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Calculated Shard: 152 (from laksa132)

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

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{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion","crawl_time":1775698645,"first_indexed_time":1426162459,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Geometric_Brownian_motion s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Geometric Brownian motion - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"From Wikipedia, the free encyclopedia\nFor the simulation generating the realizations, see below.\nA\ngeometric Brownian motion\n(\nGBM\n), also known as an\nexponential Brownian motion\n, is a continuous-time\nstochastic process\nin which the\nlogarithm\nof the randomly varying quantity follows a\nBrownian motion\nwith\ndrift\n.\n[\n1\n]\nIt is an important example of stochastic processes satisfying a\nstochastic differential equation\n(SDE); in particular, it is used in\nmathematical finance\nto model stock prices in the\nBlack–Scholes model\n.\nStochastical differential equation\n[\nedit\n]\nA stochastic process\nS\nt\nis said to follow a GBM if it satisfies the following\nstochastic differential equation\n(SDE):\nwhere\nis a\nWiener process or Brownian motion\n, and\n('the percentage drift') and\n('the percentage volatility') are constants.\nThe former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion.\nFor an arbitrary initial value\nS\n0\nthe above SDE has the analytic solution (under\nItô's interpretation\n):\nThe derivation requires the use of\nItô calculus\n. Applying\nItô's formula\nleads to\nwhere\nis the\nquadratic variation\nof the SDE.\nWhen\n,\nconverges to 0 faster than\n, \nsince\n. So the above infinitesimal can be simplified by\nPlugging the value of\nin the above equation and simplifying we obtain\nTaking the exponential and multiplying both sides by\ngives the solution claimed above.\nArithmetic Brownian motion\n[\nedit\n]\nThe process for\n, satisfying the SDE\nor more generally the process solving the SDE\nwhere\nand\nare real constants and for an initial condition\n, is called an Arithmetic Brownian Motion (ABM). This was the model postulated by\nLouis Bachelier\nin 1900 for stock prices, in the first published attempt to model Brownian motion, known today as\nBachelier model\n. As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula. Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula.\nThe above solution\n(for any value of t) is a\nlog-normally distributed\nrandom variable\nwith\nexpected value\nand\nvariance\ngiven by\n[\n2\n]\nThey can be derived using the fact that\nis a\nmartingale\n, and that\nThe\nprobability density function\nof\nis:\nDerivation of GBM probability density function\nTo derive the probability density function for GBM, we must use the\nFokker–Planck equation\nto evaluate the time evolution of the PDF:\nwhere\nis the\nDirac delta function\n. To simplify the computation, we may introduce a logarithmic transform\n, leading to the form of GBM:\nThen the equivalent Fokker–Planck equation for the evolution of the PDF becomes:\nDefine\nand\n. By introducing the new variables\nand\n, the derivatives in the Fokker–Planck equation may be transformed as:\nLeading to the new form of the Fokker–Planck equation:\nHowever, this is the canonical form of the\nheat equation\n. which has the solution given by the\nheat kernel\n:\nPlugging in the original variables leads to the PDF for GBM:\nWhen deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log(\nS\nt\n). This is an interesting process, because in the  Black–Scholes model it is related to the\nlog return\nof the stock price. Using\nItô's lemma\nwith\nf\n(\nS\n) = log(\nS\n) gives\nIt follows that\n.\nThis result can also be derived by applying the logarithm to the explicit solution of GBM:\nTaking the expectation yields the same result as above:\n.\nMultivariate version\n[\nedit\n]\nGBM can be extended to the case where there are multiple correlated price paths.\n[\n3\n]\nEach price path follows the underlying process\nwhere the Wiener processes are correlated such that\nwhere\n.\nFor the multivariate case, this implies that\nA multivariate formulation that maintains the driving Brownian motions\nindependent is\nwhere the correlation between\nand\nis now expressed through the\nterms.\nGeometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.\n[\n4\n]\nSome of the arguments for using GBM to model stock prices are:\nThe expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality.\n[\n4\n]\nA GBM process only assumes positive values, just like real stock prices.\nA GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices.\nCalculations with GBM processes are relatively easy.\nHowever, GBM is not a completely realistic model, in particular it falls short of reality in the following points:\nIn real stock prices, volatility changes over time (possibly\nstochastically\n), but in GBM, volatility is assumed constant.\nIn real life, stock prices often show jumps caused by unpredictable events or news, but in GBM, the path is continuous (no discontinuity).\nApart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies.\n[\n5\n]\nIn an attempt to make GBM more realistic as a model for stock prices, also in relation to the\nvolatility smile\nproblem, one can drop the assumption that the volatility (\n) is constant. If we assume that the volatility is a\ndeterministic\nfunction of the stock price and time, this is called a\nlocal volatility\nmodel. A straightforward extension of the Black Scholes GBM is a local volatility SDE whose distribution is a mixture of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options.\n[\n3\n]\n[\n6\n]\n[\n7\n]\n[\n8\n]\nIf instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a\nstochastic volatility\nmodel, see for example the\nHeston model\n.\n[\n9\n]\nBrownian surface\nFeynman–Kac formula\n^\nRoss, Sheldon M. (2014).\n\"Variations on Brownian Motion\"\n.\nIntroduction to Probability Models\n(11th ed.). Amsterdam: Elsevier. pp. \n612–\n14.\nISBN\n \n978-0-12-407948-9\n.\n^\nØksendal, Bernt K. (2002),\nStochastic Differential Equations: An Introduction with Applications\n, Springer, p. 326,\nISBN\n \n3-540-63720-6\n^\na\nb\nMusiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.\n^\na\nb\nHull, John (2009). \"12.3\".\nOptions, Futures, and other Derivatives\n(7 ed.).\n^\nRej, A.; Seager, P.; Bouchaud, J.-P. (January 2018).\n\"You are in a drawdown. When should you start worrying?\"\n.\nWilmott\n.\n2018\n(93):\n56–\n59.\narXiv\n:\n1707.01457\n.\ndoi\n:\n10.1002\/wilm.10646\n.\nS2CID\n \n157827746\n.\n^\nFengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI\nhttps:\/\/doi.org\/10.1007\/3-540-30591-2\n^\nBrigo, Damiano\n;\nMercurio, Fabio\n(2002). \"Lognormal-mixture dynamics and calibration to market volatility smiles\".\nInternational Journal of Theoretical and Applied Finance\n.\n5\n(4):\n427–\n446.\ndoi\n:\n10.1142\/S0219024902001511\n.\n^\nBrigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183,\nISSN\n \n1469-7688\n^\nHeston, Steven L.\n(1993). \"A closed-form solution for options with stochastic volatility with applications to bond and currency options\".\nReview of Financial Studies\n.\n6\n(2):\n327–\n343.\ndoi\n:\n10.1093\/rfs\/6.2.327\n.\nJSTOR\n \n2962057\n.\nS2CID\n \n16091300\n.\nGeometric Brownian motion models for stock movement except in rare events.\nExcel Simulation of a Geometric Brownian Motion to simulate Stock Prices\n\"Interactive Web Application: Stochastic Processes used in Quantitative Finance\"\n. Archived from\nthe original\non 2015-09-20\n. Retrieved\n2015-07-03\n.","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#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=Geometric+Brownian+motion \"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=Geometric+Brownian+motion \"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\/Geometric_Brownian_motion)\n- [1 Stochastical differential equation](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#Stochastical_differential_equation)\n  Toggle Stochastical differential equation subsection\n  - [1\\.1 Solution](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#Solution)\n- [2 Arithmetic Brownian motion](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#Arithmetic_Brownian_motion)\n- [3 Properties](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#Properties)\n- [4 Multivariate version](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#Multivariate_version)\n- [5 Use in finance](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#Use_in_finance)\n- [6 Extensions](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#Extensions)\n- [7 See also](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#See_also)\n- [8 References](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#References)\n- [9 External links](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#External_links)\n\nToggle the table of contents\n\n# Geometric Brownian motion\n12 languages\n\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Moviment_browni%C3%A0_geom%C3%A8tric \"Moviment brownià geomètric – Catalan\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Geometrische_brownsche_Bewegung \"Geometrische brownsche Bewegung – German\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Movimiento_browniano_geom%C3%A9trico \"Movimiento browniano geométrico – Spanish\")\n- [עברית](https:\/\/he.wikipedia.org\/wiki\/%D7%AA%D7%A0%D7%95%D7%A2%D7%94_%D7%91%D7%A8%D7%90%D7%95%D7%A0%D7%99%D7%AA_%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%AA \"תנועה בראונית גאומטרית – Hebrew\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Moto_browniano_geometrico \"Moto browniano geometrico – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E5%B9%BE%E4%BD%95%E3%83%96%E3%83%A9%E3%82%A6%E3%83%B3%E9%81%8B%E5%8B%95 \"幾何ブラウン運動 – Japanese\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/Movimento_browniano_geom%C3%A9trico \"Movimento browniano geométrico – Portuguese\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B1%D1%80%D0%BE%D1%83%D0%BD%D0%BE%D0%B2%D1%81%D0%BA%D0%BE%D0%B5_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5 \"Геометрическое броуновское движение – Russian\")\n- [Svenska](https:\/\/sv.wikipedia.org\/wiki\/Geometrisk_brownsk_r%C3%B6relse \"Geometrisk brownsk rörelse – Swedish\")\n- [Türkçe](https:\/\/tr.wikipedia.org\/wiki\/Geometrik_Brown_hareketi \"Geometrik Brown hareketi – Turkish\")\n- [Українська](https:\/\/uk.wikipedia.org\/wiki\/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%B1%D1%80%D0%BE%D1%83%D0%BD%D1%96%D0%B2%D1%81%D1%8C%D0%BA%D0%B8%D0%B9_%D1%80%D1%83%D1%85 \"Геометричний броунівський рух – Ukrainian\")\n- [中文](https:\/\/zh.wikipedia.org\/wiki\/%E5%87%A0%E4%BD%95%E5%B8%83%E6%9C%97%E8%BF%90%E5%8A%A8 \"几何布朗运动 – Chinese\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q1503307#sitelinks-wikipedia \"Edit interlanguage links\")\n\n- [Article](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion \"View the