a2x2+\u2026, that is, a formal power series where each coefficient is an element of the sequence. Fibonacci numbers have the generating function:\u00a0<br>\u00a0<\/li>\n<\/ol>\n\n\n\n<p>f(x)=x1\u2212x\u2212x2f(x)=x1\u2212x\u2212x2&nbsp;<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li><strong>Explicit formula<\/strong>,\u202fthis way of calculating Fibonacci numbers uses the golden number expression:\u00a0<br>\u00a0<\/li>\n<\/ol>\n\n\n\n<p>fn=(1+5\u221a2)n\u2212(1\u22121+5\u221a2)n5\u2013\u221a=(1+5\u221a2)n\u2212(1\u22125\u221a2)n5\u2013\u221afn=(1+52)n\u2212(1\u22121+52)n5=(1+52)n\u2212(1\u221252)n5&nbsp;<\/p>\n\n\n\n<p><strong>Fibonacci numbers in Mathematics<\/strong>&nbsp;<\/p>\n\n\n\n<p><strong>Golden numbers<\/strong>&nbsp;<\/p>\n\n\n\n<p>The golden number, gold number or divine proportion, is the numerical value of the proportion held by two segments of line\u202faa\u202fand\u202fbb\u202f(with\u202faa\u202flonger than\u202fbb): the total length is to segment\u202faa, as\u202faa\u202fis to segment\u202fbb.&nbsp;<\/p>\n\n\n\n<p>One property stands out among many: the number itself, its square and its inverse, have the same decimal figures:&nbsp;<\/p>\n\n\n\n<p>\u03d5\u03d521\u03d5=1.6180339887\u2026=\u03d5+1=2.6180339887\u2026=\u03d5\u22121=0.6180339887\u2026\u03d5=1.6180339887\u2026\u03d52=\u03d5+1=2.6180339887\u20261\u03d5=\u03d5\u22121=0.6180339887\u2026&nbsp;<\/p>\n\n\n\n<p>The ratio or quotient between Fibonacci terms and the immediately preceding one varies continuously, but stabilizes in the golden number:&nbsp;<\/p>\n\n\n\n<p>limn\u2192\u221efn+1fn=\u03d5\u22481.6180339887limn\u2192\u221efn+1fn=\u03d5\u22481.6180339887&nbsp;<\/p>\n\n\n\n<p><strong>Pascal\u2019s triangle<\/strong>&nbsp;<\/p>\n\n\n\n<p>Pascal\u2019s triangle is\u202f<strong>a representation of the binomial coefficients ordered in a triangle form<\/strong>. That is, each row of the triangle represents the coefficients of the monomials that appear in the development of the binomial\u202f(a+b)n(a+b)n\u202f(taking the top\u202f11as the power\u202fn=0n=0) or, in the same way, the coefficients appear in Newton\u2019s binomial coincide with the elements appearing in each row of the Pascal triangle.&nbsp;<\/p>\n\n\n\n<p>The triangle\u2019s construction is as follows:&nbsp;<\/p>\n\n\n\n<p>We put a\u202f11\u202fin the triangle\u2019s top vertex. Then, in the next row, we place a\u202f11\u202fon the right and a\u202f11\u202fon the left. In the lower rows, place 1st at the ends and for others, the sum of the numbers directly above on either side.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"162\" height=\"154\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/08\/image-34-1.jpg\" alt=\"\" class=\"wp-image-211767 lazyload\" style=\"--smush-placeholder-width: 162px; aspect-ratio: 162\/154;width:184px;height:auto\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>This triangle has&nbsp;a number of&nbsp;curious properties:&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>If we add the elements of each row, we get the powers of\u202f2:1,2,4,8,16,2:1,2,4,8,16,\u202fetc.\u00a0<\/li>\n<\/ol>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>Adding two consecutive elements of the diagonal\u202f1\u22123\u22126\u221210\u221215,1\u22123\u22126\u221210\u221215,\u202fetc., we get a perfect square:\u202f1,4,9,16,25,1,4,9,16,25,\u202fetc.\u00a0<\/li>\n<\/ol>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li>If the first number in a row (after\u202f11) is a prime number, then all other numbers are divisible by that prime number (excluding the 1s). For example, in row\u202f1\u22127\u221221\u221235\u221235\u221232\u221271\u22127\u221221\u221235\u221235\u221232\u22127, the first number is\u202f77, which is prime. The rest\u202f(7,21,35)(7,21,35)\u202fare all divisible by\u202f77.