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Calculated Shard: 152 (from laksa109)

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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/Volatility_(finance)\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Volatility_(finance)\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Volatility_(finance)","crawl_time":1775315876,"first_indexed_time":1375985279,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Volatility_(finance) s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Volatility (finance) - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"From Wikipedia, the free encyclopedia\nCBOE Volatility Index (VIX) from December 1985 to May 2012 (daily closings)\nIn\nfinance\n,\nvolatility\n(usually denoted by \"\nσ\n\") is the\ndegree of variation\nof a trading price series over time, usually measured by the\nstandard deviation\nof\nlogarithmic returns\n.\nHistoric volatility measures a time series of past market prices.\nImplied volatility\nlooks forward in time, being derived from the market price of a market-traded derivative (in particular, an option).\nVolatility terminology\n[\nedit\n]\nVolatility as described here refers to the\nactual volatility\n, more specifically:\nactual current volatility\nof a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price.\nactual historical volatility\nwhich refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past\nnear synonymous is\nrealized volatility\n, the\nsquare root\nof the\nrealized variance\n, in turn calculated using the sum of squared returns divided by the number of observations.\nactual future volatility\nwhich refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an\noption\n)\nNow turning to\nimplied volatility\n, we have:\nhistorical implied volatility\nwhich refers to the implied volatility observed from historical prices of the financial instrument (normally options)\ncurrent implied volatility\nwhich refers to the implied volatility observed from current prices of the financial instrument\nfuture implied volatility\nwhich refers to the implied volatility observed from future prices of the financial instrument\nFor a financial instrument whose price follows a\nGaussian\nrandom walk\n, or\nWiener process\n, the width of the distribution increases as time increases.  This is because there is an increasing\nprobability\nthat the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.\nSince observed price changes do not follow Gaussian distributions, others such as the\nLévy distribution\nare often used.\n[\n1\n]\nThese can capture attributes such as \"\nfat tails\n\". Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc.\nMathematical definition\n[\nedit\n]\nFor any fund that evolves randomly with time, volatility is defined as the\nstandard deviation\nof a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times.\nThus, \"annualized\" volatility\nσ\nannually\nis the standard deviation of an instrument's yearly\nlogarithmic returns\n.\n[\n2\n]\nThe generalized volatility\nσ\nT\nfor\ntime horizon\nT\nin years is expressed as:\nTherefore, if the daily logarithmic returns of a stock have a standard deviation of\nσ\ndaily\nand the time period of returns is\nP\nin trading days, the annualized volatility is\nso\nA common assumption is that\nP\n= 252 trading days in any given year. Then, if\nσ\ndaily\n= 0.01, the annualized volatility is\nThe monthly volatility (i.e.\nof a year) is\nThe formulas used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a\nrandom walk\n, or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent\nα\nto extrapolate natural processes:\nIf\nα\n = 2 the\nWiener process\nscaling relation is obtained, but some people believe\nα\n < 2 for financial activities such as stocks, indexes and so on. This was discovered by\nBenoît Mandelbrot\n, who looked at cotton prices and found that they followed a\nLévy alpha-stable distribution\nwith\nα\n = 1.7. (See New Scientist, 19 April 1997.)\nMuch research has been devoted to modelling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place.\nRoll (1984) shows that volatility is affected by\nmarket microstructure\n.\n[\n3\n]\nGlosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of\nadverse selection\n, they adjust their trading ranges, which in turn increases the band of price oscillation.\n[\n4\n]\nIn September 2019,\nJPMorgan Chase\ndetermined the effect of\nUS President\nDonald Trump\n's\ntweets\n, and called it the\nVolfefe index\ncombining volatility and the\ncovfefe\nmeme\n.\nVolatility for investors\n[\nedit\n]\nVolatility matters to investors for at least eight reasons,\n[\ncitation needed\n]\nseveral of which are alternative statements of the same feature or are directly consequent on each other:\nThe wider the swings in an investment's price, the harder emotionally it is to not worry;\nPrice volatility of a trading instrument can help to determine position sizing in a portfolio;\nWhen cash flows from selling a security are needed at a specific future date to meet a known fixed liability, higher volatility means a greater chance of a shortfall;\nHigher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values;\nHigher volatility of returns after retirement may result in withdrawals having a larger permanent impact on the portfolio's value;\nPrice volatility presents opportunities to anyone with inside information to buy assets cheaply and sell when overpriced;\nVolatility affects pricing of\noptions\n, being a parameter of the\nBlack–Scholes model\n.\nVolatility versus direction\n[\nedit\n]\nVolatility does not measure the direction of price changes, merely their dispersion. This is because when calculating\nstandard deviation\n(or\nvariance\n), all differences are squared, so that negative and positive differences are combined into one quantity.  Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time.\nFor example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. Ignoring compounding effects, this would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a\nnormal distribution\n; in reality stock price movements are found to be\nleptokurtotic\n(fat-tailed).\nVolatility over time\n[\nedit\n]\nAlthough the\nBlack-Scholes\nequation assumes predictable constant volatility, this is not observed in real markets. Amongst more realistic models are\nEmanuel Derman\nand\nIraj Kani\n's\n[\n5\n]\nand\nBruno Dupire\n's\nlocal volatility\n,\nPoisson process\nwhere volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of\nstochastic volatility\n.\n[\n6\n]\nIt is common knowledge that many types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all.\n[\n7\n]\nIn\nforeign exchange market\n, price changes are seasonally\nheteroskedastic\nwith periods of one day and one week.\n[\n8\n]\n[\n9\n]\nPeriods when prices fall quickly (a\ncrash\n) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible\nbubble\n) may often be followed by prices going up even more, or going down by an unusual amount.\nMost typically, extreme movements do not appear 'out of nowhere'; they are presaged by larger movements than usual or by known uncertainty in specific future events. This is termed\nautoregressive conditional heteroskedasticity\n. Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again.\nMeasures of volatility depend not only on the period over which it is measured, but also on the selected time resolution, as the information flow between short-term and long-term traders is asymmetric.\n[\nclarification needed\n]\nAs a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa.\n[\n10\n]\nAlternative measures of volatility\n[\nedit\n]\nSome authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns.\n[\n11\n]\nAnother considers the regular sequence of directional-changes as the proxy for the instantaneous volatility.\n[\n12\n]\nVolatility as it Relates to Options Trading\n[\nedit\n]\nOne method of measuring Volatility, often used by quant option trading firms, divides up volatility into two components. Clean volatility - the amount of volatility caused standard events like daily transactions and general noise -  and dirty vol, the amount caused by specific events like earnings or policy announcements.\n[\n13\n]\nFor instance, a company like\nMicrosoft\nwould have clean volatility caused by people buying and selling on a daily basis but dirty (or event vol) events like quarterly earnings or a possibly anti-trust announcement.\nBreaking down volatility into two components is useful in order to accurately price how much an option is worth, especially when identifying what events may contribute to a swing. The job of fundamental analysts at market makers and option trading boutique firms typically entails trying to assign numeric values to these numbers.\nCrude volatility estimation\n[\nedit\n]\nUsing a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days.  This would constitute a 1% daily movement, up or down.\nTo annualize this, you can use the \"rule of 16\", that is, multiply by 16 to get 16% as the annual volatility.  The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of\nn\nindependent variables (with equal standard deviations) is √n times the standard deviation of the individual variables.\nHowever importantly this does not capture (or in some cases may give excessive weight to) occasional large movements in market price which occur less frequently than once a year.\nThe average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation \nσ\n, the expected value of the\nmagnitude of the observations\nis √(2\/\nπ\n)\nσ\n= 0.798\nσ\n. The net effect is that this crude approach underestimates the true volatility by about 20%.\nEstimate of compound annual growth rate (CAGR)\n[\nedit\n]\nConsider the\nTaylor series\n:\nTaking only the first two terms one has:\nVolatility thus mathematically represents a drag on the CAGR (formalized as the \"\nvolatility tax\n\"). Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula:\nfor a rough estimate, where\nk\nis an empirical factor (typically five to ten).\n[\ncitation needed\n]\nCriticisms of volatility forecasting models\n[\nedit\n]\nPerformance of\nVIX\n(left) compared to past volatility (right) as 30-day volatility predictors, for the period of Jan 1990-Sep 2009. Volatility is measured as the standard deviation of S&P500 one-day returns over a month's period. The blue lines indicate\nlinear regressions\n, resulting in the\ncorrelation coefficients\nr\nshown. Note that VIX has virtually the same predictive power as past volatility, insofar as the shown correlation coefficients are nearly identical.\nDespite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility\n[\n14\n]\n[\n15\n]\nespecially out-of-sample, where different data are used to estimate the models and to test them.\n[\n16\n]\nOther works have agreed, but claim critics failed to correctly implement the more complicated models.\n[\n17\n]\nSome practitioners and\nportfolio managers\nseem to completely ignore or dismiss volatility forecasting models. For example,\nNassim Taleb\nfamously titled one of his\nJournal of Portfolio Management\npapers \"We Don't Quite Know What We are Talking About When We Talk About Volatility\".\n[\n18\n]\nIn a similar note,\nEmanuel Derman\nexpressed his disillusion with the enormous supply of empirical models unsupported by theory.\n[\n19\n]\nHe argues that, while \"theories are attempts to uncover the hidden principles underpinning the world around us, as Albert Einstein did with his theory of relativity\", we should remember that \"models are metaphors – analogies that describe one thing relative to another\". Machine learning and deep learning methods have also been applied to realized volatility forecasting.\n[\n20\n]\nBeta (finance)\n – Expected change in price of a stock relative to the whole market\nDispersion\n – Statistical property quantifying how much a collection of data is spread out\nExcess volatility puzzle\nFinancial economics\n – Academic discipline concerned with the exchange of money\nIVX\n – Intraday, VIX-like volatility index\nJules Regnault\n – French stock broker's assistant\nLow volatility anomaly\nRisk\n – Possibility of something bad happening\nVIX\n – Chicago Board Options Exchange Volatility index\nVolatility smile\n – Implied volatility patterns that arise in pricing financial options\nVolatility tax\n – Mathematical finance term\nVolatility risk\nVolatility beta\n^\n\"Levy distribution\"\n.\nwilmottwiki.com\n.\n^\n\"Calculating Historical Volatility: Step-by-Step Example\"\n(PDF)\n. Archived from the original on 30 March 2012\n. Retrieved\n18 August\n2011\n.\n{{\ncite web\n}}\n:  CS1 maint: bot: original URL status unknown (\nlink\n)\n^\nRoll, R. (1984): \"A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market\",\nJournal of Finance\n39\n(4), 1127–1139\n^\nGlosten, L. R. and P. R. Milgrom (1985): \"Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders\",\nJournal of Financial Economics\n14\n(1), 71–100\n^\nDerman, E.; Iraj Kani (1994).\n\"\n\"Riding on a Smile.\" RISK, 7(2) Feb.1994, pp. 139–145, pp. 32–39\"\n(PDF)\n. Risk. Archived from\nthe original\n(PDF)\non 10 July 2011\n. Retrieved\n1 June\n2007\n.\n^\n\"Volatility\"\n.\nwilmottwiki.com\n.\n^\n\"Taking Advantage Of Volatility Spikes With Credit Spreads\"\n.\n^\nMüller, Ulrich A.; Dacorogna, Michel M.; Olsen, Richard B.; Pictet, Olivier V.; Schwarz, Matthias; Morgenegg, Claude (1 December 1990). \"Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis\".\nJournal of Banking & Finance\n.\n14\n(6):\n1189–\n1208.\ndoi\n:\n10.1016\/0378-4266(90)90009-Q\n.\nISSN\n \n0378-4266\n.\n^\nPetrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019).\n\"Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time\"\n.\nJournal of Risk and Financial Management\n.\n12\n(2): 54.\ndoi\n:\n10.3390\/jrfm12020054\n.\nhdl\n:\n10419\/239003\n.\n^\nMuller, Ulrich A.; Dacorogna, Michel; Dave, Rakhal D.; Olsen, Richard; Pictet, Olivier V.; von Weizsäcker, Jakob (1997).\n\"Volatilities of different time resolutions -- Analyzing the dynamics of market components\"\n.\nJournal of Empirical Finance\n.\n4\n(\n2–\n3):\n213–\n239.\ndoi\n:\n10.1016\/S0927-5398(97)00007-8\n.\nISSN\n \n0927-5398\n.\n^\nSarkissian, Jack (2016). \"Express Measurement of Market Volatility Using Ergodicity Concept\".\ndoi\n:\n10.2139\/ssrn.2812353\n.\nS2CID\n \n168496910\n.\nSSRN\n \n2812353\n.\n^\nPetrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019).\n\"Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time\"\n.\nJournal of Risk and Financial Management\n.\n12\n(2): 54.\ndoi\n:\n10.3390\/jrfm12020054\n.\nhdl\n:\n10419\/239003\n.\n^\n\"Cleaning Implied Vols\"\n.\nMoontowermeta\n. Retrieved\n26 June\n2024\n.\n^\nCumby, R.; Figlewski, S.; Hasbrouck, J. (1993). \"Forecasting Volatility and Correlations with EGARCH models\".\nJournal of Derivatives\n.\n1\n(2):\n51–\n63.\ndoi\n:\n10.3905\/jod.1993.407877\n.\nS2CID\n \n154028452\n.\n^\nJorion, P. (1995). \"Predicting Volatility in Foreign Exchange Market\".\nJournal of Finance\n.\n50\n(2):\n507–\n528.\ndoi\n:\n10.1111\/j.1540-6261.1995.tb04793.x\n.\nJSTOR\n \n2329417\n.\n^\nBrooks, Chris\n; Persand, Gita (2003). \"Volatility forecasting for risk management\".\nJournal of Forecasting\n.\n22\n(1):\n1–\n22.\nCiteSeerX\n \n10.1.1.595.9113\n.\ndoi\n:\n10.1002\/for.841\n.\nISSN\n \n1099-131X\n.\nS2CID\n \n154615850\n.\n^\nAndersen, Torben G.; Bollerslev, Tim (1998). \"Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts\".\nInternational Economic Review\n.\n39\n(4):\n885–\n905.\nCiteSeerX\n \n10.1.1.28.454\n.\ndoi\n:\n10.2307\/2527343\n.\nJSTOR\n \n2527343\n.\n^\nGoldstein, Daniel and Taleb, Nassim, (28 March 2007)\n\"We Don't Quite Know What We are Talking About When We Talk About Volatility\"\n.\nJournal of Portfolio Management\n33\n(4), 2007.\n^\nDerman, Emanuel (2011): Models.Behaving.Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life”, Ed. Free Press.\n^\nLeushuis, Radmir M.; Petkov, Nicolai (2026).\n\"Advances in forecasting realized volatility: a review of methodologies\"\n.\nFinancial Innovation\n.\n12\n: 14.\ndoi\n:\n10.1186\/s40854-025-00809-5\n.\nGraphical Comparison of Implied and Historical Volatility\n, video\nDiebold, Francis X.; Hickman, Andrew; Inoue, Atsushi & Schuermannm, Til (1996) \"Converting 1-Day Volatility to h-Day Volatility: Scaling by sqrt(h) is Worse than You Think\"\nA short introduction to alternative mathematical concepts of volatility\nVolatility estimation from predicted return density\nExample based on Google daily return distribution using standard density function\nResearch paper including excerpt from report entitled Identifying Rich and Cheap Volatility\nExcerpt from Enhanced Call Overwriting, a report by Ryan Renicker and Devapriya Mallick at Lehman Brothers (2005).\nBartram, Söhnke M.; Brown, Gregory W.; Stulz, Rene M. (August 2012).\n\"Why Are U.S. Stocks More Volatile?\"\n(PDF)\n.\nJournal of Finance\n.\n67\n(4):\n1329–\n1370.\ndoi\n:\n10.1111\/j.1540-6261.2012.01749.x\n.\nS2CID\n \n18587238\n.\nSSRN\n \n2257549\n.\nNatenberg, Sheldon (2015).\nOption Volatility and Pricing: Advanced Trading Strategies and Techniques\n(Second ed.). New York.\nISBN\n \n978-0071818773\n.\n{{\ncite book\n}}\n:  CS1 maint: location missing publisher (\nlink\n)\nFassas, Athanasios P.; Siriopoulos, Costas (1 February 2021).\n\"Implied volatility indices – A review\"\n.\nThe Quarterly Review of Economics and Finance\n.\n79\n:\n303–\n329.\ndoi\n:\n10.1016\/j.qref.2020.07.004\n.\nISSN\n \n1062-9769\n.","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#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=Volatility+%28finance%29 \"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=Volatility+%28finance%29 \"You're encouraged to log in; however, it's not mandatory. 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[o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\))\n- [1 Volatility terminology](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Volatility_terminology)\n- [2 Mathematical definition](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Mathematical_definition)\n- [3 Volatility origin](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Volatility_origin)\n- [4 Volatility for investors](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Volatility_for_investors)\n- [5 Volatility versus direction](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Volatility_versus_direction)\n- [6 Volatility over time](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Volatility_over_time)\n- [7 Alternative measures of volatility](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Alternative_measures_of_volatility)\n  Toggle Alternative measures of volatility subsection\n  - [7\\.1 Volatility as it Relates to Options Trading](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Volatility_as_it_Relates_to_Options_Trading)\n- [8 Crude volatility estimation](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Crude_volatility_estimation)\n- [9 Estimate of compound annual growth rate (CAGR)](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Estimate_of_compound_annual_growth_rate_\\(CAGR\\))\n- [10 Criticisms of volatility forecasting models](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Criticisms_of_volatility_forecasting_models)\n- [11 See also](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#See_also)\n- [12 References](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#References)\n- [13 External links](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#External_links)\n- [14 Further reading](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#Further_reading)\n\nToggle the table of contents\n\n# Volatility (finance)\n33 languages\n\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D8%AA%D9%82%D9%84%D8%A8_\\(%D8%A7%D9%84%D9%85%D8%A7%D9%84\\) \"تقلب (المال) – Arabic\")\n- [الدارجة](https:\/\/ary.wikipedia.org\/wiki\/%D8%AA%D8%B7%D8%A7%D9%8A%D8%B1%D9%8A%D8%AA_\\(%D9%81%D9%8A%D9%86%D9%88%D9%86%D8%B5\\) \"تطايريت (فينونص) – Moroccan Arabic\")\n- [Azərbaycanca](https:\/\/az.wikipedia.org\/wiki\/Volatillik \"Volatillik – Azerbaijani\")\n- [Башҡортса](https:\/\/ba.wikipedia.org\/wiki\/%D0%92%D0%BE%D0%BB%D0%B0%D1%82%D0%B8%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C \"Волатильность – Bashkir\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Volatilitat_\\(finances\\) \"Volatilitat (finances) – Catalan\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/Volatilita \"Volatilita – Czech\")\n- [Dansk](https:\/\/da.wikipedia.org\/wiki\/Volatilitet \"Volatilitet – Danish\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Volatilit%C3%A4t \"Volatilität – German\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Volatilidad_\\(finanzas\\) \"Volatilidad (finanzas) – Spanish\")\n- [Eesti](https:\/\/et.wikipedia.org\/wiki\/Volatiilsus \"Volatiilsus – Estonian\")\n- [Euskara](https:\/\/eu.wikipedia.org\/wiki\/Hegazkortasun_\\(finantzak\\) \"Hegazkortasun (finantzak) – Basque\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D9%86%D9%88%D8%B3%D8%A7%D9%86_\\(%D8%A7%D9%85%D9%88%D8%B1_%D9%85%D8%A7%D9%84%DB%8C\\) \"نوسان (امور مالی) – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Volatiliteetti \"Volatiliteetti – Finnish\")\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/Volatilit%C3%A9_\\(finance\\) \"Volatilité (finance) – French\")\n- [Magyar](https:\/\/hu.wikipedia.org\/wiki\/Ill%C3%A9konys%C3%A1g \"Illékonyság – Hungarian\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Volatilit%C3%A0_\\(economia\\) \"Volatilità (economia) – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E3%83%9C%E3%83%A9%E3%83%86%E3%82%A3%E3%83%AA%E3%83%86%E3%82%A3 \"ボラティリティ – Japanese\")\n- [Қазақша](https:\/\/kk.wikipedia.org\/wiki\/%D2%9A%D2%B1%D0%B1%D1%8B%D0%BB%D0%BC%D0%B0%D0%BB%D1%8B%D0%BB%D1%8B%D2%9B \"Құбылмалылық – Kazakh\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EB%B3%80%EB%8F%99%EC%84%B1 \"변동성 – Korean\")\n- [Lietuvių](https:\/\/lt.wikipedia.org\/wiki\/Kintamumas_\\(rinka\\) \"Kintamumas (rinka) – Lithuanian\")\n- [Latviešu](https:\/\/lv.wikipedia.org\/wiki\/Sv%C4%81rst%C4%ABgums \"Svārstīgums – Latvian\")\n- [Nederlands](https:\/\/nl.wikipedia.org\/wiki\/Volatiliteit \"Volatiliteit – Dutch\")\n- [Norsk bokmål](https:\/\/no.wikipedia.org\/wiki\/Volatilitet \"Volatilitet – Norwegian Bokmål\")\n- [پښتو](https:\/\/ps.wikipedia.org\/wiki\/%D9%86%D9%88%D8%B3%D8%A7%D9%86_\\(%D9%85%D8%A7%D9%84%D9%8A\\) \"نوسان (مالي) – Pashto\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/Volatilidade_\\(finan%C3%A7as\\) \"Volatilidade (finanças) – Portuguese\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%92%D0%BE%D0%BB%D0%B0%D1%82%D0%B8%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C \"Волатильность – Russian\")\n- [Simple English](https:\/\/simple.wikipedia.org\/wiki\/Volatility \"Volatility – Simple English\")\n- [Slovenčina](https:\/\/sk.wikipedia.org\/wiki\/Volatilita_\\(financie\\) \"Volatilita (financie) – Slovak\")\n- [Svenska](https:\/\/sv.wikipedia.org\/wiki\/Volatilitet \"Volatilitet – Swedish\")\n- [Türkçe](https:\/\/tr.wikipedia.org\/wiki\/Volatilite_\\(finans\\) \"Volatilite (finans) – Turkish\")\n- [Українська](https:\/\/uk.wikipedia.org\/wiki\/%D0%92%D0%BE%D0%BB%D0%B0%D1%82%D0%B8%D0%BB%D1%8C%D0%BD%D1%96%D1%81%D1%82%D1%8C \"Волатильність – Ukrainian\")\n- [粵語](https:\/\/zh-yue.wikipedia.org\/wiki\/%E6%B3%A2%E5%8B%95%E6%80%A7 \"波動性 – Cantonese\")\n- [中文](https:\/\/zh.wikipedia.org\/wiki\/%E6%B3%A2%E5%8A%A8%E6%80%A7 \"波动性 – Chinese\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q756115#sitelinks-wikipedia 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URL](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FVolatility_%28finance%29)\n\nPrint\/export\n\n- [Download as PDF](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:DownloadAsPdf&page=Volatility_%28finance%29&action=show-download-screen \"Download this page as a PDF file\")\n- [Printable version](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q756115 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nDegree of variation of a trading price series over time\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/9b\/VIX.png\/250px-VIX.png)](https:\/\/en.wikipedia.org\/wiki\/File:VIX.png)\n\nCBOE Volatility Index (VIX) from December 1985 to May 2012 (daily closings)\n\nIn [finance](https:\/\/en.wikipedia.org\/wiki\/Finance \"Finance\"), **volatility** (usually denoted by \"[σ](https:\/\/en.wikipedia.org\/wiki\/Sigma \"Sigma\")\") is the [degree of variation](https:\/\/en.wikipedia.org\/wiki\/Variability_\\(statistics\\) \"Variability (statistics)\") of a trading price series over time, usually measured by the [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") of [logarithmic returns](https:\/\/en.wikipedia.org\/wiki\/Logarithmic_return \"Logarithmic return\").