content page [c]\")\n- [Talk](https:\/\/en.wikipedia.org\/wiki\/Talk:Geometric_Brownian_motion \"Discuss improvements to the content page [t]\")\n\nEnglish\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=history \"Past revisions of this page [h]\")\n\nTools\n\nTools\n\nmove to sidebar\n\nhide\n\nActions\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=history)\n\nGeneral\n\n- [What links here](https:\/\/en.wikipedia.org\/wiki\/Special:WhatLinksHere\/Geometric_Brownian_motion \"List of all English Wikipedia pages containing links to this page [j]\")\n- [Related changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChangesLinked\/Geometric_Brownian_motion \"Recent changes in pages linked from this page [k]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_Upload_Wizard \"Upload files [u]\")\n- [Permanent link](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&oldid=1347612612 \"Permanent link to this revision of this page\")\n- [Page information](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=info \"More information about this page\")\n- [Cite this page](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CiteThisPage&page=Geometric_Brownian_motion&id=1347612612&wpFormIdentifier=titleform \"Information on how to cite this page\")\n- [Get shortened URL](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGeometric_Brownian_motion)\n\nPrint\/export\n\n- [Download as PDF](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:DownloadAsPdf&page=Geometric_Brownian_motion&action=show-download-screen \"Download this page as a PDF file\")\n- [Printable version](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikimedia Commons](https:\/\/commons.wikimedia.org\/wiki\/Category:Brownian_motion)\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q1503307 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nContinuous stochastic process\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1c\/GBM2.png\/500px-GBM2.png)](https:\/\/en.wikipedia.org\/wiki\/File:GBM2.png)\n\nFor the simulation generating the realizations, see below.\n\nA **geometric Brownian motion** (**GBM**), also known as an **exponential Brownian motion**, is a continuous-time [stochastic process](https:\/\/en.wikipedia.org\/wiki\/Stochastic_process \"Stochastic process\") in which the [logarithm](https:\/\/en.wikipedia.org\/wiki\/Logarithm \"Logarithm\") of the randomly varying quantity follows a [Brownian motion](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\") with [drift](https:\/\/en.wikipedia.org\/wiki\/Stochastic_drift \"Stochastic drift\").[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-1) It is an important example of stochastic processes satisfying a [stochastic differential equation](https:\/\/en.wikipedia.org\/wiki\/Stochastic_differential_equation \"Stochastic differential equation\") (SDE); in particular, it is used in [mathematical finance](https:\/\/en.wikipedia.org\/wiki\/Mathematical_finance \"Mathematical finance\") to model stock prices in the [Black–Scholes model](https:\/\/en.wikipedia.org\/wiki\/Black%E2%80%93Scholes_model \"Black–Scholes model\").\n\n## Stochastical differential equation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=1 \"Edit section: Stochastical differential equation\")\\]\n\nA stochastic process *S**t* is said to follow a GBM if it satisfies the following [stochastic differential equation](https:\/\/en.wikipedia.org\/wiki\/Stochastic_differential_equation \"Stochastic differential equation\") (SDE):\n\nd\n\nS\n\nt\n\n\\=\n\nμ\n\nS\n\nt\n\nd\n\nt\n\n\\+\n\nσ\n\nS\n\nt\n\nd\n\nW\n\nt\n\n{\\\\displaystyle dS\\_{t}=\\\\mu S\\_{t}\\\\,dt+\\\\sigma S\\_{t}\\\\,dW\\_{t}}\n\n![{\\\\displaystyle dS\\_{t}=\\\\mu S\\_{t}\\\\,dt+\\\\sigma S\\_{t}\\\\,dW\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f2a3c0d1dcb510719effde045c26f1a9d0b9cb2d)\n\nwhere W t {\\\\displaystyle W\\_{t}} ![{\\\\displaystyle W\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/50680c5535c83badfa630dba63b583d2eeaa2977) is a [Wiener process or Brownian motion](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\"), and μ {\\\\displaystyle \\\\mu } ![{\\\\displaystyle \\\\mu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9fd47b2a39f7a7856952afec1f1db72c67af6161) ('the percentage drift') and σ {\\\\displaystyle \\\\sigma } ![{\\\\displaystyle \\\\sigma }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/59f59b7c3e6fdb1d0365a494b81fb9a696138c36) ('the percentage volatility') are constants.\n\nThe former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion.\n\n### Solution\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=2 \"Edit section: Solution\")\\]\n\nFor an arbitrary initial value *S*0 the above SDE has the analytic solution (under [Itô's interpretation](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4_calculus \"Itô calculus\")):\n\nS\n\nt\n\n\\=\n\nS\n\n0\n\nexp\n\n⁡\n\n(\n\n(\n\nμ\n\n−\n\nσ\n\n2\n\n2\n\n)\n\nt\n\n\\+\n\nσ\n\nW\n\nt\n\n)\n\n.\n\n{\\\\displaystyle S\\_{t}=S\\_{0}\\\\exp \\\\left(\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}\\\\right).}\n\n![{\\\\displaystyle S\\_{t}=S\\_{0}\\\\exp \\\\left(\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/266807b65fd50635526a766c0c89a2913085d0c2)\n\nThe derivation requires the use of [Itô calculus](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4_calculus \"Itô calculus\"). Applying [Itô's formula](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4%27s_formula \"Itô's formula\") leads to\n\nd\n\n(\n\nln\n\n⁡\n\nS\n\nt\n\n)\n\n\\=\n\n(\n\nln\n\n⁡\n\nS\n\nt\n\n)\n\n′\n\nd\n\nS\n\nt\n\n\\+\n\n1\n\n2\n\n(\n\nln\n\n⁡\n\nS\n\nt\n\n)\n\n″\n\nd\n\nS\n\nt\n\nd\n\nS\n\nt\n\n\\=\n\nd\n\nS\n\nt\n\nS\n\nt\n\n−\n\n1\n\n2\n\n1\n\nS\n\nt\n\n2\n\nd\n\nS\n\nt\n\nd\n\nS\n\nt\n\n{\\\\displaystyle d(\\\\ln S\\_{t})=(\\\\ln S\\_{t})'dS\\_{t}+{\\\\frac {1}{2}}(\\\\ln S\\_{t})''\\\\,dS\\_{t}\\\\,dS\\_{t}={\\\\frac {dS\\_{t}}{S\\_{t}}}-{\\\\frac {1}{2}}\\\\,{\\\\frac {1}{S\\_{t}^{2}}}\\\\,dS\\_{t}\\\\,dS\\_{t}}\n\n![{\\\\displaystyle d(\\\\ln S\\_{t})=(\\\\ln S\\_{t})'dS\\_{t}+{\\\\frac {1}{2}}(\\\\ln S\\_{t})''\\\\,dS\\_{t}\\\\,dS\\_{t}={\\\\frac {dS\\_{t}}{S\\_{t}}}-{\\\\frac {1}{2}}\\\\,{\\\\frac {1}{S\\_{t}^{2}}}\\\\,dS\\_{t}\\\\,dS\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cbef6e7e7776ca5963d9179d9d0e1d198e26fd63)\n\nwhere d S t d S t {\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}} ![{\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d7dd8f0d0a80348c6459b950635d83110c4453ae) is the [quadratic variation](https:\/\/en.wikipedia.org\/wiki\/Quadratic_variation \"Quadratic variation\") of the SDE.\n\nd\n\nS\n\nt\n\nd\n\nS\n\nt\n\n\\=\n\nσ\n\n2\n\nS\n\nt\n\n2\n\nd\n\nW\n\nt\n\n2\n\n\\+\n\n2\n\nσ\n\nS\n\nt\n\n2\n\nμ\n\nd\n\nW\n\nt\n\nd\n\nt\n\n\\+\n\nμ\n\n2\n\nS\n\nt\n\n2\n\nd\n\nt\n\n2\n\n{\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}\\\\,=\\\\,\\\\sigma ^{2}\\\\,S\\_{t}^{2}\\\\,dW\\_{t}^{2}+2\\\\sigma S\\_{t}^{2}\\\\mu \\\\,dW\\_{t}\\\\,dt+\\\\mu ^{2}S\\_{t}^{2}\\\\,dt^{2}}\n\n![{\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}\\\\,=\\\\,\\\\sigma ^{2}\\\\,S\\_{t}^{2}\\\\,dW\\_{t}^{2}+2\\\\sigma S\\_{t}^{2}\\\\mu \\\\,dW\\_{t}\\\\,dt+\\\\mu ^{2}S\\_{t}^{2}\\\\,dt^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/54a2a5473dd3d9151bdc6f88a4eb51ed055c1797)\n\nWhen d t → 0 {\\\\displaystyle dt\\\\to 0} ![{\\\\displaystyle dt\\\\to 0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/02f957c6106281a61de6adf0ec77aaf6ea42aba9), d t {\\\\displaystyle dt} ![{\\\\displaystyle dt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ebee76a835701fd1f26047a09855f2ea36bb08fc) converges to 0 faster than d W t {\\\\displaystyle dW\\_{t}} ![{\\\\displaystyle dW\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cc431f447c4bc6c9063431fe612625992c3cca91), since d W t 2 \\= O ( d t ) {\\\\displaystyle dW\\_{t}^{2}=O(dt)} ![{\\\\displaystyle dW\\_{t}^{2}=O(dt)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f1f08e336e1625340ddcf50ada36b7bf8cd083b1). So the above infinitesimal can be simplified by\n\nd\n\nS\n\nt\n\nd\n\nS\n\nt\n\n\\=\n\nσ\n\n2\n\nS\n\nt\n\n2\n\nd\n\nt\n\n{\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}\\\\,=\\\\,\\\\sigma ^{2}\\\\,S\\_{t}^{2}\\\\,dt}\n\n![{\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}\\\\,=\\\\,\\\\sigma ^{2}\\\\,S\\_{t}^{2}\\\\,dt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dceb3d38f0dd8adce9525559a5d9c0ee497b137f)\n\nPlugging the value of d S t {\\\\displaystyle dS\\_{t}} ![{\\\\displaystyle dS\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c796c605ee3a300e8b295d4225bb163e899af5ca) in the above equation and simplifying we obtain\n\nln\n\n⁡\n\nS\n\nt\n\nS\n\n0\n\n\\=\n\n(\n\nμ\n\n−\n\nσ\n\n2\n\n2\n\n)\n\nt\n\n\\+\n\nσ\n\nW\n\nt\n\n.\n\n{\\\\displaystyle \\\\ln {\\\\frac {S\\_{t}}{S\\_{0}}}=\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\,\\\\right)t+\\\\sigma W\\_{t}\\\\,.}\n\n![{\\\\displaystyle \\\\ln {\\\\frac {S\\_{t}}{S\\_{0}}}=\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\,\\\\right)t+\\\\sigma W\\_{t}\\\\,.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/adee3ba14c6e0d50918fefe1f352991b13631265)\n\nTaking the exponential and multiplying both sides by S 0 {\\\\displaystyle S\\_{0}} ![{\\\\displaystyle S\\_{0}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ebe0ac45a38c4437bd2689a14ec434cd499e7e49) gives the solution claimed above.\n\n## Arithmetic Brownian motion\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=3 \"Edit section: Arithmetic Brownian motion\")\\]\n\nThe process for X t \\= ln ⁡ S t S 0 {\\\\displaystyle X\\_{t}=\\\\ln {\\\\frac {S\\_{t}}{S\\_{0}}}} ![{\\\\displaystyle X\\_{t}=\\\\ln {\\\\frac {S\\_{t}}{S\\_{0}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7184af654d06bc2453b2bc139a81ad9458ca897b), satisfying the SDE\n\nd\n\nX\n\nt\n\n\\=\n\n(\n\nμ\n\n−\n\nσ\n\n2\n\n2\n\n)\n\nd\n\nt\n\n\\+\n\nσ\n\nd\n\nW\n\nt\n\n,\n\n{\\\\displaystyle dX\\_{t}=\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\,\\\\right)dt+\\\\sigma dW\\_{t}\\\\,,}\n\n![{\\\\displaystyle dX\\_{t}=\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\,\\\\right)dt+\\\\sigma dW\\_{t}\\\\,,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/387886bce7774547c64416a186455791dda2c4e5)\n\nor more generally the process solving the SDE\n\nd\n\nX\n\nt\n\n\\=\n\nm\n\nd\n\nt\n\n\\+\n\nv\n\nd\n\nW\n\nt\n\n,\n\n{\\\\displaystyle dX\\_{t}=m\\\\,dt+v\\\\,dW\\_{t}\\\\,,}\n\n![{\\\\displaystyle dX\\_{t}=m\\\\,dt+v\\\\,dW\\_{t}\\\\,,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/98f3370b73be582ffd0d49245c6cfc05801e91df)\n\nwhere m {\\\\displaystyle m} ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) and v \\> 0 {\\\\displaystyle v\\>0} ![{\\\\displaystyle v\\>0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c314fc908a83c555d34968d25e86c5ae0b76ef6f) are real constants and for an initial condition X 0 {\\\\displaystyle X\\_{0}} ![{\\\\displaystyle X\\_{0}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6381fdad2b9f11954b1fc2db08bbaccf634ededa), is called an Arithmetic Brownian Motion (ABM). This was the model postulated by [Louis Bachelier](https:\/\/en.wikipedia.org\/wiki\/Louis_Bachelier \"Louis Bachelier\") in 1900 for stock prices, in the first published attempt to model Brownian motion, known today as [Bachelier model](https:\/\/en.wikipedia.org\/wiki\/Bachelier_model \"Bachelier model\"). As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula. Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula.\n\n## Properties\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=4 \"Edit section: Properties\")\\]\n\nThe above solution S t {\\\\displaystyle S\\_{t}} ![{\\\\displaystyle S\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7e2391e6e796fbf718be3828080775ac2ac3d3d4) (for any value of t) is a [log-normally distributed](https:\/\/en.wikipedia.org\/wiki\/Log-normal_distribution \"Log-normal distribution\") [random variable](https:\/\/en.wikipedia.org\/wiki\/Random_variable \"Random variable\") with [expected value](https:\/\/en.wikipedia.org\/wiki\/Expected_value \"Expected value\") and [variance](https:\/\/en.wikipedia.org\/wiki\/Variance \"Variance\") given by[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-2)\n\nE\n\n⁡\n\n(\n\nS\n\nt\n\n)\n\n\\=\n\nS\n\n0\n\ne\n\nμ\n\nt\n\n,\n\n{\\\\displaystyle \\\\operatorname {E} (S\\_{t})=S\\_{0}e^{\\\\mu t},}\n\n![{\\\\displaystyle \\\\operatorname {E} (S\\_{t})=S\\_{0}e^{\\\\mu t},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/029367c196d22b8abeeb9e362cfe5cf10d505b5c)\n\nVar\n\n⁡\n\n(\n\nS\n\nt\n\n)\n\n\\=\n\nS\n\n0\n\n2\n\ne\n\n2\n\nμ\n\nt\n\n(\n\ne\n\nσ\n\n2\n\nt\n\n−\n\n1\n\n)\n\n.\n\n{\\\\displaystyle \\\\operatorname {Var} (S\\_{t})=S\\_{0}^{2}e^{2\\\\mu t}\\\\left(e^{\\\\sigma ^{2}t}-1\\\\right).}\n\n![{\\\\displaystyle \\\\operatorname {Var} (S\\_{t})=S\\_{0}^{2}e^{2\\\\mu t}\\\\left(e^{\\\\sigma ^{2}t}-1\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b6035329909c079220525421bada1f1b2d67f8d0)\n\nThey can be derived using the fact that Z t \\= exp ⁡ ( σ W t − 1 2 σ 2 t ) {\\\\displaystyle Z\\_{t}=\\\\exp \\\\left(\\\\sigma W\\_{t}-{\\\\frac {1}{2}}\\\\sigma ^{2}t\\\\right)} ![{\\\\displaystyle Z\\_{t}=\\\\exp \\\\left(\\\\sigma W\\_{t}-{\\\\frac {1}{2}}\\\\sigma ^{2}t\\\\right)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c181320a17700bcd9460adcb293ad092f50eccc4) is a [martingale](https:\/\/en.wikipedia.org\/wiki\/Martingale_\\(probability_theory\\) \"Martingale (probability theory)\"), and that\n\nE\n\n⁡\n\n\\[\n\nexp\n\n⁡\n\n(\n\n2\n\nσ\n\nW\n\nt\n\n−\n\nσ\n\n2\n\nt\n\n)\n\n∣\n\nF\n\ns\n\n\\]\n\n\\=\n\ne\n\nσ\n\n2\n\n(\n\nt\n\n−\n\ns\n\n)\n\nexp\n\n⁡\n\n(\n\n2\n\nσ\n\nW\n\ns\n\n−\n\nσ\n\n2\n\ns\n\n)\n\n,\n\n∀\n\n0\n\n≤\n\ns\n\n\\<\n\nt\n\n.\n\n{\\\\displaystyle \\\\operatorname {E} \\\\left\\[\\\\exp \\\\left(2\\\\sigma W\\_{t}-\\\\sigma ^{2}t\\\\right)\\\\mid {\\\\mathcal {F}}\\_{s}\\\\right\\]=e^{\\\\sigma ^{2}(t-s)}\\\\exp \\\\left(2\\\\sigma W\\_{s}-\\\\sigma ^{2}s\\\\right),\\\\quad \\\\forall 0\\\\leq s\\<t.}\n\n![{\\\\displaystyle \\\\operatorname {E} \\\\left\\[\\\\exp \\\\left(2\\\\sigma W\\_{t}-\\\\sigma ^{2}t\\\\right)\\\\mid {\\\\mathcal {F}}\\_{s}\\\\right\\]=e^{\\\\sigma ^{2}(t-s)}\\\\exp \\\\left(2\\\\sigma W\\_{s}-\\\\sigma ^{2}s\\\\right),\\\\quad \\\\forall 0\\\\leq s\\<t.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/121f4efbdcac966a20535657392d5de0345d6f9b)\n\nThe [probability density function](https:\/\/en.wikipedia.org\/wiki\/Probability_density_function \"Probability density function\") of S t {\\\\displaystyle S\\_{t}} ![{\\\\displaystyle S\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7e2391e6e796fbf718be3828080775ac2ac3d3d4) is:\n\nf\n\nS\n\nt\n\n(\n\ns\n\n;\n\nμ\n\n,\n\nσ\n\n,\n\nt\n\n)\n\n\\=\n\n1\n\n2\n\nπ\n\n1\n\ns\n\nσ\n\nt\n\nexp\n\n⁡\n\n(\n\n−\n\n(\n\nln\n\n⁡\n\ns\n\n−\n\nln\n\n⁡\n\nS\n\n0\n\n−\n\n(\n\nμ\n\n−\n\n1\n\n2\n\nσ\n\n2\n\n)\n\nt\n\n)\n\n2\n\n2\n\nσ\n\n2\n\nt\n\n)\n\n.\n\n{\\\\displaystyle f\\_{S\\_{t}}(s;\\\\mu ,\\\\sigma ,t)={\\\\frac {1}{\\\\sqrt {2\\\\pi }}}\\\\,{\\\\frac {1}{s\\\\sigma {\\\\sqrt {t}}}}\\\\,\\\\exp \\\\left(-{\\\\frac {\\\\left(\\\\ln s-\\\\ln S\\_{0}-\\\\left(\\\\mu -{\\\\frac {1}{2}}\\\\sigma ^{2}\\\\right)t\\\\right)^{2}}{2\\\\sigma ^{2}t}}\\\\right).}\n\n![{\\\\displaystyle f\\_{S\\_{t}}(s;\\\\mu ,\\\\sigma ,t)={\\\\frac {1}{\\\\sqrt {2\\\\pi }}}\\\\,{\\\\frac {1}{s\\\\sigma {\\\\sqrt {t}}}}\\\\,\\\\exp \\\\left(-{\\\\frac {\\\\left(\\\\ln s-\\\\ln S\\_{0}-\\\\left(\\\\mu -{\\\\frac {1}{2}}\\\\sigma ^{2}\\\\right)t\\\\right)^{2}}{2\\\\sigma ^{2}t}}\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/90a9074c173307646da549284ffc5d19c8b85b07)\n\n| Derivation of GBM probability density function |\n|---|\n| To derive the probability density function for GBM, we must use the [Fokker–Planck equation](https:\/\/en.wikipedia.org\/wiki\/Fokker%E2%80%93Planck_equation \"Fokker–Planck equation\") to evaluate the time evolution of the PDF: ∂ p ∂ t \\+ ∂ ∂ S \\[ μ ( t , S ) p ( t , S ) \\] \\= 1 2 ∂ 2 ∂ S 2 \\[ σ 2 ( t , S ) p ( t , S ) \\] , p ( 0 , S ) \\= δ ( S − S 0 ) {\\\\displaystyle {\\\\partial p \\\\over {\\\\partial t}}+{\\\\partial \\\\over {\\\\partial S}}\\[\\\\mu (t,S)p(t,S)\\]={1 \\\\over {2}}{\\\\partial ^{2} \\\\over {\\\\partial S^{2}}}\\[\\\\sigma ^{2}(t,S)p(t,S)\\],\\\\quad p(0,S)=\\\\delta (S-S\\_{0})} ![{\\\\displaystyle {\\\\partial p \\\\over {\\\\partial t}}+{\\\\partial \\\\over {\\\\partial S}}\\[\\\\mu (t,S)p(t,S)\\]={1 \\\\over {2}}{\\\\partial ^{2} \\\\over {\\\\partial S^{2}}}\\[\\\\sigma ^{2}(t,S)p(t,S)\\],\\\\quad p(0,S)=\\\\delta (S-S\\_{0})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c6827fa340748de87d1aa6a2b7d99d356ef38e6f) where δ ( S ) {\\\\displaystyle \\\\delta (S)} ![{\\\\displaystyle \\\\delta (S)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f29eec89c4928b10514c6993b942ea248b3b5a57) is the [Dirac delta function](https:\/\/en.wikipedia.org\/wiki\/Dirac_delta_function \"Dirac delta function\"). To simplify the computation, we may introduce a logarithmic transform x \\= log ⁡ ( S \/ S 0 ) {\\\\displaystyle x=\\\\log(S\/S\\_{0})} ![{\\\\displaystyle x=\\\\log(S\/S\\_{0})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/84aae4276b1b7e359e466df00fc19d2786f95717), leading to the form of GBM: d x \\= ( μ − 1 2 σ 2 ) d t \\+ σ d W {\\\\displaystyle dx=\\\\left(\\\\mu -{1 \\\\over {2}}\\\\sigma ^{2}\\\\right)dt+\\\\sigma \\\\,dW} ![{\\\\displaystyle dx=\\\\left(\\\\mu -{1 \\\\over {2}}\\\\sigma ^{2}\\\\right)dt+\\\\sigma \\\\,dW}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7b0f82ac78a1936876587eb26c15fa7b2f32e64a) Then the equivalent Fokker–Planck equation for the evolution of the PDF becomes: ∂ p ∂ t \\+ ( μ − 1 2 σ 2 ) ∂ p ∂ x \\= 1 2 σ 2 ∂ 2 p ∂ x 2 , p ( 0 , x ) \\= δ ( x ) {\\\\displaystyle {\\\\partial p \\\\over {\\\\partial t}}+\\\\left(\\\\mu -{1 \\\\over {2}}\\\\sigma ^{2}\\\\right){\\\\partial p \\\\over {\\\\partial x}}={1 \\\\over {2}}\\\\sigma ^{2}{\\\\partial ^{2}p \\\\over {\\\\partial x^{2}}},\\\\quad p(0,x)=\\\\delta (x)} ![{\\\\displaystyle {\\\\partial p \\\\over {\\\\partial t}}+\\\\left(\\\\mu -{1 \\\\over {2}}\\\\sigma ^{2}\\\\right){\\\\partial p \\\\over {\\\\partial x}}={1 \\\\over {2}}\\\\sigma ^{2}{\\\\partial ^{2}p \\\\over {\\\\partial x^{2}}},\\\\quad p(0,x)=\\\\delta (x)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/18070926ea29bd1d7df4d204932749ba98c4d6f5) Define V \\= μ − σ 2 \/ 2 {\\\\displaystyle V=\\\\mu -\\\\sigma ^{2}\/2} ![{\\\\displaystyle V=\\\\mu -\\\\sigma ^{2}\/2}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f993e6cf8ae456b42f51d4bee95123a852990451) and D \\= σ 2 \/ 2 {\\\\displaystyle D=\\\\sigma ^{2}\/2} ![{\\\\displaystyle D=\\\\sigma ^{2}\/2}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/657c30593c183d872f0b006925f3266080c180fe). By introducing the new variables ξ \\= x − V t {\\\\displaystyle \\\\xi =x-Vt} ![{\\\\displaystyle \\\\xi =x-Vt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/061ed424493e686baedd2938042ca86a38f6bb62) and τ \\= D t {\\\\displaystyle \\\\tau =Dt} ![{\\\\displaystyle \\\\tau =Dt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fad3874220f39bba512b64c56aa1ddb442039787), the derivatives in the Fokker–Planck equation may be transformed as: ∂ t p \\= D ∂ τ p − V ∂ ξ p ∂ x p \\= ∂ ξ p ∂ x 2 p \\= ∂ ξ 2 p {\\\\displaystyle {\\\\begin{aligned}\\\\partial \\_{t}p&=D\\\\partial \\_{\\\\tau }p-V\\\\partial \\_{\\\\xi }p\\\\\\\\\\\\partial \\_{x}p&=\\\\partial \\_{\\\\xi }p\\\\\\\\\\\\partial \\_{x}^{2}p&=\\\\partial \\_{\\\\xi }^{2}p\\\\end{aligned}}} ![{\\\\displaystyle {\\\\begin{aligned}\\\\partial \\_{t}p&=D\\\\partial \\_{\\\\tau }p-V\\\\partial \\_{\\\\xi }p\\\\\\\\\\\\partial \\_{x}p&=\\\\partial \\_{\\\\xi }p\\\\\\\\\\\\partial \\_{x}^{2}p&=\\\\partial \\_{\\\\xi }^{2}p\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dae8470306338fcfcfcfb5c35b0509309525d195) Leading to the new form of the Fokker–Planck equation: ∂ p ∂ τ \\= ∂ 2 p ∂ ξ 2 , p ( 0 , ξ ) \\= δ ( ξ ) {\\\\displaystyle {\\\\partial p \\\\over {\\\\partial \\\\tau }}={\\\\partial ^{2}p \\\\over {\\\\partial \\\\xi ^{2}}},\\\\quad p(0,\\\\xi )=\\\\delta (\\\\xi )} ![{\\\\displaystyle {\\\\partial p \\\\over {\\\\partial \\\\tau }}={\\\\partial ^{2}p \\\\over {\\\\partial \\\\xi ^{2}}},\\\\quad p(0,\\\\xi )=\\\\delta (\\\\xi )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3524160633d67751dbd36d9320a3d0d9cac24350) However, this is the canonical form of the [heat equation](https:\/\/en.wikipedia.org\/wiki\/Heat_equation \"Heat equation\"). which has the solution given by the [heat kernel](https:\/\/en.wikipedia.org\/wiki\/Heat_kernel \"Heat kernel\"): p ( τ , ξ ) \\= 1 4 π τ exp ⁡ ( − ξ 2 4 τ ) {\\\\displaystyle p(\\\\tau ,\\\\xi )={1 \\\\over {\\\\sqrt {4\\\\pi \\\\tau }}}\\\\exp \\\\left(-{\\\\xi ^{2} \\\\over 4\\\\tau }\\\\right)} ![{\\\\displaystyle p(\\\\tau ,\\\\xi )={1 \\\\over {\\\\sqrt {4\\\\pi \\\\tau }}}\\\\exp \\\\left(-{\\\\xi ^{2} \\\\over 4\\\\tau }\\\\right)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/39476d718fdd67a80bb7626ccb7dcf07e548158f) Plugging in the original variables leads to the PDF for GBM: p ( t , S ) \\= 1 S 2 π σ 2 t exp ⁡ { − \\[ log ⁡ ( S \/ S 0 ) − ( μ − 1 2 σ 2 ) t \\] 2 2 σ 2 t } {\\\\displaystyle p(t,S)={1 \\\\over {S{\\\\sqrt {2\\\\pi \\\\sigma ^{2}t}}}}\\\\exp \\\\left\\\\{-{\\\\left\\[\\\\log(S\/S\\_{0})-\\\\left(\\\\mu -{1 \\\\over 2}\\\\sigma ^{2}\\\\right)t\\\\right\\]^{2} \\\\over {2\\\\sigma ^{2}t}}\\\\right\\\\}} ![{\\\\displaystyle p(t,S)={1 \\\\over {S{\\\\sqrt {2\\\\pi \\\\sigma ^{2}t}}}}\\\\exp \\\\left\\\\{-{\\\\left\\[\\\\log(S\/S\\_{0})-\\\\left(\\\\mu -{1 \\\\over 2}\\\\sigma ^{2}\\\\right)t\\\\right\\]^{2} \\\\over {2\\\\sigma ^{2}t}}\\\\right\\\\}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e4a7389649e234f4214a50946c1928ea272d0d2b) |\n\nWhen deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log(*S**t*). This is an interesting process, because in the Black–Scholes model it is related to the [log return](https:\/\/en.wikipedia.org\/wiki\/Log_return \"Log return\") of the stock price. Using [Itô's lemma](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4%27s_lemma \"Itô's lemma\") with *f*(*S*) = log(*S*) gives\n\nd\n\nlog\n\n⁡\n\n(\n\nS\n\n)\n\n\\=\n\nf\n\n′\n\n(\n\nS\n\n)\n\nd\n\nS\n\n\\+\n\n1\n\n2\n\nf\n\n″\n\n(\n\nS\n\n)\n\nS\n\n2\n\nσ\n\n2\n\nd\n\nt\n\n\\=\n\n1\n\nS\n\n(\n\nσ\n\nS\n\nd\n\nW\n\nt\n\n\\+\n\nμ\n\nS\n\nd\n\nt\n\n)\n\n−\n\n1\n\n2\n\nσ\n\n2\n\nd\n\nt\n\n\\=\n\nσ\n\nd\n\nW\n\nt\n\n\\+\n\n(\n\nμ\n\n−\n\nσ\n\n2\n\n\/\n\n2\n\n)\n\nd\n\nt\n\n.