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Calculated Shard: 152 (from laksa131)

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Query:

curl -X POST \
  'http://laksa152.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Bollinger_Bands\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Bollinger_Bands\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands","crawl_time":1773196494,"first_indexed_time":1384997231,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Bollinger_Bands s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Bollinger Bands - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"From Wikipedia, the free encyclopedia\nS&P 500 with 20-day, two-standard-deviation Bollinger Bands, %b and bandwidth\nBollinger Bands\n(\n) are a type of\nstatistical\nchart\ncharacterizing the prices and\nvolatility\nover time of a financial instrument or commodity, using a\nformulaic\nmethod propounded by\nJohn Bollinger\nin the 1980s. Financial traders employ these charts as a methodical tool to inform trading decisions, control\nautomated trading systems\n, or as a component of\ntechnical analysis\n. Bollinger Bands display a graphical band (the\nenvelope\nmaximum and minimum of\nmoving averages\n, similar to\nKeltner\nor\nDonchian channels\n) and volatility (expressed by the width of the envelope) in one two-dimensional chart.\nTwo input parameters chosen independently by the user govern how a given chart summarizes the known historical price data, allowing the user to vary the response of the chart to the magnitude and frequency of price changes, similar to\nparametric equations\nin\nsignal processing\nor\ncontrol systems\n. Bollinger Bands consist of an\nN\n-period\nmoving average\n(MA), an upper band at\nK\ntimes an\nN\n-period\nstandard deviation\nabove the moving average (MA + \nKσ\n), and a lower band at\nK\ntimes an\nN\n-period\nstandard deviation\nbelow the moving average (MA − \nKσ\n). The chart thus expresses arbitrary choices or assumptions of the user, and is not strictly about the price data alone.\nTypical values for\nN\nand\nK\nare 20 days and 2, respectively. The default choice for the average is a simple\nmoving average\n, but other types of averages can be employed as needed.\nExponential moving averages\nare a common second choice.\n[\nnote 1\n]\nUsually the same period is used for both the middle band and the calculation of standard deviation.\n[\nnote 2\n]\nBollinger registered the words \"Bollinger Bands\" as a U.S.\ntrademark\nin 2011.\n[\n2\n]\nThe purpose of Bollinger Bands is to provide a relative definition of high and low prices of a market. By definition, prices are high at the upper band and low at the lower band. This definition can aid in rigorous\npattern recognition\nand is useful in comparing price action to the action of indicators to arrive at\nsystematic trading\ndecisions.\n[\n3\n]\nIndicators derived from Bollinger Bands\n[\nedit\n]\nBBImpulse\nmeasures price change as a function of the bands;\npercent bandwidth\n(%b) normalizes the width of the bands over time; and\nbandwidth delta\nquantifies the changing width of the bands.\n%b\n(pronounced \"percent b\") is derived from the formula for\nstochastics\nand shows where price is in relation to the bands. %\nb\nequals 1 at the upper band and 0 at the lower band. Writing\nupperBB\nfor the upper Bollinger Band,\nlowerBB\nfor the lower Bollinger Band, and\nlast\nfor the last (price) value:\n%\nb\n= (last − lowerBB) \/ (upperBB − lowerBB)\nBandwidth\ntells how wide the Bollinger Bands are on a normalized basis. Writing the same symbols as before, and\nmiddleBB\nfor the moving average, or middle Bollinger Band:\nBandwidth = (upperBB − lowerBB) \/ middleBB\nUsing the default parameters of a 20-period look back and plus\/minus two standard deviations,\nbandwidth\nis equal to four times the 20-period\ncoefficient of variation\n.\nUses for %\nb\ninclude system building and\npattern recognition\n. Uses for\nbandwidth\ninclude identification of opportunities arising from relative extremes in\nvolatility\nand trend identification.\nThe use of Bollinger Bands varies widely among traders. Some traders buy when price touches the lower Bollinger Band and exit when price touches the moving average in the center of the bands. Other traders buy when price breaks above the upper Bollinger Band or sell when price falls below the lower Bollinger Band.\n[\n4\n]\nMoreover, the use of Bollinger Bands is not confined to stock traders;\noptions\ntraders, most notably\nimplied volatility\ntraders, often sell options when Bollinger Bands are historically far apart or buy options when the Bollinger Bands are historically close together, in both instances, expecting volatility to revert towards the average historical volatility level for the stock.\nWhen the bands lie close together, a period of low\nvolatility\nis indicated.\n[\n5\n]\nConversely, as the bands expand, an increase in price action\/market volatility is indicated.\n[\n5\n]\nWhen the bands have only a slight slope and track approximately parallel for an extended time, the price will generally be found to oscillate between the bands as though in a channel.\nTraders are often inclined to use Bollinger Bands with other indicators to confirm price action. In particular, the use of oscillator-like Bollinger Bands will often be coupled with a non-oscillator indicator-like\nchart patterns\nor a\ntrendline\n. If these indicators confirm the recommendation of the Bollinger Bands, the trader will have greater conviction that the bands are predicting correct price action in relation to market volatility.\nVarious studies of the effectiveness of the Bollinger Band strategy have been performed with mixed results. In 2007, Lento\net al.\npublished an analysis using a variety of formats (different moving average timescales, and standard deviation ranges) and markets (e.g.,\nDow Jones\nand\nForex\n).\n[\n6\n]\nAnalysis of the trades, spanning a decade from 1995 onwards, found no evidence of consistent performance over the standard \"\nbuy and hold\n\" approach. The authors did, however, find that a simple reversal of the strategy (\"contrarian Bollinger Band\") produced positive returns in a variety of markets.\nSimilar results were found in another study, which concluded that Bollinger Band trading strategies may be effective in the Chinese marketplace, stating: \"we find significant positive returns on buy trades generated by the contrarian version of the\nmoving-average crossover\nrule, the channel breakout rule, and the Bollinger Band trading rule, after accounting for transaction costs of 0.50 percent.\"\n[\n7\n]\n(By \"the contrarian version\", they mean buying when the conventional rule mandates selling, and vice versa.) A recent study examined the application of Bollinger Band trading strategies combined with the\nADX\nfor\nEquity Market\nindices with similar results.\n[\n8\n]\nIn 2012, Butler\net al.\npublished an approach to fitting the parameters of Bollinger Bands using\nparticle swarm optimization\nmethod. Their results indicated that by tuning the parameters to a particular asset for a particular market environment, the\nout-of-sample\ntrading signals were improved compared to the default parameters.\n[\n9\n]\nStatistical properties\n[\nedit\n]\nSecurity price returns have no known\nstatistical distribution\n,\nnormal\nor otherwise; they are known to have\nfat tails\n, compared to a normal distribution.\n[\n10\n]\nThe\nsample size\ntypically used, 20, is too small for conclusions derived from statistical techniques like the\ncentral limit theorem\nto be reliable. Such techniques usually require the sample to be independent and identically distributed, which is not the case for a\ntime series\nlike security prices. Just the opposite is true; it is well recognized by practitioners that such price series are very commonly serially correlated\n[\ncitation needed\n]\n—that is, each price will be closely related to its ancestor \"most of the time\". Adjusting for\nserial correlation\nis the purpose of moving\nstandard deviations\n, which use deviations from the\nmoving average\n, but the possibility remains of high order price\nautocorrelation\nnot accounted for by simple differencing from the moving average.\nFor such reasons, it is incorrect to assume that the long-term percentage of the data that will be observed in the future outside the Bollinger Bands range will always be constrained to a certain amount. Instead of finding about 95% of the data inside the bands, as would be the expectation with the default parameters if the data were normally distributed, studies have found that only about 88% of security prices (85–90%) remain within the bands.\n[\n11\n]\nFor an individual security, one can always find factors for which certain percentages of data are contained by the factor defined bands for a certain period of time. Practitioners may also use related measures such as the\nKeltner channels\n, or the related Stoller average range channels, which base their band widths on different measures of price volatility, such as the difference between daily high and low prices, rather than on standard deviation.\nBollinger Bands outside of finance\n[\nedit\n]\nBollinger Bands have been applied to manufacturing data to detect defects (anomalies) in patterned fabrics.\n[\n12\n]\nIn this application, the upper and lower bands of Bollinger Bands are sensitive to subtle changes in the input data obtained from samples.\nThe\nInternational Civil Aviation Organization\nused Bollinger Bands to measure the accident rate as a safety indicator to measure efficacy of global safety initiatives.\n[\n13\n]\n%b and bandwidth are also used in this analysis.\n[\n14\n]\nBollinger Bands have been applied to a \"Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real-Time.\"\n[\n15\n]\n^\nWhen the average used in the calculation of Bollinger Bands is changed from a simple moving average to an exponential or weighted moving average, it must be changed for both the calculation of the middle band and the calculation of standard deviation.\n[\n1\n]\n^\nSince Bollinger Bands use the population method of calculating standard deviation, the proper divisor for the sigma calculation is\nn\n, not\nn\n − 1.\n^\nBollinger On Bollinger Bands – The Seminar, DVD I\nISBN\n \n978-0-9726111-0-7\n^\n\"Bollinger Bands – Trademark Details\"\n. Justia.com. 2011-12-20.\n^\n[1]\nsecond paragraph, center column\n^\nTechnical Analysis: The Complete Resource for Financial Market Technicians by Charles D. Kirkpatrick and Julie R. Dahlquist Chapter 14\n^\na\nb\nBaiynd, Anne-Marie\n(2011).\nThe Trading Book: A Complete Solution to Mastering Technical Systems and Trading Psychology\n.\nMcGraw-Hill\n. p. 272.\nISBN\n \n9780071766494\n. Archived from\nthe original\non 2012-03-25\n. Retrieved\n2013-04-30\n.\n^\nLento, C.; Gradojevic, N.; Wright, C. S. (2007). \"Investment information content in Bollinger Bands?\".\nApplied Financial Economics Letters\n.\n3\n(4):\n263–\n267.\ndoi\n:\n10.1080\/17446540701206576\n.\nISSN\n \n1744-6546\n.\nS2CID\n \n153603674\n.\n^\nBalsara, Nauzer J.; Chen, Gary; Zheng, Lin (2007). \"The Chinese Stock Market: An Examination of the Random Walk Model and Technical Trading Rules\".\nQuarterly Journal of Business and Economics\n.\n46\n(2):\n43–\n63.\nJSTOR\n \n40473435\n.\n^\nLim, Shawn; Hisarli, Tilman; Shi He, Ng (2014). \"The Profitability of a Combined Signal Approach: Bollinger Bands and the ADX\".\nInternational Federation of Technical Analysts Journal\n:\n23–\n29.\nSSRN\n \n2230499\n.\n^\nButler, Matthew; Kazakov, Dimitar (2010). \"Particle Swarm Optimization of Bollinger Bands\". In Dorigo, Marco; Birattari, Mauro; Caro, Gianni A. Di; Doursat, René; Engelbrecht, Andries P.; Floreano, Dario; Gambardella, Luca Maria; Groß, Roderich; Sahin, Erol; Sayama, Hiroki; Stützle, Thomas (eds.).\nSwarm Intelligence - 7th International Conference, ANTS 2010, Brussels, Belgium, September 8-10, 2010. Proceedings\n. Lecture Notes in Computer Science. Vol. 6234. Springer. pp. \n504–\n511.\ndoi\n:\n10.1007\/978-3-642-15461-4_50\n.\nISBN\n \n978-3-642-15460-7\n.\n^\nRachev; Svetlozar T., Menn, Christian; Fabozzi, Frank J. (2005), Fat Tailed and Skewed Asset Return Distributions, Implications for Risk Management, Portfolio Selection, and Option Pricing, John Wiley, New York\n^\nAdam Grimes (2012).\nThe Art & Science of Technical Analysis: Market Structure, Price Action & Trading Strategies\n. John Wiley & Sons. pp. \n196–\n198.\nISBN\n \n9781118224274\n.\n^\nPang, Grantham K. H. (2006-08-01). \"Novel method for patterned fabric inspection using Bollinger bands\".\nOptical Engineering\n.\n45\n(8): 087202.\nBibcode\n:\n2006OptEn..45h7202N\n.\ndoi\n:\n10.1117\/1.2345189\n.\nhdl\n:\n10722\/44829\n.\nISSN\n \n0091-3286\n.\n^\n\"ICAO Methodology for Accident Rate Calculation and Trending - SKYbrary Aviation Safety\"\n.\nSKYbrary\n. Retrieved\n2019-03-12\n.\n^\nJohn., Bollinger (2002).\nBollinger on Bollinger bands\n. New York: McGraw-Hill.\nISBN\n \n0071373683\n.\nOCLC\n \n46634029\n.\n^\nPagel, Christina (2015-11-16).\n\"A Novel Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real Time\"\n(PDF)\n.\nPediatr Crit Care Med\n.\n16\n(9):\n821–\n827.\ndoi\n:\n10.1097\/PCC.0000000000000540\n.\nPMID\n \n26536545\n.\nS2CID\n \n41502207\n.\nAchelis, Steve.\nTechnical Analysis from A to Z\n(pp. 71–73). Irwin, 1995.\nISBN\n \n978-0-07-136348-8\nBollinger, John.\nBollinger on Bollinger Bands\n. McGraw Hill, 2002.\nISBN\n \n978-0-07-137368-5\nCahen, Philippe.\nDynamic Technical Analysis\n. Wiley, 2001.\nISBN\n \n978-0-471-89947-1\nKirkpatrick, Charles D. II; Dahlquist, Julie R.\nTechnical Analysis: The Complete Resource for Financial Market Technicians\n, FT Press, 2006.\nISBN\n \n0-13-153113-1\nMurphy, John J.\nTechnical Analysis of the Financial Markets\n(pp. 209–211). New York Institute of Finance, 1999.\nISBN\n \n0-7352-0066-1\nJohn Bollinger's website\nJohn Bollinger's website for Bollinger Band analysis","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages \"A list of all special pages [q]\")\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/enwiki-25.svg) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en-25.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Bollinger+Bands \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Bollinger+Bands \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\nPersonal tools\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Bollinger+Bands \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Bollinger+Bands \"You're encouraged to log in; however, it's not mandatory. [o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands)\n- [1 Purpose](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#Purpose)\n- [2 Indicators derived from Bollinger Bands](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#Indicators_derived_from_Bollinger_Bands)\n- [3 Interpretation](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#Interpretation)\n- [4 Effectiveness](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#Effectiveness)\n- [5 Statistical properties](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#Statistical_properties)\n- [6 Bollinger Bands outside of finance](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#Bollinger_Bands_outside_of_finance)\n- [7 Notes](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#Notes)\n- [8 References](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#References)\n- [9 Further reading](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#Further_reading)\n- [10 External links](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#External_links)\n\nToggle the table of contents\n\n# Bollinger Bands\n18 languages\n\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D9%85%D8%A4%D8%B4%D8%B1_%D8%A8%D9%88%D9%84%D9%8A%D9%86%D8%AC%D8%B1_%D8%A8%D8%A7%D9%86%D8%AF \"مؤشر بولينجر باند – Arabic\")\n- [भोजपुरी](https:\/\/bh.wikipedia.org\/wiki\/%E0%A4%AC%E0%A5%8B%E0%A4%B2%E0%A4%BF%E0%A4%82%E0%A4%9C%E0%A4%B0_%E0%A4%AC%E0%A5%88%E0%A4%82%E0%A4%A1%E0%A5%8D%E0%A4%B8 \"बोलिंजर बैंड्स – Bhojpuri\")\n- [বাংলা](https:\/\/bn.wikipedia.org\/wiki\/%E0%A6%AC%E0%A7%8B%E0%A6%B2%E0%A6%BF%E0%A6%9E%E0%A7%8D%E0%A6%9C%E0%A6%BE%E0%A6%B0_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%A1%E0%A6%B8%E0%A6%AE%E0%A7%82%E0%A6%B9 \"বোলিঞ্জার ব্যান্ডসমূহ – Bangla\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/Bollingerova_p%C3%A1sma \"Bollingerova pásma – Czech\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Bollinger-B%C3%A4nder \"Bollinger-Bänder – German\")\n- 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[Read](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=history \"Past revisions of this page [h]\")\n\nTools\n\nTools\n\nmove to sidebar\n\nhide\n\nActions\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=history)\n\nGeneral\n\n- [What links here](https:\/\/en.wikipedia.org\/wiki\/Special:WhatLinksHere\/Bollinger_Bands \"List of all English Wikipedia pages containing links to this page [j]\")\n- [Related changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChangesLinked\/Bollinger_Bands \"Recent changes in pages linked from this page [k]\")\n- [Upload 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version](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikimedia Commons](https:\/\/commons.wikimedia.org\/wiki\/Category:Bollinger_Bands)\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q891459 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nStatistical price volatility chart\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1a\/BollingerBandsSPX.svg\/500px-BollingerBandsSPX.