\n\nHistoric volatility measures a time series of past market prices. [Implied volatility](https:\/\/en.wikipedia.org\/wiki\/Implied_volatility \"Implied volatility\") looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option).\n\n## Volatility terminology\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=1 \"Edit section: Volatility terminology\")\\]\n\nVolatility as described here refers to the **actual volatility**, more specifically:\n\n- **actual current volatility** of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price.\n- **actual historical volatility** which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past\n  - near synonymous is **realized volatility**, the [square root](https:\/\/en.wikipedia.org\/wiki\/Square_root \"Square root\") of the [realized variance](https:\/\/en.wikipedia.org\/wiki\/Realized_variance \"Realized variance\"), in turn calculated using the sum of squared returns divided by the number of observations.\n- **actual future volatility** which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an [option](https:\/\/en.wikipedia.org\/wiki\/Option_\\(finance\\) \"Option (finance)\"))\n\nNow turning to [implied volatility](https:\/\/en.wikipedia.org\/wiki\/Implied_volatility \"Implied volatility\"), we have:\n\n- **historical implied volatility** which refers to the implied volatility observed from historical prices of the financial instrument (normally options)\n- **current implied volatility** which refers to the implied volatility observed from current prices of the financial instrument\n- **future implied volatility** which refers to the implied volatility observed from future prices of the financial instrument\n\nFor a financial instrument whose price follows a [Gaussian](https:\/\/en.wikipedia.org\/wiki\/Gaussian \"Gaussian\") [random walk](https:\/\/en.wikipedia.org\/wiki\/Random_walk \"Random walk\"), or [Wiener process](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\"), the width of the distribution increases as time increases. This is because there is an increasing [probability](https:\/\/en.wikipedia.org\/wiki\/Probability \"Probability\") that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.\n\nSince observed price changes do not follow Gaussian distributions, others such as the [Lévy distribution](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9vy_distribution \"Lévy distribution\") are often used.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-1) These can capture attributes such as \"[fat tails](https:\/\/en.wikipedia.org\/wiki\/Fat_tail \"Fat tail\")\". Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc.\n\n## Mathematical definition\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=2 \"Edit section: Mathematical definition\")\\]\n\nFor any fund that evolves randomly with time, volatility is defined as the [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") of a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times.\n\nThus, \"annualized\" volatility *σ*annually is the standard deviation of an instrument's yearly [logarithmic returns](https:\/\/en.wikipedia.org\/wiki\/Rate_of_return#Logarithmic_or_continuously_compounded_return \"Rate of return\").[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-2)\n\nThe generalized volatility *σ**T* for [time horizon](https:\/\/en.wikipedia.org\/wiki\/Time_horizon \"Time horizon\") *T* in years is expressed as:\n\nσ\n\nT\n\n\\=\n\nσ\n\nannually\n\nT\n\n.\n\n{\\\\displaystyle \\\\sigma \\_{\\\\text{T}}=\\\\sigma \\_{\\\\text{annually}}{\\\\sqrt {T}}.}\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{T}}=\\\\sigma \\_{\\\\text{annually}}{\\\\sqrt {T}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/49e88f2ee170dbab0671e56003059a9ff2e632f3)\n\nTherefore, if the daily logarithmic returns of a stock have a standard deviation of *σ*daily and the time period of returns is *P* in trading days, the annualized volatility is\n\nσ\n\nannually\n\n\\=\n\nσ\n\ndaily\n\nP\n\n.\n\n{\\\\displaystyle \\\\sigma \\_{\\\\text{annually}}=\\\\sigma \\_{\\\\text{daily}}{\\\\sqrt {P}}.}\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{annually}}=\\\\sigma \\_{\\\\text{daily}}{\\\\sqrt {P}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/36940dadd58fbbfadcb59140749af3a4b3c09eaa)\n\nso\n\nσ\n\nT\n\n\\=\n\nσ\n\ndaily\n\nP\n\nT\n\n.\n\n{\\\\displaystyle \\\\sigma \\_{\\\\text{T}}=\\\\sigma \\_{\\\\text{daily}}{\\\\sqrt {PT}}.}\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{T}}=\\\\sigma \\_{\\\\text{daily}}{\\\\sqrt {PT}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b715335b9d0cbeb41da30e66b16d1a95097fc6f3)\n\nA common assumption is that *P* = 252 trading days in any given year. Then, if *σ*daily = 0.01, the annualized volatility is\n\nσ\n\nannually\n\n\\=\n\n0\\.01\n\n252\n\n\\=\n\n0\\.1587.\n\n{\\\\displaystyle \\\\sigma \\_{\\\\text{annually}}=0.01{\\\\sqrt {252}}=0.1587.}\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{annually}}=0.01{\\\\sqrt {252}}=0.1587.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77022ad5aadc3db0ebfe97fc132581e3f68042af)\n\nThe monthly volatility (i.e. T \\= 1 12 {\\\\displaystyle T={\\\\tfrac {1}{12}}} ![{\\\\displaystyle T={\\\\tfrac {1}{12}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/858a58f00d971d2a99c0adae52b82982ba310e25) of a year) is\n\nσ\n\nmonthly\n\n\\=\n\n0\\.01\n\n252\n\n12\n\n\\=\n\n0\\.0458.\n\n{\\\\displaystyle \\\\sigma \\_{\\\\text{monthly}}=0.01{\\\\sqrt {\\\\tfrac {252}{12}}}=0.0458.}\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{monthly}}=0.01{\\\\sqrt {\\\\tfrac {252}{12}}}=0.0458.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4f8a9560640b89becb2624657dea5e0f92da32fe)\n\nThe formulas used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a [random walk](https:\/\/en.wikipedia.org\/wiki\/Random_walk \"Random walk\"), or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent *α* to extrapolate natural processes:\n\nσ\n\nT\n\n\\=\n\nT\n\n1\n\n\/\n\nα\n\nσ\n\n.\n\n{\\\\displaystyle \\\\sigma \\_{T}=T^{1\/\\\\alpha }\\\\sigma .\\\\,}\n\n![{\\\\displaystyle \\\\sigma \\_{T}=T^{1\/\\\\alpha }\\\\sigma .\\\\,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f0a94ec1c310a51eba65f801a92e89b85b5729c8)\n\nIf *α* = 2 the [Wiener process](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\") scaling relation is obtained, but some people believe *α* \\< 2 for financial activities such as stocks, indexes and so on. This was discovered by [Benoît Mandelbrot](https:\/\/en.wikipedia.org\/wiki\/Beno%C3%AEt_Mandelbrot \"Benoît Mandelbrot\"), who looked at cotton prices and found that they followed a [Lévy alpha-stable distribution](https:\/\/en.wikipedia.org\/wiki\/Stable_distribution \"Stable distribution\") with *α* = 1.7. (See New Scientist, 19 April 1997.)\n\n## Volatility origin\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=3 \"Edit section: Volatility origin\")\\]\n\nMuch research has been devoted to modelling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place.\n\nRoll (1984) shows that volatility is affected by [market microstructure](https:\/\/en.wikipedia.org\/wiki\/Market_microstructure \"Market microstructure\").[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-3) Glosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of [adverse selection](https:\/\/en.wikipedia.org\/wiki\/Adverse_selection \"Adverse selection\"), they adjust their trading ranges, which in turn increases the band of price oscillation.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-4)\n\nIn September 2019, [JPMorgan Chase](https:\/\/en.wikipedia.org\/wiki\/JPMorgan_Chase \"JPMorgan Chase\") determined the effect of [US President](https:\/\/en.wikipedia.org\/wiki\/US_President \"US President\") [Donald Trump](https:\/\/en.wikipedia.org\/wiki\/Donald_Trump \"Donald Trump\")'s [tweets](https:\/\/en.wikipedia.org\/wiki\/Twitter#Tweets \"Twitter\"), and called it the [Volfefe index](https:\/\/en.wikipedia.org\/wiki\/Volfefe_index \"Volfefe index\") combining volatility and the [covfefe](https:\/\/en.wikipedia.org\/wiki\/Covfefe \"Covfefe\") [meme](https:\/\/en.wikipedia.org\/wiki\/Meme \"Meme\").\n\n## Volatility for investors\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=4 \"Edit section: Volatility for investors\")\\]\n\nVolatility matters to investors for at least eight reasons,\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\] several of which are alternative statements of the same feature or are directly consequent on each other:\n\n1. The wider the swings in an investment's price, the harder emotionally it is to not worry;\n2. Price volatility of a trading instrument can help to determine position sizing in a portfolio;\n3. When cash flows from selling a security are needed at a specific future date to meet a known fixed liability, higher volatility means a greater chance of a shortfall;\n4. Higher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values;\n5. Higher volatility of returns after retirement may result in withdrawals having a larger permanent impact on the portfolio's value;\n6. Price volatility presents opportunities to anyone with inside information to buy assets cheaply and sell when overpriced;\n7. Volatility affects pricing of [options](https:\/\/en.wikipedia.org\/wiki\/Option_\\(finance\\) \"Option (finance)\"), being a parameter of the [Black–Scholes model](https:\/\/en.wikipedia.org\/wiki\/Black%E2%80%93Scholes_model \"Black–Scholes model\").\n\n## Volatility versus direction\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=5 \"Edit section: Volatility versus direction\")\\]\n\nVolatility does not measure the direction of price changes, merely their dispersion. This is because when calculating [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") (or [variance](https:\/\/en.wikipedia.org\/wiki\/Variance \"Variance\")), all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time.\n\nFor example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. Ignoring compounding effects, this would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a [normal distribution](https:\/\/en.wikipedia.org\/wiki\/Normal_distribution \"Normal distribution\"); in reality stock price movements are found to be [leptokurtotic](https:\/\/en.wikipedia.org\/wiki\/Kurtosis \"Kurtosis\") (fat-tailed).\n\n## Volatility over time\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=6 \"Edit section: Volatility over time\")\\]\n\nAlthough the [Black-Scholes](https:\/\/en.wikipedia.org\/wiki\/Black-Scholes \"Black-Scholes\") equation assumes predictable constant volatility, this is not observed in real markets. Amongst more realistic models are [Emanuel Derman](https:\/\/en.wikipedia.org\/wiki\/Emanuel_Derman \"Emanuel Derman\") and [Iraj Kani](https:\/\/en.wikipedia.org\/w\/index.php?title=Iraj_Kani&action=edit&redlink=1 \"Iraj Kani (page does not exist)\")'s[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-derman-5) and [Bruno Dupire](https:\/\/en.wikipedia.org\/wiki\/Bruno_Dupire \"Bruno Dupire\")'s [local volatility](https:\/\/en.wikipedia.org\/wiki\/Local_volatility \"Local volatility\"), [Poisson process](https:\/\/en.wikipedia.org\/wiki\/Poisson_process \"Poisson process\") where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of [stochastic volatility](https:\/\/en.wikipedia.org\/wiki\/Stochastic_volatility \"Stochastic volatility\").[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-6)\n\nIt is common knowledge that many types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-7) In [foreign exchange market](https:\/\/en.wikipedia.org\/wiki\/Foreign_exchange_market \"Foreign exchange market\"), price changes are seasonally [heteroskedastic](https:\/\/en.wikipedia.org\/wiki\/Heteroscedasticity \"Heteroscedasticity\") with periods of one day and one week.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-8)[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-9)\n\nPeriods when prices fall quickly (a [crash](https:\/\/en.wikipedia.org\/wiki\/Stock_market_crash \"Stock market crash\")) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible [bubble](https:\/\/en.wikipedia.org\/wiki\/Bubble_\\(economics\\) \"Bubble (economics)\")) may often be followed by prices going up even more, or going down by an unusual amount.\n\nMost typically, extreme movements do not appear 'out of nowhere'; they are presaged by larger movements than usual or by known uncertainty in specific future events. This is termed [autoregressive conditional heteroskedasticity](https:\/\/en.wikipedia.org\/wiki\/Autoregressive_conditional_heteroskedasticity \"Autoregressive conditional heteroskedasticity\"). Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again.\n\nMeasures of volatility depend not only on the period over which it is measured, but also on the selected time resolution, as the information flow between short-term and long-term traders is asymmetric.\\[*[clarification needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Please_clarify \"Wikipedia:Please clarify\")*\\] As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-10)\n\n## Alternative measures of volatility\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=7 \"Edit section: Alternative measures of volatility\")\\]\n\nSome authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns.[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-11) Another considers the regular sequence of directional-changes as the proxy for the instantaneous volatility.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-12)\n\n### Volatility as it Relates to Options Trading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=8 \"Edit section: Volatility as it Relates to Options Trading\")\\]\n\nOne method of measuring Volatility, often used by quant option trading firms, divides up volatility into two components. Clean volatility - the amount of volatility caused standard events like daily transactions and general noise - and dirty vol, the amount caused by specific events like earnings or policy announcements.[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-13) For instance, a company like [Microsoft](https:\/\/en.wikipedia.org\/wiki\/Microsoft \"Microsoft\") would have clean volatility caused by people buying and selling on a daily basis but dirty (or event vol) events like quarterly earnings or a possibly anti-trust announcement.\n\nBreaking down volatility into two components is useful in order to accurately price how much an option is worth, especially when identifying what events may contribute to a swing. The job of fundamental analysts at market makers and option trading boutique firms typically entails trying to assign numeric values to these numbers.\n\n## Crude volatility estimation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=9 \"Edit section: Crude volatility estimation\")\\]\n\nUsing a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down.\n\nTo annualize this, you can use the \"rule of 16\", that is, multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of *n* independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables.\n\nHowever importantly this does not capture (or in some cases may give excessive weight to) occasional large movements in market price which occur less frequently than once a year.\n\nThe average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation *σ*, the expected value of the [magnitude of the observations](https:\/\/en.wikipedia.org\/wiki\/Absolute_deviation \"Absolute deviation\") is √(2\/π)*σ* = 0.798*σ*. The net effect is that this crude approach underestimates the true volatility by about 20%.\n\n## Estimate of compound annual growth rate (CAGR)\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=10 \"Edit section: Estimate of compound annual growth rate (CAGR)\")\\]\n\nConsider the [Taylor series](https:\/\/en.wikipedia.org\/wiki\/Taylor_series \"Taylor series\"):\n\nlog\n\n⁡\n\n(\n\n1\n\n\\+\n\ny\n\n)\n\n\\=\n\ny\n\n−\n\n1\n\n2\n\ny\n\n2\n\n\\+\n\n1\n\n3\n\ny\n\n3\n\n−\n\n1\n\n4\n\ny\n\n4\n\n\\+\n\n⋯\n\n{\\\\displaystyle \\\\log(1+y)=y-{\\\\tfrac {1}{2}}y^{2}+{\\\\tfrac {1}{3}}y^{3}-{\\\\tfrac {1}{4}}y^{4}+\\\\cdots }\n\n![{\\\\displaystyle \\\\log(1+y)=y-{\\\\tfrac {1}{2}}y^{2}+{\\\\tfrac {1}{3}}y^{3}-{\\\\tfrac {1}{4}}y^{4}+\\\\cdots }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/496a72d2a2d6e2c835aff5e506331ff0ce9309fc)\n\nTaking only the first two terms one has:\n\nC\n\nA\n\nG\n\nR\n\n≈\n\nA\n\nR\n\n−\n\n1\n\n2\n\nσ\n\n2\n\n{\\\\displaystyle \\\\mathrm {CAGR} \\\\approx \\\\mathrm {AR} -{\\\\tfrac {1}{2}}\\\\sigma ^{2}}\n\n![{\\\\displaystyle \\\\mathrm {CAGR} \\\\approx \\\\mathrm {AR} -{\\\\tfrac {1}{2}}\\\\sigma ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bfd77bb23265be12dea88d2d34234f8094cb8d25)\n\nVolatility thus mathematically represents a drag on the CAGR (formalized as the \"[volatility tax](https:\/\/en.wikipedia.org\/wiki\/Volatility_tax \"Volatility tax\")\"). Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula:\n\nC\n\nA\n\nG\n\nR\n\n≈\n\nA\n\nR\n\n−\n\n1\n\n2\n\nk\n\nσ\n\n2\n\n{\\\\displaystyle \\\\mathrm {CAGR} \\\\approx \\\\mathrm {AR} -{\\\\tfrac {1}{2}}k\\\\sigma ^{2}}\n\n![{\\\\displaystyle \\\\mathrm {CAGR} \\\\approx \\\\mathrm {AR} -{\\\\tfrac {1}{2}}k\\\\sigma ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/823832d028777a59c7a6bd47bf9d5b9f5e63c0a1)\n\nfor a rough estimate, where *k* is an empirical factor (typically five to ten).\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]\n\n## Criticisms of volatility forecasting models\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=11 \"Edit section: Criticisms of volatility forecasting models\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/ed\/Vix.png\/500px-Vix.png)](https:\/\/en.wikipedia.org\/wiki\/File:Vix.png)\n\nPerformance of [VIX](https:\/\/en.wikipedia.org\/wiki\/VIX \"VIX\") (left) compared to past volatility (right) as 30-day volatility predictors, for the period of Jan 1990-Sep 2009. Volatility is measured as the standard deviation of S\\&P500 one-day returns over a month's period. The blue lines indicate [linear regressions](https:\/\/en.wikipedia.org\/wiki\/Linear_regression \"Linear regression\"), resulting in the [correlation coefficients](https:\/\/en.wikipedia.org\/wiki\/Correlation \"Correlation\") *r* shown. Note that VIX has virtually the same predictive power as past volatility, insofar as the shown correlation coefficients are nearly identical.\n\nDespite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-14)[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-15) especially out-of-sample, where different data are used to estimate the models and to test them.[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-16) Other works have agreed, but claim critics failed to correctly implement the more complicated models.[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-17) Some practitioners and [portfolio managers](https:\/\/en.wikipedia.org\/wiki\/Portfolio_managers \"Portfolio managers\") seem to completely ignore or dismiss volatility forecasting models. For example, [Nassim Taleb](https:\/\/en.wikipedia.org\/wiki\/Nassim_Taleb \"Nassim Taleb\") famously titled one of his *[Journal of Portfolio Management](https:\/\/en.wikipedia.org\/wiki\/Journal_of_Portfolio_Management \"Journal of Portfolio Management\")* papers \"We Don't Quite Know What We are Talking About When We Talk About Volatility\".[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-18) In a similar note, [Emanuel Derman](https:\/\/en.wikipedia.org\/wiki\/Emanuel_Derman \"Emanuel Derman\") expressed his disillusion with the enormous supply of empirical models unsupported by theory.[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-19) He argues that, while \"theories are attempts to uncover the hidden principles underpinning the world around us, as Albert Einstein did with his theory of relativity\", we should remember that \"models are metaphors – analogies that describe one thing relative to another\". Machine learning and deep learning methods have also been applied to realized volatility forecasting.[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-20)\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=12 \"Edit section: See also\")\\]\n\n- [Beta (finance)](https:\/\/en.wikipedia.org\/wiki\/Beta_\\(finance\\) \"Beta (finance)\") – Expected change in price of a stock relative to the whole market\n- [Dispersion](https:\/\/en.wikipedia.org\/wiki\/Statistical_dispersion \"Statistical dispersion\") – Statistical property quantifying how much a collection of data is spread out\n- [Excess volatility puzzle](https:\/\/en.wikipedia.org\/wiki\/Excess_volatility_puzzle \"Excess volatility puzzle\")\n- [Financial economics](https:\/\/en.wikipedia.org\/wiki\/Financial_economics \"Financial economics\") – Academic discipline concerned with the exchange of money\n- [IVX](https:\/\/en.wikipedia.org\/wiki\/IVX \"IVX\") – Intraday, VIX-like volatility index\n- [Jules Regnault](https:\/\/en.wikipedia.org\/wiki\/Jules_Regnault \"Jules Regnault\") – French stock broker's assistant\n- [Low volatility anomaly](https:\/\/en.wikipedia.org\/wiki\/Low_volatility_anomaly \"Low volatility anomaly\")\n- [Risk](https:\/\/en.wikipedia.org\/wiki\/Risk \"Risk\") – Possibility of something bad happening\n- [VIX](https:\/\/en.wikipedia.org\/wiki\/VIX \"VIX\") – Chicago Board Options Exchange Volatility index\n- [Volatility smile](https:\/\/en.wikipedia.org\/wiki\/Volatility_smile \"Volatility smile\") – Implied volatility patterns that arise in pricing financial options\n- [Volatility tax](https:\/\/en.wikipedia.org\/wiki\/Volatility_tax \"Volatility tax\") – Mathematical finance term\n- [Volatility risk](https:\/\/en.wikipedia.org\/wiki\/Volatility_risk \"Volatility risk\")\n- [Volatility beta](https:\/\/en.wikipedia.org\/wiki\/Volatility_beta \"Volatility beta\")\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=13 \"Edit section: References\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-1)**\n   [\"Levy distribution\"](http:\/\/www.wilmottwiki.com\/wiki\/index.php?title=Levy_distribution). *wilmottwiki.com*.\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-2)**\n   [\"Calculating Historical Volatility: Step-by-Step Example\"](https:\/\/web.archive.org\/web\/20120330224816\/http:\/\/www.lfrankcabrera.com\/calc-hist-vol.pdf) (PDF). Archived from the original on 30 March 2012. Retrieved 18 August 2011.\n   \n   `{{cite web}}`: CS1 maint: bot: original URL status unknown ([link](https:\/\/en.wikipedia.org\/wiki\/Category:CS1_maint:_bot:_original_URL_status_unknown \"Category:CS1 maint: bot: original URL status unknown\"))\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-3)** Roll, R. (1984): \"A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market\", *Journal of Finance* **39** (4), 1127–1139\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-4)** Glosten, L. R. and P. R. Milgrom (1985): \"Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders\", *[Journal of Financial Economics](https:\/\/en.wikipedia.org\/wiki\/Journal_of_Financial_Economics \"Journal of Financial Economics\")* **14** (1), 71–100\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-derman_5-0)**\n   Derman, E.; Iraj Kani (1994). [\"\"Riding on a Smile.\" RISK, 7(2) Feb.1994, pp. 139–145, pp. 32–39\"](https:\/\/web.archive.org\/web\/20110710170610\/http:\/\/www.ederman.com\/new\/docs\/gs-volatility_smile.pdf) (PDF). Risk. Archived from [the original](http:\/\/www.ederman.com\/new\/docs\/gs-volatility_smile.pdf) (PDF) on 10 July 2011. Retrieved 1 June 2007.\n   \n   `{{cite journal}}`: Cite journal requires `|journal=` ([help](https:\/\/en.wikipedia.org\/wiki\/Help:CS1_errors#missing_periodical \"Help:CS1 errors\"))\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-6)**\n   [\"Volatility\"](http:\/\/www.wilmottwiki.com\/wiki\/index.php?title=Volatility). *wilmottwiki.com*.\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-7)**\n   [\"Taking Advantage Of Volatility Spikes With Credit Spreads\"](http:\/\/www.investopedia.com\/articles\/optioninvestor\/10\/volatility-spikes-credit-spreads.asp).\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-8)**\n   Müller, Ulrich A.; Dacorogna, Michel M.; Olsen, Richard B.; Pictet, Olivier V.; Schwarz, Matthias; Morgenegg, Claude (1 December 1990). \"Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis\". *Journal of Banking & Finance*. **14** (6): 1189–1208\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/0378-4266(90)90009-Q](https:\/\/doi.org\/10.1016%2F0378-4266%2890%2990009-Q). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0378-4266](https:\/\/search.worldcat.org\/issn\/0378-4266).\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-9)**\n   Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). [\"Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time\"](https:\/\/doi.org\/10.3390%2Fjrfm12020054). *Journal of Risk and Financial Management*. **12** (2): 54. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3390\/jrfm12020054](https:\/\/doi.org\/10.3390%2Fjrfm12020054). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10419\/239003](https:\/\/hdl.handle.net\/10419%2F239003).\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-10)**\n    Muller, Ulrich A.; Dacorogna, Michel; Dave, Rakhal D.; Olsen, Richard; Pictet, Olivier V.; von Weizsäcker, Jakob (1997). [\"Volatilities of different time resolutions -- Analyzing the dynamics of market components\"](https:\/\/econpapers.repec.org\/article\/eeeempfin\/v_3a4_3ay_3a1997_3ai_3a2-3_3ap_3a213-239.htm). *Journal of Empirical Finance*. **4** (2–3\\): 213–239\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/S0927-5398(97)00007-8](https:\/\/doi.org\/10.1016%2FS0927-5398%2897%2900007-8). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0927-5398](https:\/\/search.worldcat.org\/issn\/0927-5398).\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-11)**\n    Sarkissian, Jack (2016). \"Express Measurement of Market Volatility Using Ergodicity Concept\". [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.2139\/ssrn.2812353](https:\/\/doi.org\/10.2139%2Fssrn.2812353). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [168496910](https:\/\/api.semanticscholar.org\/CorpusID:168496910). [SSRN](https:\/\/en.wikipedia.org\/wiki\/SSRN_\\(identifier\\) \"SSRN (identifier)\") [2812353](https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=2812353).\n    \n    `{{cite journal}}`: Cite journal requires `|journal=` ([help](https:\/\/en.wikipedia.org\/wiki\/Help:CS1_errors#missing_periodical \"Help:CS1 errors\"))\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-12)**\n    Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). [\"Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time\"](https:\/\/doi.org\/10.3390%2Fjrfm12020054). *Journal of Risk and Financial Management*. **12** (2): 54. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3390\/jrfm12020054](https:\/\/doi.org\/10.3390%2Fjrfm12020054). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10419\/239003](https:\/\/hdl.handle.net\/10419%2F239003).\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-13)**\n    [\"Cleaning Implied Vols\"](https:\/\/notion.moontowermeta.com\/cleaning-implied-vols). *Moontowermeta*. Retrieved 26 June 2024.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-14)**\n    Cumby, R.; Figlewski, S.; Hasbrouck, J. (1993). \"Forecasting Volatility and Correlations with EGARCH models\". *Journal of Derivatives*. **1** (2): 51–63\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3905\/jod.1993.407877](https:\/\/doi.org\/10.3905%2Fjod.1993.407877). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [154028452](https:\/\/api.semanticscholar.org\/CorpusID:154028452).\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-15)**\n    Jorion, P. (1995). \"Predicting Volatility in Foreign Exchange Market\". *[Journal of Finance](https:\/\/en.wikipedia.org\/wiki\/Journal_of_Finance \"Journal of Finance\")*. **50** (2): 507–528\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1111\/j.1540-6261.1995.tb04793.x](https:\/\/doi.org\/10.1111%2Fj.1540-6261.1995.tb04793.x). [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [2329417](https:\/\/www.jstor.org\/stable\/2329417).\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-16)**\n    [Brooks, Chris](https:\/\/en.wikipedia.org\/wiki\/Chris_Brooks_\\(academic\\) \"Chris Brooks (academic)\"); Persand, Gita (2003). \"Volatility forecasting for risk management\". *Journal of Forecasting*. **22** (1): 1–22\\. [CiteSeerX](https:\/\/en.wikipedia.org\/wiki\/CiteSeerX_\\(identifier\\) \"CiteSeerX (identifier)\") [10\\.1.1.595.9113](https:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.595.9113). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/for.841](https:\/\/doi.org\/10.1002%2Ffor.841). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1099-131X](https:\/\/search.worldcat.org\/issn\/1099-131X). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [154615850](https:\/\/api.semanticscholar.org\/CorpusID:154615850).\n17. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-17)**\n    Andersen, Torben G.; Bollerslev, Tim (1998). \"Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts\". *[International Economic Review](https:\/\/en.wikipedia.org\/wiki\/International_Economic_Review \"International Economic Review\")*. **39** (4): 885–905\\. [CiteSeerX](https:\/\/en.wikipedia.org\/wiki\/CiteSeerX_\\(identifier\\) \"CiteSeerX (identifier)\") [10\\.1.1.28.454](https:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.28.454). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.2307\/2527343](https:\/\/doi.org\/10.2307%2F2527343). [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [2527343](https:\/\/www.jstor.org\/stable\/2527343).\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-18)** Goldstein, Daniel and Taleb, Nassim, (28 March 2007) [\"We Don't Quite Know What We are Talking About When We Talk About Volatility\"](https:\/\/ssrn.com\/abstract=970480). *Journal of Portfolio Management* **33** (4), 2007.\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-19)** Derman, Emanuel (2011): Models.Behaving.Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life”, Ed. Free Press.\n20. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-20)**\n    Leushuis, Radmir M.; Petkov, Nicolai (2026). [\"Advances in forecasting realized volatility: a review of methodologies\"](https:\/\/doi.org\/10.1186%2Fs40854-025-00809-5). *Financial Innovation*. **12**: 14. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1186\/s40854-025-00809-5](https:\/\/doi.org\/10.1186%2Fs40854-025-00809-5).\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=14 \"Edit section: External links\")\\]\n\n- [Graphical Comparison of Implied and Historical Volatility](https:\/\/archive.today\/20130204084449\/http:\/\/training.thomsonreuters.com\/video\/v.php?v=273), video\n- [Diebold, Francis X.; Hickman, Andrew; Inoue, Atsushi & Schuermannm, Til (1996) \"Converting 1-Day Volatility to h-Day Volatility: Scaling by sqrt(h) is Worse than You Think\"](http:\/\/citeseer.ist.psu.edu\/244698.html)\n- [A short introduction to alternative mathematical concepts of volatility](http:\/\/staff.science.uva.nl\/~marvisse\/volatility.html)\n- [Volatility estimation from predicted return density](http:\/\/www.macroaxis.com\/invest\/market\/GOOG--symbolVolatility) Example based on Google daily return distribution using standard density function\n- [Research paper including excerpt from report entitled Identifying Rich and Cheap Volatility](http:\/\/www.iijournals.com\/doi\/abs\/10.3905\/JOT.2010.5.2.035) Excerpt from Enhanced Call Overwriting, a report by Ryan Renicker and Devapriya Mallick at Lehman Brothers (2005).\n\n## Further reading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=15 \"Edit section: Further reading\")\\]\n\n- Bartram, Söhnke M.; Brown, Gregory W.; Stulz, Rene M. (August 2012). [\"Why Are U.S. Stocks More Volatile?\"](https:\/\/mpra.ub.uni-muenchen.de\/47341\/2\/MPRA_paper_47341.pdf) (PDF). *Journal of Finance*. **67** (4): 1329–1370\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1111\/j.1540-6261.2012.01749.x](https:\/\/doi.org\/10.1111%2Fj.1540-6261.2012.01749.x). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [18587238](https:\/\/api.semanticscholar.org\/CorpusID:18587238). [SSRN](https:\/\/en.wikipedia.org\/wiki\/SSRN_\\(identifier\\) \"SSRN (identifier)\") [2257549](https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=2257549).\n- Natenberg, Sheldon (2015). *Option Volatility and Pricing: Advanced Trading Strategies and Techniques* (Second ed.). New York. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0071818773](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0071818773 \"Special:BookSources\/978-0071818773\")\n  \n  .\n  `{{cite book}}`: CS1 maint: location missing publisher ([link](https:\/\/en.wikipedia.org\/wiki\/Category:CS1_maint:_location_missing_publisher \"Category:CS1 maint: location missing publisher\"))\n- Fassas, Athanasios P.; Siriopoulos, Costas (1 February 2021). [\"Implied volatility indices – A review\"](https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1062976920300855). *The Quarterly Review of Economics and Finance*. **79**: 303–329\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/j.qref.2020.07.004](https:\/\/doi.org\/10.1016%2Fj.qref.2020.07.004). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1062-9769](https:\/\/search.worldcat.org\/issn\/1062-9769).\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Volatility \"Template:Volatility\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Volatility \"Template talk:Volatility\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Volatility \"Special:EditPage\/Template:Volatility\")[Volatility]() |   |\n|---|---|\n| Modelling volatility | [Implied volatility](https:\/\/en.wikipedia.org\/wiki\/Implied_volatility \"Implied volatility\") [Volatility smile](https:\/\/en.wikipedia.org\/wiki\/Volatility_smile \"Volatility smile\") [Volatility clustering](https:\/\/en.wikipedia.org\/wiki\/Volatility_clustering \"Volatility clustering\") [Local volatility](https:\/\/en.wikipedia.org\/wiki\/Local_volatility \"Local volatility\") [Stochastic volatility](https:\/\/en.wikipedia.org\/wiki\/Stochastic_volatility \"Stochastic volatility\") [Jump-diffusion models](https:\/\/en.wikipedia.org\/wiki\/Jump-diffusion_models \"Jump-diffusion models\") [ARCH and GARCH](https:\/\/en.wikipedia.org\/wiki\/Autoregressive_conditional_heteroskedasticity \"Autoregressive conditional heteroskedasticity\") |\n| Trading volatility | [Volatility arbitrage](https:\/\/en.wikipedia.org\/wiki\/Volatility_arbitrage \"Volatility arbitrage\") [Straddle](https:\/\/en.wikipedia.org\/wiki\/Straddle \"Straddle\") [Volatility swap](https:\/\/en.wikipedia.org\/wiki\/Volatility_swap \"Volatility swap\") [IVX](https:\/\/en.wikipedia.org\/wiki\/IVX \"IVX\") [VIX](https:\/\/en.wikipedia.org\/wiki\/VIX \"VIX\") |\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Financial_markets_navigation \"Template:Financial markets navigation\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Financial_markets_navigation \"Template talk:Financial markets navigation\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Financial_markets_navigation \"Special:EditPage\/Template:Financial markets navigation\")[Financial markets](https:\/\/en.wikipedia.org\/wiki\/Financial_market \"Financial market\") |   |\n|---|---|\n| Types of [markets](https:\/\/en.wikipedia.org\/wiki\/Capital_market \"Capital market\") | [Primary market](https:\/\/en.wikipedia.org\/wiki\/Primary_market \"Primary market\") [Secondary market](https:\/\/en.wikipedia.org\/wiki\/Secondary_market \"Secondary market\") [Third market](https:\/\/en.wikipedia.org\/wiki\/Third_market \"Third market\") [Fourth market](https:\/\/en.wikipedia.org\/wiki\/Fourth_market \"Fourth market\") |\n| Types of [stocks](https:\/\/en.wikipedia.org\/wiki\/Stock \"Stock\") | [Common stock](https:\/\/en.wikipedia.org\/wiki\/Common_stock \"Common stock\") [Golden share](https:\/\/en.wikipedia.org\/wiki\/Golden_share \"Golden share\") [Preferred stock](https:\/\/en.wikipedia.org\/wiki\/Preferred_stock \"Preferred stock\") [Restricted stock](https:\/\/en.wikipedia.org\/wiki\/Restricted_stock \"Restricted stock\") [Tracking stock](https:\/\/en.wikipedia.org\/wiki\/Tracking_stock \"Tracking stock\") |\n| [Share capital](https:\/\/en.wikipedia.org\/wiki\/Share_capital \"Share capital\") | [Authorised capital](https:\/\/en.wikipedia.org\/wiki\/Authorised_capital \"Authorised capital\") [Issued shares](https:\/\/en.wikipedia.org\/wiki\/Issued_shares \"Issued shares\") [Shares outstanding](https:\/\/en.wikipedia.org\/wiki\/Shares_outstanding \"Shares outstanding\") [Treasury stock](https:\/\/en.wikipedia.org\/wiki\/Treasury_stock \"Treasury stock\") |\n| Participants | [Broker](https:\/\/en.wikipedia.org\/wiki\/Broker \"Broker\") [Floor broker](https:\/\/en.wikipedia.org\/wiki\/Floor_broker \"Floor broker\") [Inter-dealer broker](https:\/\/en.wikipedia.org\/wiki\/Inter-dealer_broker \"Inter-dealer broker\") [Broker-dealer](https:\/\/en.wikipedia.org\/wiki\/Broker-dealer \"Broker-dealer\") [Market maker](https:\/\/en.wikipedia.org\/wiki\/Market_maker \"Market maker\") [Trader](https:\/\/en.wikipedia.org\/wiki\/Trader_\\(finance\\) \"Trader (finance)\") [Floor trader](https:\/\/en.wikipedia.org\/wiki\/Floor_trader \"Floor trader\") [Proprietary trader](https:\/\/en.wikipedia.org\/wiki\/Proprietary_trading \"Proprietary trading\") [Quantitative analyst](https:\/\/en.wikipedia.org\/wiki\/Quantitative_analysis_\\(finance\\) \"Quantitative analysis (finance)\") [Investor](https:\/\/en.wikipedia.org\/wiki\/Investor \"Investor\") [Hedger](https:\/\/en.wikipedia.org\/wiki\/Hedge_\\(finance\\) \"Hedge (finance)\") [Speculator](https:\/\/en.wikipedia.org\/wiki\/Speculation \"Speculation\") [Arbitrager](https:\/\/en.wikipedia.org\/wiki\/Arbitrage \"Arbitrage\") [Scalper](https:\/\/en.wikipedia.org\/wiki\/Scalping_\\(trading\\) \"Scalping (trading)\") [Regulator](https:\/\/en.wikipedia.org\/wiki\/Financial_regulation \"Financial regulation\") |\n| Trading venues | [Exchange](https:\/\/en.wikipedia.org\/wiki\/Exchange_\\(organized_market\\) \"Exchange (organized market)\") [List of major stock exchanges](https:\/\/en.wikipedia.org\/wiki\/List_of_major_stock_exchanges \"List of major stock exchanges\") [Over-the-counter](https:\/\/en.wikipedia.org\/wiki\/Over-the-counter_\\(finance\\) \"Over-the-counter (finance)\") (off-exchange) [Alternative trading system](https:\/\/en.wikipedia.org\/wiki\/Alternative_trading_system \"Alternative trading system\") (ATS) [Multilateral trading facility](https:\/\/en.wikipedia.org\/wiki\/Multilateral_trading_facility \"Multilateral trading facility\") (MTF) [Electronic communication network](https:\/\/en.wikipedia.org\/wiki\/Electronic_communication_network \"Electronic communication network\") (ECN) [Direct market access](https:\/\/en.wikipedia.org\/wiki\/Direct_market_access \"Direct market access\") (DMA) [Straight-through processing](https:\/\/en.wikipedia.org\/wiki\/Straight-through_processing \"Straight-through processing\") (STP) [Dark pool](https:\/\/en.wikipedia.org\/wiki\/Dark_pool \"Dark pool\") (private exchange) [Crossing network](https:\/\/en.wikipedia.org\/wiki\/Crossing_network \"Crossing network\") [Liquidity aggregator](https:\/\/en.wikipedia.org\/wiki\/Foreign_exchange_aggregator \"Foreign exchange aggregator\") |\n| [Stock valuation](https:\/\/en.wikipedia.org\/wiki\/Stock_valuation \"Stock valuation\") | [Alpha](https:\/\/en.wikipedia.org\/wiki\/Alpha_\\(finance\\) \"Alpha (finance)\") [Arbitrage pricing theory](https:\/\/en.wikipedia.org\/wiki\/Arbitrage_pricing_theory \"Arbitrage pricing theory\") (APT) [Beta](https:\/\/en.wikipedia.org\/wiki\/Beta_\\(finance\\) \"Beta (finance)\") [Buffett indicator](https:\/\/en.wikipedia.org\/wiki\/Buffett_indicator \"Buffett indicator\") (Cap-to-GDP) [Book value](https:\/\/en.wikipedia.org\/wiki\/Book_value \"Book value\") (BV) [Capital asset pricing model](https:\/\/en.wikipedia.org\/wiki\/Capital_asset_pricing_model \"Capital asset pricing model\") (CAPM) [Capital market line](https:\/\/en.wikipedia.org\/wiki\/Capital_market_line \"Capital market line\") (CML) [Dividend discount model](https:\/\/en.wikipedia.org\/wiki\/Dividend_discount_model \"Dividend discount model\") (DDM) [Dividend yield](https:\/\/en.wikipedia.org\/wiki\/Dividend_yield \"Dividend yield\") [Earnings yield](https:\/\/en.wikipedia.org\/wiki\/Earnings_yield \"Earnings yield\") [EV\/EBITDA](https:\/\/en.wikipedia.org\/wiki\/EV\/EBITDA \"EV\/EBITDA\") [Fed model](https:\/\/en.wikipedia.org\/wiki\/Fed_model \"Fed model\") [Net asset value](https:\/\/en.wikipedia.org\/wiki\/Net_asset_value \"Net asset value\") (NAV) [Security characteristic line](https:\/\/en.wikipedia.org\/wiki\/Security_characteristic_line \"Security characteristic line\") [Security market line](https:\/\/en.wikipedia.org\/wiki\/Security_market_line \"Security market line\") (SML) [T-model](https:\/\/en.wikipedia.org\/wiki\/T-model \"T-model\") |\n| Trading theories  and [strategies](https:\/\/en.wikipedia.org\/wiki\/Trading_strategy \"Trading strategy\") | [Algorithmic trading](https:\/\/en.wikipedia.org\/wiki\/Algorithmic_trading \"Algorithmic trading\") [Buy and hold](https:\/\/en.wikipedia.org\/wiki\/Buy_and_hold \"Buy and hold\") [Contrarian investing](https:\/\/en.wikipedia.org\/wiki\/Contrarian_investing \"Contrarian investing\") [Dollar cost averaging](https:\/\/en.wikipedia.org\/wiki\/Dollar_cost_averaging \"Dollar cost averaging\") [Efficient-market hypothesis](https:\/\/en.wikipedia.org\/wiki\/Efficient-market_hypothesis \"Efficient-market hypothesis\") (EMH) [Fundamental analysis](https:\/\/en.wikipedia.org\/wiki\/Fundamental_analysis \"Fundamental analysis\") [Growth stock](https:\/\/en.wikipedia.org\/wiki\/Growth_stock \"Growth stock\") [Market timing](https:\/\/en.wikipedia.org\/wiki\/Market_timing \"Market timing\") [Modern portfolio theory](https:\/\/en.wikipedia.org\/wiki\/Modern_portfolio_theory \"Modern portfolio theory\") (MPT) [Momentum investing](https:\/\/en.wikipedia.org\/wiki\/Momentum_investing \"Momentum investing\") [Mosaic theory](https:\/\/en.wikipedia.org\/wiki\/Mosaic_theory_\\(investments\\) \"Mosaic theory (investments)\") [Pairs trade](https:\/\/en.wikipedia.org\/wiki\/Pairs_trade \"Pairs trade\") [Post-modern portfolio theory](https:\/\/en.wikipedia.org\/wiki\/Post-modern_portfolio_theory \"Post-modern portfolio theory\") (PMPT) [Random walk hypothesis](https:\/\/en.wikipedia.org\/wiki\/Random_walk_hypothesis \"Random walk hypothesis\") (RMH) [Sector rotation](https:\/\/en.wikipedia.org\/wiki\/Sector_rotation \"Sector rotation\") [Style investing](https:\/\/en.wikipedia.org\/wiki\/Style_investing \"Style investing\") [Swing trading](https:\/\/en.wikipedia.org\/wiki\/Swing_trading \"Swing trading\") [Technical analysis](https:\/\/en.wikipedia.org\/wiki\/Technical_analysis \"Technical analysis\") [Trend following](https:\/\/en.wikipedia.org\/wiki\/Trend_following \"Trend following\") [Value averaging](https:\/\/en.wikipedia.org\/wiki\/Value_averaging \"Value averaging\") [Value investing](https:\/\/en.wikipedia.org\/wiki\/Value_investing \"Value investing\") |\n| Related terms | [Bid–ask spread](https:\/\/en.wikipedia.org\/wiki\/Bid%E2%80%93ask_spread \"Bid–ask spread\") [Block trade](https:\/\/en.wikipedia.org\/wiki\/Block_trade \"Block trade\") [Cross listing](https:\/\/en.wikipedia.org\/wiki\/Cross_listing \"Cross listing\") [Dividend](https:\/\/en.wikipedia.org\/wiki\/Dividend \"Dividend\") [Dual-listed company](https:\/\/en.wikipedia.org\/wiki\/Dual-listed_company \"Dual-listed company\") [DuPont analysis](https:\/\/en.wikipedia.org\/wiki\/DuPont_analysis \"DuPont analysis\") [Efficient frontier](https:\/\/en.wikipedia.org\/wiki\/Efficient_frontier \"Efficient frontier\") [Financial law](https:\/\/en.wikipedia.org\/wiki\/Financial_law \"Financial law\") [Flight-to-quality](https:\/\/en.wikipedia.org\/wiki\/Flight-to-quality \"Flight-to-quality\") [Government bond](https:\/\/en.wikipedia.org\/wiki\/Government_bond \"Government bond\") [Greenspan put](https:\/\/en.wikipedia.org\/wiki\/Greenspan_put \"Greenspan put\") [Haircut](https:\/\/en.wikipedia.org\/wiki\/Haircut_\\(finance\\) \"Haircut (finance)\") [Initial public offering](https:\/\/en.wikipedia.org\/wiki\/Initial_public_offering \"Initial public offering\") (IPO) [Long](https:\/\/en.wikipedia.org\/wiki\/Long_\\(finance\\) \"Long (finance)\") [Mandatory offer](https:\/\/en.wikipedia.org\/wiki\/Mandatory_offer \"Mandatory offer\") [Margin](https:\/\/en.wikipedia.org\/wiki\/Margin_\\(finance\\) \"Margin (finance)\") [Market anomaly](https:\/\/en.wikipedia.org\/wiki\/Market_anomaly \"Market anomaly\") [Market capitalization](https:\/\/en.wikipedia.org\/wiki\/Market_capitalization \"Market capitalization\") [Market depth](https:\/\/en.wikipedia.org\/wiki\/Market_depth \"Market depth\") [Market manipulation](https:\/\/en.wikipedia.org\/wiki\/Market_manipulation \"Market manipulation\") [Market trend](https:\/\/en.wikipedia.org\/wiki\/Market_trend \"Market trend\") [Mean reversion](https:\/\/en.wikipedia.org\/wiki\/Mean_reversion_\\(finance\\) \"Mean reversion (finance)\") [Momentum](https:\/\/en.wikipedia.org\/wiki\/Momentum_\\(finance\\) \"Momentum (finance)\") [Open outcry](https:\/\/en.wikipedia.org\/wiki\/Open_outcry \"Open outcry\") [Order book](https:\/\/en.wikipedia.org\/wiki\/Order_book \"Order book\") [Position](https:\/\/en.wikipedia.org\/wiki\/Position_\\(finance\\) \"Position (finance)\") [Public float](https:\/\/en.wikipedia.org\/wiki\/Public_float \"Public float\") [Public offering](https:\/\/en.wikipedia.org\/wiki\/Public_offering \"Public offering\") [Rally](https:\/\/en.wikipedia.org\/wiki\/Rally_\\(stock_market\\) \"Rally (stock market)\") [Returns-based style analysis](https:\/\/en.wikipedia.org\/wiki\/Returns-based_style_analysis \"Returns-based style analysis\") [Reverse stock split](https:\/\/en.wikipedia.org\/wiki\/Reverse_stock_split \"Reverse stock split\") [Share repurchase](https:\/\/en.wikipedia.org\/wiki\/Share_repurchase \"Share repurchase\") [Short selling](https:\/\/en.wikipedia.org\/wiki\/Short_\\(finance\\) \"Short (finance)\") [Short squeeze](https:\/\/en.wikipedia.org\/wiki\/Short_squeeze \"Short squeeze\") [Slippage](https:\/\/en.wikipedia.org\/wiki\/Slippage_\\(finance\\) \"Slippage (finance)\") [Speculation](https:\/\/en.wikipedia.org\/wiki\/Speculation \"Speculation\") [Squeeze-out](https:\/\/en.wikipedia.org\/wiki\/Squeeze-out \"Squeeze-out\") [Stock dilution](https:\/\/en.wikipedia.org\/wiki\/Stock_dilution \"Stock dilution\") [Stock exchange](https:\/\/en.wikipedia.org\/wiki\/Stock_exchange \"Stock exchange\") [Stock market index](https:\/\/en.wikipedia.org\/wiki\/Stock_market_index \"Stock market index\") [Stock split](https:\/\/en.wikipedia.org\/wiki\/Stock_split \"Stock split\") [Stock swap](https:\/\/en.wikipedia.org\/wiki\/Stock_swap \"Stock swap\") [Trade](https:\/\/en.wikipedia.org\/wiki\/Trade_\\(finance\\) \"Trade (finance)\") [Tender offer](https:\/\/en.wikipedia.org\/wiki\/Tender_offer \"Tender offer\") [Uptick rule](https:\/\/en.wikipedia.org\/wiki\/Uptick_rule \"Uptick rule\") [Volatility]() [Voting interest](https:\/\/en.wikipedia.org\/wiki\/Voting_interest \"Voting interest\") [Yield](https:\/\/en.wikipedia.org\/wiki\/Yield_\\(finance\\) \"Yield (finance)\") |\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]() | [Average true range](https:\/\/en.wikipedia.org\/wiki\/Average_true_range \"Average true range\") (ATR) [Bollinger Bands](https:\/\/en.wikipedia.org\/wiki\/Bollinger_Bands \"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 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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=Volatility_\\(finance\\)&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\nVolatility (finance)\n\n33 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\))","attrs_readable_markdown":"From Wikipedia, the free encyclopedia\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/9\/9b\/VIX.png\/250px-VIX.png)](https:\/\/en.wikipedia.org\/wiki\/File:VIX.png)\n\nCBOE Volatility Index (VIX) from December 1985 to May 2012 (daily closings)\n\nIn [finance](https:\/\/en.wikipedia.org\/wiki\/Finance \"Finance\"), **volatility** (usually denoted by \"[σ](https:\/\/en.wikipedia.org\/wiki\/Sigma \"Sigma\")\") is the [degree of variation](https:\/\/en.wikipedia.org\/wiki\/Variability_\\(statistics\\) \"Variability (statistics)\") of a trading price series over time, usually measured by the [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") of [logarithmic returns](https:\/\/en.wikipedia.org\/wiki\/Logarithmic_return \"Logarithmic return\").\n\nHistoric volatility measures a time series of past market prices. [Implied volatility](https:\/\/en.wikipedia.org\/wiki\/Implied_volatility \"Implied volatility\") looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option).\n\n## Volatility terminology\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=1 \"Edit section: Volatility terminology\")\\]\n\nVolatility as described here refers to the **actual volatility**, more specifically:\n\n- **actual current volatility** of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price.\n- **actual historical volatility** which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past\n  - near synonymous is **realized volatility**, the [square root](https:\/\/en.wikipedia.org\/wiki\/Square_root \"Square root\") of the [realized variance](https:\/\/en.wikipedia.org\/wiki\/Realized_variance \"Realized variance\"), in turn calculated using the sum of squared returns divided by the number of observations.