\n\n{\\\\displaystyle {\\\\begin{alignedat}{2}d\\\\log(S)&=f'(S)\\\\,dS+{\\\\frac {1}{2}}f''(S)S^{2}\\\\sigma ^{2}\\\\,dt\\\\\\\\\\[6pt\\]&={\\\\frac {1}{S}}\\\\left(\\\\sigma S\\\\,dW\\_{t}+\\\\mu S\\\\,dt\\\\right)-{\\\\frac {1}{2}}\\\\sigma ^{2}\\\\,dt\\\\\\\\\\[6pt\\]&=\\\\sigma \\\\,dW\\_{t}+(\\\\mu -\\\\sigma ^{2}\/2)\\\\,dt.\\\\end{alignedat}}}\n\n![{\\\\displaystyle {\\\\begin{alignedat}{2}d\\\\log(S)&=f'(S)\\\\,dS+{\\\\frac {1}{2}}f''(S)S^{2}\\\\sigma ^{2}\\\\,dt\\\\\\\\\\[6pt\\]&={\\\\frac {1}{S}}\\\\left(\\\\sigma S\\\\,dW\\_{t}+\\\\mu S\\\\,dt\\\\right)-{\\\\frac {1}{2}}\\\\sigma ^{2}\\\\,dt\\\\\\\\\\[6pt\\]&=\\\\sigma \\\\,dW\\_{t}+(\\\\mu -\\\\sigma ^{2}\/2)\\\\,dt.\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/95b72bd76b7255e8dbd8e7c224df8ec2ac15ba9d)\n\nIt follows that E ⁡ log ⁡ ( S t ) \\= log ⁡ ( S 0 ) \\+ ( μ − σ 2 \/ 2 ) t {\\\\displaystyle \\\\operatorname {E} \\\\log(S\\_{t})=\\\\log(S\\_{0})+(\\\\mu -\\\\sigma ^{2}\/2)t} ![{\\\\displaystyle \\\\operatorname {E} \\\\log(S\\_{t})=\\\\log(S\\_{0})+(\\\\mu -\\\\sigma ^{2}\/2)t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/16ac3c7e020b0fbbea10e3ae8ced60fe9212604f).\n\nThis result can also be derived by applying the logarithm to the explicit solution of GBM:\n\nlog\n\n⁡\n\n(\n\nS\n\nt\n\n)\n\n\\=\n\nlog\n\n⁡\n\n(\n\nS\n\n0\n\nexp\n\n⁡\n\n(\n\n(\n\nμ\n\n−\n\nσ\n\n2\n\n2\n\n)\n\nt\n\n\\+\n\nσ\n\nW\n\nt\n\n)\n\n)\n\n\\=\n\nlog\n\n⁡\n\n(\n\nS\n\n0\n\n)\n\n\\+\n\n(\n\nμ\n\n−\n\nσ\n\n2\n\n2\n\n)\n\nt\n\n\\+\n\nσ\n\nW\n\nt\n\n.\n\n{\\\\displaystyle {\\\\begin{alignedat}{2}\\\\log(S\\_{t})&=\\\\log \\\\left(S\\_{0}\\\\exp \\\\left(\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}\\\\right)\\\\right)\\\\\\\\\\[6pt\\]&=\\\\log(S\\_{0})+\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}.\\\\end{alignedat}}}\n\n![{\\\\displaystyle {\\\\begin{alignedat}{2}\\\\log(S\\_{t})&=\\\\log \\\\left(S\\_{0}\\\\exp \\\\left(\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}\\\\right)\\\\right)\\\\\\\\\\[6pt\\]&=\\\\log(S\\_{0})+\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}.\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/592321ca2ac7a8f230e2854bef22804a8c3d1064)\n\nTaking the expectation yields the same result as above: E ⁡ log ⁡ ( S t ) \\= log ⁡ ( S 0 ) \\+ ( μ − σ 2 \/ 2 ) t {\\\\displaystyle \\\\operatorname {E} \\\\log(S\\_{t})=\\\\log(S\\_{0})+(\\\\mu -\\\\sigma ^{2}\/2)t} ![{\\\\displaystyle \\\\operatorname {E} \\\\log(S\\_{t})=\\\\log(S\\_{0})+(\\\\mu -\\\\sigma ^{2}\/2)t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/16ac3c7e020b0fbbea10e3ae8ced60fe9212604f).\n\n## Multivariate version\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=5 \"Edit section: Multivariate version\")\\]\n\nGBM can be extended to the case where there are multiple correlated price paths.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-musielarutkowski-3)\n\nEach price path follows the underlying process\n\nd\n\nS\n\nt\n\ni\n\n\\=\n\nμ\n\ni\n\nS\n\nt\n\ni\n\nd\n\nt\n\n\\+\n\nσ\n\ni\n\nS\n\nt\n\ni\n\nd\n\nW\n\nt\n\ni\n\n,\n\n{\\\\displaystyle dS\\_{t}^{i}=\\\\mu \\_{i}S\\_{t}^{i}\\\\,dt+\\\\sigma \\_{i}S\\_{t}^{i}\\\\,dW\\_{t}^{i},}\n\n![{\\\\displaystyle dS\\_{t}^{i}=\\\\mu \\_{i}S\\_{t}^{i}\\\\,dt+\\\\sigma \\_{i}S\\_{t}^{i}\\\\,dW\\_{t}^{i},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/41c7438fc68d06af0978e28caeb7f4f2ea4588cc)\n\nwhere the Wiener processes are correlated such that E ⁡ ( d W t i d W t j ) \\= ρ i , j d t {\\\\displaystyle \\\\operatorname {E} (dW\\_{t}^{i}\\\\,dW\\_{t}^{j})=\\\\rho \\_{i,j}\\\\,dt} ![{\\\\displaystyle \\\\operatorname {E} (dW\\_{t}^{i}\\\\,dW\\_{t}^{j})=\\\\rho \\_{i,j}\\\\,dt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d59a9358bd9a5d6d00a1f1610f58f3244be0761c) where ρ i , i \\= 1 {\\\\displaystyle \\\\rho \\_{i,i}=1} ![{\\\\displaystyle \\\\rho \\_{i,i}=1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/05c5293f972f2fb115ca086541bbeed11a7f2c3d).\n\nFor the multivariate case, this implies that\n\nCov\n\n⁡\n\n(\n\nS\n\nt\n\ni\n\n,\n\nS\n\nt\n\nj\n\n)\n\n\\=\n\nS\n\n0\n\ni\n\nS\n\n0\n\nj\n\ne\n\n(\n\nμ\n\ni\n\n\\+\n\nμ\n\nj\n\n)\n\nt\n\n(\n\ne\n\nρ\n\ni\n\n,\n\nj\n\nσ\n\ni\n\nσ\n\nj\n\nt\n\n−\n\n1\n\n)\n\n.\n\n{\\\\displaystyle \\\\operatorname {Cov} (S\\_{t}^{i},S\\_{t}^{j})=S\\_{0}^{i}S\\_{0}^{j}e^{(\\\\mu \\_{i}+\\\\mu \\_{j})t}\\\\left(e^{\\\\rho \\_{i,j}\\\\sigma \\_{i}\\\\sigma \\_{j}t}-1\\\\right).}\n\n![{\\\\displaystyle \\\\operatorname {Cov} (S\\_{t}^{i},S\\_{t}^{j})=S\\_{0}^{i}S\\_{0}^{j}e^{(\\\\mu \\_{i}+\\\\mu \\_{j})t}\\\\left(e^{\\\\rho \\_{i,j}\\\\sigma \\_{i}\\\\sigma \\_{j}t}-1\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1f8f47f21d96baa1dbcd2ac72458f427304aec24)\n\nA multivariate formulation that maintains the driving Brownian motions W t i {\\\\displaystyle W\\_{t}^{i}} ![{\\\\displaystyle W\\_{t}^{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3514f7cf4ee08e76eed5be82f080bdc9768727a0) independent is\n\nd\n\nS\n\nt\n\ni\n\n\\=\n\nμ\n\ni\n\nS\n\nt\n\ni\n\nd\n\nt\n\n\\+\n\n∑\n\nj\n\n\\=\n\n1\n\nd\n\nσ\n\ni\n\n,\n\nj\n\nS\n\nt\n\ni\n\nd\n\nW\n\nt\n\nj\n\n,\n\n{\\\\displaystyle dS\\_{t}^{i}=\\\\mu \\_{i}S\\_{t}^{i}\\\\,dt+\\\\sum \\_{j=1}^{d}\\\\sigma \\_{i,j}S\\_{t}^{i}\\\\,dW\\_{t}^{j},}\n\n![{\\\\displaystyle dS\\_{t}^{i}=\\\\mu \\_{i}S\\_{t}^{i}\\\\,dt+\\\\sum \\_{j=1}^{d}\\\\sigma \\_{i,j}S\\_{t}^{i}\\\\,dW\\_{t}^{j},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7d592333694e71c54ab6c9ad941b816f4727c165)\n\nwhere the correlation between S t i {\\\\displaystyle S\\_{t}^{i}} ![{\\\\displaystyle S\\_{t}^{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/816d2d0f0589121853483186d52a881bb2f16dfa) and S t j {\\\\displaystyle S\\_{t}^{j}} ![{\\\\displaystyle S\\_{t}^{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5aa071c0b70847f867a603e15ef699e1c0b9a593) is now expressed through the σ i , j \\= ρ i , j σ i σ j {\\\\displaystyle \\\\sigma \\_{i,j}=\\\\rho \\_{i,j}\\\\,\\\\sigma \\_{i}\\\\,\\\\sigma \\_{j}} ![{\\\\displaystyle \\\\sigma \\_{i,j}=\\\\rho \\_{i,j}\\\\,\\\\sigma \\_{i}\\\\,\\\\sigma \\_{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7e2e8fbcde3e6eb651edf66912e0d67881ce52cf) terms.\n\n## Use in finance\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=6 \"Edit section: Use in finance\")\\]\n\nMain article: [Black–Scholes model](https:\/\/en.wikipedia.org\/wiki\/Black%E2%80%93Scholes_model \"Black–Scholes model\")\n\nGeometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-Hull-4)\n\nSome of the arguments for using GBM to model stock prices are:\n\n- The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-Hull-4)\n- A GBM process only assumes positive values, just like real stock prices.\n- A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices.\n- Calculations with GBM processes are relatively easy.\n\nHowever, GBM is not a completely realistic model, in particular it falls short of reality in the following points:\n\n- In real stock prices, volatility changes over time (possibly [stochastically](https:\/\/en.wikipedia.org\/wiki\/Stochastic_volatility \"Stochastic volatility\")), but in GBM, volatility is assumed constant.\n- In real life, stock prices often show jumps caused by unpredictable events or news, but in GBM, the path is continuous (no discontinuity).\n\nApart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-5)\n\n## Extensions\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=7 \"Edit section: Extensions\")\\]\n\nIn an attempt to make GBM more realistic as a model for stock prices, also in relation to the [volatility smile](https:\/\/en.wikipedia.org\/wiki\/Volatility_smile \"Volatility smile\") problem, one can drop the assumption that the volatility (σ {\\\\displaystyle \\\\sigma } ![{\\\\displaystyle \\\\sigma }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)) is constant. If we assume that the volatility is a [deterministic](https:\/\/en.wikipedia.org\/wiki\/Deterministic_system \"Deterministic system\") function of the stock price and time, this is called a [local volatility](https:\/\/en.wikipedia.org\/wiki\/Local_volatility \"Local volatility\") model. A straightforward extension of the Black Scholes GBM is a local volatility SDE whose distribution is a mixture of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-musielarutkowski-3)[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-6)[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-7)[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-8) If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a [stochastic volatility](https:\/\/en.wikipedia.org\/wiki\/Stochastic_volatility \"Stochastic volatility\") model, see for example the [Heston model](https:\/\/en.wikipedia.org\/wiki\/Heston_model \"Heston model\").[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-9)\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=8 \"Edit section: See also\")\\]\n\n- [Brownian surface](https:\/\/en.wikipedia.org\/wiki\/Brownian_surface \"Brownian surface\")\n- [Feynman–Kac formula](https:\/\/en.wikipedia.org\/wiki\/Feynman%E2%80%93Kac_formula \"Feynman–Kac formula\")\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=9 \"Edit section: References\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-1)**\n   Ross, Sheldon M. (2014). [\"Variations on Brownian Motion\"](https:\/\/books.google.com\/books?id=A3YpAgAAQBAJ&pg=PA612). *Introduction to Probability Models* (11th ed.). Amsterdam: Elsevier. pp. 612–14\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-12-407948-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-12-407948-9 \"Special:BookSources\/978-0-12-407948-9\")\n   \n   .\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-2)**\n   Øksendal, Bernt K. (2002), *Stochastic Differential Equations: An Introduction with Applications*, Springer, p. 326, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [3-540-63720-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/3-540-63720-6 \"Special:BookSources\/3-540-63720-6\")\n3. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-musielarutkowski_3-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-musielarutkowski_3-1) Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.\n4. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-Hull_4-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-Hull_4-1)\n   Hull, John (2009). \"12.3\". *Options, Futures, and other Derivatives* (7 ed.).\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-5)**\n   Rej, A.; Seager, P.; Bouchaud, J.-P. (January 2018). [\"You are in a drawdown. When should you start worrying?