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:BollingerBandsSPX.svg)\n\nS\\&P 500 with 20-day, two-standard-deviation Bollinger Bands, %b and bandwidth\n\n**Bollinger Bands** ([\/ˈbɒlɪndʒər\/](https:\/\/en.wikipedia.org\/wiki\/Help:IPA\/English \"Help:IPA\/English\")) are a type of [statistical](https:\/\/en.wikipedia.org\/wiki\/Statistical \"Statistical\") [chart](https:\/\/en.wikipedia.org\/wiki\/Chart \"Chart\") characterizing the prices and [volatility](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\) \"Volatility (finance)\") over time of a financial instrument or commodity, using a [formulaic](https:\/\/en.wikipedia.org\/wiki\/Formula \"Formula\") method propounded by [John Bollinger](https:\/\/en.wikipedia.org\/wiki\/John_Bollinger \"John Bollinger\") in the 1980s. Financial traders employ these charts as a methodical tool to inform trading decisions, control [automated trading systems](https:\/\/en.wikipedia.org\/wiki\/Automated_trading_system \"Automated trading system\"), or as a component of [technical analysis](https:\/\/en.wikipedia.org\/wiki\/Technical_analysis \"Technical analysis\"). Bollinger Bands display a graphical band (the [envelope](https:\/\/en.wikipedia.org\/wiki\/Envelope_\\(mathematics\\) \"Envelope (mathematics)\") maximum and minimum of [moving averages](https:\/\/en.wikipedia.org\/wiki\/Moving_average_\\(finance\\) \"Moving average (finance)\"), similar to [Keltner](https:\/\/en.wikipedia.org\/wiki\/Keltner_channel \"Keltner channel\") or [Donchian channels](https:\/\/en.wikipedia.org\/wiki\/Donchian_channel \"Donchian channel\")) and volatility (expressed by the width of the envelope) in one two-dimensional chart.\n\nTwo input parameters chosen independently by the user govern how a given chart summarizes the known historical price data, allowing the user to vary the response of the chart to the magnitude and frequency of price changes, similar to [parametric equations](https:\/\/en.wikipedia.org\/wiki\/Parametric_equation \"Parametric equation\") in [signal processing](https:\/\/en.wikipedia.org\/wiki\/Signal_processing \"Signal processing\") or [control systems](https:\/\/en.wikipedia.org\/wiki\/Control_system \"Control system\"). Bollinger Bands consist of an *[N](https:\/\/en.wikipedia.org\/wiki\/Variable_\\(mathematics\\)#Naming_conventions \"Variable (mathematics)\")*\\-period [moving average](https:\/\/en.wikipedia.org\/wiki\/Moving_average_\\(finance\\) \"Moving average (finance)\") (MA), an upper band at *K* times an *N*\\-period [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") above the moving average (MA + *Kσ*), and a lower band at *K* times an *N*\\-period [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") below the moving average (MA − *Kσ*). The chart thus expresses arbitrary choices or assumptions of the user, and is not strictly about the price data alone.\n\nTypical values for *N* and *K* are 20 days and 2, respectively. The default choice for the average is a simple [moving average](https:\/\/en.wikipedia.org\/wiki\/Moving_average \"Moving average\"), but other types of averages can be employed as needed. [Exponential moving averages](https:\/\/en.wikipedia.org\/wiki\/Moving_average_\\(finance\\)#Exponential_moving_average \"Moving average (finance)\") are a common second choice.[\\[note 1\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-2) Usually the same period is used for both the middle band and the calculation of standard deviation.[\\[note 2\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-3)\n\nBollinger registered the words \"Bollinger Bands\" as a U.S. [trademark](https:\/\/en.wikipedia.org\/wiki\/Trademark \"Trademark\") in 2011.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-4)\n\n## Purpose\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=1 \"Edit section: Purpose\")\\]\n\nThe purpose of Bollinger Bands is to provide a relative definition of high and low prices of a market. By definition, prices are high at the upper band and low at the lower band. This definition can aid in rigorous [pattern recognition](https:\/\/en.wikipedia.org\/wiki\/Pattern_recognition \"Pattern recognition\") and is useful in comparing price action to the action of indicators to arrive at [systematic trading](https:\/\/en.wikipedia.org\/wiki\/Systematic_trading \"Systematic trading\") decisions.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-5)\n\n## Indicators derived from Bollinger Bands\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=2 \"Edit section: Indicators derived from Bollinger Bands\")\\]\n\n*BBImpulse* measures price change as a function of the bands; *percent bandwidth* (%b) normalizes the width of the bands over time; and *bandwidth delta* quantifies the changing width of the bands.\n\n*%b* (pronounced \"percent b\") is derived from the formula for [stochastics](https:\/\/en.wikipedia.org\/wiki\/Stochastic_oscillator \"Stochastic oscillator\") and shows where price is in relation to the bands. %*b* equals 1 at the upper band and 0 at the lower band. Writing *upperBB* for the upper Bollinger Band, *lowerBB* for the lower Bollinger Band, and *last* for the last (price) value:\n\n%*b* = (last − lowerBB) \/ (upperBB − lowerBB)\n\n*Bandwidth* tells how wide the Bollinger Bands are on a normalized basis. Writing the same symbols as before, and *middleBB* for the moving average, or middle Bollinger Band:\n\nBandwidth = (upperBB − lowerBB) \/ middleBB\n\nUsing the default parameters of a 20-period look back and plus\/minus two standard deviations, *bandwidth* is equal to four times the 20-period [coefficient of variation](https:\/\/en.wikipedia.org\/wiki\/Coefficient_of_variation \"Coefficient of variation\").\n\nUses for %*b* include system building and [pattern recognition](https:\/\/en.wikipedia.org\/wiki\/Pattern_recognition \"Pattern recognition\"). Uses for *bandwidth* include identification of opportunities arising from relative extremes in [volatility](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\) \"Volatility (finance)\") and trend identification.\n\n## Interpretation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=3 \"Edit section: Interpretation\")\\]\n\nThe use of Bollinger Bands varies widely among traders. Some traders buy when price touches the lower Bollinger Band and exit when price touches the moving average in the center of the bands. Other traders buy when price breaks above the upper Bollinger Band or sell when price falls below the lower Bollinger Band.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-6) Moreover, the use of Bollinger Bands is not confined to stock traders; [options](https:\/\/en.wikipedia.org\/wiki\/Option_\\(finance\\) \"Option (finance)\") traders, most notably [implied volatility](https:\/\/en.wikipedia.org\/wiki\/Implied_volatility \"Implied volatility\") traders, often sell options when Bollinger Bands are historically far apart or buy options when the Bollinger Bands are historically close together, in both instances, expecting volatility to revert towards the average historical volatility level for the stock.\n\nWhen the bands lie close together, a period of low [volatility](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\) \"Volatility (finance)\") is indicated.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-Baiynd2011-7) Conversely, as the bands expand, an increase in price action\/market volatility is indicated.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-Baiynd2011-7) When the bands have only a slight slope and track approximately parallel for an extended time, the price will generally be found to oscillate between the bands as though in a channel.\n\nTraders are often inclined to use Bollinger Bands with other indicators to confirm price action. In particular, the use of oscillator-like Bollinger Bands will often be coupled with a non-oscillator indicator-like [chart patterns](https:\/\/en.wikipedia.org\/wiki\/Chart_patterns \"Chart patterns\") or a [trendline](https:\/\/en.wikipedia.org\/wiki\/Trend_line_\\(technical_analysis\\) \"Trend line (technical analysis)\"). If these indicators confirm the recommendation of the Bollinger Bands, the trader will have greater conviction that the bands are predicting correct price action in relation to market volatility.\n\n## Effectiveness\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=4 \"Edit section: Effectiveness\")\\]\n\nVarious studies of the effectiveness of the Bollinger Band strategy have been performed with mixed results. In 2007, Lento *et al.* published an analysis using a variety of formats (different moving average timescales, and standard deviation ranges) and markets (e.g., [Dow Jones](https:\/\/en.wikipedia.org\/wiki\/Dow_Jones_%26_Company \"Dow Jones & Company\") and [Forex](https:\/\/en.wikipedia.org\/wiki\/Foreign_exchange_market \"Foreign exchange market\")).[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-LentoGradojevic2007-8) Analysis of the trades, spanning a decade from 1995 onwards, found no evidence of consistent performance over the standard \"[buy and hold](https:\/\/en.wikipedia.org\/wiki\/Buy_and_hold \"Buy and hold\")\" approach. The authors did, however, find that a simple reversal of the strategy (\"contrarian Bollinger Band\") produced positive returns in a variety of markets.\n\nSimilar results were found in another study, which concluded that Bollinger Band trading strategies may be effective in the Chinese marketplace, stating: \"we find significant positive returns on buy trades generated by the contrarian version of the [moving-average crossover](https:\/\/en.wikipedia.org\/wiki\/Moving-average_crossover \"Moving-average crossover\") rule, the channel breakout rule, and the Bollinger Band trading rule, after accounting for transaction costs of 0.50 percent.\"[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-9) (By \"the contrarian version\", they mean buying when the conventional rule mandates selling, and vice versa.) A recent study examined the application of Bollinger Band trading strategies combined with the [ADX](https:\/\/en.wikipedia.org\/wiki\/Average_directional_movement_index \"Average directional movement index\") for [Equity Market](https:\/\/en.wikipedia.org\/wiki\/Stock_market \"Stock market\") indices with similar results.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-10)\n\nIn 2012, Butler *et al.* published an approach to fitting the parameters of Bollinger Bands using [particle swarm optimization](https:\/\/en.wikipedia.org\/wiki\/Particle_swarm_optimization \"Particle swarm optimization\") method. Their results indicated that by tuning the parameters to a particular asset for a particular market environment, the [out-of-sample](https:\/\/en.wikipedia.org\/wiki\/Cross-validation_\\(statistics\\) \"Cross-validation (statistics)\") trading signals were improved compared to the default parameters.[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-ButlerKazakov2012-11)\n\n## Statistical properties\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=5 \"Edit section: Statistical properties\")\\]\n\nSecurity price returns have no known [statistical distribution](https:\/\/en.wikipedia.org\/wiki\/Statistical_distribution \"Statistical distribution\"), [normal](https:\/\/en.wikipedia.org\/wiki\/Normally_distributed \"Normally distributed\") or otherwise; they are known to have [fat tails](https:\/\/en.wikipedia.org\/wiki\/Fat_tail \"Fat tail\"), compared to a normal distribution.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-12) The [sample size](https:\/\/en.wikipedia.org\/wiki\/Sample_size \"Sample size\") typically used, 20, is too small for conclusions derived from statistical techniques like the [central limit theorem](https:\/\/en.wikipedia.org\/wiki\/Central_limit_theorem \"Central limit theorem\") to be reliable. Such techniques usually require the sample to be independent and identically distributed, which is not the case for a [time series](https:\/\/en.wikipedia.org\/wiki\/Time_series \"Time series\") like security prices. Just the opposite is true; it is well recognized by practitioners that such price series are very commonly serially correlated\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]—that is, each price will be closely related to its ancestor \"most of the time\". Adjusting for [serial correlation](https:\/\/en.wikipedia.org\/wiki\/Serial_correlation \"Serial correlation\") is the purpose of moving [standard deviations](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\"), which use deviations from the [moving average](https:\/\/en.wikipedia.org\/wiki\/Moving_average \"Moving average\"), but the possibility remains of high order price [autocorrelation](https:\/\/en.wikipedia.org\/wiki\/Autocorrelation \"Autocorrelation\") not accounted for by simple differencing from the moving average.\n\nFor such reasons, it is incorrect to assume that the long-term percentage of the data that will be observed in the future outside the Bollinger Bands range will always be constrained to a certain amount. Instead of finding about 95% of the data inside the bands, as would be the expectation with the default parameters if the data were normally distributed, studies have found that only about 88% of security prices (85–90%) remain within the bands.[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-grimes-13) For an individual security, one can always find factors for which certain percentages of data are contained by the factor defined bands for a certain period of time. Practitioners may also use related measures such as the [Keltner channels](https:\/\/en.wikipedia.org\/wiki\/Keltner_channel \"Keltner channel\"), or the related Stoller average range channels, which base their band widths on different measures of price volatility, such as the difference between daily high and low prices, rather than on standard deviation.\n\n## Bollinger Bands outside of finance\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=6 \"Edit section: Bollinger Bands outside of finance\")\\]\n\nBollinger Bands have been applied to manufacturing data to detect defects (anomalies) in patterned fabrics.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-14) In this application, the upper and lower bands of Bollinger Bands are sensitive to subtle changes in the input data obtained from samples.\n\nThe [International Civil Aviation Organization](https:\/\/en.wikipedia.org\/wiki\/International_Civil_Aviation_Organization \"International Civil Aviation Organization\") used Bollinger Bands to measure the accident rate as a safety indicator to measure efficacy of global safety initiatives.[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-15) %b and bandwidth are also used in this analysis.[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-16)\n\nBollinger Bands have been applied to a \"Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real-Time.\"[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-17)\n\n## Notes\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=7 \"Edit section: Notes\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-2)** When the average used in the calculation of Bollinger Bands is changed from a simple moving average to an exponential or weighted moving average, it must be changed for both the calculation of the middle band and the calculation of standard deviation.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-1)\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-3)** Since Bollinger Bands use the population method of calculating standard deviation, the proper divisor for the sigma calculation is *n*, not *n* − 1.\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=8 \"Edit section: References\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-1)**\n   Bollinger On Bollinger Bands – The Seminar, DVD I [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-9726111-0-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-9726111-0-7 \"Special:BookSources\/978-0-9726111-0-7\")\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-4)**\n   [\"Bollinger Bands – Trademark Details\"](http:\/\/trademarks.justia.com\/852\/32\/bollinger-bands-85232573.html). Justia.com. 2011-12-20.\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-5)** [\\[1\\]](http:\/\/www.bollingerbands.com\/) second paragraph, center column\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-6)** Technical Analysis: The Complete Resource for Financial Market Technicians by Charles D. Kirkpatrick and Julie R. Dahlquist Chapter 14\n5. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-Baiynd2011_7-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-Baiynd2011_7-1)\n   [Baiynd, Anne-Marie](https:\/\/en.wikipedia.org\/wiki\/Anne-Marie_Baiynd \"Anne-Marie Baiynd\") (2011). [*The Trading Book: A Complete Solution to Mastering Technical Systems and Trading Psychology*](https:\/\/web.archive.org\/web\/20120325050543\/http:\/\/mcgraw-hill.com.au\/html\/9780071766494.html). [McGraw-Hill](https:\/\/en.wikipedia.org\/wiki\/McGraw-Hill \"McGraw-Hill\"). p. 272. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [9780071766494](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9780071766494 \"Special:BookSources\/9780071766494\")\n   \n   . Archived from [the original](http:\/\/www.mcgraw-hill.com.au\/html\/9780071766494.html) on 2012-03-25. Retrieved 2013-04-30.\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-LentoGradojevic2007_8-0)**\n   Lento, C.; Gradojevic, N.; Wright, C. S. (2007). \"Investment information content in Bollinger Bands?\". *Applied Financial Economics Letters*. **3** (4): 263–267\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1080\/17446540701206576](https:\/\/doi.org\/10.1080%2F17446540701206576). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1744-6546](https:\/\/search.worldcat.org\/issn\/1744-6546). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [153603674](https:\/\/api.semanticscholar.org\/CorpusID:153603674).\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-9)**\n   Balsara, Nauzer J.; Chen, Gary; Zheng, Lin (2007). \"The Chinese Stock Market: An Examination of the Random Walk Model and Technical Trading Rules\". *Quarterly Journal of Business and Economics*. **46** (2): 43–63\\. [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [40473435](https:\/\/www.jstor.org\/stable\/40473435).\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-10)**\n   Lim, Shawn; Hisarli, Tilman; Shi He, Ng (2014). \"The Profitability of a Combined Signal Approach: Bollinger Bands and the ADX\". *International Federation of Technical Analysts Journal*: 23–29\\. [SSRN](https:\/\/en.wikipedia.org\/wiki\/SSRN_\\(identifier\\) \"SSRN (identifier)\") [2230499](https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=2230499).