\n- **actual future volatility** which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an [option](https:\/\/en.wikipedia.org\/wiki\/Option_\\(finance\\) \"Option (finance)\"))\n\nNow turning to [implied volatility](https:\/\/en.wikipedia.org\/wiki\/Implied_volatility \"Implied volatility\"), we have:\n\n- **historical implied volatility** which refers to the implied volatility observed from historical prices of the financial instrument (normally options)\n- **current implied volatility** which refers to the implied volatility observed from current prices of the financial instrument\n- **future implied volatility** which refers to the implied volatility observed from future prices of the financial instrument\n\nFor a financial instrument whose price follows a [Gaussian](https:\/\/en.wikipedia.org\/wiki\/Gaussian \"Gaussian\") [random walk](https:\/\/en.wikipedia.org\/wiki\/Random_walk \"Random walk\"), or [Wiener process](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\"), the width of the distribution increases as time increases. This is because there is an increasing [probability](https:\/\/en.wikipedia.org\/wiki\/Probability \"Probability\") that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.\n\nSince observed price changes do not follow Gaussian distributions, others such as the [Lévy distribution](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9vy_distribution \"Lévy distribution\") are often used.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-1) These can capture attributes such as \"[fat tails](https:\/\/en.wikipedia.org\/wiki\/Fat_tail \"Fat tail\")\". Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc.\n\n## Mathematical definition\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=2 \"Edit section: Mathematical definition\")\\]\n\nFor any fund that evolves randomly with time, volatility is defined as the [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") of a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times.\n\nThus, \"annualized\" volatility *σ*annually is the standard deviation of an instrument's yearly [logarithmic returns](https:\/\/en.wikipedia.org\/wiki\/Rate_of_return#Logarithmic_or_continuously_compounded_return \"Rate of return\").[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-2)\n\nThe generalized volatility *σ**T* for [time horizon](https:\/\/en.wikipedia.org\/wiki\/Time_horizon \"Time horizon\") *T* in years is expressed as:\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{T}}=\\\\sigma \\_{\\\\text{annually}}{\\\\sqrt {T}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/49e88f2ee170dbab0671e56003059a9ff2e632f3)\n\nTherefore, if the daily logarithmic returns of a stock have a standard deviation of *σ*daily and the time period of returns is *P* in trading days, the annualized volatility is\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{annually}}=\\\\sigma \\_{\\\\text{daily}}{\\\\sqrt {P}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/36940dadd58fbbfadcb59140749af3a4b3c09eaa)\n\nso\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{T}}=\\\\sigma \\_{\\\\text{daily}}{\\\\sqrt {PT}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b715335b9d0cbeb41da30e66b16d1a95097fc6f3)\n\nA common assumption is that *P* = 252 trading days in any given year. Then, if *σ*daily = 0.01, the annualized volatility is\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{annually}}=0.01{\\\\sqrt {252}}=0.1587.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77022ad5aadc3db0ebfe97fc132581e3f68042af)\n\nThe monthly volatility (i.e. ![{\\\\displaystyle T={\\\\tfrac {1}{12}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/858a58f00d971d2a99c0adae52b82982ba310e25) of a year) is\n\n![{\\\\displaystyle \\\\sigma \\_{\\\\text{monthly}}=0.01{\\\\sqrt {\\\\tfrac {252}{12}}}=0.0458.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4f8a9560640b89becb2624657dea5e0f92da32fe)\n\nThe formulas used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a [random walk](https:\/\/en.wikipedia.org\/wiki\/Random_walk \"Random walk\"), or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent *α* to extrapolate natural processes:\n\n![{\\\\displaystyle \\\\sigma \\_{T}=T^{1\/\\\\alpha }\\\\sigma .\\\\,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f0a94ec1c310a51eba65f801a92e89b85b5729c8)\n\nIf *α* = 2 the [Wiener process](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\") scaling relation is obtained, but some people believe *α* \\< 2 for financial activities such as stocks, indexes and so on. This was discovered by [Benoît Mandelbrot](https:\/\/en.wikipedia.org\/wiki\/Beno%C3%AEt_Mandelbrot \"Benoît Mandelbrot\"), who looked at cotton prices and found that they followed a [Lévy alpha-stable distribution](https:\/\/en.wikipedia.org\/wiki\/Stable_distribution \"Stable distribution\") with *α* = 1.7. (See New Scientist, 19 April 1997.)\n\nMuch research has been devoted to modelling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place.\n\nRoll (1984) shows that volatility is affected by [market microstructure](https:\/\/en.wikipedia.org\/wiki\/Market_microstructure \"Market microstructure\").[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-3) Glosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of [adverse selection](https:\/\/en.wikipedia.org\/wiki\/Adverse_selection \"Adverse selection\"), they adjust their trading ranges, which in turn increases the band of price oscillation.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-4)\n\nIn September 2019, [JPMorgan Chase](https:\/\/en.wikipedia.org\/wiki\/JPMorgan_Chase \"JPMorgan Chase\") determined the effect of [US President](https:\/\/en.wikipedia.org\/wiki\/US_President \"US President\") [Donald Trump](https:\/\/en.wikipedia.org\/wiki\/Donald_Trump \"Donald Trump\")'s [tweets](https:\/\/en.wikipedia.org\/wiki\/Twitter#Tweets \"Twitter\"), and called it the [Volfefe index](https:\/\/en.wikipedia.org\/wiki\/Volfefe_index \"Volfefe index\") combining volatility and the [covfefe](https:\/\/en.wikipedia.org\/wiki\/Covfefe \"Covfefe\") [meme](https:\/\/en.wikipedia.org\/wiki\/Meme \"Meme\").\n\n## Volatility for investors\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=4 \"Edit section: Volatility for investors\")\\]\n\nVolatility matters to investors for at least eight reasons,\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\] several of which are alternative statements of the same feature or are directly consequent on each other:\n\n1. The wider the swings in an investment's price, the harder emotionally it is to not worry;\n2. Price volatility of a trading instrument can help to determine position sizing in a portfolio;\n3. When cash flows from selling a security are needed at a specific future date to meet a known fixed liability, higher volatility means a greater chance of a shortfall;\n4. Higher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values;\n5. Higher volatility of returns after retirement may result in withdrawals having a larger permanent impact on the portfolio's value;\n6. Price volatility presents opportunities to anyone with inside information to buy assets cheaply and sell when overpriced;\n7. Volatility affects pricing of [options](https:\/\/en.wikipedia.org\/wiki\/Option_\\(finance\\) \"Option (finance)\"), being a parameter of the [Black–Scholes model](https:\/\/en.wikipedia.org\/wiki\/Black%E2%80%93Scholes_model \"Black–Scholes model\").\n\n## Volatility versus direction\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=5 \"Edit section: Volatility versus direction\")\\]\n\nVolatility does not measure the direction of price changes, merely their dispersion. This is because when calculating [standard deviation](https:\/\/en.wikipedia.org\/wiki\/Standard_deviation \"Standard deviation\") (or [variance](https:\/\/en.wikipedia.org\/wiki\/Variance \"Variance\")), all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time.\n\nFor example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. Ignoring compounding effects, this would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a [normal distribution](https:\/\/en.wikipedia.org\/wiki\/Normal_distribution \"Normal distribution\"); in reality stock price movements are found to be [leptokurtotic](https:\/\/en.wikipedia.org\/wiki\/Kurtosis \"Kurtosis\") (fat-tailed).\n\n## Volatility over time\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=6 \"Edit section: Volatility over time\")\\]\n\nAlthough the [Black-Scholes](https:\/\/en.wikipedia.org\/wiki\/Black-Scholes \"Black-Scholes\") equation assumes predictable constant volatility, this is not observed in real markets. Amongst more realistic models are [Emanuel Derman](https:\/\/en.wikipedia.org\/wiki\/Emanuel_Derman \"Emanuel Derman\") and [Iraj Kani](https:\/\/en.wikipedia.org\/w\/index.php?title=Iraj_Kani&action=edit&redlink=1 \"Iraj Kani (page does not exist)\")'s[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-derman-5) and [Bruno Dupire](https:\/\/en.wikipedia.org\/wiki\/Bruno_Dupire \"Bruno Dupire\")'s [local volatility](https:\/\/en.wikipedia.org\/wiki\/Local_volatility \"Local volatility\"), [Poisson process](https:\/\/en.wikipedia.org\/wiki\/Poisson_process \"Poisson process\") where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of [stochastic volatility](https:\/\/en.wikipedia.org\/wiki\/Stochastic_volatility \"Stochastic volatility\").[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-6)\n\nIt is common knowledge that many types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-7) In [foreign exchange market](https:\/\/en.wikipedia.org\/wiki\/Foreign_exchange_market \"Foreign exchange market\"), price changes are seasonally [heteroskedastic](https:\/\/en.wikipedia.org\/wiki\/Heteroscedasticity \"Heteroscedasticity\") with periods of one day and one week.[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-8)[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-9)\n\nPeriods when prices fall quickly (a [crash](https:\/\/en.wikipedia.org\/wiki\/Stock_market_crash \"Stock market crash\")) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible [bubble](https:\/\/en.wikipedia.org\/wiki\/Bubble_\\(economics\\) \"Bubble (economics)\")) may often be followed by prices going up even more, or going down by an unusual amount.\n\nMost typically, extreme movements do not appear 'out of nowhere'; they are presaged by larger movements than usual or by known uncertainty in specific future events. This is termed [autoregressive conditional heteroskedasticity](https:\/\/en.wikipedia.org\/wiki\/Autoregressive_conditional_heteroskedasticity \"Autoregressive conditional heteroskedasticity\"). Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again.\n\nMeasures of volatility depend not only on the period over which it is measured, but also on the selected time resolution, as the information flow between short-term and long-term traders is asymmetric.\\[*[clarification needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Please_clarify \"Wikipedia:Please clarify\")*\\] As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-10)\n\n## Alternative measures of volatility\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=7 \"Edit section: Alternative measures of volatility\")\\]\n\nSome authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns.[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-11) Another considers the regular sequence of directional-changes as the proxy for the instantaneous volatility.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-12)\n\n### Volatility as it Relates to Options Trading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=8 \"Edit section: Volatility as it Relates to Options Trading\")\\]\n\nOne method of measuring Volatility, often used by quant option trading firms, divides up volatility into two components. Clean volatility - the amount of volatility caused standard events like daily transactions and general noise - and dirty vol, the amount caused by specific events like earnings or policy announcements.[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-13) For instance, a company like [Microsoft](https:\/\/en.wikipedia.org\/wiki\/Microsoft \"Microsoft\") would have clean volatility caused by people buying and selling on a daily basis but dirty (or event vol) events like quarterly earnings or a possibly anti-trust announcement.\n\nBreaking down volatility into two components is useful in order to accurately price how much an option is worth, especially when identifying what events may contribute to a swing. The job of fundamental analysts at market makers and option trading boutique firms typically entails trying to assign numeric values to these numbers.\n\n## Crude volatility estimation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=9 \"Edit section: Crude volatility estimation\")\\]\n\nUsing a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down.\n\nTo annualize this, you can use the \"rule of 16\", that is, multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of *n* independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables.\n\nHowever importantly this does not capture (or in some cases may give excessive weight to) occasional large movements in market price which occur less frequently than once a year.\n\nThe average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation *σ*, the expected value of the [magnitude of the observations](https:\/\/en.wikipedia.org\/wiki\/Absolute_deviation \"Absolute deviation\") is √(2\/π)*σ* = 0.798*σ*. The net effect is that this crude approach underestimates the true volatility by about 20%.\n\n## Estimate of compound annual growth rate (CAGR)\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=10 \"Edit section: Estimate of compound annual growth rate (CAGR)\")\\]\n\nConsider the [Taylor series](https:\/\/en.wikipedia.org\/wiki\/Taylor_series \"Taylor series\"):\n\n![{\\\\displaystyle \\\\log(1+y)=y-{\\\\tfrac {1}{2}}y^{2}+{\\\\tfrac {1}{3}}y^{3}-{\\\\tfrac {1}{4}}y^{4}+\\\\cdots }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/496a72d2a2d6e2c835aff5e506331ff0ce9309fc)\n\nTaking only the first two terms one has:\n\n![{\\\\displaystyle \\\\mathrm {CAGR} \\\\approx \\\\mathrm {AR} -{\\\\tfrac {1}{2}}\\\\sigma ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bfd77bb23265be12dea88d2d34234f8094cb8d25)\n\nVolatility thus mathematically represents a drag on the CAGR (formalized as the \"[volatility tax](https:\/\/en.wikipedia.org\/wiki\/Volatility_tax \"Volatility tax\")\"). Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula:\n\n![{\\\\displaystyle \\\\mathrm {CAGR} \\\\approx \\\\mathrm {AR} -{\\\\tfrac {1}{2}}k\\\\sigma ^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/823832d028777a59c7a6bd47bf9d5b9f5e63c0a1)\n\nfor a rough estimate, where *k* is an empirical factor (typically five to ten).\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]\n\n## Criticisms of volatility forecasting models\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Volatility_\\(finance\\)&action=edit&section=11 \"Edit section: Criticisms of volatility forecasting models\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/ed\/Vix.png\/500px-Vix.png)](https:\/\/en.wikipedia.org\/wiki\/File:Vix.png)\n\nPerformance of [VIX](https:\/\/en.wikipedia.org\/wiki\/VIX \"VIX\") (left) compared to past volatility (right) as 30-day volatility predictors, for the period of Jan 1990-Sep 2009. Volatility is measured as the standard deviation of S\\&P500 one-day returns over a month's period. The blue lines indicate [linear regressions](https:\/\/en.wikipedia.org\/wiki\/Linear_regression \"Linear regression\"), resulting in the [correlation coefficients](https:\/\/en.wikipedia.org\/wiki\/Correlation \"Correlation\") *r* shown. Note that VIX has virtually the same predictive power as past volatility, insofar as the shown correlation coefficients are nearly identical.\n\nDespite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-14)[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-15) especially out-of-sample, where different data are used to estimate the models and to test them.[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-16) Other works have agreed, but claim critics failed to correctly implement the more complicated models.[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-17) Some practitioners and [portfolio managers](https:\/\/en.wikipedia.org\/wiki\/Portfolio_managers \"Portfolio managers\") seem to completely ignore or dismiss volatility forecasting models. For example, [Nassim Taleb](https:\/\/en.wikipedia.org\/wiki\/Nassim_Taleb \"Nassim Taleb\") famously titled one of his *[Journal of Portfolio Management](https:\/\/en.wikipedia.org\/wiki\/Journal_of_Portfolio_Management \"Journal of Portfolio Management\")* papers \"We Don't Quite Know What We are Talking About When We Talk About Volatility\".[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-18) In a similar note, [Emanuel Derman](https:\/\/en.wikipedia.org\/wiki\/Emanuel_Derman \"Emanuel Derman\") expressed his disillusion with the enormous supply of empirical models unsupported by theory.[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-19) He argues that, while \"theories are attempts to uncover the hidden principles underpinning the world around us, as Albert Einstein did with his theory of relativity\", we should remember that \"models are metaphors – analogies that describe one thing relative to another\". Machine learning and deep learning methods have also been applied to realized volatility forecasting.[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_note-20)\n\n- [Beta (finance)](https:\/\/en.wikipedia.org\/wiki\/Beta_\\(finance\\) \"Beta (finance)\") – Expected change in price of a stock relative to the whole market\n- [Dispersion](https:\/\/en.wikipedia.org\/wiki\/Statistical_dispersion \"Statistical dispersion\") – Statistical property quantifying how much a collection of data is spread out\n- [Excess volatility puzzle](https:\/\/en.wikipedia.org\/wiki\/Excess_volatility_puzzle \"Excess volatility puzzle\")\n- [Financial economics](https:\/\/en.wikipedia.org\/wiki\/Financial_economics \"Financial economics\") – Academic discipline concerned with the exchange of money\n- [IVX](https:\/\/en.wikipedia.org\/wiki\/IVX \"IVX\") – Intraday, VIX-like volatility index\n- [Jules Regnault](https:\/\/en.wikipedia.org\/wiki\/Jules_Regnault \"Jules Regnault\") – French stock broker's assistant\n- [Low volatility anomaly](https:\/\/en.wikipedia.org\/wiki\/Low_volatility_anomaly \"Low volatility anomaly\")\n- [Risk](https:\/\/en.wikipedia.org\/wiki\/Risk \"Risk\") – Possibility of something bad happening\n- [VIX](https:\/\/en.wikipedia.org\/wiki\/VIX \"VIX\") – Chicago Board Options Exchange Volatility index\n- [Volatility smile](https:\/\/en.wikipedia.org\/wiki\/Volatility_smile \"Volatility smile\") – Implied volatility patterns that arise in pricing financial options\n- [Volatility tax](https:\/\/en.wikipedia.org\/wiki\/Volatility_tax \"Volatility tax\") – Mathematical finance term\n- [Volatility risk](https:\/\/en.wikipedia.org\/wiki\/Volatility_risk \"Volatility risk\")\n- [Volatility beta](https:\/\/en.wikipedia.org\/wiki\/Volatility_beta \"Volatility beta\")\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-1)**\n   [\"Levy distribution\"](http:\/\/www.wilmottwiki.com\/wiki\/index.php?title=Levy_distribution). *wilmottwiki.com*.\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-2)**\n   [\"Calculating Historical Volatility: Step-by-Step Example\"](https:\/\/web.archive.org\/web\/20120330224816\/http:\/\/www.lfrankcabrera.com\/calc-hist-vol.pdf) (PDF). Archived from the original on 30 March 2012. Retrieved 18 August 2011.\n   \n   `{{cite web}}`: CS1 maint: bot: original URL status unknown ([link](https:\/\/en.wikipedia.org\/wiki\/Category:CS1_maint:_bot:_original_URL_status_unknown \"Category:CS1 maint: bot: original URL status unknown\"))\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-3)** Roll, R. (1984): \"A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market\", *Journal of Finance* **39** (4), 1127–1139\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-4)** Glosten, L. R. and P. R. Milgrom (1985): \"Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders\", *[Journal of Financial Economics](https:\/\/en.wikipedia.org\/wiki\/Journal_of_Financial_Economics \"Journal of Financial Economics\")* **14** (1), 71–100\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-derman_5-0)**\n   Derman, E.; Iraj Kani (1994). [\"\"Riding on a Smile.\" RISK, 7(2) Feb.1994, pp. 139–145, pp. 32–39\"](https:\/\/web.archive.org\/web\/20110710170610\/http:\/\/www.ederman.com\/new\/docs\/gs-volatility_smile.pdf) (PDF). Risk. Archived from [the original](http:\/\/www.ederman.com\/new\/docs\/gs-volatility_smile.pdf) (PDF) on 10 July 2011. Retrieved 1 June 2007.\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-6)**\n   [\"Volatility\"](http:\/\/www.wilmottwiki.com\/wiki\/index.php?title=Volatility). *wilmottwiki.com*.\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-7)**\n   [\"Taking Advantage Of Volatility Spikes With Credit Spreads\"](http:\/\/www.investopedia.com\/articles\/optioninvestor\/10\/volatility-spikes-credit-spreads.asp).\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-8)**\n   Müller, Ulrich A.; Dacorogna, Michel M.; Olsen, Richard B.; Pictet, Olivier V.; Schwarz, Matthias; Morgenegg, Claude (1 December 1990). \"Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis\". *Journal of Banking & Finance*. **14** (6): 1189–1208\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/0378-4266(90)90009-Q](https:\/\/doi.org\/10.1016%2F0378-4266%2890%2990009-Q). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0378-4266](https:\/\/search.worldcat.org\/issn\/0378-4266).\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-9)**\n   Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). [\"Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time\"](https:\/\/doi.org\/10.3390%2Fjrfm12020054). *Journal of Risk and Financial Management*. **12** (2): 54. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3390\/jrfm12020054](https:\/\/doi.org\/10.3390%2Fjrfm12020054). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10419\/239003](https:\/\/hdl.handle.net\/10419%2F239003).\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-10)**\n    Muller, Ulrich A.; Dacorogna, Michel; Dave, Rakhal D.; Olsen, Richard; Pictet, Olivier V.; von Weizsäcker, Jakob (1997). [\"Volatilities of different time resolutions -- Analyzing the dynamics of market components\"](https:\/\/econpapers.repec.org\/article\/eeeempfin\/v_3a4_3ay_3a1997_3ai_3a2-3_3ap_3a213-239.htm). *Journal of Empirical Finance*. **4** (2–3\\): 213–239\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/S0927-5398(97)00007-8](https:\/\/doi.org\/10.1016%2FS0927-5398%2897%2900007-8). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0927-5398](https:\/\/search.worldcat.org\/issn\/0927-5398).\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-11)**\n    Sarkissian, Jack (2016). \"Express Measurement of Market Volatility Using Ergodicity Concept\". [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.2139\/ssrn.2812353](https:\/\/doi.org\/10.2139%2Fssrn.2812353). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [168496910](https:\/\/api.semanticscholar.org\/CorpusID:168496910). [SSRN](https:\/\/en.wikipedia.org\/wiki\/SSRN_\\(identifier\\) \"SSRN (identifier)\") [2812353](https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=2812353).\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-12)**\n    Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). [\"Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time\"](https:\/\/doi.org\/10.3390%2Fjrfm12020054). *Journal of Risk and Financial Management*. **12** (2): 54. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3390\/jrfm12020054](https:\/\/doi.org\/10.3390%2Fjrfm12020054). [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10419\/239003](https:\/\/hdl.handle.net\/10419%2F239003).\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-13)**\n    [\"Cleaning Implied Vols\"](https:\/\/notion.moontowermeta.com\/cleaning-implied-vols). *Moontowermeta*. Retrieved 26 June 2024.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-14)**\n    Cumby, R.; Figlewski, S.; Hasbrouck, J. (1993). \"Forecasting Volatility and Correlations with EGARCH models\". *Journal of Derivatives*. **1** (2): 51–63\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.3905\/jod.1993.407877](https:\/\/doi.org\/10.3905%2Fjod.1993.407877). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [154028452](https:\/\/api.semanticscholar.org\/CorpusID:154028452).\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-15)**\n    Jorion, P. (1995). \"Predicting Volatility in Foreign Exchange Market\". *[Journal of Finance](https:\/\/en.wikipedia.org\/wiki\/Journal_of_Finance \"Journal of Finance\")*. **50** (2): 507–528\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1111\/j.1540-6261.1995.tb04793.x](https:\/\/doi.org\/10.1111%2Fj.1540-6261.1995.tb04793.x). [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [2329417](https:\/\/www.jstor.org\/stable\/2329417).