\"](https:\/\/onlinelibrary.wiley.com\/doi\/abs\/10.1002\/wilm.10646). *Wilmott*. **2018** (93): 56–59\\. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1707\\.01457](https:\/\/arxiv.org\/abs\/1707.01457). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/wilm.10646](https:\/\/doi.org\/10.1002%2Fwilm.10646). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [157827746](https:\/\/api.semanticscholar.org\/CorpusID:157827746).\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-6)** Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI <https:\/\/doi.org\/10.1007\/3-540-30591-2>\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-7)**\n   [Brigo, Damiano](https:\/\/en.wikipedia.org\/wiki\/Damiano_Brigo \"Damiano Brigo\"); [Mercurio, Fabio](https:\/\/en.wikipedia.org\/wiki\/Fabio_Mercurio \"Fabio Mercurio\") (2002). \"Lognormal-mixture dynamics and calibration to market volatility smiles\". *International Journal of Theoretical and Applied Finance*. **5** (4): 427–446\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1142\/S0219024902001511](https:\/\/doi.org\/10.1142%2FS0219024902001511).\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-8)**\n   Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183,\n   \n   [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1469-7688](https:\/\/search.worldcat.org\/issn\/1469-7688)\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-9)**\n   [Heston, Steven L.](https:\/\/en.wikipedia.org\/wiki\/Steven_L._Heston \"Steven L. Heston\") (1993). \"A closed-form solution for options with stochastic volatility with applications to bond and currency options\". *Review of Financial Studies*. **6** (2): 327–343\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1093\/rfs\/6.2.327](https:\/\/doi.org\/10.1093%2Frfs%2F6.2.327). [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [2962057](https:\/\/www.jstor.org\/stable\/2962057). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [16091300](https:\/\/api.semanticscholar.org\/CorpusID:16091300).\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=10 \"Edit section: External links\")\\]\n\n- [Geometric Brownian motion models for stock movement except in rare events.](https:\/\/web.archive.org\/web\/20120130222949\/http:\/\/math.nyu.edu\/financial_mathematics\/content\/02_financial\/02.html)\n- [Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices](https:\/\/web.archive.org\/web\/20170402233858\/http:\/\/excelandfinance.com\/simulation-of-stock-prices\/brownian-motion\/)\n- [\"Interactive Web Application: Stochastic Processes used in Quantitative Finance\"](https:\/\/web.archive.org\/web\/20150920231636\/http:\/\/turingfinance.com\/interactive-stochastic-processes\/). Archived from [the original](http:\/\/turingfinance.com\/interactive-stochastic-processes\/) on 2015-09-20. Retrieved 2015-07-03.\n\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Stochastic_processes \"Template:Stochastic processes\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Stochastic_processes \"Template talk:Stochastic processes\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Stochastic_processes \"Special:EditPage\/Template:Stochastic processes\")[Stochastic processes](https:\/\/en.wikipedia.org\/wiki\/Stochastic_process \"Stochastic process\") |   |\n|---|---|\n| [Discrete time](https:\/\/en.wikipedia.org\/wiki\/Discrete-time_stochastic_process \"Discrete-time stochastic process\") | [Bernoulli process](https:\/\/en.wikipedia.org\/wiki\/Bernoulli_process \"Bernoulli process\") [Branching process](https:\/\/en.wikipedia.org\/wiki\/Branching_process \"Branching process\") [Chinese restaurant process](https:\/\/en.wikipedia.org\/wiki\/Chinese_restaurant_process \"Chinese restaurant process\") [Galton–Watson process](https:\/\/en.wikipedia.org\/wiki\/Galton%E2%80%93Watson_process \"Galton–Watson process\") [Independent and identically distributed random variables](https:\/\/en.wikipedia.org\/wiki\/Independent_and_identically_distributed_random_variables \"Independent and identically distributed random variables\") [Markov chain](https:\/\/en.wikipedia.org\/wiki\/Markov_chain \"Markov chain\") [Moran process](https:\/\/en.wikipedia.org\/wiki\/Moran_process \"Moran process\") [Random walk](https:\/\/en.wikipedia.org\/wiki\/Random_walk \"Random walk\") [Loop-erased](https:\/\/en.wikipedia.org\/wiki\/Loop-erased_random_walk \"Loop-erased random walk\") [Self-avoiding](https:\/\/en.wikipedia.org\/wiki\/Self-avoiding_walk \"Self-avoiding walk\") [Biased](https:\/\/en.wikipedia.org\/wiki\/Biased_random_walk_on_a_graph \"Biased random walk on a graph\") [Maximal entropy](https:\/\/en.wikipedia.org\/wiki\/Maximal_entropy_random_walk \"Maximal entropy random walk\") |\n| [Continuous time](https:\/\/en.wikipedia.org\/wiki\/Continuous-time_stochastic_process \"Continuous-time stochastic process\") | [Additive process](https:\/\/en.wikipedia.org\/wiki\/Additive_process \"Additive process\") [Airy process](https:\/\/en.wikipedia.org\/wiki\/Airy_process \"Airy process\") [Bessel process](https:\/\/en.wikipedia.org\/wiki\/Bessel_process \"Bessel process\") [Birth–death process](https:\/\/en.wikipedia.org\/wiki\/Birth%E2%80%93death_process \"Birth–death process\") [pure birth](https:\/\/en.wikipedia.org\/wiki\/Birth_process \"Birth process\") [Brownian motion](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\") [Bridge](https:\/\/en.wikipedia.org\/wiki\/Brownian_bridge \"Brownian bridge\") [Dyson](https:\/\/en.wikipedia.org\/wiki\/Dyson_Brownian_motion \"Dyson Brownian motion\") [Excursion](https:\/\/en.wikipedia.org\/wiki\/Brownian_excursion \"Brownian excursion\") [Fractional](https:\/\/en.wikipedia.org\/wiki\/Fractional_Brownian_motion \"Fractional Brownian motion\") [Geometric]() [Meander](https:\/\/en.wikipedia.org\/wiki\/Brownian_meander \"Brownian meander\") [Cauchy process](https:\/\/en.wikipedia.org\/wiki\/Cauchy_process \"Cauchy process\") [Contact process](https:\/\/en.wikipedia.org\/wiki\/Contact_process_\\(mathematics\\) \"Contact process (mathematics)\") [Continuous-time random walk](https:\/\/en.wikipedia.org\/wiki\/Continuous-time_random_walk \"Continuous-time random walk\") [Cox process](https:\/\/en.wikipedia.org\/wiki\/Cox_process \"Cox process\") [Diffusion process](https:\/\/en.wikipedia.org\/wiki\/Diffusion_process \"Diffusion process\") [Empirical process](https:\/\/en.wikipedia.org\/wiki\/Empirical_process \"Empirical process\") [Feller process](https:\/\/en.wikipedia.org\/wiki\/Feller_process \"Feller process\") [Fleming–Viot process](https:\/\/en.wikipedia.org\/wiki\/Fleming%E2%80%93Viot_process \"Fleming–Viot process\") [Gamma process](https:\/\/en.wikipedia.org\/wiki\/Gamma_process \"Gamma process\") [Geometric process](https:\/\/en.wikipedia.org\/wiki\/Geometric_process \"Geometric process\") [Hawkes process](https:\/\/en.wikipedia.org\/wiki\/Hawkes_process \"Hawkes process\") [Hunt process](https:\/\/en.wikipedia.org\/wiki\/Hunt_process \"Hunt process\") [Interacting particle systems](https:\/\/en.wikipedia.org\/wiki\/Interacting_particle_system \"Interacting particle system\") [Itô diffusion](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4_diffusion \"Itô diffusion\") [Itô process](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4_process \"Itô process\") [Jump diffusion](https:\/\/en.wikipedia.org\/wiki\/Jump_diffusion \"Jump diffusion\") [Jump process](https:\/\/en.wikipedia.org\/wiki\/Jump_process \"Jump process\") [Lévy process](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9vy_process \"Lévy process\") [Local time](https:\/\/en.wikipedia.org\/wiki\/Local_time_\\(mathematics\\) \"Local time (mathematics)\") [Markov additive process](https:\/\/en.wikipedia.org\/wiki\/Markov_additive_process \"Markov additive process\") [McKean–Vlasov process](https:\/\/en.wikipedia.org\/wiki\/McKean%E2%80%93Vlasov_process \"McKean–Vlasov process\") [Ornstein–Uhlenbeck process](https:\/\/en.wikipedia.org\/wiki\/Ornstein%E2%80%93Uhlenbeck_process \"Ornstein–Uhlenbeck process\") [Poisson process](https:\/\/en.wikipedia.org\/wiki\/Poisson_point_process \"Poisson point process\") [Compound](https:\/\/en.wikipedia.org\/wiki\/Compound_Poisson_process \"Compound Poisson process\") [Non-homogeneous](https:\/\/en.wikipedia.org\/wiki\/Non-homogeneous_Poisson_process \"Non-homogeneous Poisson process\") [Quasimartingale](https:\/\/en.wikipedia.org\/wiki\/Quasimartingale \"Quasimartingale\") [Schramm–Loewner evolution](https:\/\/en.wikipedia.org\/wiki\/Schramm%E2%80%93Loewner_evolution \"Schramm–Loewner evolution\") [Semimartingale](https:\/\/en.wikipedia.org\/wiki\/Semimartingale \"Semimartingale\") [Sigma-martingale](https:\/\/en.wikipedia.org\/wiki\/Sigma-martingale \"Sigma-martingale\") [Stable process](https:\/\/en.wikipedia.org\/wiki\/Stable_process \"Stable process\") [Superprocess](https:\/\/en.wikipedia.org\/wiki\/Superprocess \"Superprocess\") [Telegraph process](https:\/\/en.wikipedia.org\/wiki\/Telegraph_process \"Telegraph process\") [Variance gamma process](https:\/\/en.wikipedia.org\/wiki\/Variance_gamma_process \"Variance gamma process\") [Wiener process](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\") [Wiener sausage](https:\/\/en.wikipedia.org\/wiki\/Wiener_sausage \"Wiener sausage\") |\n| Both | [Branching process](https:\/\/en.wikipedia.org\/wiki\/Branching_process \"Branching process\") [Gaussian process](https:\/\/en.wikipedia.org\/wiki\/Gaussian_process \"Gaussian process\") [Hidden Markov model (HMM)](https:\/\/en.wikipedia.org\/wiki\/Hidden_Markov_model \"Hidden Markov model\") [Markov process](https:\/\/en.wikipedia.org\/wiki\/Markov_process \"Markov process\") [Martingale](https:\/\/en.wikipedia.org\/wiki\/Martingale_\\(probability_theory\\) \"Martingale (probability theory)\") [Differences](https:\/\/en.wikipedia.org\/wiki\/Martingale_difference_sequence \"Martingale difference sequence\") [Local](https:\/\/en.wikipedia.org\/wiki\/Local_martingale \"Local martingale\") [Sub-](https:\/\/en.wikipedia.org\/wiki\/Submartingale \"Submartingale\") [Super-](https:\/\/en.wikipedia.org\/wiki\/Supermartingale \"Supermartingale\") [Random dynamical system](https:\/\/en.wikipedia.org\/wiki\/Random_dynamical_system \"Random dynamical system\") [Regenerative process](https:\/\/en.wikipedia.org\/wiki\/Regenerative_process \"Regenerative process\") [Renewal process](https:\/\/en.wikipedia.org\/wiki\/Renewal_process \"Renewal process\") [Stochastic chains with memory of variable length](https:\/\/en.wikipedia.org\/wiki\/Stochastic_chains_with_memory_of_variable_length \"Stochastic chains with memory of variable length\") [White noise](https:\/\/en.wikipedia.org\/wiki\/White_noise \"White noise\") |\n| Fields and other | [Dirichlet process](https:\/\/en.wikipedia.org\/wiki\/Dirichlet_process \"Dirichlet process\") [Gaussian random field](https:\/\/en.wikipedia.org\/wiki\/Gaussian_random_field \"Gaussian random field\") [Gibbs measure](https:\/\/en.wikipedia.org\/wiki\/Gibbs_measure \"Gibbs measure\") [Hopfield model](https:\/\/en.wikipedia.org\/wiki\/Hopfield_model \"Hopfield model\") [Ising model](https:\/\/en.wikipedia.org\/wiki\/Ising_model \"Ising model\") [Potts model](https:\/\/en.wikipedia.org\/wiki\/Potts_model \"Potts model\") [Boolean network](https:\/\/en.wikipedia.org\/wiki\/Boolean_network \"Boolean network\") [Markov random field](https:\/\/en.wikipedia.org\/wiki\/Markov_random_field \"Markov random field\") [Percolation](https:\/\/en.wikipedia.org\/wiki\/Percolation_theory \"Percolation theory\") [Pitman–Yor process](https:\/\/en.wikipedia.org\/wiki\/Pitman%E2%80%93Yor_process \"Pitman–Yor process\") [Point process](https:\/\/en.wikipedia.org\/wiki\/Point_process \"Point process\") [Cox](https:\/\/en.wikipedia.org\/wiki\/Point_process#Cox_point_process \"Point process\") [Determinantal](https:\/\/en.wikipedia.org\/wiki\/Determinantal_point_process \"Determinantal point process\") [Poisson](https:\/\/en.wikipedia.org\/wiki\/Poisson_point_process \"Poisson point process\") [Random field](https:\/\/en.wikipedia.org\/wiki\/Random_field \"Random field\") [Random graph](https:\/\/en.wikipedia.org\/wiki\/Random_graph \"Random graph\") |\n| [Time series models](https:\/\/en.wikipedia.org\/wiki\/Time_series \"Time series\") | [Autoregressive conditional heteroskedasticity (ARCH) model](https:\/\/en.wikipedia.org\/wiki\/Autoregressive_conditional_heteroskedasticity \"Autoregressive conditional heteroskedasticity\") [Autoregressive integrated moving average (ARIMA) model](https:\/\/en.wikipedia.org\/wiki\/Autoregressive_integrated_moving_average \"Autoregressive integrated moving average\") [Autoregressive (AR) model](https:\/\/en.wikipedia.org\/wiki\/Autoregressive_model \"Autoregressive model\") [Autoregressive moving-average (ARMA) model](https:\/\/en.wikipedia.org\/wiki\/Autoregressive_moving-average_model \"Autoregressive moving-average model\") [Generalized autoregressive conditional heteroskedasticity (GARCH) model](https:\/\/en.wikipedia.org\/wiki\/Autoregressive_conditional_heteroskedasticity \"Autoregressive conditional heteroskedasticity\") [Moving-average (MA) model](https:\/\/en.wikipedia.org\/wiki\/Moving-average_model \"Moving-average model\") |\n| [Financial models](https:\/\/en.wikipedia.org\/wiki\/Asset_pricing_model \"Asset pricing model\") | [Binomial options pricing model](https:\/\/en.wikipedia.org\/wiki\/Binomial_options_pricing_model \"Binomial options pricing model\") [Black–Derman–Toy](https:\/\/en.wikipedia.org\/wiki\/Black%E2%80%93Derman%E2%80%93Toy_model \"Black–Derman–Toy model\") [Black–Karasinski](https:\/\/en.wikipedia.org\/wiki\/Black%E2%80%93Karasinski_model \"Black–Karasinski model\") [Black–Scholes](https:\/\/en.wikipedia.org\/wiki\/Black%E2%80%93Scholes_model \"Black–Scholes model\") [Chan–Karolyi–Longstaff–Sanders (CKLS)](https:\/\/en.wikipedia.org\/wiki\/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process \"Chan–Karolyi–Longstaff–Sanders process\") [Chen](https:\/\/en.wikipedia.org\/wiki\/Chen_model \"Chen model\") [Constant elasticity of variance (CEV)](https:\/\/en.wikipedia.org\/wiki\/Constant_elasticity_of_variance_model \"Constant elasticity of variance model\") [Cox–Ingersoll–Ross (CIR)](https:\/\/en.wikipedia.org\/wiki\/Cox%E2%80%93Ingersoll%E2%80%93Ross_model \"Cox–Ingersoll–Ross model\") [Garman–Kohlhagen](https:\/\/en.wikipedia.org\/wiki\/Garman%E2%80%93Kohlhagen_model \"Garman–Kohlhagen model\") [Heath–Jarrow–Morton (HJM)](https:\/\/en.wikipedia.org\/wiki\/Heath%E2%80%93Jarrow%E2%80%93Morton_framework \"Heath–Jarrow–Morton framework\") [Heston](https:\/\/en.wikipedia.org\/wiki\/Heston_model \"Heston model\") [Ho–Lee](https:\/\/en.wikipedia.org\/wiki\/Ho%E2%80%93Lee_model \"Ho–Lee model\") [Hull–White](https:\/\/en.wikipedia.org\/wiki\/Hull%E2%80%93White_model \"Hull–White model\") [Korn-Kreer-Lenssen](https:\/\/en.wikipedia.org\/wiki\/Korn%E2%80%93Kreer%E2%80%93Lenssen_model \"Korn–Kreer–Lenssen model\") [LIBOR market](https:\/\/en.wikipedia.org\/wiki\/LIBOR_market_model \"LIBOR market model\") [Rendleman–Bartter](https:\/\/en.wikipedia.org\/wiki\/Rendleman%E2%80%93Bartter_model \"Rendleman–Bartter model\") [SABR volatility](https:\/\/en.wikipedia.org\/wiki\/SABR_volatility_model \"SABR volatility model\") [Vašíček](https:\/\/en.wikipedia.org\/wiki\/Vasicek_model \"Vasicek model\") [Wilkie](https:\/\/en.wikipedia.org\/wiki\/Wilkie_investment_model \"Wilkie investment model\") |\n| [Actuarial models](https:\/\/en.wikipedia.org\/wiki\/Actuarial_mathematics \"Actuarial mathematics\") | [Bühlmann](https:\/\/en.wikipedia.org\/wiki\/B%C3%BChlmann_model \"Bühlmann model\") [Cramér–Lundberg](https:\/\/en.wikipedia.org\/wiki\/Cram%C3%A9r%E2%80%93Lundberg_model \"Cramér–Lundberg model\") [Risk process](https:\/\/en.wikipedia.org\/wiki\/Risk_process \"Risk process\") [Sparre–Anderson](https:\/\/en.wikipedia.org\/wiki\/Sparre%E2%80%93Anderson_model \"Sparre–Anderson model\") |\n| [Queueing models](https:\/\/en.wikipedia.org\/wiki\/Queueing_model \"Queueing model\") | [Bulk](https:\/\/en.wikipedia.org\/wiki\/Bulk_queue \"Bulk queue\") [Fluid](https:\/\/en.wikipedia.org\/wiki\/Fluid_queue \"Fluid queue\") [Generalized queueing network](https:\/\/en.wikipedia.org\/wiki\/G-network \"G-network\") [M\/G\/1](https:\/\/en.wikipedia.org\/wiki\/M\/G\/1_queue \"M\/G\/1 queue\") [M\/M\/1](https:\/\/en.wikipedia.org\/wiki\/M\/M\/1_queue \"M\/M\/1 queue\") [M\/M\/c](https:\/\/en.wikipedia.org\/wiki\/M\/M\/c_queue \"M\/M\/c queue\") |\n| Properties | [Càdlàg paths](https:\/\/en.wikipedia.org\/wiki\/C%C3%A0dl%C3%A0g \"Càdlàg\") [Continuous](https:\/\/en.wikipedia.org\/wiki\/Continuous_stochastic_process \"Continuous stochastic process\") [Continuous paths](https:\/\/en.wikipedia.org\/wiki\/Sample-continuous_process \"Sample-continuous process\") [Ergodic](https:\/\/en.wikipedia.org\/wiki\/Ergodicity \"Ergodicity\") [Exchangeable](https:\/\/en.wikipedia.org\/wiki\/Exchangeable_random_variables \"Exchangeable random variables\") [Feller-continuous](https:\/\/en.wikipedia.org\/wiki\/Feller-continuous_process \"Feller-continuous process\") [Gauss–Markov](https:\/\/en.wikipedia.org\/wiki\/Gauss%E2%80%93Markov_process \"Gauss–Markov process\") [Markov](https:\/\/en.wikipedia.org\/wiki\/Markov_property \"Markov property\") [Mixing](https:\/\/en.wikipedia.org\/wiki\/Mixing_\\(mathematics\\) \"Mixing (mathematics)\") [Piecewise-deterministic](https:\/\/en.wikipedia.org\/wiki\/Piecewise-deterministic_Markov_process \"Piecewise-deterministic Markov process\") [Predictable](https:\/\/en.wikipedia.org\/wiki\/Predictable_process \"Predictable process\") [Progressively measurable](https:\/\/en.wikipedia.org\/wiki\/Progressively_measurable_process \"Progressively measurable process\") [Self-similar](https:\/\/en.wikipedia.org\/wiki\/Self-similar_process \"Self-similar process\") [Stationary](https:\/\/en.wikipedia.org\/wiki\/Stationary_process \"Stationary process\") [Time-reversible](https:\/\/en.wikipedia.org\/wiki\/Time_reversibility \"Time reversibility\") |\n| Limit theorems | [Central limit theorem](https:\/\/en.wikipedia.org\/wiki\/Central_limit_theorem \"Central limit theorem\") [Donsker's theorem](https:\/\/en.wikipedia.org\/wiki\/Donsker%27s_theorem \"Donsker's theorem\") [Doob's martingale convergence theorems](https:\/\/en.wikipedia.org\/wiki\/Doob%27s_martingale_convergence_theorems \"Doob's martingale convergence theorems\") [Ergodic theorem](https:\/\/en.wikipedia.org\/wiki\/Ergodic_theory \"Ergodic theory\") [Fisher–Tippett–Gnedenko theorem](https:\/\/en.wikipedia.org\/wiki\/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem \"Fisher–Tippett–Gnedenko theorem\") [Large deviation principle](https:\/\/en.wikipedia.org\/wiki\/Large_deviation_principle \"Large deviation principle\") [Law of large numbers (weak\/strong)](https:\/\/en.wikipedia.org\/wiki\/Law_of_large_numbers \"Law of large numbers\") [Law of the iterated logarithm](https:\/\/en.wikipedia.org\/wiki\/Law_of_the_iterated_logarithm \"Law of the iterated logarithm\") [Maximal ergodic theorem](https:\/\/en.wikipedia.org\/wiki\/Maximal_ergodic_theorem \"Maximal ergodic theorem\") [Sanov's theorem](https:\/\/en.wikipedia.org\/wiki\/Sanov%27s_theorem \"Sanov's theorem\") [Zero–one laws](https:\/\/en.wikipedia.org\/wiki\/Zero%E2%80%93one_law \"Zero–one law\") ([Blumenthal](https:\/\/en.wikipedia.org\/wiki\/Blumenthal%27s_zero%E2%80%93one_law \"Blumenthal's zero–one law\"), [Borel–Cantelli](https:\/\/en.wikipedia.org\/wiki\/Borel%E2%80%93Cantelli_lemma \"Borel–Cantelli lemma\"), [Engelbert–Schmidt](https:\/\/en.wikipedia.org\/wiki\/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law \"Engelbert–Schmidt zero–one law\"), [Hewitt–Savage](https:\/\/en.wikipedia.org\/wiki\/Hewitt%E2%80%93Savage_zero%E2%80%93one_law \"Hewitt–Savage zero–one law\"),  [Kolmogorov](https:\/\/en.wikipedia.org\/wiki\/Kolmogorov%27s_zero%E2%80%93one_law \"Kolmogorov's zero–one law\"), [Lévy](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9vy%27s_zero%E2%80%93one_law \"Lévy's zero–one law\")) |\n| [Inequalities](https:\/\/en.wikipedia.org\/wiki\/List_of_inequalities#Probability_theory_and_statistics \"List of inequalities\") | [Burkholder–Davis–Gundy](https:\/\/en.wikipedia.org\/wiki\/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities \"Burkholder–Davis–Gundy inequalities\") [Doob's martingale](https:\/\/en.wikipedia.org\/wiki\/Doob%27s_martingale_inequality \"Doob's martingale inequality\") [Doob's upcrossing](https:\/\/en.wikipedia.org\/wiki\/Doob%27s_upcrossing_inequality \"Doob's upcrossing inequality\") [Kunita–Watanabe](https:\/\/en.wikipedia.org\/wiki\/Kunita%E2%80%93Watanabe_inequality \"Kunita–Watanabe inequality\") [Marcinkiewicz–Zygmund](https:\/\/en.wikipedia.org\/wiki\/Marcinkiewicz%E2%80%93Zygmund_inequality \"Marcinkiewicz–Zygmund inequality\") |\n| Tools | [Cameron–Martin theorem](https:\/\/en.wikipedia.org\/wiki\/Cameron%E2%80%93Martin_theorem \"Cameron–Martin theorem\") [Convergence of random variables](https:\/\/en.wikipedia.org\/wiki\/Convergence_of_random_variables \"Convergence of random variables\") [Doléans-Dade exponential](https:\/\/en.wikipedia.org\/wiki\/Dol%C3%A9ans-Dade_exponential \"Doléans-Dade exponential\") [Doob decomposition theorem](https:\/\/en.wikipedia.org\/wiki\/Doob_decomposition_theorem \"Doob decomposition theorem\") [Doob–Meyer decomposition theorem](https:\/\/en.wikipedia.org\/wiki\/Doob%E2%80%93Meyer_decomposition_theorem \"Doob–Meyer decomposition theorem\") [Doob's optional stopping theorem](https:\/\/en.wikipedia.org\/wiki\/Doob%27s_optional_stopping_theorem \"Doob's optional stopping theorem\") [Dynkin's formula](https:\/\/en.wikipedia.org\/wiki\/Dynkin%27s_formula \"Dynkin's formula\") [Feynman–Kac formula](https:\/\/en.wikipedia.org\/wiki\/Feynman%E2%80%93Kac_formula \"Feynman–Kac formula\") [Filtration](https:\/\/en.wikipedia.org\/wiki\/Filtration_\\(probability_theory\\) \"Filtration (probability theory)\") [Girsanov theorem](https:\/\/en.wikipedia.org\/wiki\/Girsanov_theorem \"Girsanov theorem\") [Infinitesimal generator](https:\/\/en.wikipedia.org\/wiki\/Infinitesimal_generator_\\(stochastic_processes\\) \"Infinitesimal