\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-ButlerKazakov2012_11-0)**\n   Butler, Matthew; Kazakov, Dimitar (2010). \"Particle Swarm Optimization of Bollinger Bands\". In Dorigo, Marco; Birattari, Mauro; Caro, Gianni A. Di; Doursat, René; Engelbrecht, Andries P.; Floreano, Dario; Gambardella, Luca Maria; Groß, Roderich; Sahin, Erol; Sayama, Hiroki; Stützle, Thomas (eds.). *Swarm Intelligence - 7th International Conference, ANTS 2010, Brussels, Belgium, September 8-10, 2010. Proceedings*. Lecture Notes in Computer Science. Vol. 6234. Springer. pp. 504–511\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/978-3-642-15461-4\\_50](https:\/\/doi.org\/10.1007%2F978-3-642-15461-4_50). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-3-642-15460-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-642-15460-7 \"Special:BookSources\/978-3-642-15460-7\")\n   \n   .\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-12)** Rachev; Svetlozar T., Menn, Christian; Fabozzi, Frank J. (2005), Fat Tailed and Skewed Asset Return Distributions, Implications for Risk Management, Portfolio Selection, and Option Pricing, John Wiley, New York\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-grimes_13-0)**\n    Adam Grimes (2012). [*The Art & Science of Technical Analysis: Market Structure, Price Action & Trading Strategies*](https:\/\/books.google.com\/books?id=Yzs_0ZkXnyAC&pg=PA196). John Wiley & Sons. pp. 196–198\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [9781118224274](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781118224274 \"Special:BookSources\/9781118224274\")\n    \n    .\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-14)**\n    Pang, Grantham K. H. (2006-08-01). \"Novel method for patterned fabric inspection using Bollinger bands\". *Optical Engineering*. **45** (8): 087202. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2006OptEn..45h7202N](https:\/\/ui.adsabs.harvard.edu\/abs\/2006OptEn..45h7202N). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1117\/1.2345189](https:\/\/doi.org\/10.1117%2F1.2345189). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10722\/44829](https:\/\/hdl.handle.net\/10722%2F44829). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0091-3286](https:\/\/search.worldcat.org\/issn\/0091-3286).\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-15)**\n    [\"ICAO Methodology for Accident Rate Calculation and Trending - SKYbrary Aviation Safety\"](https:\/\/www.skybrary.aero\/index.php\/ICAO_Methodology_for_Accident_Rate_Calculation_and_Trending). [SKYbrary](https:\/\/en.wikipedia.org\/wiki\/SKYbrary \"SKYbrary\"). Retrieved 2019-03-12.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-16)**\n    John., Bollinger (2002). [*Bollinger on Bollinger bands*](https:\/\/archive.org\/details\/bollingeronbolli00john). New York: McGraw-Hill. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0071373683](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0071373683 \"Special:BookSources\/0071373683\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [46634029](https:\/\/search.worldcat.org\/oclc\/46634029).\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-17)**\n    Pagel, Christina (2015-11-16). [\"A Novel Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real Time\"](https:\/\/discovery.ucl.ac.uk\/1468466\/3\/PCCM%20accepted%20April%202015.pdf) (PDF). *Pediatr Crit Care Med*. **16** (9): 821–827\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1097\/PCC.0000000000000540](https:\/\/doi.org\/10.1097%2FPCC.0000000000000540). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [26536545](https:\/\/pubmed.ncbi.nlm.nih.gov\/26536545). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [41502207](https:\/\/api.semanticscholar.org\/CorpusID:41502207).\n\n## Further reading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=9 \"Edit section: Further reading\")\\]\n\n- Achelis, Steve. *Technical Analysis from A to Z* (pp. 71–73). Irwin, 1995.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-07-136348-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-07-136348-8 \"Special:BookSources\/978-0-07-136348-8\")\n- Bollinger, John. *Bollinger on Bollinger Bands*. McGraw Hill, 2002.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-07-137368-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-07-137368-5 \"Special:BookSources\/978-0-07-137368-5\")\n- Cahen, Philippe. *Dynamic Technical Analysis*. Wiley, 2001.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-471-89947-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-471-89947-1 \"Special:BookSources\/978-0-471-89947-1\")\n- Kirkpatrick, Charles D. II; Dahlquist, Julie R. *Technical Analysis: The Complete Resource for Financial Market Technicians*, FT Press, 2006.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-13-153113-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-13-153113-1 \"Special:BookSources\/0-13-153113-1\")\n- Murphy, John J. *Technical Analysis of the Financial Markets* (pp. 209–211). New York Institute of Finance, 1999.\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-7352-0066-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-7352-0066-1 \"Special:BookSources\/0-7352-0066-1\")\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=10 \"Edit section: External links\")\\]\n\n- [John Bollinger's website](https:\/\/www.bollingerbands.com\/)\n- [John Bollinger's website for Bollinger Band analysis](https:\/\/bollingerbands.us\/)\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Technical_analysis \"Template:Technical analysis\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Technical_analysis \"Template talk:Technical analysis\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Technical_analysis \"Special:EditPage\/Template:Technical analysis\")[Technical analysis](https:\/\/en.wikipedia.org\/wiki\/Technical_analysis \"Technical analysis\") |   |\n|---|---|\n| Concepts | [Breakout](https:\/\/en.wikipedia.org\/wiki\/Breakout_\\(technical_analysis\\) \"Breakout (technical analysis)\") [Dead cat bounce](https:\/\/en.wikipedia.org\/wiki\/Dead_cat_bounce \"Dead cat bounce\") [Dow theory](https:\/\/en.wikipedia.org\/wiki\/Dow_theory \"Dow theory\") [Elliott wave principle](https:\/\/en.wikipedia.org\/wiki\/Elliott_wave_principle \"Elliott wave principle\") [Market trend](https:\/\/en.wikipedia.org\/wiki\/Market_trend \"Market trend\") |\n| Charts | [Candlestick](https:\/\/en.wikipedia.org\/wiki\/Candlestick_chart \"Candlestick chart\") [Renko](https:\/\/en.wikipedia.org\/wiki\/Renko_chart \"Renko chart\") [Kagi](https:\/\/en.wikipedia.org\/wiki\/Kagi_chart \"Kagi chart\") [Line](https:\/\/en.wikipedia.org\/wiki\/Line_chart \"Line chart\") [Open-high-low-close](https:\/\/en.wikipedia.org\/wiki\/Open-high-low-close_chart \"Open-high-low-close chart\") [Point and figure](https:\/\/en.wikipedia.org\/wiki\/Point_and_figure_chart \"Point and figure chart\") [Line break](https:\/\/en.wikipedia.org\/wiki\/Line_break_chart \"Line break chart\") |\n| Patterns |   |\n|   |   |\n| [Chart](https:\/\/en.wikipedia.org\/wiki\/Chart_pattern \"Chart pattern\") | [Broadening top](https:\/\/en.wikipedia.org\/wiki\/Broadening_top \"Broadening top\") [Cup and handle](https:\/\/en.wikipedia.org\/wiki\/Cup_and_handle \"Cup and handle\") [Double top and double bottom](https:\/\/en.wikipedia.org\/wiki\/Double_top_and_double_bottom \"Double top and double bottom\") [Flag and pennant](https:\/\/en.wikipedia.org\/wiki\/Flag_and_pennant_patterns \"Flag and pennant patterns\") [Gap](https:\/\/en.wikipedia.org\/wiki\/Gap_\\(chart_pattern\\) \"Gap (chart pattern)\") [Head and shoulders](https:\/\/en.wikipedia.org\/wiki\/Head_and_shoulders_\\(chart_pattern\\) \"Head and shoulders (chart pattern)\") [Island reversal](https:\/\/en.wikipedia.org\/wiki\/Island_reversal \"Island reversal\") [Price channels](https:\/\/en.wikipedia.org\/wiki\/Price_channels \"Price channels\") [Triangle](https:\/\/en.wikipedia.org\/wiki\/Triangle_\\(chart_pattern\\) \"Triangle (chart pattern)\") [Triple top and triple bottom](https:\/\/en.wikipedia.org\/wiki\/Triple_top_and_triple_bottom \"Triple top and triple bottom\") [Wedge pattern](https:\/\/en.wikipedia.org\/wiki\/Wedge_pattern \"Wedge pattern\") |\n| [Candlestick](https:\/\/en.wikipedia.org\/wiki\/Candlestick_pattern \"Candlestick pattern\") |   |\n|   |   |\n| [Simple](https:\/\/en.wikipedia.org\/wiki\/Candlestick_pattern#Simple_patterns \"Candlestick pattern\") | [Doji](https:\/\/en.wikipedia.org\/wiki\/Doji \"Doji\") |\n| [Complex](https:\/\/en.wikipedia.org\/wiki\/Candlestick_pattern#Complex_patterns \"Candlestick pattern\") | [Hikkake pattern](https:\/\/en.wikipedia.org\/wiki\/Hikkake_pattern \"Hikkake pattern\") [Morning star](https:\/\/en.wikipedia.org\/wiki\/Morning_star_\\(candlestick_pattern\\) \"Morning star (candlestick pattern)\") [Three black crows](https:\/\/en.wikipedia.org\/wiki\/Three_black_crows \"Three black crows\") [Three white soldiers](https:\/\/en.wikipedia.org\/wiki\/Three_white_soldiers \"Three white soldiers\") |\n| [Point and figure](https:\/\/en.wikipedia.org\/wiki\/Point_and_figure_chart \"Point and figure chart\") | [Bull trap](https:\/\/en.wikipedia.org\/wiki\/Bull_trap \"Bull trap\") [Bear trap](https:\/\/en.wikipedia.org\/wiki\/Bear_market \"Bear market\") |\n| [Indicators](https:\/\/en.wikipedia.org\/wiki\/Technical_indicator \"Technical indicator\") |   |\n|   |   |\n| [Support & resistance](https:\/\/en.wikipedia.org\/wiki\/Support_and_resistance \"Support and resistance\") | [Fibonacci retracement](https:\/\/en.wikipedia.org\/wiki\/Fibonacci_retracement \"Fibonacci retracement\") [Pivot point](https:\/\/en.wikipedia.org\/wiki\/Pivot_point_\\(technical_analysis\\) \"Pivot point (technical analysis)\") (PP) |\n| [Trend](https:\/\/en.wikipedia.org\/wiki\/Market_trend \"Market trend\") | [Average directional index](https:\/\/en.wikipedia.org\/wiki\/Average_directional_movement_index \"Average directional movement index\") (A.D.X.) [Commodity channel index](https:\/\/en.wikipedia.org\/wiki\/Commodity_channel_index \"Commodity channel index\") (CCI) [Detrended price oscillator](https:\/\/en.wikipedia.org\/wiki\/Detrended_price_oscillator \"Detrended price oscillator\") (DPO) [Know sure thing oscillator](https:\/\/en.wikipedia.org\/wiki\/KST_oscillator \"KST oscillator\") (KST) [Ichimoku Kinkō Hyō](https:\/\/en.wikipedia.org\/wiki\/Ichimoku_Kink%C5%8D_Hy%C5%8D \"Ichimoku Kinkō Hyō\") [Moving average convergence\/divergence](https:\/\/en.wikipedia.org\/wiki\/MACD \"MACD\") (MACD) [Mass index](https:\/\/en.wikipedia.org\/wiki\/Mass_index \"Mass index\") [Moving average](https:\/\/en.wikipedia.org\/wiki\/Moving_average \"Moving average\") (MA) [Parabolic SAR](https:\/\/en.wikipedia.org\/wiki\/Parabolic_SAR \"Parabolic SAR\") (SAR) [Smart money index](https:\/\/en.wikipedia.org\/wiki\/Smart_money_index \"Smart money index\") (SMI) [Trend line](https:\/\/en.wikipedia.org\/wiki\/Trend_line_\\(technical_analysis\\) \"Trend line (technical analysis)\") [Trix](https:\/\/en.wikipedia.org\/wiki\/Trix_\\(technical_analysis\\) \"Trix (technical analysis)\") [Vortex indicator](https:\/\/en.wikipedia.org\/wiki\/Vortex_indicator \"Vortex indicator\") (VI) |\n| [Momentum](https:\/\/en.wikipedia.org\/wiki\/Momentum_\\(finance\\) \"Momentum (finance)\") | [Money flow index](https:\/\/en.wikipedia.org\/wiki\/Money_flow_index \"Money flow index\") (MFI) [Relative strength index](https:\/\/en.wikipedia.org\/wiki\/Relative_strength_index \"Relative strength index\") (RSI) [Stochastic oscillator](https:\/\/en.wikipedia.org\/wiki\/Stochastic_oscillator \"Stochastic oscillator\") [True strength index](https:\/\/en.wikipedia.org\/wiki\/True_strength_index \"True strength index\") (TSI) [Ultimate oscillator](https:\/\/en.wikipedia.org\/wiki\/Ultimate_oscillator \"Ultimate oscillator\") [Williams %R](https:\/\/en.wikipedia.org\/wiki\/Williams_%25R \"Williams %R\") (%R) |\n| [Volume](https:\/\/en.wikipedia.org\/wiki\/Volume_\\(finance\\) \"Volume (finance)\") | [Accumulation\/distribution line](https:\/\/en.wikipedia.org\/wiki\/Accumulation\/distribution_index \"Accumulation\/distribution index\") [Ease of movement](https:\/\/en.wikipedia.org\/wiki\/Ease_of_movement \"Ease of movement\") (EMV) [Force index](https:\/\/en.wikipedia.org\/wiki\/Force_index \"Force index\") (FI) [Negative volume index](https:\/\/en.wikipedia.org\/wiki\/Negative_volume_index \"Negative volume index\") (NVI) [On-balance volume](https:\/\/en.wikipedia.org\/wiki\/On-balance_volume \"On-balance volume\") (OBV) [Put\/call ratio](https:\/\/en.wikipedia.org\/wiki\/Put\/call_ratio \"Put\/call ratio\") (PCR) [Volume–price trend](https:\/\/en.wikipedia.org\/wiki\/Volume%E2%80%93price_trend \"Volume–price trend\") (VPT) |\n| [Volatility](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\) \"Volatility (finance)\") | [Average true range](https:\/\/en.wikipedia.org\/wiki\/Average_true_range \"Average true range\") (ATR) [Bollinger Bands]() (BB) [Donchian channel](https:\/\/en.wikipedia.org\/wiki\/Donchian_channel \"Donchian channel\") [Keltner channel](https:\/\/en.wikipedia.org\/wiki\/Keltner_channel \"Keltner channel\") [CBOE Market Volatility Index](https:\/\/en.wikipedia.org\/wiki\/VIX \"VIX\") (VIX) [Standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") (σ) |\n| [Breadth](https:\/\/en.wikipedia.org\/wiki\/Breadth_of_market \"Breadth of market\") | [Advance–decline line](https:\/\/en.wikipedia.org\/wiki\/Advance%E2%80%93decline_line \"Advance–decline line\") (ADL) [Arms index](https:\/\/en.wikipedia.org\/wiki\/TRIN_\\(finance\\) \"TRIN (finance)\") (TRIN) [McClellan oscillator](https:\/\/en.wikipedia.org\/wiki\/McClellan_oscillator \"McClellan oscillator\") |\n| Other | [Coppock curve](https:\/\/en.wikipedia.org\/wiki\/Coppock_curve \"Coppock curve\") [Ulcer index](https:\/\/en.wikipedia.org\/wiki\/Ulcer_index \"Ulcer index\") |\n| Analysts | [John Bollinger](https:\/\/en.wikipedia.org\/wiki\/John_Bollinger \"John Bollinger\") [Ned Davis](https:\/\/en.wikipedia.org\/wiki\/Ned_Davis_\\(analyst\\) \"Ned Davis (analyst)\") [Charles Dow](https:\/\/en.wikipedia.org\/wiki\/Charles_Dow \"Charles Dow\") [Ralph Nelson Elliott](https:\/\/en.wikipedia.org\/wiki\/Ralph_Nelson_Elliott \"Ralph Nelson Elliott\") [Bob Farrell](https:\/\/en.wikipedia.org\/wiki\/Robert_Farrell_\\(technical_analyst\\) \"Robert Farrell (technical analyst)\") [John Murphy](https:\/\/en.wikipedia.org\/wiki\/John_Murphy_\\(technical_analyst\\) \"John Murphy (technical analyst)\") [Mark Hulbert](https:\/\/en.wikipedia.org\/wiki\/Mark_Hulbert \"Mark Hulbert\") |\n\n![](https:\/\/en.wikipedia.org\/wiki\/Special:CentralAutoLogin\/start?useformat=desktop&type=1x1&usesul3=1)\n\nRetrieved from \"<https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&oldid=1341622285>\"\n\n[Categories](https:\/\/en.wikipedia.org\/wiki\/Help:Category \"Help:Category\"):\n\n- [Chart overlays](https:\/\/en.wikipedia.org\/wiki\/Category:Chart_overlays \"Category:Chart overlays\")\n- [Technical indicators](https:\/\/en.wikipedia.org\/wiki\/Category:Technical_indicators \"Category:Technical indicators\")\n- [Statistical deviation and dispersion](https:\/\/en.wikipedia.org\/wiki\/Category:Statistical_deviation_and_dispersion \"Category:Statistical deviation and dispersion\")\n\nHidden categories:\n\n- [Articles with short description](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_short_description \"Category:Articles with short description\")\n- [Short description is different from Wikidata](https:\/\/en.wikipedia.org\/wiki\/Category:Short_description_is_different_from_Wikidata \"Category:Short description is different from Wikidata\")\n- [All articles with unsourced statements](https:\/\/en.wikipedia.org\/wiki\/Category:All_articles_with_unsourced_statements \"Category:All articles with unsourced statements\")\n- [Articles with unsourced statements from August 2014](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_unsourced_statements_from_August_2014 \"Category:Articles with unsourced statements from August 2014\")\n\n- This page was last edited on 4 March 2026, at 04:01 (UTC).\n- Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License \"Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License\"); additional terms may apply. By using this site, you agree to the [Terms of Use](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Terms_of_Use \"foundation:Special:MyLanguage\/Policy:Terms of Use\") and [Privacy Policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy \"foundation:Special:MyLanguage\/Policy:Privacy policy\"). Wikipedia® is a registered trademark of the [Wikimedia Foundation, Inc.](https:\/\/wikimediafoundation.org\/), a non-profit organization.\n\n- [Privacy policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy)\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About)\n- [Disclaimers](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:General_disclaimer)\n- [Contact Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us)\n- [Legal & safety contacts](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Legal:Wikimedia_Foundation_Legal_and_Safety_Contact_Information)\n- [Code of Conduct](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Universal_Code_of_Conduct)\n- [Developers](https:\/\/developer.wikimedia.org\/)\n- [Statistics](https:\/\/stats.wikimedia.org\/#\/en.wikipedia.org)\n- [Cookie statement](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Cookie_statement)\n- [Mobile view](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&mobileaction=toggle_view_mobile)\n\n- [![Wikimedia Foundation](https:\/\/en.wikipedia.org\/static\/images\/footer\/wikimedia.svg)](https:\/\/www.wikimedia.org\/)\n- [![Powered by MediaWiki](https:\/\/en.wikipedia.org\/w\/resources\/assets\/mediawiki_compact.svg)](https:\/\/www.mediawiki.org\/)\n\nSearch\n\nToggle the table of contents\n\nBollinger Bands\n\n18 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands)","attrs_readable_markdown":"From Wikipedia, the free encyclopedia\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1a\/BollingerBandsSPX.svg\/500px-BollingerBandsSPX.