\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-16)**\n    [Brooks, Chris](https:\/\/en.wikipedia.org\/wiki\/Chris_Brooks_\\(academic\\) \"Chris Brooks (academic)\"); Persand, Gita (2003). \"Volatility forecasting for risk management\". *Journal of Forecasting*. **22** (1): 1–22\\. [CiteSeerX](https:\/\/en.wikipedia.org\/wiki\/CiteSeerX_\\(identifier\\) \"CiteSeerX (identifier)\") [10\\.1.1.595.9113](https:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.595.9113). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/for.841](https:\/\/doi.org\/10.1002%2Ffor.841). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1099-131X](https:\/\/search.worldcat.org\/issn\/1099-131X). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [154615850](https:\/\/api.semanticscholar.org\/CorpusID:154615850).\n17. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-17)**\n    Andersen, Torben G.; Bollerslev, Tim (1998). \"Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts\". *[International Economic Review](https:\/\/en.wikipedia.org\/wiki\/International_Economic_Review \"International Economic Review\")*. **39** (4): 885–905\\. [CiteSeerX](https:\/\/en.wikipedia.org\/wiki\/CiteSeerX_\\(identifier\\) \"CiteSeerX (identifier)\") [10\\.1.1.28.454](https:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.28.454). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.2307\/2527343](https:\/\/doi.org\/10.2307%2F2527343). [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [2527343](https:\/\/www.jstor.org\/stable\/2527343).\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-18)** Goldstein, Daniel and Taleb, Nassim, (28 March 2007) [\"We Don't Quite Know What We are Talking About When We Talk About Volatility\"](https:\/\/ssrn.com\/abstract=970480). *Journal of Portfolio Management* **33** (4), 2007.\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-19)** Derman, Emanuel (2011): Models.Behaving.Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life”, Ed. Free Press.\n20. **[^](https:\/\/en.wikipedia.org\/wiki\/Volatility_\\(finance\\)#cite_ref-20)**\n    Leushuis, Radmir M.; Petkov, Nicolai (2026). [\"Advances in forecasting realized volatility: a review of methodologies\"](https:\/\/doi.org\/10.1186%2Fs40854-025-00809-5). *Financial Innovation*. **12**: 14. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1186\/s40854-025-00809-5](https:\/\/doi.org\/10.1186%2Fs40854-025-00809-5).\n\n- [Graphical Comparison of Implied and Historical Volatility](https:\/\/archive.today\/20130204084449\/http:\/\/training.thomsonreuters.com\/video\/v.php?v=273), video\n- [Diebold, Francis X.; Hickman, Andrew; Inoue, Atsushi & Schuermannm, Til (1996) \"Converting 1-Day Volatility to h-Day Volatility: Scaling by sqrt(h) is Worse than You Think\"](http:\/\/citeseer.ist.psu.edu\/244698.html)\n- [A short introduction to alternative mathematical concepts of volatility](http:\/\/staff.science.uva.nl\/~marvisse\/volatility.html)\n- [Volatility estimation from predicted return density](http:\/\/www.macroaxis.com\/invest\/market\/GOOG--symbolVolatility) Example based on Google daily return distribution using standard density function\n- [Research paper including excerpt from report entitled Identifying Rich and Cheap Volatility](http:\/\/www.iijournals.com\/doi\/abs\/10.3905\/JOT.2010.5.2.035) Excerpt from Enhanced Call Overwriting, a report by Ryan Renicker and Devapriya Mallick at Lehman Brothers (2005).\n\n- Bartram, Söhnke M.; Brown, Gregory W.; Stulz, Rene M. (August 2012). [\"Why Are U.S. Stocks More Volatile?\"](https:\/\/mpra.ub.uni-muenchen.de\/47341\/2\/MPRA_paper_47341.pdf) (PDF). *Journal of Finance*. **67** (4): 1329–1370\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1111\/j.1540-6261.2012.01749.x](https:\/\/doi.org\/10.1111%2Fj.1540-6261.2012.01749.x). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [18587238](https:\/\/api.semanticscholar.org\/CorpusID:18587238). [SSRN](https:\/\/en.wikipedia.org\/wiki\/SSRN_\\(identifier\\) \"SSRN (identifier)\") [2257549](https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=2257549).\n- Natenberg, Sheldon (2015). *Option Volatility and Pricing: Advanced Trading Strategies and Techniques* (Second ed.). New York. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [978-0071818773](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0071818773 \"Special:BookSources\/978-0071818773\")\n  \n  .\n  `{{cite book}}`: CS1 maint: location missing publisher ([link](https:\/\/en.wikipedia.org\/wiki\/Category:CS1_maint:_location_missing_publisher \"Category:CS1 maint: location missing publisher\"))\n- Fassas, Athanasios P.; Siriopoulos, Costas (1 February 2021). [\"Implied volatility indices – A review\"](https:\/\/www.sciencedirect.com\/science\/article\/pii\/S1062976920300855). *The Quarterly Review of Economics and Finance*. **79**: 303–329\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/j.qref.2020.07.004](https:\/\/doi.org\/10.1016%2Fj.qref.2020.07.004). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1062-9769](https:\/\/search.worldcat.org\/issn\/1062-9769).","meta_canonical":null}

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Boilerpipe Text	From Wikipedia, the free encyclopedia CBOE Volatility Index (VIX) from December 1985 to May 2012 (daily closings) In finance , volatility (usually denoted by " σ ") is the degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns . Historic volatility measures a time series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). Volatility terminology [ edit ] Volatility as described here refers to the actual volatility , more specifically: actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price. actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past near synonymous is realized volatility , the square root of the realized variance , in turn calculated using the sum of squared returns divided by the number of observations. actual future volatility which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an option ) Now turning to implied volatility , we have: historical implied volatility which refers to the implied volatility observed from historical prices of the financial instrument (normally options) current implied volatility which refers to the implied volatility observed from current prices of the financial instrument future implied volatility which refers to the implied volatility observed from future prices of the financial instrument For a financial instrument whose price follows a Gaussian random walk , or Wiener process , the width of the distribution increases as time increases. This is because there is an increasing probability that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero. Since observed price changes do not follow Gaussian distributions, others such as the Lévy distribution are often used. [ 1 ] These can capture attributes such as " fat tails ". Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc. Mathematical definition [ edit ] For any fund that evolves randomly with time, volatility is defined as the standard deviation of a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times. Thus, "annualized" volatility σ annually is the standard deviation of an instrument's yearly logarithmic returns . [ 2 ] The generalized volatility σ T for time horizon T in years is expressed as: Therefore, if the daily logarithmic returns of a stock have a standard deviation of σ daily and the time period of returns is P in trading days, the annualized volatility is so A common assumption is that P = 252 trading days in any given year. Then, if σ daily = 0.01, the annualized volatility is The monthly volatility (i.e. of a year) is The formulas used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a random walk , or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent α to extrapolate natural processes: If α = 2 the Wiener process scaling relation is obtained, but some people believe α < 2 for financial activities such as stocks, indexes and so on. This was discovered by Benoît Mandelbrot , who looked at cotton prices and found that they followed a Lévy alpha-stable distribution with α = 1.7. (See New Scientist, 19 April 1997.) Much research has been devoted to modelling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place. Roll (1984) shows that volatility is affected by market microstructure . [ 3 ] Glosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of adverse selection , they adjust their trading ranges, which in turn increases the band of price oscillation. [ 4 ] In September 2019, JPMorgan Chase determined the effect of US President Donald Trump 's tweets , and called it the Volfefe index combining volatility and the covfefe meme . Volatility for investors [ edit ] Volatility matters to investors for at least eight reasons, [ citation needed ] several of which are alternative statements of the same feature or are directly consequent on each other: The wider the swings in an investment's price, the harder emotionally it is to not worry; Price volatility of a trading instrument can help to determine position sizing in a portfolio; When cash flows from selling a security are needed at a specific future date to meet a known fixed liability, higher volatility means a greater chance of a shortfall; Higher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values; Higher volatility of returns after retirement may result in withdrawals having a larger permanent impact on the portfolio's value; Price volatility presents opportunities to anyone with inside information to buy assets cheaply and sell when overpriced; Volatility affects pricing of options , being a parameter of the Black–Scholes model . Volatility versus direction [ edit ] Volatility does not measure the direction of price changes, merely their dispersion. This is because when calculating standard deviation (or variance ), all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time. For example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. Ignoring compounding effects, this would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a normal distribution ; in reality stock price movements are found to be leptokurtotic (fat-tailed). Volatility over time [ edit ] Although the Black-Scholes equation assumes predictable constant volatility, this is not observed in real markets. Amongst more realistic models are Emanuel Derman and Iraj Kani 's [ 5 ] and Bruno Dupire 's local volatility , Poisson process where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of stochastic volatility . [ 6 ] It is common knowledge that many types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all. [ 7 ] In foreign exchange market , price changes are seasonally heteroskedastic with periods of one day and one week. [ 8 ] [ 9 ] Periods when prices fall quickly (a crash ) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible bubble ) may often be followed by prices going up even more, or going down by an unusual amount. Most typically, extreme movements do not appear 'out of nowhere'; they are presaged by larger movements than usual or by known uncertainty in specific future events. This is termed autoregressive conditional heteroskedasticity . Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again. Measures of volatility depend not only on the period over which it is measured, but also on the selected time resolution, as the information flow between short-term and long-term traders is asymmetric. [ clarification needed ] As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa. [ 10 ] Alternative measures of volatility [ edit ] Some authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns. [ 11 ] Another considers the regular sequence of directional-changes as the proxy for the instantaneous volatility. [ 12 ] Volatility as it Relates to Options Trading [ edit ] One method of measuring Volatility, often used by quant option trading firms, divides up volatility into two components. Clean volatility - the amount of volatility caused standard events like daily transactions and general noise - and dirty vol, the amount caused by specific events like earnings or policy announcements. [ 13 ] For instance, a company like Microsoft would have clean volatility caused by people buying and selling on a daily basis but dirty (or event vol) events like quarterly earnings or a possibly anti-trust announcement. Breaking down volatility into two components is useful in order to accurately price how much an option is worth, especially when identifying what events may contribute to a swing. The job of fundamental analysts at market makers and option trading boutique firms typically entails trying to assign numeric values to these numbers. Crude volatility estimation [ edit ] Using a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down. To annualize this, you can use the "rule of 16", that is, multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of n independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables. However importantly this does not capture (or in some cases may give excessive weight to) occasional large movements in market price which occur less frequently than once a year. The average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation σ , the expected value of the magnitude of the observations is √(2/ π ) σ = 0.798 σ . The net effect is that this crude approach underestimates the true volatility by about 20%. Estimate of compound annual growth rate (CAGR) [ edit ] Consider the Taylor series : Taking only the first two terms one has: Volatility thus mathematically represents a drag on the CAGR (formalized as the " volatility tax "). Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula: for a rough estimate, where k is an empirical factor (typically five to ten). [ citation needed ] Criticisms of volatility forecasting models [ edit ] Performance of VIX (left) compared to past volatility (right) as 30-day volatility predictors, for the period of Jan 1990-Sep 2009. Volatility is measured as the standard deviation of S&P500 one-day returns over a month's period. The blue lines indicate linear regressions , resulting in the correlation coefficients r shown. Note that VIX has virtually the same predictive power as past volatility, insofar as the shown correlation coefficients are nearly identical. Despite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility [ 14 ] [ 15 ] especially out-of-sample, where different data are used to estimate the models and to test them. [ 16 ] Other works have agreed, but claim critics failed to correctly implement the more complicated models. [ 17 ] Some practitioners and portfolio managers seem to completely ignore or dismiss volatility forecasting models. For example, Nassim Taleb famously titled one of his Journal of Portfolio Management papers "We Don't Quite Know What We are Talking About When We Talk About Volatility". [ 18 ] In a similar note, Emanuel Derman expressed his disillusion with the enormous supply of empirical models unsupported by theory. [ 19 ] He argues that, while "theories are attempts to uncover the hidden principles underpinning the world around us, as Albert Einstein did with his theory of relativity", we should remember that "models are metaphors – analogies that describe one thing relative to another". Machine learning and deep learning methods have also been applied to realized volatility forecasting. [ 20 ] Beta (finance) – Expected change in price of a stock relative to the whole market Dispersion – Statistical property quantifying how much a collection of data is spread out Excess volatility puzzle Financial economics – Academic discipline concerned with the exchange of money IVX – Intraday, VIX-like volatility index Jules Regnault – French stock broker's assistant Low volatility anomaly Risk – Possibility of something bad happening VIX – Chicago Board Options Exchange Volatility index Volatility smile – Implied volatility patterns that arise in pricing financial options Volatility tax – Mathematical finance term Volatility risk Volatility beta ^ "Levy distribution" . wilmottwiki.com . ^ "Calculating Historical Volatility: Step-by-Step Example" (PDF) . Archived from the original on 30 March 2012 . Retrieved 18 August 2011 . {{ cite web }} : CS1 maint: bot: original URL status unknown ( link ) ^ Roll, R. (1984): "A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market", Journal of Finance 39 (4), 1127–1139 ^ Glosten, L. R. and P. R. Milgrom (1985): "Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders", Journal of Financial Economics 14 (1), 71–100 ^ Derman, E.; Iraj Kani (1994). " "Riding on a Smile." RISK, 7(2) Feb.1994, pp. 139–145, pp. 32–39" (PDF) . Risk. Archived from the original (PDF) on 10 July 2011 . Retrieved 1 June 2007 . ^ "Volatility" . wilmottwiki.com . ^ "Taking Advantage Of Volatility Spikes With Credit Spreads" . ^ Müller, Ulrich A.; Dacorogna, Michel M.; Olsen, Richard B.; Pictet, Olivier V.; Schwarz, Matthias; Morgenegg, Claude (1 December 1990). "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis". Journal of Banking & Finance . 14 (6): 1189– 1208. doi : 10.1016/0378-4266(90)90009-Q . ISSN 0378-4266 . ^ Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). "Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time" . Journal of Risk and Financial Management . 12 (2): 54. doi : 10.3390/jrfm12020054 . hdl : 10419/239003 . ^ Muller, Ulrich A.; Dacorogna, Michel; Dave, Rakhal D.; Olsen, Richard; Pictet, Olivier V.; von Weizsäcker, Jakob (1997). "Volatilities of different time resolutions -- Analyzing the dynamics of market components" . Journal of Empirical Finance . 4 ( 2– 3): 213– 239. doi : 10.1016/S0927-5398(97)00007-8 . ISSN 0927-5398 . ^ Sarkissian, Jack (2016). "Express Measurement of Market Volatility Using Ergodicity Concept". doi : 10.2139/ssrn.2812353 . S2CID 168496910 . SSRN 2812353 . ^ Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). "Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time" . Journal of Risk and Financial Management . 12 (2): 54. doi : 10.3390/jrfm12020054 . hdl : 10419/239003 . ^ "Cleaning Implied Vols" . Moontowermeta . Retrieved 26 June 2024 . ^ Cumby, R.; Figlewski, S.; Hasbrouck, J. (1993). "Forecasting Volatility and Correlations with EGARCH models". Journal of Derivatives . 1 (2): 51– 63. doi : 10.3905/jod.1993.407877 . S2CID 154028452 . ^ Jorion, P. (1995). "Predicting Volatility in Foreign Exchange Market". Journal of Finance . 50 (2): 507– 528. doi : 10.1111/j.1540-6261.1995.tb04793.x . JSTOR 2329417 . ^ Brooks, Chris ; Persand, Gita (2003). "Volatility forecasting for risk management". Journal of Forecasting . 22 (1): 1– 22. CiteSeerX 10.1.1.595.9113 . doi : 10.1002/for.841 . ISSN 1099-131X . S2CID 154615850 . ^ Andersen, Torben G.; Bollerslev, Tim (1998). "Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts". International Economic Review . 39 (4): 885– 905. CiteSeerX 10.1.1.28.454 . doi : 10.2307/2527343 . JSTOR 2527343 . ^ Goldstein, Daniel and Taleb, Nassim, (28 March 2007) "We Don't Quite Know What We are Talking About When We Talk About Volatility" . Journal of Portfolio Management 33 (4), 2007. ^ Derman, Emanuel (2011): Models.Behaving.Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life”, Ed. Free Press. ^ Leushuis, Radmir M.; Petkov, Nicolai (2026). "Advances in forecasting realized volatility: a review of methodologies" . Financial Innovation . 12 : 14. doi : 10.1186/s40854-025-00809-5 . Graphical Comparison of Implied and Historical Volatility , video Diebold, Francis X.; Hickman, Andrew; Inoue, Atsushi & Schuermannm, Til (1996) "Converting 1-Day Volatility to h-Day Volatility: Scaling by sqrt(h) is Worse than You Think" A short introduction to alternative mathematical concepts of volatility Volatility estimation from predicted return density Example based on Google daily return distribution using standard density function Research paper including excerpt from report entitled Identifying Rich and Cheap Volatility Excerpt from Enhanced Call Overwriting, a report by Ryan Renicker and Devapriya Mallick at Lehman Brothers (2005). Bartram, Söhnke M.; Brown, Gregory W.; Stulz, Rene M. (August 2012). "Why Are U.S. Stocks More Volatile?" (PDF) . Journal of Finance . 67 (4): 1329– 1370. doi : 10.1111/j.1540-6261.2012.01749.x . S2CID 18587238 . SSRN 2257549 . Natenberg, Sheldon (2015). Option Volatility and Pricing: Advanced Trading Strategies and Techniques (Second ed.). New York. ISBN 978-0071818773 . {{ cite book }} : CS1 maint: location missing publisher ( link ) Fassas, Athanasios P.; Siriopoulos, Costas (1 February 2021). "Implied volatility indices – A review" . The Quarterly Review of Economics and Finance . 79 : 303– 329. doi : 10.1016/j.qref.2020.07.004 . ISSN 1062-9769 .
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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Volatility_\(finance\)) - [1 Volatility terminology](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Volatility_terminology) - [2 Mathematical definition](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Mathematical_definition) - [3 Volatility origin](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Volatility_origin) - [4 Volatility for investors](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Volatility_for_investors) - [5 Volatility versus direction](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Volatility_versus_direction) - [6 Volatility over time](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Volatility_over_time) - [7 Alternative measures of volatility](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Alternative_measures_of_volatility) Toggle Alternative measures of volatility subsection - [7\.1 Volatility as it Relates to Options Trading](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Volatility_as_it_Relates_to_Options_Trading) - [8 Crude volatility estimation](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Crude_volatility_estimation) - [9 Estimate of compound annual growth rate (CAGR)](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Estimate_of_compound_annual_growth_rate_\(CAGR\)) - [10 Criticisms of volatility forecasting models](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Criticisms_of_volatility_forecasting_models) - [11 See also](https://en.wikipedia.org/wiki/Volatility_\(finance\)#See_also) - [12 References](https://en.wikipedia.org/wiki/Volatility_\(finance\)#References) - [13 External links](https://en.wikipedia.org/wiki/Volatility_\(finance\)#External_links) - [14 Further reading](https://en.wikipedia.org/wiki/Volatility_\(finance\)#Further_reading) Toggle the table of contents # Volatility (finance) 33 languages - [العربية](https://ar.wikipedia.org/wiki/%D8%AA%D9%82%D9%84%D8%A8_\(%D8%A7%D9%84%D9%85%D8%A7%D9%84\) "تقلب (المال) – Arabic") - [الدارجة](https://ary.wikipedia.org/wiki/%D8%AA%D8%B7%D8%A7%D9%8A%D8%B1%D9%8A%D8%AA_\(%D9%81%D9%8A%D9%86%D9%88%D9%86%D8%B5\) "تطايريت (فينونص) – Moroccan Arabic") - 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[Get shortened URL](https://en.wikipedia.org/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FVolatility_%28finance%29) Print/export - [Download as PDF](https://en.wikipedia.org/w/index.php?title=Special:DownloadAsPdf&page=Volatility_%28finance%29&action=show-download-screen "Download this page as a PDF file") - [Printable version](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&printable=yes "Printable version of this page [p]") In other projects - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q756115 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Degree of variation of a trading price series over time [![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9b/VIX.png/250px-VIX.png)](https://en.wikipedia.org/wiki/File:VIX.png) CBOE Volatility Index (VIX) from December 1985 to May 2012 (daily closings) In [finance](https://en.wikipedia.org/wiki/Finance "Finance"), volatility (usually denoted by "[σ](https://en.wikipedia.org/wiki/Sigma "Sigma")") is the [degree of variation](https://en.wikipedia.org/wiki/Variability_\(statistics\) "Variability (statistics)") of a trading price series over time, usually measured by the [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") of [logarithmic returns](https://en.wikipedia.org/wiki/Logarithmic_return "Logarithmic return"). Historic volatility measures a time series of past market prices. [Implied volatility](https://en.wikipedia.org/wiki/Implied_volatility "Implied volatility") looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). ## Volatility terminology \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=1 "Edit section: Volatility terminology")\] Volatility as described here refers to the actual volatility, more specifically: - actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price. - actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past - near synonymous is realized volatility, the [square root](https://en.wikipedia.org/wiki/Square_root "Square root") of the [realized variance](https://en.wikipedia.org/wiki/Realized_variance "Realized variance"), in turn calculated using the sum of squared returns divided by the number of observations. - actual future volatility which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an [option](https://en.wikipedia.org/wiki/Option_\(finance\) "Option (finance)")) Now turning to [implied volatility](https://en.wikipedia.org/wiki/Implied_volatility "Implied volatility"), we have: - historical implied volatility which refers to the implied volatility observed from historical prices of the financial instrument (normally options) - current implied volatility which refers to the implied volatility observed from current prices of the financial instrument - future implied volatility which refers to the implied volatility observed from future prices of the financial instrument For a financial instrument whose price follows a [Gaussian](https://en.wikipedia.org/wiki/Gaussian "Gaussian") [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk"), or [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process"), the width of the distribution increases as time increases. This is because there is an increasing [probability](https://en.wikipedia.org/wiki/Probability "Probability") that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero. Since observed price changes do not follow Gaussian distributions, others such as the [Lévy distribution](https://en.wikipedia.org/wiki/L%C3%A9vy_distribution "Lévy distribution") are often used.