generator (stochastic processes)\") [Itô integral](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4_integral \"Itô integral\") [Itô's lemma](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4%27s_lemma \"Itô's lemma\") [Kolmogorov continuity theorem](https:\/\/en.wikipedia.org\/wiki\/Kolmogorov_continuity_theorem \"Kolmogorov continuity theorem\") [Kolmogorov extension theorem](https:\/\/en.wikipedia.org\/wiki\/Kolmogorov_extension_theorem \"Kolmogorov extension theorem\") [Kosambi–Karhunen–Loève theorem](https:\/\/en.wikipedia.org\/wiki\/Kosambi%E2%80%93Karhunen%E2%80%93Lo%C3%A8ve_theorem \"Kosambi–Karhunen–Loève theorem\") [Lévy–Prokhorov metric](https:\/\/en.wikipedia.org\/wiki\/L%C3%A9vy%E2%80%93Prokhorov_metric \"Lévy–Prokhorov metric\") [Malliavin calculus](https:\/\/en.wikipedia.org\/wiki\/Malliavin_calculus \"Malliavin calculus\") [Martingale representation theorem](https:\/\/en.wikipedia.org\/wiki\/Martingale_representation_theorem \"Martingale representation theorem\") [Optional stopping theorem](https:\/\/en.wikipedia.org\/wiki\/Optional_stopping_theorem \"Optional stopping theorem\") [Prokhorov's theorem](https:\/\/en.wikipedia.org\/wiki\/Prokhorov%27s_theorem \"Prokhorov's theorem\") [Quadratic variation](https:\/\/en.wikipedia.org\/wiki\/Quadratic_variation \"Quadratic variation\") [Reflection principle](https:\/\/en.wikipedia.org\/wiki\/Reflection_principle_\\(Wiener_process\\) \"Reflection principle (Wiener process)\") [Skorokhod integral](https:\/\/en.wikipedia.org\/wiki\/Skorokhod_integral \"Skorokhod integral\") [Skorokhod's representation theorem](https:\/\/en.wikipedia.org\/wiki\/Skorokhod%27s_representation_theorem \"Skorokhod's representation theorem\") [Skorokhod space](https:\/\/en.wikipedia.org\/wiki\/Skorokhod_space \"Skorokhod space\") [Snell envelope](https:\/\/en.wikipedia.org\/wiki\/Snell_envelope \"Snell envelope\") [Stochastic differential equation](https:\/\/en.wikipedia.org\/wiki\/Stochastic_differential_equation \"Stochastic differential equation\") [Tanaka](https:\/\/en.wikipedia.org\/wiki\/Tanaka_equation \"Tanaka equation\") [Stopping time](https:\/\/en.wikipedia.org\/wiki\/Stopping_time \"Stopping time\") [Stratonovich integral](https:\/\/en.wikipedia.org\/wiki\/Stratonovich_integral \"Stratonovich integral\") [Uniform integrability](https:\/\/en.wikipedia.org\/wiki\/Uniform_integrability \"Uniform integrability\") [Usual hypotheses](https:\/\/en.wikipedia.org\/wiki\/Usual_hypotheses \"Usual hypotheses\") Wiener space [Classical](https:\/\/en.wikipedia.org\/wiki\/Classical_Wiener_space \"Classical Wiener space\") [Abstract](https:\/\/en.wikipedia.org\/wiki\/Abstract_Wiener_space \"Abstract Wiener space\") |\n| Disciplines | [Actuarial mathematics](https:\/\/en.wikipedia.org\/wiki\/Actuarial_mathematics \"Actuarial mathematics\") [Control theory](https:\/\/en.wikipedia.org\/wiki\/Stochastic_control \"Stochastic control\") [Econometrics](https:\/\/en.wikipedia.org\/wiki\/Econometrics \"Econometrics\") [Ergodic theory](https:\/\/en.wikipedia.org\/wiki\/Ergodic_theory \"Ergodic theory\") [Extreme value theory (EVT)](https:\/\/en.wikipedia.org\/wiki\/Extreme_value_theory \"Extreme value theory\") [Large deviations theory](https:\/\/en.wikipedia.org\/wiki\/Large_deviations_theory \"Large deviations theory\") [Mathematical finance](https:\/\/en.wikipedia.org\/wiki\/Mathematical_finance \"Mathematical finance\") [Mathematical statistics](https:\/\/en.wikipedia.org\/wiki\/Mathematical_statistics \"Mathematical statistics\") [Probability theory](https:\/\/en.wikipedia.org\/wiki\/Probability_theory \"Probability theory\") [Queueing theory](https:\/\/en.wikipedia.org\/wiki\/Queueing_theory \"Queueing theory\") [Renewal theory](https:\/\/en.wikipedia.org\/wiki\/Renewal_theory \"Renewal theory\") [Ruin theory](https:\/\/en.wikipedia.org\/wiki\/Ruin_theory \"Ruin theory\") [Signal processing](https:\/\/en.wikipedia.org\/wiki\/Signal_processing \"Signal processing\") [Statistics](https:\/\/en.wikipedia.org\/wiki\/Statistics \"Statistics\") [Stochastic analysis](https:\/\/en.wikipedia.org\/wiki\/Stochastic_analysis \"Stochastic analysis\") [Time series analysis](https:\/\/en.wikipedia.org\/wiki\/Time_series_analysis \"Time series analysis\") [Machine learning](https:\/\/en.wikipedia.org\/wiki\/Machine_learning \"Machine learning\") |\n| [List of topics](https:\/\/en.wikipedia.org\/wiki\/List_of_stochastic_processes_topics \"List of stochastic processes topics\") [Category](https:\/\/en.wikipedia.org\/wiki\/Category:Stochastic_processes \"Category:Stochastic processes\") |   |\n\n![](https:\/\/en.wikipedia.org\/wiki\/Special:CentralAutoLogin\/start?useformat=desktop&type=1x1&usesul3=1)\n\nRetrieved from \"<https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&oldid=1347612612>\"\n\n[Category](https:\/\/en.wikipedia.org\/wiki\/Help:Category \"Help:Category\"):\n\n- [Wiener process](https:\/\/en.wikipedia.org\/wiki\/Category:Wiener_process \"Category:Wiener process\")\n\nHidden categories:\n\n- [Articles with short description](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_short_description 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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=Geometric_Brownian_motion&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\nGeometric Brownian motion\n\n12 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion)","attrs_readable_markdown":"From Wikipedia, the free encyclopedia\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1c\/GBM2.png\/500px-GBM2.png)](https:\/\/en.wikipedia.org\/wiki\/File:GBM2.png)\n\nFor the simulation generating the realizations, see below.\n\nA **geometric Brownian motion** (**GBM**), also known as an **exponential Brownian motion**, is a continuous-time [stochastic process](https:\/\/en.wikipedia.org\/wiki\/Stochastic_process \"Stochastic process\") in which the [logarithm](https:\/\/en.wikipedia.org\/wiki\/Logarithm \"Logarithm\") of the randomly varying quantity follows a [Brownian motion](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\") with [drift](https:\/\/en.wikipedia.org\/wiki\/Stochastic_drift \"Stochastic drift\").[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-1) It is an important example of stochastic processes satisfying a [stochastic differential equation](https:\/\/en.wikipedia.org\/wiki\/Stochastic_differential_equation \"Stochastic differential equation\") (SDE); in particular, it is used in [mathematical finance](https:\/\/en.wikipedia.org\/wiki\/Mathematical_finance \"Mathematical finance\") to model stock prices in the [Black–Scholes model](https:\/\/en.wikipedia.org\/wiki\/Black%E2%80%93Scholes_model \"Black–Scholes model\").\n\n## Stochastical differential equation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=1 \"Edit section: Stochastical differential equation\")\\]\n\nA stochastic process *S**t* is said to follow a GBM if it satisfies the following [stochastic differential equation](https:\/\/en.wikipedia.org\/wiki\/Stochastic_differential_equation \"Stochastic differential equation\") (SDE):\n\n![{\\\\displaystyle dS\\_{t}=\\\\mu S\\_{t}\\\\,dt+\\\\sigma S\\_{t}\\\\,dW\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f2a3c0d1dcb510719effde045c26f1a9d0b9cb2d)\n\nwhere ![{\\\\displaystyle W\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/50680c5535c83badfa630dba63b583d2eeaa2977) is a [Wiener process or Brownian motion](https:\/\/en.wikipedia.org\/wiki\/Wiener_process \"Wiener process\"), and ![{\\\\displaystyle \\\\mu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9fd47b2a39f7a7856952afec1f1db72c67af6161) ('the percentage drift') and ![{\\\\displaystyle \\\\sigma }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/59f59b7c3e6fdb1d0365a494b81fb9a696138c36) ('the percentage volatility') are constants.\n\nThe former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion.\n\nFor an arbitrary initial value *S*0 the above SDE has the analytic solution (under [Itô's interpretation](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4_calculus \"Itô calculus\")):\n\n![{\\\\displaystyle S\\_{t}=S\\_{0}\\\\exp \\\\left(\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/266807b65fd50635526a766c0c89a2913085d0c2)\n\nThe derivation requires the use of [Itô calculus](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4_calculus \"Itô calculus\"). Applying [Itô's formula](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4%27s_formula \"Itô's formula\") leads to\n\n![{\\\\displaystyle d(\\\\ln S\\_{t})=(\\\\ln S\\_{t})'dS\\_{t}+{\\\\frac {1}{2}}(\\\\ln S\\_{t})''\\\\,dS\\_{t}\\\\,dS\\_{t}={\\\\frac {dS\\_{t}}{S\\_{t}}}-{\\\\frac {1}{2}}\\\\,{\\\\frac {1}{S\\_{t}^{2}}}\\\\,dS\\_{t}\\\\,dS\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cbef6e7e7776ca5963d9179d9d0e1d198e26fd63)\n\nwhere ![{\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d7dd8f0d0a80348c6459b950635d83110c4453ae) is the [quadratic variation](https:\/\/en.wikipedia.org\/wiki\/Quadratic_variation \"Quadratic variation\") of the SDE.\n\n![{\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}\\\\,=\\\\,\\\\sigma ^{2}\\\\,S\\_{t}^{2}\\\\,dW\\_{t}^{2}+2\\\\sigma S\\_{t}^{2}\\\\mu \\\\,dW\\_{t}\\\\,dt+\\\\mu ^{2}S\\_{t}^{2}\\\\,dt^{2}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/54a2a5473dd3d9151bdc6f88a4eb51ed055c1797)\n\nWhen ![{\\\\displaystyle dt\\\\to 0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/02f957c6106281a61de6adf0ec77aaf6ea42aba9), ![{\\\\displaystyle dt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ebee76a835701fd1f26047a09855f2ea36bb08fc) converges to 0 faster than ![{\\\\displaystyle dW\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cc431f447c4bc6c9063431fe612625992c3cca91), since ![{\\\\displaystyle dW\\_{t}^{2}=O(dt)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f1f08e336e1625340ddcf50ada36b7bf8cd083b1). So the above infinitesimal can be simplified by\n\n![{\\\\displaystyle dS\\_{t}\\\\,dS\\_{t}\\\\,=\\\\,\\\\sigma ^{2}\\\\,S\\_{t}^{2}\\\\,dt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dceb3d38f0dd8adce9525559a5d9c0ee497b137f)\n\nPlugging the value of ![{\\\\displaystyle dS\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c796c605ee3a300e8b295d4225bb163e899af5ca) in the above equation and simplifying we obtain\n\n![{\\\\displaystyle \\\\ln {\\\\frac {S\\_{t}}{S\\_{0}}}=\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\,\\\\right)t+\\\\sigma W\\_{t}\\\\,.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/adee3ba14c6e0d50918fefe1f352991b13631265)\n\nTaking the exponential and multiplying both sides by ![{\\\\displaystyle S\\_{0}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ebe0ac45a38c4437bd2689a14ec434cd499e7e49) gives the solution claimed above.