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:BollingerBandsSPX.svg)\n\nS\\&P 500 with 20-day, two-standard-deviation Bollinger Bands, %b and bandwidth\n\n**Bollinger Bands** () are a type of [statistical](https:\/\/en.wikipedia.org\/wiki\/Statistical \"Statistical\") [chart](https:\/\/en.wikipedia.org\/wiki\/Chart \"Chart\") characterizing the prices and [volatility](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\) \"Volatility (finance)\") over time of a financial instrument or commodity, using a [formulaic](https:\/\/en.wikipedia.org\/wiki\/Formula \"Formula\") method propounded by [John Bollinger](https:\/\/en.wikipedia.org\/wiki\/John_Bollinger \"John Bollinger\") in the 1980s. Financial traders employ these charts as a methodical tool to inform trading decisions, control [automated trading systems](https:\/\/en.wikipedia.org\/wiki\/Automated_trading_system \"Automated trading system\"), or as a component of [technical analysis](https:\/\/en.wikipedia.org\/wiki\/Technical_analysis \"Technical analysis\"). Bollinger Bands display a graphical band (the [envelope](https:\/\/en.wikipedia.org\/wiki\/Envelope_\\(mathematics\\) \"Envelope (mathematics)\") maximum and minimum of [moving averages](https:\/\/en.wikipedia.org\/wiki\/Moving_average_\\(finance\\) \"Moving average (finance)\"), similar to [Keltner](https:\/\/en.wikipedia.org\/wiki\/Keltner_channel \"Keltner channel\") or [Donchian channels](https:\/\/en.wikipedia.org\/wiki\/Donchian_channel \"Donchian channel\")) and volatility (expressed by the width of the envelope) in one two-dimensional chart.\n\nTwo input parameters chosen independently by the user govern how a given chart summarizes the known historical price data, allowing the user to vary the response of the chart to the magnitude and frequency of price changes, similar to [parametric equations](https:\/\/en.wikipedia.org\/wiki\/Parametric_equation \"Parametric equation\") in [signal processing](https:\/\/en.wikipedia.org\/wiki\/Signal_processing \"Signal processing\") or [control systems](https:\/\/en.wikipedia.org\/wiki\/Control_system \"Control system\"). Bollinger Bands consist of an *[N](https:\/\/en.wikipedia.org\/wiki\/Variable_\\(mathematics\\)#Naming_conventions \"Variable (mathematics)\")*\\-period [moving average](https:\/\/en.wikipedia.org\/wiki\/Moving_average_\\(finance\\) \"Moving average (finance)\") (MA), an upper band at *K* times an *N*\\-period [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") above the moving average (MA + *Kσ*), and a lower band at *K* times an *N*\\-period [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") below the moving average (MA − *Kσ*). The chart thus expresses arbitrary choices or assumptions of the user, and is not strictly about the price data alone.\n\nTypical values for *N* and *K* are 20 days and 2, respectively. The default choice for the average is a simple [moving average](https:\/\/en.wikipedia.org\/wiki\/Moving_average \"Moving average\"), but other types of averages can be employed as needed. [Exponential moving averages](https:\/\/en.wikipedia.org\/wiki\/Moving_average_\\(finance\\)#Exponential_moving_average \"Moving average (finance)\") are a common second choice.[\\[note 1\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-2) Usually the same period is used for both the middle band and the calculation of standard deviation.[\\[note 2\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-3)\n\nBollinger registered the words \"Bollinger Bands\" as a U.S. [trademark](https:\/\/en.wikipedia.org\/wiki\/Trademark \"Trademark\") in 2011.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-4)\n\nThe purpose of Bollinger Bands is to provide a relative definition of high and low prices of a market. By definition, prices are high at the upper band and low at the lower band. This definition can aid in rigorous [pattern recognition](https:\/\/en.wikipedia.org\/wiki\/Pattern_recognition \"Pattern recognition\") and is useful in comparing price action to the action of indicators to arrive at [systematic trading](https:\/\/en.wikipedia.org\/wiki\/Systematic_trading \"Systematic trading\") decisions.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-5)\n\n## Indicators derived from Bollinger Bands\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=2 \"Edit section: Indicators derived from Bollinger Bands\")\\]\n\n*BBImpulse* measures price change as a function of the bands; *percent bandwidth* (%b) normalizes the width of the bands over time; and *bandwidth delta* quantifies the changing width of the bands.\n\n*%b* (pronounced \"percent b\") is derived from the formula for [stochastics](https:\/\/en.wikipedia.org\/wiki\/Stochastic_oscillator \"Stochastic oscillator\") and shows where price is in relation to the bands. %*b* equals 1 at the upper band and 0 at the lower band. Writing *upperBB* for the upper Bollinger Band, *lowerBB* for the lower Bollinger Band, and *last* for the last (price) value:\n\n%*b* = (last − lowerBB) \/ (upperBB − lowerBB)\n\n*Bandwidth* tells how wide the Bollinger Bands are on a normalized basis. Writing the same symbols as before, and *middleBB* for the moving average, or middle Bollinger Band:\n\nBandwidth = (upperBB − lowerBB) \/ middleBB\n\nUsing the default parameters of a 20-period look back and plus\/minus two standard deviations, *bandwidth* is equal to four times the 20-period [coefficient of variation](https:\/\/en.wikipedia.org\/wiki\/Coefficient_of_variation \"Coefficient of variation\").\n\nUses for %*b* include system building and [pattern recognition](https:\/\/en.wikipedia.org\/wiki\/Pattern_recognition \"Pattern recognition\"). Uses for *bandwidth* include identification of opportunities arising from relative extremes in [volatility](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\) \"Volatility (finance)\") and trend identification.\n\nThe use of Bollinger Bands varies widely among traders. Some traders buy when price touches the lower Bollinger Band and exit when price touches the moving average in the center of the bands. Other traders buy when price breaks above the upper Bollinger Band or sell when price falls below the lower Bollinger Band.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-6) Moreover, the use of Bollinger Bands is not confined to stock traders; [options](https:\/\/en.wikipedia.org\/wiki\/Option_\\(finance\\) \"Option (finance)\") traders, most notably [implied volatility](https:\/\/en.wikipedia.org\/wiki\/Implied_volatility \"Implied volatility\") traders, often sell options when Bollinger Bands are historically far apart or buy options when the Bollinger Bands are historically close together, in both instances, expecting volatility to revert towards the average historical volatility level for the stock.\n\nWhen the bands lie close together, a period of low [volatility](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\) \"Volatility (finance)\") is indicated.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-Baiynd2011-7) Conversely, as the bands expand, an increase in price action\/market volatility is indicated.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-Baiynd2011-7) When the bands have only a slight slope and track approximately parallel for an extended time, the price will generally be found to oscillate between the bands as though in a channel.\n\nTraders are often inclined to use Bollinger Bands with other indicators to confirm price action. In particular, the use of oscillator-like Bollinger Bands will often be coupled with a non-oscillator indicator-like [chart patterns](https:\/\/en.wikipedia.org\/wiki\/Chart_patterns \"Chart patterns\") or a [trendline](https:\/\/en.wikipedia.org\/wiki\/Trend_line_\\(technical_analysis\\) \"Trend line (technical analysis)\"). If these indicators confirm the recommendation of the Bollinger Bands, the trader will have greater conviction that the bands are predicting correct price action in relation to market volatility.\n\nVarious studies of the effectiveness of the Bollinger Band strategy have been performed with mixed results. In 2007, Lento *et al.* published an analysis using a variety of formats (different moving average timescales, and standard deviation ranges) and markets (e.g., [Dow Jones](https:\/\/en.wikipedia.org\/wiki\/Dow_Jones_%26_Company \"Dow Jones & Company\") and [Forex](https:\/\/en.wikipedia.org\/wiki\/Foreign_exchange_market \"Foreign exchange market\")).[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-LentoGradojevic2007-8) Analysis of the trades, spanning a decade from 1995 onwards, found no evidence of consistent performance over the standard \"[buy and hold](https:\/\/en.wikipedia.org\/wiki\/Buy_and_hold \"Buy and hold\")\" approach. The authors did, however, find that a simple reversal of the strategy (\"contrarian Bollinger Band\") produced positive returns in a variety of markets.\n\nSimilar results were found in another study, which concluded that Bollinger Band trading strategies may be effective in the Chinese marketplace, stating: \"we find significant positive returns on buy trades generated by the contrarian version of the [moving-average crossover](https:\/\/en.wikipedia.org\/wiki\/Moving-average_crossover \"Moving-average crossover\") rule, the channel breakout rule, and the Bollinger Band trading rule, after accounting for transaction costs of 0.50 percent.\"[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-9) (By \"the contrarian version\", they mean buying when the conventional rule mandates selling, and vice versa.) A recent study examined the application of Bollinger Band trading strategies combined with the [ADX](https:\/\/en.wikipedia.org\/wiki\/Average_directional_movement_index \"Average directional movement index\") for [Equity Market](https:\/\/en.wikipedia.org\/wiki\/Stock_market \"Stock market\") indices with similar results.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-10)\n\nIn 2012, Butler *et al.* published an approach to fitting the parameters of Bollinger Bands using [particle swarm optimization](https:\/\/en.wikipedia.org\/wiki\/Particle_swarm_optimization \"Particle swarm optimization\") method. Their results indicated that by tuning the parameters to a particular asset for a particular market environment, the [out-of-sample](https:\/\/en.wikipedia.org\/wiki\/Cross-validation_\\(statistics\\) \"Cross-validation (statistics)\") trading signals were improved compared to the default parameters.[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-ButlerKazakov2012-11)\n\n## Statistical properties\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=5 \"Edit section: Statistical properties\")\\]\n\nSecurity price returns have no known [statistical distribution](https:\/\/en.wikipedia.org\/wiki\/Statistical_distribution \"Statistical distribution\"), [normal](https:\/\/en.wikipedia.org\/wiki\/Normally_distributed \"Normally distributed\") or otherwise; they are known to have [fat tails](https:\/\/en.wikipedia.org\/wiki\/Fat_tail \"Fat tail\"), compared to a normal distribution.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-12) The [sample size](https:\/\/en.wikipedia.org\/wiki\/Sample_size \"Sample size\") typically used, 20, is too small for conclusions derived from statistical techniques like the [central limit theorem](https:\/\/en.wikipedia.org\/wiki\/Central_limit_theorem \"Central limit theorem\") to be reliable. Such techniques usually require the sample to be independent and identically distributed, which is not the case for a [time series](https:\/\/en.wikipedia.org\/wiki\/Time_series \"Time series\") like security prices. Just the opposite is true; it is well recognized by practitioners that such price series are very commonly serially correlated\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]—that is, each price will be closely related to its ancestor \"most of the time\". Adjusting for [serial correlation](https:\/\/en.wikipedia.org\/wiki\/Serial_correlation \"Serial correlation\") is the purpose of moving [standard deviations](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\"), which use deviations from the [moving average](https:\/\/en.wikipedia.org\/wiki\/Moving_average \"Moving average\"), but the possibility remains of high order price [autocorrelation](https:\/\/en.wikipedia.org\/wiki\/Autocorrelation \"Autocorrelation\") not accounted for by simple differencing from the moving average.\n\nFor such reasons, it is incorrect to assume that the long-term percentage of the data that will be observed in the future outside the Bollinger Bands range will always be constrained to a certain amount. Instead of finding about 95% of the data inside the bands, as would be the expectation with the default parameters if the data were normally distributed, studies have found that only about 88% of security prices (85–90%) remain within the bands.[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-grimes-13) For an individual security, one can always find factors for which certain percentages of data are contained by the factor defined bands for a certain period of time. Practitioners may also use related measures such as the [Keltner channels](https:\/\/en.wikipedia.org\/wiki\/Keltner_channel \"Keltner channel\"), or the related Stoller average range channels, which base their band widths on different measures of price volatility, such as the difference between daily high and low prices, rather than on standard deviation.\n\n## Bollinger Bands outside of finance\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Bollinger_Bands&action=edit&section=6 \"Edit section: Bollinger Bands outside of finance\")\\]\n\nBollinger Bands have been applied to manufacturing data to detect defects (anomalies) in patterned fabrics.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-14) In this application, the upper and lower bands of Bollinger Bands are sensitive to subtle changes in the input data obtained from samples.\n\nThe [International Civil Aviation Organization](https:\/\/en.wikipedia.org\/wiki\/International_Civil_Aviation_Organization \"International Civil Aviation Organization\") used Bollinger Bands to measure the accident rate as a safety indicator to measure efficacy of global safety initiatives.[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-15) %b and bandwidth are also used in this analysis.[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-16)\n\nBollinger Bands have been applied to a \"Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real-Time.\"[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-17)\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-2)** When the average used in the calculation of Bollinger Bands is changed from a simple moving average to an exponential or weighted moving average, it must be changed for both the calculation of the middle band and the calculation of standard deviation.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_note-1)\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-3)** Since Bollinger Bands use the population method of calculating standard deviation, the proper divisor for the sigma calculation is *n*, not *n* − 1.\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-1)**\n   Bollinger On Bollinger Bands – The Seminar, DVD I [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-9726111-0-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-9726111-0-7 \"Special:BookSources\/978-0-9726111-0-7\")\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-4)**\n   [\"Bollinger Bands – Trademark Details\"](http:\/\/trademarks.justia.com\/852\/32\/bollinger-bands-85232573.html). Justia.com. 2011-12-20.\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-5)** [\\[1\\]](http:\/\/www.bollingerbands.com\/) second paragraph, center column\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-6)** Technical Analysis: The Complete Resource for Financial Market Technicians by Charles D. Kirkpatrick and Julie R. Dahlquist Chapter 14\n5. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-Baiynd2011_7-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-Baiynd2011_7-1)\n   [Baiynd, Anne-Marie](https:\/\/en.wikipedia.org\/wiki\/Anne-Marie_Baiynd \"Anne-Marie Baiynd\") (2011). [*The Trading Book: A Complete Solution to Mastering Technical Systems and Trading Psychology*](https:\/\/web.archive.org\/web\/20120325050543\/http:\/\/mcgraw-hill.com.au\/html\/9780071766494.html). [McGraw-Hill](https:\/\/en.wikipedia.org\/wiki\/McGraw-Hill \"McGraw-Hill\"). p. 272. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [9780071766494](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9780071766494 \"Special:BookSources\/9780071766494\")\n   \n   . Archived from [the original](http:\/\/www.mcgraw-hill.com.au\/html\/9780071766494.html) on 2012-03-25. Retrieved 2013-04-30.\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-LentoGradojevic2007_8-0)**\n   Lento, C.; Gradojevic, N.; Wright, C. S. (2007). \"Investment information content in Bollinger Bands?\". *Applied Financial Economics Letters*. **3** (4): 263–267\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1080\/17446540701206576](https:\/\/doi.org\/10.1080%2F17446540701206576). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1744-6546](https:\/\/search.worldcat.org\/issn\/1744-6546). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [153603674](https:\/\/api.semanticscholar.org\/CorpusID:153603674).\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-9)**\n   Balsara, Nauzer J.; Chen, Gary; Zheng, Lin (2007). \"The Chinese Stock Market: An Examination of the Random Walk Model and Technical Trading Rules\". *Quarterly Journal of Business and Economics*. **46** (2): 43–63\\. [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [40473435](https:\/\/www.jstor.org\/stable\/40473435).\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-10)**\n   Lim, Shawn; Hisarli, Tilman; Shi He, Ng (2014). \"The Profitability of a Combined Signal Approach: Bollinger Bands and the ADX\". *International Federation of Technical Analysts Journal*: 23–29\\. [SSRN](https:\/\/en.wikipedia.org\/wiki\/SSRN_\\(identifier\\) \"SSRN (identifier)\") [2230499](https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=2230499).\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-ButlerKazakov2012_11-0)**\n   Butler, Matthew; Kazakov, Dimitar (2010). \"Particle Swarm Optimization of Bollinger Bands\". In Dorigo, Marco; Birattari, Mauro; Caro, Gianni A. Di; Doursat, René; Engelbrecht, Andries P.; Floreano, Dario; Gambardella, Luca Maria; Groß, Roderich; Sahin, Erol; Sayama, Hiroki; Stützle, Thomas (eds.). *Swarm Intelligence - 7th International Conference, ANTS 2010, Brussels, Belgium, September 8-10, 2010. Proceedings*. Lecture Notes in Computer Science. Vol. 6234. Springer. pp. 504–511\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1007\/978-3-642-15461-4\\_50](https:\/\/doi.org\/10.1007%2F978-3-642-15461-4_50). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-3-642-15460-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-642-15460-7 \"Special:BookSources\/978-3-642-15460-7\")\n   \n   .\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-12)** Rachev; Svetlozar T., Menn, Christian; Fabozzi, Frank J. (2005), Fat Tailed and Skewed Asset Return Distributions, Implications for Risk Management, Portfolio Selection, and Option Pricing, John Wiley, New York\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-grimes_13-0)**\n    Adam Grimes (2012). [*The Art & Science of Technical Analysis: Market Structure, Price Action & Trading Strategies*](https:\/\/books.google.com\/books?id=Yzs_0ZkXnyAC&pg=PA196). John Wiley & Sons. pp. 196–198\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [9781118224274](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781118224274 \"Special:BookSources\/9781118224274\")\n    \n    .\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-14)**\n    Pang, Grantham K. H. (2006-08-01). \"Novel method for patterned fabric inspection using Bollinger bands\". *Optical Engineering*. **45** (8): 087202. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2006OptEn..45h7202N](https:\/\/ui.adsabs.harvard.edu\/abs\/2006OptEn..45h7202N). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1117\/1.2345189](https:\/\/doi.org\/10.1117%2F1.2345189). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10722\/44829](https:\/\/hdl.handle.net\/10722%2F44829). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0091-3286](https:\/\/search.worldcat.org\/issn\/0091-3286).\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-15)**\n    [\"ICAO Methodology for Accident Rate Calculation and Trending - SKYbrary Aviation Safety\"](https:\/\/www.skybrary.aero\/index.php\/ICAO_Methodology_for_Accident_Rate_Calculation_and_Trending). [SKYbrary](https:\/\/en.wikipedia.org\/wiki\/SKYbrary \"SKYbrary\"). Retrieved 2019-03-12.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-16)**\n    John., Bollinger (2002). [*Bollinger on Bollinger bands*](https:\/\/archive.org\/details\/bollingeronbolli00john). New York: McGraw-Hill. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [0071373683](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0071373683 \"Special:BookSources\/0071373683\")\n    \n    . [OCLC](https:\/\/en.wikipedia.org\/wiki\/OCLC_\\(identifier\\) \"OCLC (identifier)\") [46634029](https:\/\/search.worldcat.org\/oclc\/46634029).\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands#cite_ref-17)**\n    Pagel, Christina (2015-11-16). [\"A Novel Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real Time\"](https:\/\/discovery.ucl.ac.uk\/1468466\/3\/PCCM%20accepted%20April%202015.pdf) (PDF). *Pediatr Crit Care Med*. **16** (9): 821–827\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1097\/PCC.0000000000000540](https:\/\/doi.org\/10.1097%2FPCC.0000000000000540). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [26536545](https:\/\/pubmed.ncbi.nlm.nih.gov\/26536545). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [41502207](https:\/\/api.semanticscholar.org\/CorpusID:41502207).\n\n- Achelis, Steve. *Technical Analysis from A to Z* (pp. 71–73). Irwin, 1995. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-07-136348-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-07-136348-8 \"Special:BookSources\/978-0-07-136348-8\")\n- Bollinger, John. *Bollinger on Bollinger Bands*. McGraw Hill, 2002. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-07-137368-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-07-137368-5 \"Special:BookSources\/978-0-07-137368-5\")\n- Cahen, Philippe. *Dynamic Technical Analysis*. Wiley, 2001. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [978-0-471-89947-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-471-89947-1 \"Special:BookSources\/978-0-471-89947-1\")\n- Kirkpatrick, Charles D. II; Dahlquist, Julie R. *Technical Analysis: The Complete Resource for Financial Market Technicians*, FT Press, 2006. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-13-153113-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-13-153113-1 \"Special:BookSources\/0-13-153113-1\")\n- Murphy, John J. *Technical Analysis of the Financial Markets* (pp. 209–211). New York Institute of Finance, 1999. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-7352-0066-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-7352-0066-1 \"Special:BookSources\/0-7352-0066-1\")\n\n- [John Bollinger's website](https:\/\/www.bollingerbands.com\/)\n- [John Bollinger's website for Bollinger Band analysis](https:\/\/bollingerbands.us\/)","meta_canonical":null}

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Boilerpipe Text	From Wikipedia, the free encyclopedia S&P 500 with 20-day, two-standard-deviation Bollinger Bands, %b and bandwidth Bollinger Bands ( ) are a type of statistical chart characterizing the prices and volatility over time of a financial instrument or commodity, using a formulaic method propounded by John Bollinger in the 1980s. Financial traders employ these charts as a methodical tool to inform trading decisions, control automated trading systems , or as a component of technical analysis . Bollinger Bands display a graphical band (the envelope maximum and minimum of moving averages , similar to Keltner or Donchian channels ) and volatility (expressed by the width of the envelope) in one two-dimensional chart. Two input parameters chosen independently by the user govern how a given chart summarizes the known historical price data, allowing the user to vary the response of the chart to the magnitude and frequency of price changes, similar to parametric equations in signal processing or control systems . Bollinger Bands consist of an N -period moving average (MA), an upper band at K times an N -period standard deviation above the moving average (MA + Kσ ), and a lower band at K times an N -period standard deviation below the moving average (MA − Kσ ). The chart thus expresses arbitrary choices or assumptions of the user, and is not strictly about the price data alone. Typical values for N and K are 20 days and 2, respectively. The default choice for the average is a simple moving average , but other types of averages can be employed as needed. Exponential moving averages are a common second choice. [ note 1 ] Usually the same period is used for both the middle band and the calculation of standard deviation. [ note 2 ] Bollinger registered the words "Bollinger Bands" as a U.S. trademark in 2011. [ 2 ] The purpose of Bollinger Bands is to provide a relative definition of high and low prices of a market. By definition, prices are high at the upper band and low at the lower band. This definition can aid in rigorous pattern recognition and is useful in comparing price action to the action of indicators to arrive at systematic trading decisions. [ 3 ] Indicators derived from Bollinger Bands [ edit ] BBImpulse measures price change as a function of the bands; percent bandwidth (%b) normalizes the width of the bands over time; and bandwidth delta quantifies the changing width of the bands. %b (pronounced "percent b") is derived from the formula for stochastics and shows where price is in relation to the bands. % b equals 1 at the upper band and 0 at the lower band. Writing upperBB for the upper Bollinger Band, lowerBB for the lower Bollinger Band, and last for the last (price) value: % b = (last − lowerBB) / (upperBB − lowerBB) Bandwidth tells how wide the Bollinger Bands are on a normalized basis. Writing the same symbols as before, and middleBB for the moving average, or middle Bollinger Band: Bandwidth = (upperBB − lowerBB) / middleBB Using the default parameters of a 20-period look back and plus/minus two standard deviations, bandwidth is equal to four times the 20-period coefficient of variation . Uses for % b include system building and pattern recognition . Uses for bandwidth include identification of opportunities arising from relative extremes in volatility and trend identification. The use of Bollinger Bands varies widely among traders. Some traders buy when price touches the lower Bollinger Band and exit when price touches the moving average in the center of the bands. Other traders buy when price breaks above the upper Bollinger Band or sell when price falls below the lower Bollinger Band. [ 4 ] Moreover, the use of Bollinger Bands is not confined to stock traders; options traders, most notably implied volatility traders, often sell options when Bollinger Bands are historically far apart or buy options when the Bollinger Bands are historically close together, in both instances, expecting volatility to revert towards the average historical volatility level for the stock. When the bands lie close together, a period of low volatility is indicated. [ 5 ] Conversely, as the bands expand, an increase in price action/market volatility is indicated. [ 5 ] When the bands have only a slight slope and track approximately parallel for an extended time, the price will generally be found to oscillate between the bands as though in a channel. Traders are often inclined to use Bollinger Bands with other indicators to confirm price action. In particular, the use of oscillator-like Bollinger Bands will often be coupled with a non-oscillator indicator-like chart patterns or a trendline . If these indicators confirm the recommendation of the Bollinger Bands, the trader will have greater conviction that the bands are predicting correct price action in relation to market volatility. Various studies of the effectiveness of the Bollinger Band strategy have been performed with mixed results. In 2007, Lento et al. published an analysis using a variety of formats (different moving average timescales, and standard deviation ranges) and markets (e.g., Dow Jones and Forex ). [ 6 ] Analysis of the trades, spanning a decade from 1995 onwards, found no evidence of consistent performance over the standard " buy and hold " approach. The authors did, however, find that a simple reversal of the strategy ("contrarian Bollinger Band") produced positive returns in a variety of markets. Similar results were found in another study, which concluded that Bollinger Band trading strategies may be effective in the Chinese marketplace, stating: "we find significant positive returns on buy trades generated by the contrarian version of the moving-average crossover rule, the channel breakout rule, and the Bollinger Band trading rule, after accounting for transaction costs of 0.50 percent." [ 7 ] (By "the contrarian version", they mean buying when the conventional rule mandates selling, and vice versa.) A recent study examined the application of Bollinger Band trading strategies combined with the ADX for Equity Market indices with similar results. [ 8 ] In 2012, Butler et al. published an approach to fitting the parameters of Bollinger Bands using particle swarm optimization method. Their results indicated that by tuning the parameters to a particular asset for a particular market environment, the out-of-sample trading signals were improved compared to the default parameters. [ 9 ] Statistical properties [ edit ] Security price returns have no known statistical distribution , normal or otherwise; they are known to have fat tails , compared to a normal distribution. [ 10 ] The sample size typically used, 20, is too small for conclusions derived from statistical techniques like the central limit theorem to be reliable. Such techniques usually require the sample to be independent and identically distributed, which is not the case for a time series like security prices. Just the opposite is true; it is well recognized by practitioners that such price series are very commonly serially correlated [ citation needed ] —that is, each price will be closely related to its ancestor "most of the time". Adjusting for serial correlation is the purpose of moving standard deviations , which use deviations from the moving average , but the possibility remains of high order price autocorrelation not accounted for by simple differencing from the moving average. For such reasons, it is incorrect to assume that the long-term percentage of the data that will be observed in the future outside the Bollinger Bands range will always be constrained to a certain amount. Instead of finding about 95% of the data inside the bands, as would be the expectation with the default parameters if the data were normally distributed, studies have found that only about 88% of security prices (85–90%) remain within the bands. [ 11 ] For an individual security, one can always find factors for which certain percentages of data are contained by the factor defined bands for a certain period of time. Practitioners may also use related measures such as the Keltner channels , or the related Stoller average range channels, which base their band widths on different measures of price volatility, such as the difference between daily high and low prices, rather than on standard deviation. Bollinger Bands outside of finance [ edit ] Bollinger Bands have been applied to manufacturing data to detect defects (anomalies) in patterned fabrics. [ 12 ] In this application, the upper and lower bands of Bollinger Bands are sensitive to subtle changes in the input data obtained from samples. The International Civil Aviation Organization used Bollinger Bands to measure the accident rate as a safety indicator to measure efficacy of global safety initiatives. [ 13 ] %b and bandwidth are also used in this analysis. [ 14 ] Bollinger Bands have been applied to a "Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real-Time." [ 15 ] ^ When the average used in the calculation of Bollinger Bands is changed from a simple moving average to an exponential or weighted moving average, it must be changed for both the calculation of the middle band and the calculation of standard deviation. [ 1 ] ^ Since Bollinger Bands use the population method of calculating standard deviation, the proper divisor for the sigma calculation is n , not n − 1. ^ Bollinger On Bollinger Bands – The Seminar, DVD I ISBN 978-0-9726111-0-7 ^ "Bollinger Bands – Trademark Details" . Justia.com. 2011-12-20. ^ [1] second paragraph, center column ^ Technical Analysis: The Complete Resource for Financial Market Technicians by Charles D. Kirkpatrick and Julie R. Dahlquist Chapter 14 ^ a b Baiynd, Anne-Marie (2011). The Trading Book: A Complete Solution to Mastering Technical Systems and Trading Psychology . McGraw-Hill . p. 272. ISBN 9780071766494 . Archived from the original on 2012-03-25 . Retrieved 2013-04-30 . ^ Lento, C.; Gradojevic, N.; Wright, C. S. (2007). "Investment information content in Bollinger Bands?". Applied Financial Economics Letters . 3 (4): 263– 267. doi : 10.1080/17446540701206576 . ISSN 1744-6546 . S2CID 153603674 . ^ Balsara, Nauzer J.; Chen, Gary; Zheng, Lin (2007). "The Chinese Stock Market: An Examination of the Random Walk Model and Technical Trading Rules". Quarterly Journal of Business and Economics . 46 (2): 43– 63. JSTOR 40473435 . ^ Lim, Shawn; Hisarli, Tilman; Shi He, Ng (2014). "The Profitability of a Combined Signal Approach: Bollinger Bands and the ADX". International Federation of Technical Analysts Journal : 23– 29. SSRN 2230499 . ^ Butler, Matthew; Kazakov, Dimitar (2010). "Particle Swarm Optimization of Bollinger Bands". In Dorigo, Marco; Birattari, Mauro; Caro, Gianni A. Di; Doursat, René; Engelbrecht, Andries P.; Floreano, Dario; Gambardella, Luca Maria; Groß, Roderich; Sahin, Erol; Sayama, Hiroki; Stützle, Thomas (eds.). Swarm Intelligence - 7th International Conference, ANTS 2010, Brussels, Belgium, September 8-10, 2010. Proceedings . Lecture Notes in Computer Science. Vol. 6234. Springer. pp. 504– 511. doi : 10.1007/978-3-642-15461-4_50 . ISBN 978-3-642-15460-7 . ^ Rachev; Svetlozar T., Menn, Christian; Fabozzi, Frank J. (2005), Fat Tailed and Skewed Asset Return Distributions, Implications for Risk Management, Portfolio Selection, and Option Pricing, John Wiley, New York ^ Adam Grimes (2012). The Art & Science of Technical Analysis: Market Structure, Price Action & Trading Strategies . John Wiley & Sons. pp. 196– 198. ISBN 9781118224274 . ^ Pang, Grantham K. H. (2006-08-01). "Novel method for patterned fabric inspection using Bollinger bands". Optical Engineering . 45 (8): 087202. Bibcode : 2006OptEn..45h7202N . doi : 10.1117/1.2345189 . hdl : 10722/44829 . ISSN 0091-3286 . ^ "ICAO Methodology for Accident Rate Calculation and Trending - SKYbrary Aviation Safety" . SKYbrary . Retrieved 2019-03-12 . ^ John., Bollinger (2002). Bollinger on Bollinger bands . New York: McGraw-Hill. ISBN 0071373683 . OCLC 46634029 . ^ Pagel, Christina (2015-11-16). "A Novel Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real Time" (PDF) . Pediatr Crit Care Med . 16 (9): 821– 827. doi : 10.1097/PCC.0000000000000540 . PMID 26536545 . S2CID 41502207 . Achelis, Steve. Technical Analysis from A to Z (pp. 71–73). Irwin, 1995. ISBN 978-0-07-136348-8 Bollinger, John. Bollinger on Bollinger Bands . McGraw Hill, 2002. ISBN 978-0-07-137368-5 Cahen, Philippe. Dynamic Technical Analysis . Wiley, 2001. ISBN 978-0-471-89947-1 Kirkpatrick, Charles D. II; Dahlquist, Julie R. Technical Analysis: The Complete Resource for Financial Market Technicians , FT Press, 2006. ISBN 0-13-153113-1 Murphy, John J. Technical Analysis of the Financial Markets (pp. 209–211). New York Institute of Finance, 1999. ISBN 0-7352-0066-1 John Bollinger's website John Bollinger's website for Bollinger Band analysis