[\[1\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-1) These can capture attributes such as "[fat tails](https://en.wikipedia.org/wiki/Fat_tail "Fat tail")". Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc. ## Mathematical definition \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=2 "Edit section: Mathematical definition")\] For any fund that evolves randomly with time, volatility is defined as the [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") of a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times. Thus, "annualized" volatility σannually is the standard deviation of an instrument's yearly [logarithmic returns](https://en.wikipedia.org/wiki/Rate_of_return#Logarithmic_or_continuously_compounded_return "Rate of return").[\[2\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-2) The generalized volatility σT for [time horizon](https://en.wikipedia.org/wiki/Time_horizon "Time horizon") T in years is expressed as: σ T \= σ annually T . {\\displaystyle \\sigma \_{\\text{T}}=\\sigma \_{\\text{annually}}{\\sqrt {T}}.} ![{\\displaystyle \\sigma \_{\\text{T}}=\\sigma \_{\\text{annually}}{\\sqrt {T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e88f2ee170dbab0671e56003059a9ff2e632f3) Therefore, if the daily logarithmic returns of a stock have a standard deviation of σdaily and the time period of returns is P in trading days, the annualized volatility is σ annually \= σ daily P . {\\displaystyle \\sigma \_{\\text{annually}}=\\sigma \_{\\text{daily}}{\\sqrt {P}}.} ![{\\displaystyle \\sigma \_{\\text{annually}}=\\sigma \_{\\text{daily}}{\\sqrt {P}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36940dadd58fbbfadcb59140749af3a4b3c09eaa) so σ T \= σ daily P T . {\\displaystyle \\sigma \_{\\text{T}}=\\sigma \_{\\text{daily}}{\\sqrt {PT}}.} ![{\\displaystyle \\sigma \_{\\text{T}}=\\sigma \_{\\text{daily}}{\\sqrt {PT}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b715335b9d0cbeb41da30e66b16d1a95097fc6f3) A common assumption is that P = 252 trading days in any given year. Then, if σdaily = 0.01, the annualized volatility is σ annually \= 0\.01 252 \= 0\.1587. {\\displaystyle \\sigma \_{\\text{annually}}=0.01{\\sqrt {252}}=0.1587.} ![{\\displaystyle \\sigma \_{\\text{annually}}=0.01{\\sqrt {252}}=0.1587.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77022ad5aadc3db0ebfe97fc132581e3f68042af) The monthly volatility (i.e. T \= 1 12 {\\displaystyle T={\\tfrac {1}{12}}} ![{\\displaystyle T={\\tfrac {1}{12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/858a58f00d971d2a99c0adae52b82982ba310e25) of a year) is σ monthly \= 0\.01 252 12 \= 0\.0458. {\\displaystyle \\sigma \_{\\text{monthly}}=0.01{\\sqrt {\\tfrac {252}{12}}}=0.0458.} ![{\\displaystyle \\sigma \_{\\text{monthly}}=0.01{\\sqrt {\\tfrac {252}{12}}}=0.0458.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f8a9560640b89becb2624657dea5e0f92da32fe) The formulas used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk"), or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent α to extrapolate natural processes: σ T \= T 1 / α σ . {\\displaystyle \\sigma \_{T}=T^{1/\\alpha }\\sigma .\\,} ![{\\displaystyle \\sigma \_{T}=T^{1/\\alpha }\\sigma .\\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a94ec1c310a51eba65f801a92e89b85b5729c8) If α = 2 the [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process") scaling relation is obtained, but some people believe α \< 2 for financial activities such as stocks, indexes and so on. This was discovered by [Benoît Mandelbrot](https://en.wikipedia.org/wiki/Beno%C3%AEt_Mandelbrot "Benoît Mandelbrot"), who looked at cotton prices and found that they followed a [Lévy alpha-stable distribution](https://en.wikipedia.org/wiki/Stable_distribution "Stable distribution") with α = 1.7. (See New Scientist, 19 April 1997.) ## Volatility origin \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=3 "Edit section: Volatility origin")\] Much research has been devoted to modelling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place. Roll (1984) shows that volatility is affected by [market microstructure](https://en.wikipedia.org/wiki/Market_microstructure "Market microstructure").[\[3\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-3) Glosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of [adverse selection](https://en.wikipedia.org/wiki/Adverse_selection "Adverse selection"), they adjust their trading ranges, which in turn increases the band of price oscillation.[\[4\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-4) In September 2019, [JPMorgan Chase](https://en.wikipedia.org/wiki/JPMorgan_Chase "JPMorgan Chase") determined the effect of [US President](https://en.wikipedia.org/wiki/US_President "US President") [Donald Trump](https://en.wikipedia.org/wiki/Donald_Trump "Donald Trump")'s [tweets](https://en.wikipedia.org/wiki/Twitter#Tweets "Twitter"), and called it the [Volfefe index](https://en.wikipedia.org/wiki/Volfefe_index "Volfefe index") combining volatility and the [covfefe](https://en.wikipedia.org/wiki/Covfefe "Covfefe") [meme](https://en.wikipedia.org/wiki/Meme "Meme"). ## Volatility for investors \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=4 "Edit section: Volatility for investors")\] Volatility matters to investors for at least eight reasons,\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\] several of which are alternative statements of the same feature or are directly consequent on each other: 1. The wider the swings in an investment's price, the harder emotionally it is to not worry; 2. Price volatility of a trading instrument can help to determine position sizing in a portfolio; 3. When cash flows from selling a security are needed at a specific future date to meet a known fixed liability, higher volatility means a greater chance of a shortfall; 4. Higher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values; 5. Higher volatility of returns after retirement may result in withdrawals having a larger permanent impact on the portfolio's value; 6. Price volatility presents opportunities to anyone with inside information to buy assets cheaply and sell when overpriced; 7. Volatility affects pricing of [options](https://en.wikipedia.org/wiki/Option_\(finance\) "Option (finance)"), being a parameter of the [Black–Scholes model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model "Black–Scholes model"). ## Volatility versus direction \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=5 "Edit section: Volatility versus direction")\] Volatility does not measure the direction of price changes, merely their dispersion. This is because when calculating [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") (or [variance](https://en.wikipedia.org/wiki/Variance "Variance")), all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time. For example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. Ignoring compounding effects, this would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution"); in reality stock price movements are found to be [leptokurtotic](https://en.wikipedia.org/wiki/Kurtosis "Kurtosis") (fat-tailed). ## Volatility over time \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=6 "Edit section: Volatility over time")\] Although the [Black-Scholes](https://en.wikipedia.org/wiki/Black-Scholes "Black-Scholes") equation assumes predictable constant volatility, this is not observed in real markets. Amongst more realistic models are [Emanuel Derman](https://en.wikipedia.org/wiki/Emanuel_Derman "Emanuel Derman") and [Iraj Kani](https://en.wikipedia.org/w/index.php?title=Iraj_Kani&action=edit&redlink=1 "Iraj Kani (page does not exist)")'s[\[5\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-derman-5) and [Bruno Dupire](https://en.wikipedia.org/wiki/Bruno_Dupire "Bruno Dupire")'s [local volatility](https://en.wikipedia.org/wiki/Local_volatility "Local volatility"), [Poisson process](https://en.wikipedia.org/wiki/Poisson_process "Poisson process") where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of [stochastic volatility](https://en.wikipedia.org/wiki/Stochastic_volatility "Stochastic volatility").[\[6\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-6) It is common knowledge that many types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all.[\[7\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-7) In [foreign exchange market](https://en.wikipedia.org/wiki/Foreign_exchange_market "Foreign exchange market"), price changes are seasonally [heteroskedastic](https://en.wikipedia.org/wiki/Heteroscedasticity "Heteroscedasticity") with periods of one day and one week.[\[8\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-8)[\[9\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-9) Periods when prices fall quickly (a [crash](https://en.wikipedia.org/wiki/Stock_market_crash "Stock market crash")) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible [bubble](https://en.wikipedia.org/wiki/Bubble_\(economics\) "Bubble (economics)")) may often be followed by prices going up even more, or going down by an unusual amount. Most typically, extreme movements do not appear 'out of nowhere'; they are presaged by larger movements than usual or by known uncertainty in specific future events. This is termed [autoregressive conditional heteroskedasticity](https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity "Autoregressive conditional heteroskedasticity"). Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again. Measures of volatility depend not only on the period over which it is measured, but also on the selected time resolution, as the information flow between short-term and long-term traders is asymmetric.\[[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")\] As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa.[\[10\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-10) ## Alternative measures of volatility \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=7 "Edit section: Alternative measures of volatility")\] Some authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns.[\[11\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-11) Another considers the regular sequence of directional-changes as the proxy for the instantaneous volatility.[\[12\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-12) ### Volatility as it Relates to Options Trading \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=8 "Edit section: Volatility as it Relates to Options Trading")\] One method of measuring Volatility, often used by quant option trading firms, divides up volatility into two components. Clean volatility - the amount of volatility caused standard events like daily transactions and general noise - and dirty vol, the amount caused by specific events like earnings or policy announcements.[\[13\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-13) For instance, a company like [Microsoft](https://en.wikipedia.org/wiki/Microsoft "Microsoft") would have clean volatility caused by people buying and selling on a daily basis but dirty (or event vol) events like quarterly earnings or a possibly anti-trust announcement. Breaking down volatility into two components is useful in order to accurately price how much an option is worth, especially when identifying what events may contribute to a swing. The job of fundamental analysts at market makers and option trading boutique firms typically entails trying to assign numeric values to these numbers. ## Crude volatility estimation \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=9 "Edit section: Crude volatility estimation")\] Using a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down. To annualize this, you can use the "rule of 16", that is, multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of n independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables. However importantly this does not capture (or in some cases may give excessive weight to) occasional large movements in market price which occur less frequently than once a year. The average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation σ, the expected value of the [magnitude of the observations](https://en.wikipedia.org/wiki/Absolute_deviation "Absolute deviation") is √(2/π)σ = 0.798σ. The net effect is that this crude approach underestimates the true volatility by about 20%. ## Estimate of compound annual growth rate (CAGR) \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=10 "Edit section: Estimate of compound annual growth rate (CAGR)")\] Consider the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series"): log ⁡ ( 1 \+ y ) \= y − 1 2 y 2 \+ 1 3 y 3 − 1 4 y 4 \+ ⋯ {\\displaystyle \\log(1+y)=y-{\\tfrac {1}{2}}y^{2}+{\\tfrac {1}{3}}y^{3}-{\\tfrac {1}{4}}y^{4}+\\cdots } ![{\\displaystyle \\log(1+y)=y-{\\tfrac {1}{2}}y^{2}+{\\tfrac {1}{3}}y^{3}-{\\tfrac {1}{4}}y^{4}+\\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/496a72d2a2d6e2c835aff5e506331ff0ce9309fc) Taking only the first two terms one has: C A G R ≈ A R − 1 2 σ 2 {\\displaystyle \\mathrm {CAGR} \\approx \\mathrm {AR} -{\\tfrac {1}{2}}\\sigma ^{2}} ![{\\displaystyle \\mathrm {CAGR} \\approx \\mathrm {AR} -{\\tfrac {1}{2}}\\sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd77bb23265be12dea88d2d34234f8094cb8d25) Volatility thus mathematically represents a drag on the CAGR (formalized as the "[volatility tax](https://en.wikipedia.org/wiki/Volatility_tax "Volatility tax")"). Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula: C A G R ≈ A R − 1 2 k σ 2 {\\displaystyle \\mathrm {CAGR} \\approx \\mathrm {AR} -{\\tfrac {1}{2}}k\\sigma ^{2}} ![{\\displaystyle \\mathrm {CAGR} \\approx \\mathrm {AR} -{\\tfrac {1}{2}}k\\sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/823832d028777a59c7a6bd47bf9d5b9f5e63c0a1) for a rough estimate, where k is an empirical factor (typically five to ten).\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\] ## Criticisms of volatility forecasting models \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=11 "Edit section: Criticisms of volatility forecasting models")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Vix.png/500px-Vix.png)](https://en.wikipedia.org/wiki/File:Vix.png) Performance of [VIX](https://en.wikipedia.org/wiki/VIX "VIX") (left) compared to past volatility (right) as 30-day volatility predictors, for the period of Jan 1990-Sep 2009. Volatility is measured as the standard deviation of S\&P500 one-day returns over a month's period. The blue lines indicate [linear regressions](https://en.wikipedia.org/wiki/Linear_regression "Linear regression"), resulting in the [correlation coefficients](https://en.wikipedia.org/wiki/Correlation "Correlation") r shown. Note that VIX has virtually the same predictive power as past volatility, insofar as the shown correlation coefficients are nearly identical. Despite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility[\[14\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-15) especially out-of-sample, where different data are used to estimate the models and to test them.[\[16\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-16) Other works have agreed, but claim critics failed to correctly implement the more complicated models.[\[17\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-17) Some practitioners and [portfolio managers](https://en.wikipedia.org/wiki/Portfolio_managers "Portfolio managers") seem to completely ignore or dismiss volatility forecasting models. For example, [Nassim Taleb](https://en.wikipedia.org/wiki/Nassim_Taleb "Nassim Taleb") famously titled one of his [Journal of Portfolio Management](https://en.wikipedia.org/wiki/Journal_of_Portfolio_Management "Journal of Portfolio Management") papers "We Don't Quite Know What We are Talking About When We Talk About Volatility".[\[18\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-18) In a similar note, [Emanuel Derman](https://en.wikipedia.org/wiki/Emanuel_Derman "Emanuel Derman") expressed his disillusion with the enormous supply of empirical models unsupported by theory.[\[19\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-19) He argues that, while "theories are attempts to uncover the hidden principles underpinning the world around us, as Albert Einstein did with his theory of relativity", we should remember that "models are metaphors – analogies that describe one thing relative to another". Machine learning and deep learning methods have also been applied to realized volatility forecasting.[\[20\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-20) ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=12 "Edit section: See also")\] - [Beta (finance)](https://en.wikipedia.org/wiki/Beta_\(finance\) "Beta (finance)") – Expected change in price of a stock relative to the whole market - [Dispersion](https://en.wikipedia.org/wiki/Statistical_dispersion "Statistical dispersion") – Statistical property quantifying how much a collection of data is spread out - [Excess volatility puzzle](https://en.wikipedia.org/wiki/Excess_volatility_puzzle "Excess volatility puzzle") - [Financial economics](https://en.wikipedia.org/wiki/Financial_economics "Financial economics") – Academic discipline concerned with the exchange of money - [IVX](https://en.wikipedia.org/wiki/IVX "IVX") – Intraday, VIX-like volatility index - [Jules Regnault](https://en.wikipedia.org/wiki/Jules_Regnault "Jules Regnault") – French stock broker's assistant - [Low volatility anomaly](https://en.wikipedia.org/wiki/Low_volatility_anomaly "Low volatility anomaly") - [Risk](https://en.wikipedia.org/wiki/Risk "Risk") – Possibility of something bad happening - [VIX](https://en.wikipedia.org/wiki/VIX "VIX") – Chicago Board Options Exchange Volatility index - [Volatility smile](https://en.wikipedia.org/wiki/Volatility_smile "Volatility smile") – Implied volatility patterns that arise in pricing financial options - [Volatility tax](https://en.wikipedia.org/wiki/Volatility_tax "Volatility tax") – Mathematical finance term - [Volatility risk](https://en.wikipedia.org/wiki/Volatility_risk "Volatility risk") - [Volatility beta](https://en.wikipedia.org/wiki/Volatility_beta "Volatility beta") ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=13 "Edit section: References")\] 1. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-1) ["Levy distribution"](http://www.wilmottwiki.com/wiki/index.php?title=Levy_distribution). wilmottwiki.com. 2. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-2) ["Calculating Historical Volatility: Step-by-Step Example"](https://web.archive.org/web/20120330224816/http://www.lfrankcabrera.com/calc-hist-vol.pdf) (PDF). Archived from the original on 30 March 2012. Retrieved 18 August 2011. `{{cite web}}`: CS1 maint: bot: original URL status unknown ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown "Category:CS1 maint: bot: original URL status unknown")) 3. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-3) Roll, R. (1984): "A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market", Journal of Finance 39 (4), 1127–1139 4. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-4) Glosten, L. R. and P. R. Milgrom (1985): "Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders", [Journal of Financial Economics](https://en.wikipedia.org/wiki/Journal_of_Financial_Economics "Journal of Financial Economics") 14 (1), 71–100 5. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-derman_5-0) Derman, E.; Iraj Kani (1994). [""Riding on a Smile." RISK, 7(2) Feb.1994, pp. 139–145, pp. 32–39"](https://web.archive.org/web/20110710170610/http://www.ederman.com/new/docs/gs-volatility_smile.pdf) (PDF). Risk. Archived from [the original](http://www.ederman.com/new/docs/gs-volatility_smile.pdf) (PDF) on 10 July 2011. Retrieved 1 June 2007. `{{cite journal}}`: Cite journal requires `\|journal=` ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#missing_periodical "Help:CS1 errors")) 6. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-6) ["Volatility"](http://www.wilmottwiki.com/wiki/index.php?title=Volatility). wilmottwiki.com. 7. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-7) ["Taking Advantage Of Volatility Spikes With Credit Spreads"](http://www.investopedia.com/articles/optioninvestor/10/volatility-spikes-credit-spreads.asp). 8. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-8) Müller, Ulrich A.; Dacorogna, Michel M.; Olsen, Richard B.; Pictet, Olivier V.; Schwarz, Matthias; Morgenegg, Claude (1 December 1990). "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis". Journal of Banking & Finance. 14 (6): 1189–1208\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/0378-4266(90)90009-Q](https://doi.org/10.1016%2F0378-4266%2890%2990009-Q). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0378-4266](https://search.worldcat.org/issn/0378-4266). 9. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-9) Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). ["Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time"](https://doi.org/10.3390%2Fjrfm12020054). Journal of Risk and Financial Management. 12 (2): 54. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3390/jrfm12020054](https://doi.org/10.3390%2Fjrfm12020054). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10419/239003](https://hdl.handle.net/10419%2F239003). 10. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-10) Muller, Ulrich A.; Dacorogna, Michel; Dave, Rakhal D.; Olsen, Richard; Pictet, Olivier V.; von Weizsäcker, Jakob (1997). ["Volatilities of different time resolutions -- Analyzing the dynamics of market components"](https://econpapers.repec.org/article/eeeempfin/v_3a4_3ay_3a1997_3ai_3a2-3_3ap_3a213-239.htm). Journal of Empirical Finance. 4 (2–3\): 213–239\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/S0927-5398(97)00007-8](https://doi.org/10.1016%2FS0927-5398%2897%2900007-8). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0927-5398](https://search.worldcat.org/issn/0927-5398). 11. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-11) Sarkissian, Jack (2016). "Express Measurement of Market Volatility Using Ergodicity Concept". [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2139/ssrn.2812353](https://doi.org/10.2139%2Fssrn.2812353). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [168496910](https://api.semanticscholar.org/CorpusID:168496910). [SSRN](https://en.wikipedia.org/wiki/SSRN_\(identifier\) "SSRN (identifier)") [2812353](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2812353). `{{cite journal}}`: Cite journal requires `\|journal=` ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#missing_periodical "Help:CS1 errors")) 12. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-12) Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). ["Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time"](https://doi.org/10.3390%2Fjrfm12020054). Journal of Risk and Financial Management. 12 (2): 54. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3390/jrfm12020054](https://doi.org/10.3390%2Fjrfm12020054). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10419/239003](https://hdl.handle.net/10419%2F239003). 13. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-13) ["Cleaning Implied Vols"](https://notion.moontowermeta.com/cleaning-implied-vols). Moontowermeta. Retrieved 26 June 2024. 14. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-14) Cumby, R.; Figlewski, S.; Hasbrouck, J. (1993). "Forecasting Volatility and Correlations with EGARCH models". Journal of Derivatives. 1 (2): 51–63\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3905/jod.1993.407877](https://doi.org/10.3905%2Fjod.1993.407877). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [154028452](https://api.semanticscholar.org/CorpusID:154028452). 15. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-15) Jorion, P. (1995). "Predicting Volatility in Foreign Exchange Market". [Journal of Finance](https://en.wikipedia.org/wiki/Journal_of_Finance "Journal of Finance"). 50 (2): 507–528\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1111/j.1540-6261.1995.tb04793.x](https://doi.org/10.1111%2Fj.1540-6261.1995.tb04793.x). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2329417](https://www.jstor.org/stable/2329417). 16. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-16) [Brooks, Chris](https://en.wikipedia.org/wiki/Chris_Brooks_\(academic\) "Chris Brooks (academic)"); Persand, Gita (2003). "Volatility forecasting for risk management". Journal of Forecasting. 22 (1): 1–22\. [CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_\(identifier\) "CiteSeerX (identifier)") [10\.1.1.595.9113](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.595.9113). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/for.841](https://doi.org/10.1002%2Ffor.841). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1099-131X](https://search.worldcat.org/issn/1099-131X). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [154615850](https://api.semanticscholar.org/CorpusID:154615850). 17. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-17) Andersen, Torben G.; Bollerslev, Tim (1998). "Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts". [International Economic Review](https://en.wikipedia.org/wiki/International_Economic_Review "International Economic Review"). 39 (4): 885–905\. [CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_\(identifier\) "CiteSeerX (identifier)") [10\.1.1.28.454](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.454). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/2527343](https://doi.org/10.2307%2F2527343). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2527343](https://www.jstor.org/stable/2527343). 18. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-18) Goldstein, Daniel and Taleb, Nassim, (28 March 2007) ["We Don't Quite Know What We are Talking About When We Talk About Volatility"](https://ssrn.com/abstract=970480). Journal of Portfolio Management 33 (4), 2007. 19. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-19) Derman, Emanuel (2011): Models.Behaving.Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life”, Ed. Free Press. 20. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-20) Leushuis, Radmir M.; Petkov, Nicolai (2026). ["Advances in forecasting realized volatility: a review of methodologies"](https://doi.org/10.1186%2Fs40854-025-00809-5). Financial Innovation. 12: 14. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1186/s40854-025-00809-5](https://doi.org/10.1186%2Fs40854-025-00809-5). ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=14 "Edit section: External links")\] - [Graphical Comparison of Implied and Historical Volatility](https://archive.today/20130204084449/http://training.thomsonreuters.com/video/v.php?v=273), video - [Diebold, Francis X.; Hickman, Andrew; Inoue, Atsushi & Schuermannm, Til (1996) "Converting 1-Day Volatility to h-Day Volatility: Scaling by sqrt(h) is Worse than You Think"](http://citeseer.ist.psu.edu/244698.html) - [A short introduction to alternative mathematical concepts of volatility](http://staff.science.uva.nl/~marvisse/volatility.html) - [Volatility estimation from predicted return density](http://www.macroaxis.com/invest/market/GOOG--symbolVolatility) Example based on Google daily return distribution using standard density function - [Research paper including excerpt from report entitled Identifying Rich and Cheap Volatility](http://www.iijournals.com/doi/abs/10.3905/JOT.2010.5.2.035) Excerpt from Enhanced Call Overwriting, a report by Ryan Renicker and Devapriya Mallick at Lehman Brothers (2005). ## Further reading \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=15 "Edit section: Further reading")\] - Bartram, Söhnke M.; Brown, Gregory W.; Stulz, Rene M. (August 2012). ["Why Are U.S. Stocks More Volatile?"](https://mpra.ub.uni-muenchen.de/47341/2/MPRA_paper_47341.pdf) (PDF). Journal of Finance. 67 (4): 1329–1370\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1111/j.1540-6261.2012.01749.x](https://doi.org/10.1111%2Fj.1540-6261.2012.01749.x). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [18587238](https://api.semanticscholar.org/CorpusID:18587238). [SSRN](https://en.wikipedia.org/wiki/SSRN_\(identifier\) "SSRN (identifier)") [2257549](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2257549). - Natenberg, Sheldon (2015). Option Volatility and Pricing: Advanced Trading Strategies and Techniques (Second ed.). New York. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0071818773](https://en.wikipedia.org/wiki/Special:BookSources/978-0071818773 "Special:BookSources/978-0071818773") . `{{cite book}}`: CS1 maint: location missing publisher ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_location_missing_publisher "Category:CS1 maint: location missing publisher")) - Fassas, Athanasios P.; Siriopoulos, Costas (1 February 2021). ["Implied volatility indices – A review"](https://www.sciencedirect.com/science/article/pii/S1062976920300855). The Quarterly Review of Economics and Finance. 79: 303–329\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.qref.2020.07.004](https://doi.org/10.1016%2Fj.qref.2020.07.004). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1062-9769](https://search.worldcat.org/issn/1062-9769). \| [v](https://en.wikipedia.org/wiki/Template:Volatility "Template:Volatility") [t](https://en.wikipedia.org/wiki/Template_talk:Volatility "Template talk:Volatility") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Volatility "Special:EditPage/Template:Volatility")[Volatility]() \| \| \|---\|---\| \| Modelling volatility \| [Implied volatility](https://en.wikipedia.org/wiki/Implied_volatility "Implied volatility") [Volatility smile](https://en.wikipedia.org/wiki/Volatility_smile "Volatility smile") [Volatility clustering](https://en.wikipedia.org/wiki/Volatility_clustering "Volatility clustering") [Local volatility](https://en.wikipedia.org/wiki/Local_volatility "Local volatility") [Stochastic volatility](https://en.wikipedia.org/wiki/Stochastic_volatility "Stochastic volatility") [Jump-diffusion models](https://en.wikipedia.org/wiki/Jump-diffusion_models "Jump-diffusion models") [ARCH and GARCH](https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity "Autoregressive conditional heteroskedasticity") \| \| Trading volatility \| [Volatility arbitrage](https://en.wikipedia.org/wiki/Volatility_arbitrage "Volatility arbitrage") [Straddle](https://en.wikipedia.org/wiki/Straddle "Straddle") [Volatility swap](https://en.wikipedia.org/wiki/Volatility_swap "Volatility swap") [IVX](https://en.wikipedia.org/wiki/IVX "IVX") [VIX](https://en.wikipedia.org/wiki/VIX "VIX") \| \| [v](https://en.wikipedia.org/wiki/Template:Financial_markets_navigation "Template:Financial markets navigation") [t](https://en.wikipedia.org/wiki/Template_talk:Financial_markets_navigation "Template talk:Financial markets navigation") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Financial_markets_navigation "Special:EditPage/Template:Financial markets navigation")[Financial markets](https://en.wikipedia.org/wiki/Financial_market "Financial market") \| \| \|---\|---\| \| Types of [markets](https://en.wikipedia.org/wiki/Capital_market "Capital market") \| [Primary market](https://en.wikipedia.org/wiki/Primary_market "Primary market") [Secondary market](https://en.wikipedia.org/wiki/Secondary_market "Secondary market") [Third market](https://en.wikipedia.org/wiki/Third_market "Third market") [Fourth market](https://en.wikipedia.org/wiki/Fourth_market "Fourth market") \| \| Types of [stocks](https://en.wikipedia.org/wiki/Stock "Stock") \| [Common stock](https://en.wikipedia.org/wiki/Common_stock "Common stock") [Golden share](https://en.wikipedia.org/wiki/Golden_share "Golden share") [Preferred stock](https://en.wikipedia.org/wiki/Preferred_stock "Preferred stock") [Restricted stock](https://en.wikipedia.org/wiki/Restricted_stock "Restricted stock") [Tracking stock](https://en.wikipedia.org/wiki/Tracking_stock "Tracking stock") \| \| [Share capital](https://en.wikipedia.org/wiki/Share_capital "Share capital") \| [Authorised capital](https://en.wikipedia.org/wiki/Authorised_capital "Authorised capital") [Issued shares](https://en.wikipedia.org/wiki/Issued_shares "Issued shares") [Shares outstanding](https://en.wikipedia.org/wiki/Shares_outstanding "Shares outstanding") [Treasury stock](https://en.wikipedia.org/wiki/Treasury_stock "Treasury stock") \| \| Participants \| [Broker](https://en.wikipedia.org/wiki/Broker "Broker") [Floor broker](https://en.wikipedia.org/wiki/Floor_broker "Floor broker") [Inter-dealer broker](https://en.wikipedia.org/wiki/Inter-dealer_broker "Inter-dealer broker") [Broker-dealer](https://en.wikipedia.org/wiki/Broker-dealer "Broker-dealer") [Market maker](https://en.wikipedia.org/wiki/Market_maker "Market maker") [Trader](https://en.wikipedia.org/wiki/Trader_\(finance\) "Trader (finance)") [Floor trader](https://en.wikipedia.org/wiki/Floor_trader "Floor trader") [Proprietary trader](https://en.wikipedia.org/wiki/Proprietary_trading "Proprietary trading") [Quantitative analyst](https://en.wikipedia.org/wiki/Quantitative_analysis_\(finance\) "Quantitative analysis (finance)") [Investor](https://en.wikipedia.org/wiki/Investor "Investor") [Hedger](https://en.wikipedia.org/wiki/Hedge_\(finance\) "Hedge (finance)") [Speculator](https://en.wikipedia.org/wiki/Speculation "Speculation") [Arbitrager](https://en.wikipedia.org/wiki/Arbitrage "Arbitrage") [Scalper](https://en.wikipedia.org/wiki/Scalping_\(trading\) "Scalping (trading)") [Regulator](https://en.wikipedia.org/wiki/Financial_regulation "Financial regulation") \| \| Trading venues \| [Exchange](https://en.wikipedia.org/wiki/Exchange_\(organized_market\) "Exchange (organized market)") [List of major stock exchanges](https://en.wikipedia.org/wiki/List_of_major_stock_exchanges "List of major stock exchanges") [Over-the-counter](https://en.wikipedia.org/wiki/Over-the-counter_\(finance\) "Over-the-counter (finance)") (off-exchange) [Alternative trading system](https://en.wikipedia.org/wiki/Alternative_trading_system "Alternative trading system") (ATS) [Multilateral trading facility](https://en.wikipedia.org/wiki/Multilateral_trading_facility "Multilateral trading facility") (MTF) [Electronic communication network](https://en.wikipedia.org/wiki/Electronic_communication_network "Electronic communication network") (ECN) [Direct market access](https://en.wikipedia.org/wiki/Direct_market_access "Direct market access") (DMA) [Straight-through processing](https://en.wikipedia.org/wiki/Straight-through_processing "Straight-through processing") (STP) [Dark pool](https://en.wikipedia.org/wiki/Dark_pool "Dark pool") (private exchange) [Crossing network](https://en.wikipedia.org/wiki/Crossing_network "Crossing network") [Liquidity aggregator](https://en.wikipedia.org/wiki/Foreign_exchange_aggregator "Foreign exchange aggregator") \| \| [Stock valuation](https://en.wikipedia.org/wiki/Stock_valuation "Stock valuation") \| [Alpha](https://en.wikipedia.org/wiki/Alpha_\(finance\) "Alpha (finance)") [Arbitrage pricing theory](https://en.wikipedia.org/wiki/Arbitrage_pricing_theory "Arbitrage pricing theory") (APT) [Beta](https://en.wikipedia.org/wiki/Beta_\(finance\) "Beta (finance)") [Buffett indicator](https://en.wikipedia.org/wiki/Buffett_indicator "Buffett indicator") (Cap-to-GDP) [Book value](https://en.wikipedia.org/wiki/Book_value "Book value") (BV) [Capital asset pricing model](https://en.wikipedia.org/wiki/Capital_asset_pricing_model "Capital asset pricing model") (CAPM) [Capital market line](https://en.wikipedia.org/wiki/Capital_market_line "Capital market line") (CML) [Dividend discount model](https://en.wikipedia.org/wiki/Dividend_discount_model "Dividend discount model") (DDM) [Dividend yield](https://en.wikipedia.org/wiki/Dividend_yield "Dividend yield") [Earnings yield](https://en.wikipedia.org/wiki/Earnings_yield "Earnings yield") [EV/EBITDA](https://en.wikipedia.org/wiki/EV/EBITDA "EV/EBITDA") [Fed model](https://en.wikipedia.org/wiki/Fed_model "Fed model") [Net asset value](https://en.wikipedia.org/wiki/Net_asset_value "Net asset value") (NAV) [Security characteristic line](https://en.wikipedia.org/wiki/Security_characteristic_line "Security characteristic line") [Security market line](https://en.wikipedia.org/wiki/Security_market_line "Security market line") (SML) [T-model](https://en.wikipedia.org/wiki/T-model "T-model") \| \| Trading theories and [strategies](https://en.wikipedia.org/wiki/Trading_strategy "Trading strategy") \| [Algorithmic trading](https://en.wikipedia.org/wiki/Algorithmic_trading "Algorithmic trading") [Buy and hold](https://en.wikipedia.org/wiki/Buy_and_hold "Buy and hold") [Contrarian investing](https://en.wikipedia.org/wiki/Contrarian_investing "Contrarian investing") [Dollar cost averaging](https://en.wikipedia.org/wiki/Dollar_cost_averaging "Dollar cost averaging") [Efficient-market hypothesis](https://en.wikipedia.org/wiki/Efficient-market_hypothesis "Efficient-market hypothesis") (EMH) [Fundamental analysis](https://en.wikipedia.org/wiki/Fundamental_analysis "Fundamental analysis") [Growth stock](https://en.wikipedia.org/wiki/Growth_stock "Growth stock") [Market timing](https://en.wikipedia.org/wiki/Market_timing "Market timing") [Modern portfolio theory](https://en.wikipedia.org/wiki/Modern_portfolio_theory "Modern portfolio theory") (MPT) [Momentum investing](https://en.wikipedia.org/wiki/Momentum_investing "Momentum investing") [Mosaic theory](https://en.wikipedia.org/wiki/Mosaic_theory_\(investments\) "Mosaic theory (investments)") [Pairs trade](https://en.wikipedia.org/wiki/Pairs_trade "Pairs trade") [Post-modern portfolio theory](https://en.wikipedia.org/wiki/Post-modern_portfolio_theory "Post-modern portfolio theory") (PMPT) [Random walk hypothesis](https://en.wikipedia.org/wiki/Random_walk_hypothesis "Random walk hypothesis") (RMH) [Sector rotation](https://en.wikipedia.org/wiki/Sector_rotation "Sector rotation") [Style investing](https://en.wikipedia.org/wiki/Style_investing "Style investing") [Swing trading](https://en.wikipedia.org/wiki/Swing_trading "Swing trading") [Technical analysis](https://en.wikipedia.org/wiki/Technical_analysis "Technical analysis") [Trend following](https://en.wikipedia.org/wiki/Trend_following "Trend following") [Value averaging](https://en.wikipedia.org/wiki/Value_averaging "Value averaging") [Value investing](https://en.wikipedia.org/wiki/Value_investing "Value investing") \| \| Related terms \| [Bid–ask spread](https://en.wikipedia.org/wiki/Bid%E2%80%93ask_spread "Bid–ask spread") [Block trade](https://en.wikipedia.org/wiki/Block_trade "Block trade") [Cross listing](https://en.wikipedia.org/wiki/Cross_listing "Cross listing") [Dividend](https://en.wikipedia.org/wiki/Dividend "Dividend") [Dual-listed company](https://en.wikipedia.org/wiki/Dual-listed_company "Dual-listed company") [DuPont analysis](https://en.wikipedia.org/wiki/DuPont_analysis "DuPont analysis") [Efficient frontier](https://en.wikipedia.org/wiki/Efficient_frontier "Efficient frontier") [Financial law](https://en.wikipedia.org/wiki/Financial_law "Financial law") [Flight-to-quality](https://en.wikipedia.org/wiki/Flight-to-quality "Flight-to-quality") [Government bond](https://en.wikipedia.org/wiki/Government_bond "Government bond") [Greenspan put](https://en.wikipedia.org/wiki/Greenspan_put "Greenspan put") [Haircut](https://en.wikipedia.org/wiki/Haircut_\(finance\) "Haircut (finance)") [Initial public offering](https://en.wikipedia.org/wiki/Initial_public_offering "Initial public offering") (IPO) [Long](https://en.wikipedia.org/wiki/Long_\(finance\) "Long (finance)") [Mandatory offer](https://en.wikipedia.org/wiki/Mandatory_offer "Mandatory offer") [Margin](https://en.wikipedia.org/wiki/Margin_\(finance\) "Margin (finance)") [Market anomaly](https://en.wikipedia.org/wiki/Market_anomaly "Market anomaly") [Market capitalization](https://en.wikipedia.org/wiki/Market_capitalization "Market capitalization") [Market depth](https://en.wikipedia.org/wiki/Market_depth "Market depth") [Market manipulation](https://en.wikipedia.org/wiki/Market_manipulation "Market manipulation") [Market trend](https://en.wikipedia.org/wiki/Market_trend "Market trend") [Mean reversion](https://en.wikipedia.org/wiki/Mean_reversion_\(finance\) "Mean reversion (finance)") [Momentum](https://en.wikipedia.org/wiki/Momentum_\(finance\) "Momentum (finance)") [Open outcry](https://en.wikipedia.org/wiki/Open_outcry "Open outcry") [Order book](https://en.wikipedia.org/wiki/Order_book "Order book") [Position](https://en.wikipedia.org/wiki/Position_\(finance\) "Position (finance)") [Public float](https://en.wikipedia.org/wiki/Public_float "Public float") [Public offering](https://en.wikipedia.org/wiki/Public_offering "Public offering") [Rally](https://en.wikipedia.org/wiki/Rally_\(stock_market\) "Rally (stock market)") [Returns-based style analysis](https://en.wikipedia.org/wiki/Returns-based_style_analysis "Returns-based style analysis") [Reverse stock split](https://en.wikipedia.org/wiki/Reverse_stock_split "Reverse stock split") [Share repurchase](https://en.wikipedia.org/wiki/Share_repurchase "Share repurchase") [Short selling](https://en.wikipedia.org/wiki/Short_\(finance\) "Short (finance)") [Short squeeze](https://en.wikipedia.org/wiki/Short_squeeze "Short squeeze") [Slippage](https://en.wikipedia.org/wiki/Slippage_\(finance\) "Slippage (finance)") [Speculation](https://en.wikipedia.org/wiki/Speculation "Speculation") [Squeeze-out](https://en.wikipedia.org/wiki/Squeeze-out "Squeeze-out") [Stock dilution](https://en.wikipedia.org/wiki/Stock_dilution "Stock dilution") [Stock exchange](https://en.wikipedia.org/wiki/Stock_exchange "Stock exchange") [Stock market index](https://en.wikipedia.org/wiki/Stock_market_index "Stock market index") [Stock split](https://en.wikipedia.org/wiki/Stock_split "Stock split") [Stock swap](https://en.wikipedia.org/wiki/Stock_swap "Stock swap") [Trade](https://en.wikipedia.org/wiki/Trade_\(finance\) "Trade (finance)") [Tender offer](https://en.wikipedia.org/wiki/Tender_offer "Tender offer") [Uptick rule](https://en.wikipedia.org/wiki/Uptick_rule "Uptick rule") [Volatility]() [Voting interest](https://en.wikipedia.org/wiki/Voting_interest "Voting interest") [Yield](https://en.wikipedia.org/wiki/Yield_\(finance\) "Yield (finance)") \| \| [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]() \| [Average true range](https://en.wikipedia.org/wiki/Average_true_range "Average true range") (ATR) [Bollinger Bands](https://en.wikipedia.org/wiki/Bollinger_Bands "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=Volatility_(finance)&oldid=1343928341>" [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/9/9b/VIX.png/250px-VIX.png)](https://en.wikipedia.org/wiki/File:VIX.png) CBOE Volatility Index (VIX) from December 1985 to May 2012 (daily closings) In [finance](https://en.wikipedia.org/wiki/Finance "Finance"), volatility (usually denoted by "[σ](https://en.wikipedia.org/wiki/Sigma "Sigma")") is the [degree of variation](https://en.wikipedia.org/wiki/Variability_\(statistics\) "Variability (statistics)") of a trading price series over time, usually measured by the [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") of [logarithmic returns](https://en.wikipedia.org/wiki/Logarithmic_return "Logarithmic return"). Historic volatility measures a time series of past market prices. [Implied volatility](https://en.wikipedia.org/wiki/Implied_volatility "Implied volatility") looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). ## Volatility terminology \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=1 "Edit section: Volatility terminology")\] Volatility as described here refers to the actual volatility, more specifically: - actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price. - actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past - near synonymous is realized volatility, the [square root](https://en.wikipedia.org/wiki/Square_root "Square root") of the [realized variance](https://en.wikipedia.org/wiki/Realized_variance "Realized variance"), in turn calculated using the sum of squared returns divided by the number of observations. - actual future volatility which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an [option](https://en.wikipedia.org/wiki/Option_\(finance\) "Option (finance)")) Now turning to [implied volatility](https://en.wikipedia.org/wiki/Implied_volatility "Implied volatility"), we have: - historical implied volatility which refers to the implied volatility observed from historical prices of the financial instrument (normally options) - current implied volatility which refers to the implied volatility observed from current prices of the financial instrument - future implied volatility which refers to the implied volatility observed from future prices of the financial instrument For a financial instrument whose price follows a [Gaussian](https://en.wikipedia.org/wiki/Gaussian "Gaussian") [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk"), or [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process"), the width of the distribution increases as time increases. This is because there is an increasing [probability](https://en.wikipedia.org/wiki/Probability "Probability") that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero. Since observed price changes do not follow Gaussian distributions, others such as the [Lévy distribution](https://en.wikipedia.org/wiki/L%C3%A9vy_distribution "Lévy distribution") are often used.[\[1\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-1) These can capture attributes such as "[fat tails](https://en.wikipedia.org/wiki/Fat_tail "Fat tail")". Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc. ## Mathematical definition \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=2 "Edit section: Mathematical definition")\] For any fund that evolves randomly with time, volatility is defined as the [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") of a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times. Thus, "annualized" volatility σannually is the standard deviation of an instrument's yearly [logarithmic returns](https://en.wikipedia.org/wiki/Rate_of_return#Logarithmic_or_continuously_compounded_return "Rate of return").[\[2\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-2) The generalized volatility σT for [time horizon](https://en.wikipedia.org/wiki/Time_horizon "Time horizon") T in years is expressed as: ![{\\displaystyle \\sigma \_{\\text{T}}=\\sigma \_{\\text{annually}}{\\sqrt {T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e88f2ee170dbab0671e56003059a9ff2e632f3) Therefore, if the daily logarithmic returns of a stock have a standard deviation of σdaily and the time period of returns is P in trading days, the annualized volatility is ![{\\displaystyle \\sigma \_{\\text{annually}}=\\sigma \_{\\text{daily}}{\\sqrt {P}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36940dadd58fbbfadcb59140749af3a4b3c09eaa) so ![{\\displaystyle \\sigma \_{\\text{T}}=\\sigma \_{\\text{daily}}{\\sqrt {PT}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b715335b9d0cbeb41da30e66b16d1a95097fc6f3) A common assumption is that P = 252 trading days in any given year. Then, if σdaily = 0.01, the annualized volatility is ![{\\displaystyle \\sigma \_{\\text{annually}}=0.01{\\sqrt {252}}=0.1587.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77022ad5aadc3db0ebfe97fc132581e3f68042af) The monthly volatility (i.e. ![{\\displaystyle T={\\tfrac {1}{12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/858a58f00d971d2a99c0adae52b82982ba310e25) of a year) is ![{\\displaystyle \\sigma \_{\\text{monthly}}=0.01{\\sqrt {\\tfrac {252}{12}}}=0.0458.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f8a9560640b89becb2624657dea5e0f92da32fe) The formulas used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk"), or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent α to extrapolate natural processes: ![{\\displaystyle \\sigma \_{T}=T^{1/\\alpha }\\sigma .