\u00a0<\/li>\n<\/ol>\n\n\n\n<p>But the main curiosity is the property relating to the Fibonacci numbers:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"382\" height=\"234\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/08\/image-36.png\" alt=\"Pascal's triangle property relating to the Fibonacci numbers\" class=\"wp-image-55359 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/08\/image-36.png 382w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/08\/image-36-300x184.png 300w\" data-sizes=\"(max-width: 382px) 100vw, 382px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 382px; aspect-ratio: 382\/234;\" \/><\/figure>\n\n\n\n<p><strong>Pythagorean triples<\/strong>&nbsp;<\/p>\n\n\n\n<p>A Pythagorean triple consists of three elements (a,b,ca,b,c) that satisfy\u202fa2+b2=c2a2+b2=c2\u202f(Pythagorean theorem).&nbsp;<\/p>\n\n\n\n<p>There\u2019s a close relationship between the Fibonacci numbers and the Pythagorean triples. If we take four consecutive numbers from the Fibonacci sequence,\u202f(w,x,y,z)(w,x,y,z), we can get a Pythagorean triple if we make the following assignments:&nbsp;<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u202f\u202f\u202fLet\u202faa\u202fbe the product of the numbers belonging to the extremes\u202fa=xza=xz.\u00a0<\/li>\n<\/ol>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>\u202f\u202f\u202fLet\u202fbb\u202fbe the double of the product of the intermediate numbers\u202fc=2ywc=2yw.\u00a0<\/li>\n<\/ol>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li>\u202f\u202f\u202fLet\u202fcc\u202fbe the sum of the product of the odd numbers and the product of the even numbers\u202fc=xw+zyc=xw+zy.\u00a0<\/li>\n<\/ol>\n\n\n\n<p>Therefore (a,b,ca,b,c) is a Pythagorean triple.&nbsp;<\/p>\n\n\n\n<p><strong>Fibonacci numbers in trading techniques<\/strong>&nbsp;<\/p>\n\n\n\n<p>In trading, Fibonacci numbers appear in so-called Fibonacci studies. Fibonacci studies encompass a series of analysis tools based on sequence and Fibonacci ratios, which represent geometric laws of nature and human&nbsp;behaviour&nbsp;applied to financial markets.&nbsp;<\/p>\n\n\n\n<p>The most popular of these tools are Fibonacci retracements, extensions, arcs, fan and time zones. Other tools include the Fibonacci eclipse, spiral and canals.\u202f&nbsp;<\/p>\n\n\n\n<p>If you want to know how some of these tools work in financial markets, read our post \u201c<a href=\"https:\/\/quantdare.com\/fibonacci-retracement-and-extensions\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Fibonacci retracement and extensions<\/strong><\/a>\u201c.&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In trading, Fibonacci numbers appear in so-called Fibonacci studies. Fibonacci studies encompass a series of analysis tools based on sequence and Fibonacci ratios, which represent geometric laws of nature and human behavior applied to financial markets. <\/p>\n","protected":false},"author":465,"featured_media":48731,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,341,351,344],"tags":[3606,8167,8164,8166],"contributors-categories":[13705],"class_list":{"0":"post-55333","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-development","10":"category-quant-europe","11":"category-quant-regions","12":"tag-fibonacci","13":"tag-fibonacci-studies","14":"tag-mathematics","15":"tag-pythagorean-theorem","16":"contributors-categories-quantdare"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.3) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>The Magic of Fibonacci Numbers | IBKR Quant<\/title>\n<meta name=\"description\" content=\"In trading, Fibonacci numbers appear in so-called Fibonacci studies. 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