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[Printable version](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&printable=yes "Printable version of this page [p]") In other projects - [Wikimedia Commons](https://commons.wikimedia.org/wiki/Category:Bollinger_Bands) - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q891459 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Statistical price volatility chart [![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/1a/BollingerBandsSPX.svg/500px-BollingerBandsSPX.svg.png)](https://en.wikipedia.org/wiki/File:BollingerBandsSPX.svg) S\&P 500 with 20-day, two-standard-deviation Bollinger Bands, %b and bandwidth Bollinger Bands ([/ˈbɒlɪndʒər/](https://en.wikipedia.org/wiki/Help:IPA/English "Help:IPA/English")) are a type of [statistical](https://en.wikipedia.org/wiki/Statistical "Statistical") [chart](https://en.wikipedia.org/wiki/Chart "Chart") characterizing the prices and [volatility](https://en.wikipedia.org/wiki/Volatility_\(finance\) "Volatility (finance)") over time of a financial instrument or commodity, using a [formulaic](https://en.wikipedia.org/wiki/Formula "Formula") method propounded by [John Bollinger](https://en.wikipedia.org/wiki/John_Bollinger "John Bollinger") in the 1980s. Financial traders employ these charts as a methodical tool to inform trading decisions, control [automated trading systems](https://en.wikipedia.org/wiki/Automated_trading_system "Automated trading system"), or as a component of [technical analysis](https://en.wikipedia.org/wiki/Technical_analysis "Technical analysis"). Bollinger Bands display a graphical band (the [envelope](https://en.wikipedia.org/wiki/Envelope_\(mathematics\) "Envelope (mathematics)") maximum and minimum of [moving averages](https://en.wikipedia.org/wiki/Moving_average_\(finance\) "Moving average (finance)"), similar to [Keltner](https://en.wikipedia.org/wiki/Keltner_channel "Keltner channel") or [Donchian channels](https://en.wikipedia.org/wiki/Donchian_channel "Donchian channel")) and volatility (expressed by the width of the envelope) in one two-dimensional chart. Two input parameters chosen independently by the user govern how a given chart summarizes the known historical price data, allowing the user to vary the response of the chart to the magnitude and frequency of price changes, similar to [parametric equations](https://en.wikipedia.org/wiki/Parametric_equation "Parametric equation") in [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") or [control systems](https://en.wikipedia.org/wiki/Control_system "Control system"). Bollinger Bands consist of an [N](https://en.wikipedia.org/wiki/Variable_\(mathematics\)#Naming_conventions "Variable (mathematics)")\-period [moving average](https://en.wikipedia.org/wiki/Moving_average_\(finance\) "Moving average (finance)") (MA), an upper band at K times an N\-period [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") above the moving average (MA + Kσ), and a lower band at K times an N\-period [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") below the moving average (MA − Kσ). The chart thus expresses arbitrary choices or assumptions of the user, and is not strictly about the price data alone. Typical values for N and K are 20 days and 2, respectively. The default choice for the average is a simple [moving average](https://en.wikipedia.org/wiki/Moving_average "Moving average"), but other types of averages can be employed as needed. [Exponential moving averages](https://en.wikipedia.org/wiki/Moving_average_\(finance\)#Exponential_moving_average "Moving average (finance)") are a common second choice.[\[note 1\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-2) Usually the same period is used for both the middle band and the calculation of standard deviation.[\[note 2\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-3) Bollinger registered the words "Bollinger Bands" as a U.S. [trademark](https://en.wikipedia.org/wiki/Trademark "Trademark") in 2011.[\[2\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-4) ## Purpose \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=1 "Edit section: Purpose")\] The purpose of Bollinger Bands is to provide a relative definition of high and low prices of a market. By definition, prices are high at the upper band and low at the lower band. This definition can aid in rigorous [pattern recognition](https://en.wikipedia.org/wiki/Pattern_recognition "Pattern recognition") and is useful in comparing price action to the action of indicators to arrive at [systematic trading](https://en.wikipedia.org/wiki/Systematic_trading "Systematic trading") decisions.[\[3\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-5) ## Indicators derived from Bollinger Bands \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=2 "Edit section: Indicators derived from Bollinger Bands")\] BBImpulse measures price change as a function of the bands; percent bandwidth (%b) normalizes the width of the bands over time; and bandwidth delta quantifies the changing width of the bands. %b (pronounced "percent b") is derived from the formula for [stochastics](https://en.wikipedia.org/wiki/Stochastic_oscillator "Stochastic oscillator") and shows where price is in relation to the bands. %b equals 1 at the upper band and 0 at the lower band. Writing upperBB for the upper Bollinger Band, lowerBB for the lower Bollinger Band, and last for the last (price) value: %b = (last − lowerBB) / (upperBB − lowerBB) Bandwidth tells how wide the Bollinger Bands are on a normalized basis. Writing the same symbols as before, and middleBB for the moving average, or middle Bollinger Band: Bandwidth = (upperBB − lowerBB) / middleBB Using the default parameters of a 20-period look back and plus/minus two standard deviations, bandwidth is equal to four times the 20-period [coefficient of variation](https://en.wikipedia.org/wiki/Coefficient_of_variation "Coefficient of variation"). Uses for %b include system building and [pattern recognition](https://en.wikipedia.org/wiki/Pattern_recognition "Pattern recognition"). Uses for bandwidth include identification of opportunities arising from relative extremes in [volatility](https://en.wikipedia.org/wiki/Volatility_\(finance\) "Volatility (finance)") and trend identification. ## Interpretation \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=3 "Edit section: Interpretation")\] The use of Bollinger Bands varies widely among traders. Some traders buy when price touches the lower Bollinger Band and exit when price touches the moving average in the center of the bands. Other traders buy when price breaks above the upper Bollinger Band or sell when price falls below the lower Bollinger Band.[\[4\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-6) Moreover, the use of Bollinger Bands is not confined to stock traders; [options](https://en.wikipedia.org/wiki/Option_\(finance\) "Option (finance)") traders, most notably [implied volatility](https://en.wikipedia.org/wiki/Implied_volatility "Implied volatility") traders, often sell options when Bollinger Bands are historically far apart or buy options when the Bollinger Bands are historically close together, in both instances, expecting volatility to revert towards the average historical volatility level for the stock. When the bands lie close together, a period of low [volatility](https://en.wikipedia.org/wiki/Volatility_\(finance\) "Volatility (finance)") is indicated.[\[5\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-Baiynd2011-7) Conversely, as the bands expand, an increase in price action/market volatility is indicated.[\[5\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-Baiynd2011-7) When the bands have only a slight slope and track approximately parallel for an extended time, the price will generally be found to oscillate between the bands as though in a channel. Traders are often inclined to use Bollinger Bands with other indicators to confirm price action. In particular, the use of oscillator-like Bollinger Bands will often be coupled with a non-oscillator indicator-like [chart patterns](https://en.wikipedia.org/wiki/Chart_patterns "Chart patterns") or a [trendline](https://en.wikipedia.org/wiki/Trend_line_\(technical_analysis\) "Trend line (technical analysis)"). If these indicators confirm the recommendation of the Bollinger Bands, the trader will have greater conviction that the bands are predicting correct price action in relation to market volatility. ## Effectiveness \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=4 "Edit section: Effectiveness")\] Various studies of the effectiveness of the Bollinger Band strategy have been performed with mixed results. In 2007, Lento et al. published an analysis using a variety of formats (different moving average timescales, and standard deviation ranges) and markets (e.g., [Dow Jones](https://en.wikipedia.org/wiki/Dow_Jones_%26_Company "Dow Jones & Company") and [Forex](https://en.wikipedia.org/wiki/Foreign_exchange_market "Foreign exchange market")).[\[6\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-LentoGradojevic2007-8) Analysis of the trades, spanning a decade from 1995 onwards, found no evidence of consistent performance over the standard "[buy and hold](https://en.wikipedia.org/wiki/Buy_and_hold "Buy and hold")" approach. The authors did, however, find that a simple reversal of the strategy ("contrarian Bollinger Band") produced positive returns in a variety of markets. Similar results were found in another study, which concluded that Bollinger Band trading strategies may be effective in the Chinese marketplace, stating: "we find significant positive returns on buy trades generated by the contrarian version of the [moving-average crossover](https://en.wikipedia.org/wiki/Moving-average_crossover "Moving-average crossover") rule, the channel breakout rule, and the Bollinger Band trading rule, after accounting for transaction costs of 0.50 percent."[\[7\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-9) (By "the contrarian version", they mean buying when the conventional rule mandates selling, and vice versa.) A recent study examined the application of Bollinger Band trading strategies combined with the [ADX](https://en.wikipedia.org/wiki/Average_directional_movement_index "Average directional movement index") for [Equity Market](https://en.wikipedia.org/wiki/Stock_market "Stock market") indices with similar results.[\[8\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-10) In 2012, Butler et al. published an approach to fitting the parameters of Bollinger Bands using [particle swarm optimization](https://en.wikipedia.org/wiki/Particle_swarm_optimization "Particle swarm optimization") method. Their results indicated that by tuning the parameters to a particular asset for a particular market environment, the [out-of-sample](https://en.wikipedia.org/wiki/Cross-validation_\(statistics\) "Cross-validation (statistics)") trading signals were improved compared to the default parameters.[\[9\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-ButlerKazakov2012-11) ## Statistical properties \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=5 "Edit section: Statistical properties")\] Security price returns have no known [statistical distribution](https://en.wikipedia.org/wiki/Statistical_distribution "Statistical distribution"), [normal](https://en.wikipedia.org/wiki/Normally_distributed "Normally distributed") or otherwise; they are known to have [fat tails](https://en.wikipedia.org/wiki/Fat_tail "Fat tail"), compared to a normal distribution.[\[10\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-12) The [sample size](https://en.wikipedia.org/wiki/Sample_size "Sample size") typically used, 20, is too small for conclusions derived from statistical techniques like the [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem "Central limit theorem") to be reliable. Such techniques usually require the sample to be independent and identically distributed, which is not the case for a [time series](https://en.wikipedia.org/wiki/Time_series "Time series") like security prices. Just the opposite is true; it is well recognized by practitioners that such price series are very commonly serially correlated\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\]—that is, each price will be closely related to its ancestor "most of the time". Adjusting for [serial correlation](https://en.wikipedia.org/wiki/Serial_correlation "Serial correlation") is the purpose of moving [standard deviations](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation"), which use deviations from the [moving average](https://en.wikipedia.org/wiki/Moving_average "Moving average"), but the possibility remains of high order price [autocorrelation](https://en.wikipedia.org/wiki/Autocorrelation "Autocorrelation") not accounted for by simple differencing from the moving average. For such reasons, it is incorrect to assume that the long-term percentage of the data that will be observed in the future outside the Bollinger Bands range will always be constrained to a certain amount. Instead of finding about 95% of the data inside the bands, as would be the expectation with the default parameters if the data were normally distributed, studies have found that only about 88% of security prices (85–90%) remain within the bands.[\[11\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-grimes-13) For an individual security, one can always find factors for which certain percentages of data are contained by the factor defined bands for a certain period of time. Practitioners may also use related measures such as the [Keltner channels](https://en.wikipedia.org/wiki/Keltner_channel "Keltner channel"), or the related Stoller average range channels, which base their band widths on different measures of price volatility, such as the difference between daily high and low prices, rather than on standard deviation. ## Bollinger Bands outside of finance \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=6 "Edit section: Bollinger Bands outside of finance")\] Bollinger Bands have been applied to manufacturing data to detect defects (anomalies) in patterned fabrics.[\[12\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-14) In this application, the upper and lower bands of Bollinger Bands are sensitive to subtle changes in the input data obtained from samples. The [International Civil Aviation Organization](https://en.wikipedia.org/wiki/International_Civil_Aviation_Organization "International Civil Aviation Organization") used Bollinger Bands to measure the accident rate as a safety indicator to measure efficacy of global safety initiatives.[\[13\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-15) %b and bandwidth are also used in this analysis.[\[14\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-16) Bollinger Bands have been applied to a "Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real-Time."[\[15\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-17) ## Notes \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=7 "Edit section: Notes")\] 1. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-2) When the average used in the calculation of Bollinger Bands is changed from a simple moving average to an exponential or weighted moving average, it must be changed for both the calculation of the middle band and the calculation of standard deviation.[\[1\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-1) 2. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-3) Since Bollinger Bands use the population method of calculating standard deviation, the proper divisor for the sigma calculation is n, not n − 1. ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=8 "Edit section: References")\] 1. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-1) Bollinger On Bollinger Bands – The Seminar, DVD I [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-9726111-0-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-9726111-0-7 "Special:BookSources/978-0-9726111-0-7") 2. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-4) ["Bollinger Bands – Trademark Details"](http://trademarks.justia.com/852/32/bollinger-bands-85232573.html). Justia.com. 2011-12-20. 3. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-5) [\[1\]](http://www.bollingerbands.com/) second paragraph, center column 4. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-6) Technical Analysis: The Complete Resource for Financial Market Technicians by Charles D. Kirkpatrick and Julie R. Dahlquist Chapter 14 5. ^ [*a](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-Baiynd2011_7-0) [b](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-Baiynd2011_7-1) [Baiynd, Anne-Marie](https://en.wikipedia.org/wiki/Anne-Marie_Baiynd "Anne-Marie Baiynd") (2011). [The Trading Book: A Complete Solution to Mastering Technical Systems and Trading Psychology](https://web.archive.org/web/20120325050543/http://mcgraw-hill.com.au/html/9780071766494.html). [McGraw-Hill](https://en.wikipedia.org/wiki/McGraw-Hill "McGraw-Hill"). p. 272. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9780071766494](https://en.wikipedia.org/wiki/Special:BookSources/9780071766494 "Special:BookSources/9780071766494") . Archived from [the original](http://www.mcgraw-hill.com.au/html/9780071766494.html) on 2012-03-25. Retrieved 2013-04-30. 6. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-LentoGradojevic2007_8-0)* Lento, C.; Gradojevic, N.; Wright, C. S. (2007). "Investment information content in Bollinger Bands?". Applied Financial Economics Letters. 3 (4): 263–267\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/17446540701206576](https://doi.org/10.1080%2F17446540701206576). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1744-6546](https://search.worldcat.org/issn/1744-6546). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [153603674](https://api.semanticscholar.org/CorpusID:153603674). 7. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-9) Balsara, Nauzer J.; Chen, Gary; Zheng, Lin (2007). "The Chinese Stock Market: An Examination of the Random Walk Model and Technical Trading Rules". Quarterly Journal of Business and Economics. 46 (2): 43–63\. [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [40473435](https://www.jstor.org/stable/40473435). 8. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-10) Lim, Shawn; Hisarli, Tilman; Shi He, Ng (2014). "The Profitability of a Combined Signal Approach: Bollinger Bands and the ADX". International Federation of Technical Analysts Journal: 23–29\. [SSRN](https://en.wikipedia.org/wiki/SSRN_\(identifier\) "SSRN (identifier)") [2230499](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2230499). 9. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-ButlerKazakov2012_11-0) Butler, Matthew; Kazakov, Dimitar (2010). "Particle Swarm Optimization of Bollinger Bands". In Dorigo, Marco; Birattari, Mauro; Caro, Gianni A. Di; Doursat, René; Engelbrecht, Andries P.; Floreano, Dario; Gambardella, Luca Maria; Groß, Roderich; Sahin, Erol; Sayama, Hiroki; Stützle, Thomas (eds.). Swarm Intelligence - 7th International Conference, ANTS 2010, Brussels, Belgium, September 8-10, 2010. Proceedings. Lecture Notes in Computer Science. Vol. 6234. Springer. pp. 504–511\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-642-15461-4\_50](https://doi.org/10.1007%2F978-3-642-15461-4_50). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-642-15460-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-15460-7 "Special:BookSources/978-3-642-15460-7") . 10. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-12) Rachev; Svetlozar T., Menn, Christian; Fabozzi, Frank J. (2005), Fat Tailed and Skewed Asset Return Distributions, Implications for Risk Management, Portfolio Selection, and Option Pricing, John Wiley, New York 11. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-grimes_13-0) Adam Grimes (2012). [The Art & Science of Technical Analysis: Market Structure, Price Action & Trading Strategies](https://books.google.com/books?id=Yzs_0ZkXnyAC&pg=PA196). John Wiley & Sons. pp. 196–198\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781118224274](https://en.wikipedia.org/wiki/Special:BookSources/9781118224274 "Special:BookSources/9781118224274") . 12. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-14) Pang, Grantham K. H. (2006-08-01). "Novel method for patterned fabric inspection using Bollinger bands". Optical Engineering. 45 (8): 087202. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2006OptEn..45h7202N](https://ui.adsabs.harvard.edu/abs/2006OptEn..45h7202N). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1117/1.2345189](https://doi.org/10.1117%2F1.2345189). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10722/44829](https://hdl.handle.net/10722%2F44829). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0091-3286](https://search.worldcat.org/issn/0091-3286). 13. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-15) ["ICAO Methodology for Accident Rate Calculation and Trending - SKYbrary Aviation Safety"](https://www.skybrary.aero/index.php/ICAO_Methodology_for_Accident_Rate_Calculation_and_Trending). [SKYbrary](https://en.wikipedia.org/wiki/SKYbrary "SKYbrary"). Retrieved 2019-03-12. 14. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-16) John., Bollinger (2002). [Bollinger on Bollinger bands](https://archive.org/details/bollingeronbolli00john). New York: McGraw-Hill. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0071373683](https://en.wikipedia.org/wiki/Special:BookSources/0071373683 "Special:BookSources/0071373683") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [46634029](https://search.worldcat.org/oclc/46634029). 15. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-17) Pagel, Christina (2015-11-16). ["A Novel Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real Time"](https://discovery.ucl.ac.uk/1468466/3/PCCM%20accepted%20April%202015.pdf) (PDF). Pediatr Crit Care Med. 16 (9): 821–827\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1097/PCC.0000000000000540](https://doi.org/10.1097%2FPCC.0000000000000540). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [26536545](https://pubmed.ncbi.nlm.nih.gov/26536545). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [41502207](https://api.semanticscholar.org/CorpusID:41502207). ## Further reading \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=9 "Edit section: Further reading")\] - Achelis, Steve. Technical Analysis from A to Z (pp. 71–73). Irwin, 1995. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-07-136348-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-136348-8 "Special:BookSources/978-0-07-136348-8") - Bollinger, John. Bollinger on Bollinger Bands. McGraw Hill, 2002. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-07-137368-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-137368-5 "Special:BookSources/978-0-07-137368-5") - Cahen, Philippe. Dynamic Technical Analysis. Wiley, 2001. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-89947-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-89947-1 "Special:BookSources/978-0-471-89947-1") - Kirkpatrick, Charles D. II; Dahlquist, Julie R. Technical Analysis: The Complete Resource for Financial Market Technicians, FT Press, 2006. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-13-153113-1](https://en.wikipedia.org/wiki/Special:BookSources/0-13-153113-1 "Special:BookSources/0-13-153113-1") - Murphy, John J. Technical Analysis of the Financial Markets (pp. 209–211). New York Institute of Finance, 1999. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-7352-0066-1](https://en.wikipedia.org/wiki/Special:BookSources/0-7352-0066-1 "Special:BookSources/0-7352-0066-1") ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=10 "Edit section: External links")\] - [John Bollinger's website](https://www.bollingerbands.com/) - [John Bollinger's website for Bollinger Band analysis](https://bollingerbands.us/) \| [v](https://en.wikipedia.org/wiki/Template:Technical_analysis "Template:Technical analysis") [t](https://en.wikipedia.org/wiki/Template_talk:Technical_analysis "Template talk:Technical analysis") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Technical_analysis "Special:EditPage/Template:Technical analysis")[Technical analysis](https://en.wikipedia.org/wiki/Technical_analysis "Technical analysis") \| \| \|---\|---\| \| Concepts \| [Breakout](https://en.wikipedia.org/wiki/Breakout_\(technical_analysis\) "Breakout (technical analysis)") [Dead cat bounce](https://en.wikipedia.org/wiki/Dead_cat_bounce "Dead cat bounce") [Dow theory](https://en.wikipedia.org/wiki/Dow_theory "Dow theory") [Elliott wave principle](https://en.wikipedia.org/wiki/Elliott_wave_principle "Elliott wave principle") [Market trend](https://en.wikipedia.org/wiki/Market_trend "Market trend") \| \| Charts \| [Candlestick](https://en.wikipedia.org/wiki/Candlestick_chart "Candlestick chart") [Renko](https://en.wikipedia.org/wiki/Renko_chart "Renko chart") [Kagi](https://en.wikipedia.org/wiki/Kagi_chart "Kagi chart") [Line](https://en.wikipedia.org/wiki/Line_chart "Line chart") [Open-high-low-close](https://en.wikipedia.org/wiki/Open-high-low-close_chart "Open-high-low-close chart") [Point and figure](https://en.wikipedia.org/wiki/Point_and_figure_chart "Point and figure chart") [Line break](https://en.wikipedia.org/wiki/Line_break_chart "Line break chart") \| \| Patterns \| \| \| \| \| \| [Chart](https://en.wikipedia.org/wiki/Chart_pattern "Chart pattern") \| [Broadening top](https://en.wikipedia.org/wiki/Broadening_top "Broadening top") [Cup and handle](https://en.wikipedia.org/wiki/Cup_and_handle "Cup and handle") [Double top and double bottom](https://en.wikipedia.org/wiki/Double_top_and_double_bottom "Double top and double bottom") [Flag and pennant](https://en.wikipedia.org/wiki/Flag_and_pennant_patterns "Flag and pennant patterns") [Gap](https://en.wikipedia.org/wiki/Gap_\(chart_pattern\) "Gap (chart pattern)") [Head and shoulders](https://en.wikipedia.org/wiki/Head_and_shoulders_\(chart_pattern\) "Head and shoulders (chart pattern)") [Island reversal](https://en.wikipedia.org/wiki/Island_reversal "Island reversal") [Price channels](https://en.wikipedia.org/wiki/Price_channels "Price channels") [Triangle](https://en.wikipedia.org/wiki/Triangle_\(chart_pattern\) "Triangle (chart pattern)") [Triple top and triple bottom](https://en.wikipedia.org/wiki/Triple_top_and_triple_bottom "Triple top and triple bottom") [Wedge pattern](https://en.wikipedia.org/wiki/Wedge_pattern "Wedge pattern") \| \| [Candlestick](https://en.wikipedia.org/wiki/Candlestick_pattern "Candlestick pattern") \| \| \| \| \| \| [Simple](https://en.wikipedia.org/wiki/Candlestick_pattern#Simple_patterns "Candlestick pattern") \| [Doji](https://en.wikipedia.org/wiki/Doji "Doji") \| \| [Complex](https://en.wikipedia.org/wiki/Candlestick_pattern#Complex_patterns "Candlestick pattern") \| [Hikkake pattern](https://en.wikipedia.org/wiki/Hikkake_pattern "Hikkake pattern") [Morning star](https://en.wikipedia.org/wiki/Morning_star_\(candlestick_pattern\) "Morning star (candlestick pattern)") [Three black crows](https://en.wikipedia.org/wiki/Three_black_crows "Three black crows") [Three white soldiers](https://en.wikipedia.org/wiki/Three_white_soldiers "Three white soldiers") \| \| [Point and figure](https://en.wikipedia.org/wiki/Point_and_figure_chart "Point and figure chart") \| [Bull trap](https://en.wikipedia.org/wiki/Bull_trap "Bull trap") [Bear trap](https://en.wikipedia.org/wiki/Bear_market "Bear market") \| \| [Indicators](https://en.wikipedia.org/wiki/Technical_indicator "Technical indicator") \| \| \| \| \| \| [Support & resistance](https://en.wikipedia.org/wiki/Support_and_resistance "Support and resistance") \| [Fibonacci retracement](https://en.wikipedia.org/wiki/Fibonacci_retracement "Fibonacci retracement") [Pivot point](https://en.wikipedia.org/wiki/Pivot_point_\(technical_analysis\) "Pivot point (technical analysis)") (PP) \| \| [Trend](https://en.wikipedia.org/wiki/Market_trend "Market trend") \| [Average directional index](https://en.wikipedia.org/wiki/Average_directional_movement_index "Average directional movement index") (A.D.X.) [Commodity channel index](https://en.wikipedia.org/wiki/Commodity_channel_index "Commodity channel index") (CCI) [Detrended price oscillator](https://en.wikipedia.org/wiki/Detrended_price_oscillator "Detrended price oscillator") (DPO) [Know sure thing oscillator](https://en.wikipedia.org/wiki/KST_oscillator "KST oscillator") (KST) [Ichimoku Kinkō Hyō](https://en.wikipedia.org/wiki/Ichimoku_Kink%C5%8D_Hy%C5%8D "Ichimoku Kinkō Hyō") [Moving average convergence/divergence](https://en.wikipedia.org/wiki/MACD "MACD") (MACD) [Mass index](https://en.wikipedia.org/wiki/Mass_index "Mass index") [Moving average](https://en.wikipedia.org/wiki/Moving_average "Moving average") (MA) [Parabolic SAR](https://en.wikipedia.org/wiki/Parabolic_SAR "Parabolic SAR") (SAR) [Smart money index](https://en.wikipedia.org/wiki/Smart_money_index "Smart money index") (SMI) [Trend line](https://en.wikipedia.org/wiki/Trend_line_\(technical_analysis\) "Trend line (technical analysis)") [Trix](https://en.wikipedia.org/wiki/Trix_\(technical_analysis\) "Trix (technical analysis)") [Vortex indicator](https://en.wikipedia.org/wiki/Vortex_indicator "Vortex indicator") (VI) \| \| [Momentum](https://en.wikipedia.org/wiki/Momentum_\(finance\) "Momentum (finance)") \| [Money flow index](https://en.wikipedia.org/wiki/Money_flow_index "Money flow index") (MFI) [Relative strength index](https://en.wikipedia.org/wiki/Relative_strength_index "Relative strength index") (RSI) [Stochastic oscillator](https://en.wikipedia.org/wiki/Stochastic_oscillator "Stochastic oscillator") [True strength index](https://en.wikipedia.org/wiki/True_strength_index "True strength index") (TSI) [Ultimate oscillator](https://en.wikipedia.org/wiki/Ultimate_oscillator "Ultimate oscillator") [Williams %R](https://en.wikipedia.org/wiki/Williams_%25R "Williams %R") (%R) \| \| [Volume](https://en.wikipedia.org/wiki/Volume_\(finance\) "Volume (finance)") \| [Accumulation/distribution line](https://en.wikipedia.org/wiki/Accumulation/distribution_index "Accumulation/distribution index") [Ease of movement](https://en.wikipedia.org/wiki/Ease_of_movement "Ease of movement") (EMV) [Force index](https://en.wikipedia.org/wiki/Force_index "Force index") (FI) [Negative volume index](https://en.wikipedia.org/wiki/Negative_volume_index "Negative volume index") (NVI) [On-balance volume](https://en.wikipedia.org/wiki/On-balance_volume "On-balance volume") (OBV) [Put/call ratio](https://en.wikipedia.org/wiki/Put/call_ratio "Put/call ratio") (PCR) [Volume–price trend](https://en.wikipedia.org/wiki/Volume%E2%80%93price_trend "Volume–price trend") (VPT) \| \| [Volatility](https://en.wikipedia.org/wiki/Volatility_\(finance\) "Volatility (finance)") \| [Average true range](https://en.wikipedia.org/wiki/Average_true_range "Average true range") (ATR) [Bollinger Bands]() (BB) [Donchian channel](https://en.wikipedia.org/wiki/Donchian_channel "Donchian channel") [Keltner channel](https://en.wikipedia.org/wiki/Keltner_channel "Keltner channel") [CBOE Market Volatility Index](https://en.wikipedia.org/wiki/VIX "VIX") (VIX) [Standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") (σ) \| \| [Breadth](https://en.wikipedia.org/wiki/Breadth_of_market "Breadth of market") \| [Advance–decline line](https://en.wikipedia.org/wiki/Advance%E2%80%93decline_line "Advance–decline line") (ADL) [Arms index](https://en.wikipedia.org/wiki/TRIN_\(finance\) "TRIN (finance)") (TRIN) [McClellan oscillator](https://en.wikipedia.org/wiki/McClellan_oscillator "McClellan oscillator") \| \| Other \| [Coppock curve](https://en.wikipedia.org/wiki/Coppock_curve "Coppock curve") [Ulcer index](https://en.wikipedia.org/wiki/Ulcer_index "Ulcer index") \| \| Analysts \| [John Bollinger](https://en.wikipedia.org/wiki/John_Bollinger "John Bollinger") [Ned Davis](https://en.wikipedia.org/wiki/Ned_Davis_\(analyst\) "Ned Davis (analyst)") [Charles Dow](https://en.wikipedia.org/wiki/Charles_Dow "Charles Dow") [Ralph Nelson Elliott](https://en.wikipedia.org/wiki/Ralph_Nelson_Elliott "Ralph Nelson Elliott") [Bob Farrell](https://en.wikipedia.org/wiki/Robert_Farrell_\(technical_analyst\) "Robert Farrell (technical analyst)") [John Murphy](https://en.wikipedia.org/wiki/John_Murphy_\(technical_analyst\) "John Murphy (technical analyst)") [Mark Hulbert](https://en.wikipedia.org/wiki/Mark_Hulbert "Mark Hulbert") \| ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1) Retrieved from "<https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&oldid=1341622285>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - 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Readable Markdown	From Wikipedia, the free encyclopedia [![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/1a/BollingerBandsSPX.svg/500px-BollingerBandsSPX.svg.png)](https://en.wikipedia.org/wiki/File:BollingerBandsSPX.svg) S\&P 500 with 20-day, two-standard-deviation Bollinger Bands, %b and bandwidth Bollinger Bands () are a type of [statistical](https://en.wikipedia.org/wiki/Statistical "Statistical") [chart](https://en.wikipedia.org/wiki/Chart "Chart") characterizing the prices and [volatility](https://en.wikipedia.org/wiki/Volatility_\(finance\) "Volatility (finance)") over time of a financial instrument or commodity, using a [formulaic](https://en.wikipedia.org/wiki/Formula "Formula") method propounded by [John Bollinger](https://en.wikipedia.org/wiki/John_Bollinger "John Bollinger") in the 1980s. Financial traders employ these charts as a methodical tool to inform trading decisions, control [automated trading systems](https://en.wikipedia.org/wiki/Automated_trading_system "Automated trading system"), or as a component of [technical analysis](https://en.wikipedia.org/wiki/Technical_analysis "Technical analysis"). Bollinger Bands display a graphical band (the [envelope](https://en.wikipedia.org/wiki/Envelope_\(mathematics\) "Envelope (mathematics)") maximum and minimum of [moving averages](https://en.wikipedia.org/wiki/Moving_average_\(finance\) "Moving average (finance)"), similar to [Keltner](https://en.wikipedia.org/wiki/Keltner_channel "Keltner channel") or [Donchian channels](https://en.wikipedia.org/wiki/Donchian_channel "Donchian channel")) and volatility (expressed by the width of the envelope) in one two-dimensional chart. Two input parameters chosen independently by the user govern how a given chart summarizes the known historical price data, allowing the user to vary the response of the chart to the magnitude and frequency of price changes, similar to [parametric equations](https://en.wikipedia.org/wiki/Parametric_equation "Parametric equation") in [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") or [control systems](https://en.wikipedia.org/wiki/Control_system "Control system"). Bollinger Bands consist of an [N](https://en.wikipedia.org/wiki/Variable_\(mathematics\)#Naming_conventions "Variable (mathematics)")\-period [moving average](https://en.wikipedia.org/wiki/Moving_average_\(finance\) "Moving average (finance)") (MA), an upper band at K times an N\-period [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") above the moving average (MA + Kσ), and a lower band at K times an N\-period [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") below the moving average (MA − Kσ). The chart thus expresses arbitrary choices or assumptions of the user, and is not strictly about the price data alone. Typical values for N and K are 20 days and 2, respectively. The default choice for the average is a simple [moving average](https://en.wikipedia.org/wiki/Moving_average "Moving average"), but other types of averages can be employed as needed. [Exponential moving averages](https://en.wikipedia.org/wiki/Moving_average_\(finance\)#Exponential_moving_average "Moving average (finance)") are a common second choice.[\[note 1\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-2) Usually the same period is used for both the middle band and the calculation of standard deviation.[\[note 2\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-3) Bollinger registered the words "Bollinger Bands" as a U.S. [trademark](https://en.wikipedia.org/wiki/Trademark "Trademark") in 2011.[\[2\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-4) The purpose of Bollinger Bands is to provide a relative definition of high and low prices of a market. By definition, prices are high at the upper band and low at the lower band. This definition can aid in rigorous [pattern recognition](https://en.wikipedia.org/wiki/Pattern_recognition "Pattern recognition") and is useful in comparing price action to the action of indicators to arrive at [systematic trading](https://en.wikipedia.org/wiki/Systematic_trading "Systematic trading") decisions.