\\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a94ec1c310a51eba65f801a92e89b85b5729c8) If α = 2 the [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process") scaling relation is obtained, but some people believe α \< 2 for financial activities such as stocks, indexes and so on. This was discovered by [Benoît Mandelbrot](https://en.wikipedia.org/wiki/Beno%C3%AEt_Mandelbrot "Benoît Mandelbrot"), who looked at cotton prices and found that they followed a [Lévy alpha-stable distribution](https://en.wikipedia.org/wiki/Stable_distribution "Stable distribution") with α = 1.7. (See New Scientist, 19 April 1997.) Much research has been devoted to modelling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place. Roll (1984) shows that volatility is affected by [market microstructure](https://en.wikipedia.org/wiki/Market_microstructure "Market microstructure").[\[3\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-3) Glosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of [adverse selection](https://en.wikipedia.org/wiki/Adverse_selection "Adverse selection"), they adjust their trading ranges, which in turn increases the band of price oscillation.[\[4\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-4) In September 2019, [JPMorgan Chase](https://en.wikipedia.org/wiki/JPMorgan_Chase "JPMorgan Chase") determined the effect of [US President](https://en.wikipedia.org/wiki/US_President "US President") [Donald Trump](https://en.wikipedia.org/wiki/Donald_Trump "Donald Trump")'s [tweets](https://en.wikipedia.org/wiki/Twitter#Tweets "Twitter"), and called it the [Volfefe index](https://en.wikipedia.org/wiki/Volfefe_index "Volfefe index") combining volatility and the [covfefe](https://en.wikipedia.org/wiki/Covfefe "Covfefe") [meme](https://en.wikipedia.org/wiki/Meme "Meme"). ## Volatility for investors \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=4 "Edit section: Volatility for investors")\] Volatility matters to investors for at least eight reasons,\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\] several of which are alternative statements of the same feature or are directly consequent on each other: 1. The wider the swings in an investment's price, the harder emotionally it is to not worry; 2. Price volatility of a trading instrument can help to determine position sizing in a portfolio; 3. When cash flows from selling a security are needed at a specific future date to meet a known fixed liability, higher volatility means a greater chance of a shortfall; 4. Higher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values; 5. Higher volatility of returns after retirement may result in withdrawals having a larger permanent impact on the portfolio's value; 6. Price volatility presents opportunities to anyone with inside information to buy assets cheaply and sell when overpriced; 7. Volatility affects pricing of [options](https://en.wikipedia.org/wiki/Option_\(finance\) "Option (finance)"), being a parameter of the [Black–Scholes model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model "Black–Scholes model"). ## Volatility versus direction \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=5 "Edit section: Volatility versus direction")\] Volatility does not measure the direction of price changes, merely their dispersion. This is because when calculating [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") (or [variance](https://en.wikipedia.org/wiki/Variance "Variance")), all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time. For example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. Ignoring compounding effects, this would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution"); in reality stock price movements are found to be [leptokurtotic](https://en.wikipedia.org/wiki/Kurtosis "Kurtosis") (fat-tailed). ## Volatility over time \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=6 "Edit section: Volatility over time")\] Although the [Black-Scholes](https://en.wikipedia.org/wiki/Black-Scholes "Black-Scholes") equation assumes predictable constant volatility, this is not observed in real markets. Amongst more realistic models are [Emanuel Derman](https://en.wikipedia.org/wiki/Emanuel_Derman "Emanuel Derman") and [Iraj Kani](https://en.wikipedia.org/w/index.php?title=Iraj_Kani&action=edit&redlink=1 "Iraj Kani (page does not exist)")'s[\[5\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-derman-5) and [Bruno Dupire](https://en.wikipedia.org/wiki/Bruno_Dupire "Bruno Dupire")'s [local volatility](https://en.wikipedia.org/wiki/Local_volatility "Local volatility"), [Poisson process](https://en.wikipedia.org/wiki/Poisson_process "Poisson process") where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of [stochastic volatility](https://en.wikipedia.org/wiki/Stochastic_volatility "Stochastic volatility").[\[6\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-6) It is common knowledge that many types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all.[\[7\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-7) In [foreign exchange market](https://en.wikipedia.org/wiki/Foreign_exchange_market "Foreign exchange market"), price changes are seasonally [heteroskedastic](https://en.wikipedia.org/wiki/Heteroscedasticity "Heteroscedasticity") with periods of one day and one week.[\[8\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-8)[\[9\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-9) Periods when prices fall quickly (a [crash](https://en.wikipedia.org/wiki/Stock_market_crash "Stock market crash")) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible [bubble](https://en.wikipedia.org/wiki/Bubble_\(economics\) "Bubble (economics)")) may often be followed by prices going up even more, or going down by an unusual amount. Most typically, extreme movements do not appear 'out of nowhere'; they are presaged by larger movements than usual or by known uncertainty in specific future events. This is termed [autoregressive conditional heteroskedasticity](https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity "Autoregressive conditional heteroskedasticity"). Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again. Measures of volatility depend not only on the period over which it is measured, but also on the selected time resolution, as the information flow between short-term and long-term traders is asymmetric.\[[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify "Wikipedia:Please clarify")\] As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa.[\[10\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-10) ## Alternative measures of volatility \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=7 "Edit section: Alternative measures of volatility")\] Some authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns.[\[11\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-11) Another considers the regular sequence of directional-changes as the proxy for the instantaneous volatility.[\[12\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-12) ### Volatility as it Relates to Options Trading \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=8 "Edit section: Volatility as it Relates to Options Trading")\] One method of measuring Volatility, often used by quant option trading firms, divides up volatility into two components. Clean volatility - the amount of volatility caused standard events like daily transactions and general noise - and dirty vol, the amount caused by specific events like earnings or policy announcements.[\[13\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-13) For instance, a company like [Microsoft](https://en.wikipedia.org/wiki/Microsoft "Microsoft") would have clean volatility caused by people buying and selling on a daily basis but dirty (or event vol) events like quarterly earnings or a possibly anti-trust announcement. Breaking down volatility into two components is useful in order to accurately price how much an option is worth, especially when identifying what events may contribute to a swing. The job of fundamental analysts at market makers and option trading boutique firms typically entails trying to assign numeric values to these numbers. ## Crude volatility estimation \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=9 "Edit section: Crude volatility estimation")\] Using a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down. To annualize this, you can use the "rule of 16", that is, multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of n independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables. However importantly this does not capture (or in some cases may give excessive weight to) occasional large movements in market price which occur less frequently than once a year. The average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation σ, the expected value of the [magnitude of the observations](https://en.wikipedia.org/wiki/Absolute_deviation "Absolute deviation") is √(2/π)σ = 0.798σ. The net effect is that this crude approach underestimates the true volatility by about 20%. ## Estimate of compound annual growth rate (CAGR) \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=10 "Edit section: Estimate of compound annual growth rate (CAGR)")\] Consider the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series"): ![{\\displaystyle \\log(1+y)=y-{\\tfrac {1}{2}}y^{2}+{\\tfrac {1}{3}}y^{3}-{\\tfrac {1}{4}}y^{4}+\\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/496a72d2a2d6e2c835aff5e506331ff0ce9309fc) Taking only the first two terms one has: ![{\\displaystyle \\mathrm {CAGR} \\approx \\mathrm {AR} -{\\tfrac {1}{2}}\\sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd77bb23265be12dea88d2d34234f8094cb8d25) Volatility thus mathematically represents a drag on the CAGR (formalized as the "[volatility tax](https://en.wikipedia.org/wiki/Volatility_tax "Volatility tax")"). Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula: ![{\\displaystyle \\mathrm {CAGR} \\approx \\mathrm {AR} -{\\tfrac {1}{2}}k\\sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/823832d028777a59c7a6bd47bf9d5b9f5e63c0a1) for a rough estimate, where k is an empirical factor (typically five to ten).\[[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")\] ## Criticisms of volatility forecasting models \[[edit](https://en.wikipedia.org/w/index.php?title=Volatility_\(finance\)&action=edit&section=11 "Edit section: Criticisms of volatility forecasting models")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Vix.png/500px-Vix.png)](https://en.wikipedia.org/wiki/File:Vix.png) Performance of [VIX](https://en.wikipedia.org/wiki/VIX "VIX") (left) compared to past volatility (right) as 30-day volatility predictors, for the period of Jan 1990-Sep 2009. Volatility is measured as the standard deviation of S\&P500 one-day returns over a month's period. The blue lines indicate [linear regressions](https://en.wikipedia.org/wiki/Linear_regression "Linear regression"), resulting in the [correlation coefficients](https://en.wikipedia.org/wiki/Correlation "Correlation") r shown. Note that VIX has virtually the same predictive power as past volatility, insofar as the shown correlation coefficients are nearly identical. Despite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility[\[14\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-15) especially out-of-sample, where different data are used to estimate the models and to test them.[\[16\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-16) Other works have agreed, but claim critics failed to correctly implement the more complicated models.[\[17\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-17) Some practitioners and [portfolio managers](https://en.wikipedia.org/wiki/Portfolio_managers "Portfolio managers") seem to completely ignore or dismiss volatility forecasting models. For example, [Nassim Taleb](https://en.wikipedia.org/wiki/Nassim_Taleb "Nassim Taleb") famously titled one of his [Journal of Portfolio Management](https://en.wikipedia.org/wiki/Journal_of_Portfolio_Management "Journal of Portfolio Management") papers "We Don't Quite Know What We are Talking About When We Talk About Volatility".[\[18\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-18) In a similar note, [Emanuel Derman](https://en.wikipedia.org/wiki/Emanuel_Derman "Emanuel Derman") expressed his disillusion with the enormous supply of empirical models unsupported by theory.[\[19\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-19) He argues that, while "theories are attempts to uncover the hidden principles underpinning the world around us, as Albert Einstein did with his theory of relativity", we should remember that "models are metaphors – analogies that describe one thing relative to another". Machine learning and deep learning methods have also been applied to realized volatility forecasting.[\[20\]](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_note-20) - [Beta (finance)](https://en.wikipedia.org/wiki/Beta_\(finance\) "Beta (finance)") – Expected change in price of a stock relative to the whole market - [Dispersion](https://en.wikipedia.org/wiki/Statistical_dispersion "Statistical dispersion") – Statistical property quantifying how much a collection of data is spread out - [Excess volatility puzzle](https://en.wikipedia.org/wiki/Excess_volatility_puzzle "Excess volatility puzzle") - [Financial economics](https://en.wikipedia.org/wiki/Financial_economics "Financial economics") – Academic discipline concerned with the exchange of money - [IVX](https://en.wikipedia.org/wiki/IVX "IVX") – Intraday, VIX-like volatility index - [Jules Regnault](https://en.wikipedia.org/wiki/Jules_Regnault "Jules Regnault") – French stock broker's assistant - [Low volatility anomaly](https://en.wikipedia.org/wiki/Low_volatility_anomaly "Low volatility anomaly") - [Risk](https://en.wikipedia.org/wiki/Risk "Risk") – Possibility of something bad happening - [VIX](https://en.wikipedia.org/wiki/VIX "VIX") – Chicago Board Options Exchange Volatility index - [Volatility smile](https://en.wikipedia.org/wiki/Volatility_smile "Volatility smile") – Implied volatility patterns that arise in pricing financial options - [Volatility tax](https://en.wikipedia.org/wiki/Volatility_tax "Volatility tax") – Mathematical finance term - [Volatility risk](https://en.wikipedia.org/wiki/Volatility_risk "Volatility risk") - [Volatility beta](https://en.wikipedia.org/wiki/Volatility_beta "Volatility beta") 1. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-1) ["Levy distribution"](http://www.wilmottwiki.com/wiki/index.php?title=Levy_distribution). wilmottwiki.com. 2. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-2) ["Calculating Historical Volatility: Step-by-Step Example"](https://web.archive.org/web/20120330224816/http://www.lfrankcabrera.com/calc-hist-vol.pdf) (PDF). Archived from the original on 30 March 2012. Retrieved 18 August 2011. `{{cite web}}`: CS1 maint: bot: original URL status unknown ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown "Category:CS1 maint: bot: original URL status unknown")) 3. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-3) Roll, R. (1984): "A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market", Journal of Finance 39 (4), 1127–1139 4. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-4) Glosten, L. R. and P. R. Milgrom (1985): "Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders", [Journal of Financial Economics](https://en.wikipedia.org/wiki/Journal_of_Financial_Economics "Journal of Financial Economics") 14 (1), 71–100 5. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-derman_5-0) Derman, E.; Iraj Kani (1994). [""Riding on a Smile." RISK, 7(2) Feb.1994, pp. 139–145, pp. 32–39"](https://web.archive.org/web/20110710170610/http://www.ederman.com/new/docs/gs-volatility_smile.pdf) (PDF). Risk. Archived from [the original](http://www.ederman.com/new/docs/gs-volatility_smile.pdf) (PDF) on 10 July 2011. Retrieved 1 June 2007. 6. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-6) ["Volatility"](http://www.wilmottwiki.com/wiki/index.php?title=Volatility). wilmottwiki.com. 7. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-7) ["Taking Advantage Of Volatility Spikes With Credit Spreads"](http://www.investopedia.com/articles/optioninvestor/10/volatility-spikes-credit-spreads.asp). 8. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-8) Müller, Ulrich A.; Dacorogna, Michel M.; Olsen, Richard B.; Pictet, Olivier V.; Schwarz, Matthias; Morgenegg, Claude (1 December 1990). "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis". Journal of Banking & Finance. 14 (6): 1189–1208\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/0378-4266(90)90009-Q](https://doi.org/10.1016%2F0378-4266%2890%2990009-Q). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0378-4266](https://search.worldcat.org/issn/0378-4266). 9. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-9) Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). ["Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time"](https://doi.org/10.3390%2Fjrfm12020054). Journal of Risk and Financial Management. 12 (2): 54. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3390/jrfm12020054](https://doi.org/10.3390%2Fjrfm12020054). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10419/239003](https://hdl.handle.net/10419%2F239003). 10. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-10) Muller, Ulrich A.; Dacorogna, Michel; Dave, Rakhal D.; Olsen, Richard; Pictet, Olivier V.; von Weizsäcker, Jakob (1997). ["Volatilities of different time resolutions -- Analyzing the dynamics of market components"](https://econpapers.repec.org/article/eeeempfin/v_3a4_3ay_3a1997_3ai_3a2-3_3ap_3a213-239.htm). Journal of Empirical Finance. 4 (2–3\): 213–239\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/S0927-5398(97)00007-8](https://doi.org/10.1016%2FS0927-5398%2897%2900007-8). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0927-5398](https://search.worldcat.org/issn/0927-5398). 11. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-11) Sarkissian, Jack (2016). "Express Measurement of Market Volatility Using Ergodicity Concept". [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2139/ssrn.2812353](https://doi.org/10.2139%2Fssrn.2812353). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [168496910](https://api.semanticscholar.org/CorpusID:168496910). [SSRN](https://en.wikipedia.org/wiki/SSRN_\(identifier\) "SSRN (identifier)") [2812353](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2812353). 12. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-12) Petrov, Vladimir; Golub, Anton; Olsen, Richard (June 2019). ["Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time"](https://doi.org/10.3390%2Fjrfm12020054). Journal of Risk and Financial Management. 12 (2): 54. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3390/jrfm12020054](https://doi.org/10.3390%2Fjrfm12020054). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10419/239003](https://hdl.handle.net/10419%2F239003). 13. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-13) ["Cleaning Implied Vols"](https://notion.moontowermeta.com/cleaning-implied-vols). Moontowermeta. Retrieved 26 June 2024. 14. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-14) Cumby, R.; Figlewski, S.; Hasbrouck, J. (1993). "Forecasting Volatility and Correlations with EGARCH models". Journal of Derivatives. 1 (2): 51–63\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3905/jod.1993.407877](https://doi.org/10.3905%2Fjod.1993.407877). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [154028452](https://api.semanticscholar.org/CorpusID:154028452). 15. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-15) Jorion, P. (1995). "Predicting Volatility in Foreign Exchange Market". [Journal of Finance](https://en.wikipedia.org/wiki/Journal_of_Finance "Journal of Finance"). 50 (2): 507–528\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1111/j.1540-6261.1995.tb04793.x](https://doi.org/10.1111%2Fj.1540-6261.1995.tb04793.x). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2329417](https://www.jstor.org/stable/2329417). 16. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-16) [Brooks, Chris](https://en.wikipedia.org/wiki/Chris_Brooks_\(academic\) "Chris Brooks (academic)"); Persand, Gita (2003). "Volatility forecasting for risk management". Journal of Forecasting. 22 (1): 1–22\. [CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_\(identifier\) "CiteSeerX (identifier)") [10\.1.1.595.9113](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.595.9113). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/for.841](https://doi.org/10.1002%2Ffor.841). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1099-131X](https://search.worldcat.org/issn/1099-131X). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [154615850](https://api.semanticscholar.org/CorpusID:154615850). 17. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-17) Andersen, Torben G.; Bollerslev, Tim (1998). "Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts". [International Economic Review](https://en.wikipedia.org/wiki/International_Economic_Review "International Economic Review"). 39 (4): 885–905\. [CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_\(identifier\) "CiteSeerX (identifier)") [10\.1.1.28.454](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.454). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/2527343](https://doi.org/10.2307%2F2527343). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2527343](https://www.jstor.org/stable/2527343). 18. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-18) Goldstein, Daniel and Taleb, Nassim, (28 March 2007) ["We Don't Quite Know What We are Talking About When We Talk About Volatility"](https://ssrn.com/abstract=970480). Journal of Portfolio Management 33 (4), 2007. 19. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-19) Derman, Emanuel (2011): Models.Behaving.Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life”, Ed. Free Press. 20. [^](https://en.wikipedia.org/wiki/Volatility_\(finance\)#cite_ref-20) Leushuis, Radmir M.; Petkov, Nicolai (2026). ["Advances in forecasting realized volatility: a review of methodologies"](https://doi.org/10.1186%2Fs40854-025-00809-5). Financial Innovation. 12: 14. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1186/s40854-025-00809-5](https://doi.org/10.1186%2Fs40854-025-00809-5). - [Graphical Comparison of Implied and Historical Volatility](https://archive.today/20130204084449/http://training.thomsonreuters.com/video/v.php?v=273), video - [Diebold, Francis X.; Hickman, Andrew; Inoue, Atsushi & Schuermannm, Til (1996) "Converting 1-Day Volatility to h-Day Volatility: Scaling by sqrt(h) is Worse than You Think"](http://citeseer.ist.psu.edu/244698.html) - [A short introduction to alternative mathematical concepts of volatility](http://staff.science.uva.nl/~marvisse/volatility.html) - [Volatility estimation from predicted return density](http://www.macroaxis.com/invest/market/GOOG--symbolVolatility) Example based on Google daily return distribution using standard density function - [Research paper including excerpt from report entitled Identifying Rich and Cheap Volatility](http://www.iijournals.com/doi/abs/10.3905/JOT.2010.5.2.035) Excerpt from Enhanced Call Overwriting, a report by Ryan Renicker and Devapriya Mallick at Lehman Brothers (2005). - Bartram, Söhnke M.; Brown, Gregory W.; Stulz, Rene M. (August 2012). ["Why Are U.S. Stocks More Volatile?"](https://mpra.ub.uni-muenchen.de/47341/2/MPRA_paper_47341.pdf) (PDF). Journal of Finance. 67 (4): 1329–1370\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1111/j.1540-6261.2012.01749.x](https://doi.org/10.1111%2Fj.1540-6261.2012.01749.x). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [18587238](https://api.semanticscholar.org/CorpusID:18587238). [SSRN](https://en.wikipedia.org/wiki/SSRN_\(identifier\) "SSRN (identifier)") [2257549](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2257549). - Natenberg, Sheldon (2015). Option Volatility and Pricing: Advanced Trading Strategies and Techniques (Second ed.). New York. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0071818773](https://en.wikipedia.org/wiki/Special:BookSources/978-0071818773 "Special:BookSources/978-0071818773") . `{{cite book}}`: CS1 maint: location missing publisher ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_location_missing_publisher "Category:CS1 maint: location missing publisher")) - Fassas, Athanasios P.; Siriopoulos, Costas (1 February 2021). ["Implied volatility indices – A review"](https://www.sciencedirect.com/science/article/pii/S1062976920300855). The Quarterly Review of Economics and Finance. 79: 303–329\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.qref.2020.07.004](https://doi.org/10.1016%2Fj.qref.2020.07.004). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1062-9769](https://search.worldcat.org/issn/1062-9769).
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