[\[3\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-5) ## Indicators derived from Bollinger Bands \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=2 "Edit section: Indicators derived from Bollinger Bands")\] BBImpulse measures price change as a function of the bands; percent bandwidth (%b) normalizes the width of the bands over time; and bandwidth delta quantifies the changing width of the bands. %b (pronounced "percent b") is derived from the formula for [stochastics](https://en.wikipedia.org/wiki/Stochastic_oscillator "Stochastic oscillator") and shows where price is in relation to the bands. %b equals 1 at the upper band and 0 at the lower band. Writing upperBB for the upper Bollinger Band, lowerBB for the lower Bollinger Band, and last for the last (price) value: %b = (last − lowerBB) / (upperBB − lowerBB) Bandwidth tells how wide the Bollinger Bands are on a normalized basis. Writing the same symbols as before, and middleBB for the moving average, or middle Bollinger Band: Bandwidth = (upperBB − lowerBB) / middleBB Using the default parameters of a 20-period look back and plus/minus two standard deviations, bandwidth is equal to four times the 20-period [coefficient of variation](https://en.wikipedia.org/wiki/Coefficient_of_variation "Coefficient of variation"). Uses for %b include system building and [pattern recognition](https://en.wikipedia.org/wiki/Pattern_recognition "Pattern recognition"). Uses for bandwidth include identification of opportunities arising from relative extremes in [volatility](https://en.wikipedia.org/wiki/Volatility_\(finance\) "Volatility (finance)") and trend identification. The use of Bollinger Bands varies widely among traders. Some traders buy when price touches the lower Bollinger Band and exit when price touches the moving average in the center of the bands. Other traders buy when price breaks above the upper Bollinger Band or sell when price falls below the lower Bollinger Band.[\[4\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-6) Moreover, the use of Bollinger Bands is not confined to stock traders; [options](https://en.wikipedia.org/wiki/Option_\(finance\) "Option (finance)") traders, most notably [implied volatility](https://en.wikipedia.org/wiki/Implied_volatility "Implied volatility") traders, often sell options when Bollinger Bands are historically far apart or buy options when the Bollinger Bands are historically close together, in both instances, expecting volatility to revert towards the average historical volatility level for the stock. When the bands lie close together, a period of low [volatility](https://en.wikipedia.org/wiki/Volatility_\(finance\) "Volatility (finance)") is indicated.[\[5\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-Baiynd2011-7) Conversely, as the bands expand, an increase in price action/market volatility is indicated.[\[5\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-Baiynd2011-7) When the bands have only a slight slope and track approximately parallel for an extended time, the price will generally be found to oscillate between the bands as though in a channel. Traders are often inclined to use Bollinger Bands with other indicators to confirm price action. In particular, the use of oscillator-like Bollinger Bands will often be coupled with a non-oscillator indicator-like [chart patterns](https://en.wikipedia.org/wiki/Chart_patterns "Chart patterns") or a [trendline](https://en.wikipedia.org/wiki/Trend_line_\(technical_analysis\) "Trend line (technical analysis)"). If these indicators confirm the recommendation of the Bollinger Bands, the trader will have greater conviction that the bands are predicting correct price action in relation to market volatility. Various studies of the effectiveness of the Bollinger Band strategy have been performed with mixed results. In 2007, Lento et al. published an analysis using a variety of formats (different moving average timescales, and standard deviation ranges) and markets (e.g., [Dow Jones](https://en.wikipedia.org/wiki/Dow_Jones_%26_Company "Dow Jones & Company") and [Forex](https://en.wikipedia.org/wiki/Foreign_exchange_market "Foreign exchange market")).[\[6\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-LentoGradojevic2007-8) Analysis of the trades, spanning a decade from 1995 onwards, found no evidence of consistent performance over the standard "[buy and hold](https://en.wikipedia.org/wiki/Buy_and_hold "Buy and hold")" approach. The authors did, however, find that a simple reversal of the strategy ("contrarian Bollinger Band") produced positive returns in a variety of markets. Similar results were found in another study, which concluded that Bollinger Band trading strategies may be effective in the Chinese marketplace, stating: "we find significant positive returns on buy trades generated by the contrarian version of the [moving-average crossover](https://en.wikipedia.org/wiki/Moving-average_crossover "Moving-average crossover") rule, the channel breakout rule, and the Bollinger Band trading rule, after accounting for transaction costs of 0.50 percent."[\[7\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-9) (By "the contrarian version", they mean buying when the conventional rule mandates selling, and vice versa.) A recent study examined the application of Bollinger Band trading strategies combined with the [ADX](https://en.wikipedia.org/wiki/Average_directional_movement_index "Average directional movement index") for [Equity Market](https://en.wikipedia.org/wiki/Stock_market "Stock market") indices with similar results.[\[8\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-10) In 2012, Butler et al. published an approach to fitting the parameters of Bollinger Bands using [particle swarm optimization](https://en.wikipedia.org/wiki/Particle_swarm_optimization "Particle swarm optimization") method. Their results indicated that by tuning the parameters to a particular asset for a particular market environment, the [out-of-sample](https://en.wikipedia.org/wiki/Cross-validation_\(statistics\) "Cross-validation (statistics)") trading signals were improved compared to the default parameters.[\[9\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-ButlerKazakov2012-11) ## Statistical properties \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=5 "Edit section: Statistical properties")\] Security price returns have no known [statistical distribution](https://en.wikipedia.org/wiki/Statistical_distribution "Statistical distribution"), [normal](https://en.wikipedia.org/wiki/Normally_distributed "Normally distributed") or otherwise; they are known to have [fat tails](https://en.wikipedia.org/wiki/Fat_tail "Fat tail"), compared to a normal distribution.[\[10\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-12) The [sample size](https://en.wikipedia.org/wiki/Sample_size "Sample size") typically used, 20, is too small for conclusions derived from statistical techniques like the [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem "Central limit theorem") to be reliable. Such techniques usually require the sample to be independent and identically distributed, which is not the case for a [time series](https://en.wikipedia.org/wiki/Time_series "Time series") like security prices. Just the opposite is true; it is well recognized by practitioners that such price series are very commonly serially correlated\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\]—that is, each price will be closely related to its ancestor "most of the time". Adjusting for [serial correlation](https://en.wikipedia.org/wiki/Serial_correlation "Serial correlation") is the purpose of moving [standard deviations](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation"), which use deviations from the [moving average](https://en.wikipedia.org/wiki/Moving_average "Moving average"), but the possibility remains of high order price [autocorrelation](https://en.wikipedia.org/wiki/Autocorrelation "Autocorrelation") not accounted for by simple differencing from the moving average. For such reasons, it is incorrect to assume that the long-term percentage of the data that will be observed in the future outside the Bollinger Bands range will always be constrained to a certain amount. Instead of finding about 95% of the data inside the bands, as would be the expectation with the default parameters if the data were normally distributed, studies have found that only about 88% of security prices (85–90%) remain within the bands.[\[11\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-grimes-13) For an individual security, one can always find factors for which certain percentages of data are contained by the factor defined bands for a certain period of time. Practitioners may also use related measures such as the [Keltner channels](https://en.wikipedia.org/wiki/Keltner_channel "Keltner channel"), or the related Stoller average range channels, which base their band widths on different measures of price volatility, such as the difference between daily high and low prices, rather than on standard deviation. ## Bollinger Bands outside of finance \[[edit](https://en.wikipedia.org/w/index.php?title=Bollinger_Bands&action=edit&section=6 "Edit section: Bollinger Bands outside of finance")\] Bollinger Bands have been applied to manufacturing data to detect defects (anomalies) in patterned fabrics.[\[12\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-14) In this application, the upper and lower bands of Bollinger Bands are sensitive to subtle changes in the input data obtained from samples. The [International Civil Aviation Organization](https://en.wikipedia.org/wiki/International_Civil_Aviation_Organization "International Civil Aviation Organization") used Bollinger Bands to measure the accident rate as a safety indicator to measure efficacy of global safety initiatives.[\[13\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-15) %b and bandwidth are also used in this analysis.[\[14\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-16) Bollinger Bands have been applied to a "Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real-Time."[\[15\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-17) 1. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-2) When the average used in the calculation of Bollinger Bands is changed from a simple moving average to an exponential or weighted moving average, it must be changed for both the calculation of the middle band and the calculation of standard deviation.[\[1\]](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_note-1) 2. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-3) Since Bollinger Bands use the population method of calculating standard deviation, the proper divisor for the sigma calculation is n, not n − 1. 1. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-1) Bollinger On Bollinger Bands – The Seminar, DVD I [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-9726111-0-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-9726111-0-7 "Special:BookSources/978-0-9726111-0-7") 2. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-4) ["Bollinger Bands – Trademark Details"](http://trademarks.justia.com/852/32/bollinger-bands-85232573.html). Justia.com. 2011-12-20. 3. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-5) [\[1\]](http://www.bollingerbands.com/) second paragraph, center column 4. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-6) Technical Analysis: The Complete Resource for Financial Market Technicians by Charles D. Kirkpatrick and Julie R. Dahlquist Chapter 14 5. ^ [*a](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-Baiynd2011_7-0) [b](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-Baiynd2011_7-1) [Baiynd, Anne-Marie](https://en.wikipedia.org/wiki/Anne-Marie_Baiynd "Anne-Marie Baiynd") (2011). [The Trading Book: A Complete Solution to Mastering Technical Systems and Trading Psychology](https://web.archive.org/web/20120325050543/http://mcgraw-hill.com.au/html/9780071766494.html). [McGraw-Hill](https://en.wikipedia.org/wiki/McGraw-Hill "McGraw-Hill"). p. 272. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9780071766494](https://en.wikipedia.org/wiki/Special:BookSources/9780071766494 "Special:BookSources/9780071766494") . Archived from [the original](http://www.mcgraw-hill.com.au/html/9780071766494.html) on 2012-03-25. Retrieved 2013-04-30. 6. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-LentoGradojevic2007_8-0)* Lento, C.; Gradojevic, N.; Wright, C. S. (2007). "Investment information content in Bollinger Bands?". Applied Financial Economics Letters. 3 (4): 263–267\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/17446540701206576](https://doi.org/10.1080%2F17446540701206576). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1744-6546](https://search.worldcat.org/issn/1744-6546). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [153603674](https://api.semanticscholar.org/CorpusID:153603674). 7. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-9) Balsara, Nauzer J.; Chen, Gary; Zheng, Lin (2007). "The Chinese Stock Market: An Examination of the Random Walk Model and Technical Trading Rules". Quarterly Journal of Business and Economics. 46 (2): 43–63\. [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [40473435](https://www.jstor.org/stable/40473435). 8. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-10) Lim, Shawn; Hisarli, Tilman; Shi He, Ng (2014). "The Profitability of a Combined Signal Approach: Bollinger Bands and the ADX". International Federation of Technical Analysts Journal: 23–29\. [SSRN](https://en.wikipedia.org/wiki/SSRN_\(identifier\) "SSRN (identifier)") [2230499](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2230499). 9. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-ButlerKazakov2012_11-0) Butler, Matthew; Kazakov, Dimitar (2010). "Particle Swarm Optimization of Bollinger Bands". In Dorigo, Marco; Birattari, Mauro; Caro, Gianni A. Di; Doursat, René; Engelbrecht, Andries P.; Floreano, Dario; Gambardella, Luca Maria; Groß, Roderich; Sahin, Erol; Sayama, Hiroki; Stützle, Thomas (eds.). Swarm Intelligence - 7th International Conference, ANTS 2010, Brussels, Belgium, September 8-10, 2010. Proceedings. Lecture Notes in Computer Science. Vol. 6234. Springer. pp. 504–511\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-642-15461-4\_50](https://doi.org/10.1007%2F978-3-642-15461-4_50). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-642-15460-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-15460-7 "Special:BookSources/978-3-642-15460-7") . 10. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-12) Rachev; Svetlozar T., Menn, Christian; Fabozzi, Frank J. (2005), Fat Tailed and Skewed Asset Return Distributions, Implications for Risk Management, Portfolio Selection, and Option Pricing, John Wiley, New York 11. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-grimes_13-0) Adam Grimes (2012). [The Art & Science of Technical Analysis: Market Structure, Price Action & Trading Strategies](https://books.google.com/books?id=Yzs_0ZkXnyAC&pg=PA196). John Wiley & Sons. pp. 196–198\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781118224274](https://en.wikipedia.org/wiki/Special:BookSources/9781118224274 "Special:BookSources/9781118224274") . 12. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-14) Pang, Grantham K. H. (2006-08-01). "Novel method for patterned fabric inspection using Bollinger bands". Optical Engineering. 45 (8): 087202. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2006OptEn..45h7202N](https://ui.adsabs.harvard.edu/abs/2006OptEn..45h7202N). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1117/1.2345189](https://doi.org/10.1117%2F1.2345189). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10722/44829](https://hdl.handle.net/10722%2F44829). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0091-3286](https://search.worldcat.org/issn/0091-3286). 13. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-15) ["ICAO Methodology for Accident Rate Calculation and Trending - SKYbrary Aviation Safety"](https://www.skybrary.aero/index.php/ICAO_Methodology_for_Accident_Rate_Calculation_and_Trending). [SKYbrary](https://en.wikipedia.org/wiki/SKYbrary "SKYbrary"). Retrieved 2019-03-12. 14. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-16) John., Bollinger (2002). [Bollinger on Bollinger bands](https://archive.org/details/bollingeronbolli00john). New York: McGraw-Hill. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0071373683](https://en.wikipedia.org/wiki/Special:BookSources/0071373683 "Special:BookSources/0071373683") . [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [46634029](https://search.worldcat.org/oclc/46634029). 15. [^](https://en.wikipedia.org/wiki/Bollinger_Bands#cite_ref-17) Pagel, Christina (2015-11-16). ["A Novel Method to Identify the Start and End of the Winter Surge in Demand for Pediatric Intensive Care in Real Time"](https://discovery.ucl.ac.uk/1468466/3/PCCM%20accepted%20April%202015.pdf) (PDF). Pediatr Crit Care Med. 16 (9): 821–827\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1097/PCC.0000000000000540](https://doi.org/10.1097%2FPCC.0000000000000540). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [26536545](https://pubmed.ncbi.nlm.nih.gov/26536545). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [41502207](https://api.semanticscholar.org/CorpusID:41502207). - Achelis, Steve. Technical Analysis from A to Z (pp. 71–73). Irwin, 1995. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-07-136348-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-136348-8 "Special:BookSources/978-0-07-136348-8") - Bollinger, John. Bollinger on Bollinger Bands. McGraw Hill, 2002. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-07-137368-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-137368-5 "Special:BookSources/978-0-07-137368-5") - Cahen, Philippe. Dynamic Technical Analysis. Wiley, 2001. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-89947-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-89947-1 "Special:BookSources/978-0-471-89947-1") - Kirkpatrick, Charles D. II; Dahlquist, Julie R. Technical Analysis: The Complete Resource for Financial Market Technicians, FT Press, 2006. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-13-153113-1](https://en.wikipedia.org/wiki/Special:BookSources/0-13-153113-1 "Special:BookSources/0-13-153113-1") - Murphy, John J. Technical Analysis of the Financial Markets (pp. 209–211). New York Institute of Finance, 1999. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-7352-0066-1](https://en.wikipedia.org/wiki/Special:BookSources/0-7352-0066-1 "Special:BookSources/0-7352-0066-1") - [John Bollinger's website](https://www.bollingerbands.com/) - [John Bollinger's website for Bollinger Band analysis](https://bollingerbands.us/)
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