\n\n## Arithmetic Brownian motion\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=3 \"Edit section: Arithmetic Brownian motion\")\\]\n\nThe process for ![{\\\\displaystyle X\\_{t}=\\\\ln {\\\\frac {S\\_{t}}{S\\_{0}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7184af654d06bc2453b2bc139a81ad9458ca897b), satisfying the SDE\n\n![{\\\\displaystyle dX\\_{t}=\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\,\\\\right)dt+\\\\sigma dW\\_{t}\\\\,,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/387886bce7774547c64416a186455791dda2c4e5)\n\nor more generally the process solving the SDE\n\n![{\\\\displaystyle dX\\_{t}=m\\\\,dt+v\\\\,dW\\_{t}\\\\,,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/98f3370b73be582ffd0d49245c6cfc05801e91df)\n\nwhere ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) and ![{\\\\displaystyle v\\>0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c314fc908a83c555d34968d25e86c5ae0b76ef6f) are real constants and for an initial condition ![{\\\\displaystyle X\\_{0}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6381fdad2b9f11954b1fc2db08bbaccf634ededa), is called an Arithmetic Brownian Motion (ABM). This was the model postulated by [Louis Bachelier](https:\/\/en.wikipedia.org\/wiki\/Louis_Bachelier \"Louis Bachelier\") in 1900 for stock prices, in the first published attempt to model Brownian motion, known today as [Bachelier model](https:\/\/en.wikipedia.org\/wiki\/Bachelier_model \"Bachelier model\"). As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula. Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula.\n\nThe above solution ![{\\\\displaystyle S\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7e2391e6e796fbf718be3828080775ac2ac3d3d4) (for any value of t) is a [log-normally distributed](https:\/\/en.wikipedia.org\/wiki\/Log-normal_distribution \"Log-normal distribution\") [random variable](https:\/\/en.wikipedia.org\/wiki\/Random_variable \"Random variable\") with [expected value](https:\/\/en.wikipedia.org\/wiki\/Expected_value \"Expected value\") and [variance](https:\/\/en.wikipedia.org\/wiki\/Variance \"Variance\") given by[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-2)\n\n![{\\\\displaystyle \\\\operatorname {E} (S\\_{t})=S\\_{0}e^{\\\\mu t},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/029367c196d22b8abeeb9e362cfe5cf10d505b5c)\n\n![{\\\\displaystyle \\\\operatorname {Var} (S\\_{t})=S\\_{0}^{2}e^{2\\\\mu t}\\\\left(e^{\\\\sigma ^{2}t}-1\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b6035329909c079220525421bada1f1b2d67f8d0)\n\nThey can be derived using the fact that ![{\\\\displaystyle Z\\_{t}=\\\\exp \\\\left(\\\\sigma W\\_{t}-{\\\\frac {1}{2}}\\\\sigma ^{2}t\\\\right)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c181320a17700bcd9460adcb293ad092f50eccc4) is a [martingale](https:\/\/en.wikipedia.org\/wiki\/Martingale_\\(probability_theory\\) \"Martingale (probability theory)\"), and that\n\n![{\\\\displaystyle \\\\operatorname {E} \\\\left\\[\\\\exp \\\\left(2\\\\sigma W\\_{t}-\\\\sigma ^{2}t\\\\right)\\\\mid {\\\\mathcal {F}}\\_{s}\\\\right\\]=e^{\\\\sigma ^{2}(t-s)}\\\\exp \\\\left(2\\\\sigma W\\_{s}-\\\\sigma ^{2}s\\\\right),\\\\quad \\\\forall 0\\\\leq s\\<t.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/121f4efbdcac966a20535657392d5de0345d6f9b)\n\nThe [probability density function](https:\/\/en.wikipedia.org\/wiki\/Probability_density_function \"Probability density function\") of ![{\\\\displaystyle S\\_{t}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7e2391e6e796fbf718be3828080775ac2ac3d3d4) is:\n\n![{\\\\displaystyle f\\_{S\\_{t}}(s;\\\\mu ,\\\\sigma ,t)={\\\\frac {1}{\\\\sqrt {2\\\\pi }}}\\\\,{\\\\frac {1}{s\\\\sigma {\\\\sqrt {t}}}}\\\\,\\\\exp \\\\left(-{\\\\frac {\\\\left(\\\\ln s-\\\\ln S\\_{0}-\\\\left(\\\\mu -{\\\\frac {1}{2}}\\\\sigma ^{2}\\\\right)t\\\\right)^{2}}{2\\\\sigma ^{2}t}}\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/90a9074c173307646da549284ffc5d19c8b85b07)\n\n| Derivation of GBM probability density function |\n|---|\n| To derive the probability density function for GBM, we must use the [Fokker–Planck equation](https:\/\/en.wikipedia.org\/wiki\/Fokker%E2%80%93Planck_equation \"Fokker–Planck equation\") to evaluate the time evolution of the PDF: ![{\\\\displaystyle {\\\\partial p \\\\over {\\\\partial t}}+{\\\\partial \\\\over {\\\\partial S}}\\[\\\\mu (t,S)p(t,S)\\]={1 \\\\over {2}}{\\\\partial ^{2} \\\\over {\\\\partial S^{2}}}\\[\\\\sigma ^{2}(t,S)p(t,S)\\],\\\\quad p(0,S)=\\\\delta (S-S\\_{0})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c6827fa340748de87d1aa6a2b7d99d356ef38e6f) where ![{\\\\displaystyle \\\\delta (S)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f29eec89c4928b10514c6993b942ea248b3b5a57) is the [Dirac delta function](https:\/\/en.wikipedia.org\/wiki\/Dirac_delta_function \"Dirac delta function\"). To simplify the computation, we may introduce a logarithmic transform ![{\\\\displaystyle x=\\\\log(S\/S\\_{0})}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/84aae4276b1b7e359e466df00fc19d2786f95717), leading to the form of GBM: ![{\\\\displaystyle dx=\\\\left(\\\\mu -{1 \\\\over {2}}\\\\sigma ^{2}\\\\right)dt+\\\\sigma \\\\,dW}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7b0f82ac78a1936876587eb26c15fa7b2f32e64a) Then the equivalent Fokker–Planck equation for the evolution of the PDF becomes: ![{\\\\displaystyle {\\\\partial p \\\\over {\\\\partial t}}+\\\\left(\\\\mu -{1 \\\\over {2}}\\\\sigma ^{2}\\\\right){\\\\partial p \\\\over {\\\\partial x}}={1 \\\\over {2}}\\\\sigma ^{2}{\\\\partial ^{2}p \\\\over {\\\\partial x^{2}}},\\\\quad p(0,x)=\\\\delta (x)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/18070926ea29bd1d7df4d204932749ba98c4d6f5) Define ![{\\\\displaystyle V=\\\\mu -\\\\sigma ^{2}\/2}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f993e6cf8ae456b42f51d4bee95123a852990451) and ![{\\\\displaystyle D=\\\\sigma ^{2}\/2}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/657c30593c183d872f0b006925f3266080c180fe). By introducing the new variables ![{\\\\displaystyle \\\\xi =x-Vt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/061ed424493e686baedd2938042ca86a38f6bb62) and ![{\\\\displaystyle \\\\tau =Dt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fad3874220f39bba512b64c56aa1ddb442039787), the derivatives in the Fokker–Planck equation may be transformed as: ![{\\\\displaystyle {\\\\begin{aligned}\\\\partial \\_{t}p&=D\\\\partial \\_{\\\\tau }p-V\\\\partial \\_{\\\\xi }p\\\\\\\\\\\\partial \\_{x}p&=\\\\partial \\_{\\\\xi }p\\\\\\\\\\\\partial \\_{x}^{2}p&=\\\\partial \\_{\\\\xi }^{2}p\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dae8470306338fcfcfcfb5c35b0509309525d195) Leading to the new form of the Fokker–Planck equation: ![{\\\\displaystyle {\\\\partial p \\\\over {\\\\partial \\\\tau }}={\\\\partial ^{2}p \\\\over {\\\\partial \\\\xi ^{2}}},\\\\quad p(0,\\\\xi )=\\\\delta (\\\\xi )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3524160633d67751dbd36d9320a3d0d9cac24350) However, this is the canonical form of the [heat equation](https:\/\/en.wikipedia.org\/wiki\/Heat_equation \"Heat equation\"). which has the solution given by the [heat kernel](https:\/\/en.wikipedia.org\/wiki\/Heat_kernel \"Heat kernel\"): ![{\\\\displaystyle p(\\\\tau ,\\\\xi )={1 \\\\over {\\\\sqrt {4\\\\pi \\\\tau }}}\\\\exp \\\\left(-{\\\\xi ^{2} \\\\over 4\\\\tau }\\\\right)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/39476d718fdd67a80bb7626ccb7dcf07e548158f) Plugging in the original variables leads to the PDF for GBM: ![{\\\\displaystyle p(t,S)={1 \\\\over {S{\\\\sqrt {2\\\\pi \\\\sigma ^{2}t}}}}\\\\exp \\\\left\\\\{-{\\\\left\\[\\\\log(S\/S\\_{0})-\\\\left(\\\\mu -{1 \\\\over 2}\\\\sigma ^{2}\\\\right)t\\\\right\\]^{2} \\\\over {2\\\\sigma ^{2}t}}\\\\right\\\\}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e4a7389649e234f4214a50946c1928ea272d0d2b) |\n\nWhen deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log(*S**t*). This is an interesting process, because in the Black–Scholes model it is related to the [log return](https:\/\/en.wikipedia.org\/wiki\/Log_return \"Log return\") of the stock price. Using [Itô's lemma](https:\/\/en.wikipedia.org\/wiki\/It%C3%B4%27s_lemma \"Itô's lemma\") with *f*(*S*) = log(*S*) gives\n\n![{\\\\displaystyle {\\\\begin{alignedat}{2}d\\\\log(S)&=f'(S)\\\\,dS+{\\\\frac {1}{2}}f''(S)S^{2}\\\\sigma ^{2}\\\\,dt\\\\\\\\\\[6pt\\]&={\\\\frac {1}{S}}\\\\left(\\\\sigma S\\\\,dW\\_{t}+\\\\mu S\\\\,dt\\\\right)-{\\\\frac {1}{2}}\\\\sigma ^{2}\\\\,dt\\\\\\\\\\[6pt\\]&=\\\\sigma \\\\,dW\\_{t}+(\\\\mu -\\\\sigma ^{2}\/2)\\\\,dt.\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/95b72bd76b7255e8dbd8e7c224df8ec2ac15ba9d)\n\nIt follows that ![{\\\\displaystyle \\\\operatorname {E} \\\\log(S\\_{t})=\\\\log(S\\_{0})+(\\\\mu -\\\\sigma ^{2}\/2)t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/16ac3c7e020b0fbbea10e3ae8ced60fe9212604f).\n\nThis result can also be derived by applying the logarithm to the explicit solution of GBM:\n\n![{\\\\displaystyle {\\\\begin{alignedat}{2}\\\\log(S\\_{t})&=\\\\log \\\\left(S\\_{0}\\\\exp \\\\left(\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}\\\\right)\\\\right)\\\\\\\\\\[6pt\\]&=\\\\log(S\\_{0})+\\\\left(\\\\mu -{\\\\frac {\\\\sigma ^{2}}{2}}\\\\right)t+\\\\sigma W\\_{t}.\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/592321ca2ac7a8f230e2854bef22804a8c3d1064)\n\nTaking the expectation yields the same result as above: ![{\\\\displaystyle \\\\operatorname {E} \\\\log(S\\_{t})=\\\\log(S\\_{0})+(\\\\mu -\\\\sigma ^{2}\/2)t}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/16ac3c7e020b0fbbea10e3ae8ced60fe9212604f).\n\n## Multivariate version\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Geometric_Brownian_motion&action=edit&section=5 \"Edit section: Multivariate version\")\\]\n\nGBM can be extended to the case where there are multiple correlated price paths.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-musielarutkowski-3)\n\nEach price path follows the underlying process\n\n![{\\\\displaystyle dS\\_{t}^{i}=\\\\mu \\_{i}S\\_{t}^{i}\\\\,dt+\\\\sigma \\_{i}S\\_{t}^{i}\\\\,dW\\_{t}^{i},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/41c7438fc68d06af0978e28caeb7f4f2ea4588cc)\n\nwhere the Wiener processes are correlated such that ![{\\\\displaystyle \\\\operatorname {E} (dW\\_{t}^{i}\\\\,dW\\_{t}^{j})=\\\\rho \\_{i,j}\\\\,dt}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d59a9358bd9a5d6d00a1f1610f58f3244be0761c) where ![{\\\\displaystyle \\\\rho \\_{i,i}=1}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/05c5293f972f2fb115ca086541bbeed11a7f2c3d).\n\nFor the multivariate case, this implies that\n\n![{\\\\displaystyle \\\\operatorname {Cov} (S\\_{t}^{i},S\\_{t}^{j})=S\\_{0}^{i}S\\_{0}^{j}e^{(\\\\mu \\_{i}+\\\\mu \\_{j})t}\\\\left(e^{\\\\rho \\_{i,j}\\\\sigma \\_{i}\\\\sigma \\_{j}t}-1\\\\right).}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1f8f47f21d96baa1dbcd2ac72458f427304aec24)\n\nA multivariate formulation that maintains the driving Brownian motions ![{\\\\displaystyle W\\_{t}^{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3514f7cf4ee08e76eed5be82f080bdc9768727a0) independent is\n\n![{\\\\displaystyle dS\\_{t}^{i}=\\\\mu \\_{i}S\\_{t}^{i}\\\\,dt+\\\\sum \\_{j=1}^{d}\\\\sigma \\_{i,j}S\\_{t}^{i}\\\\,dW\\_{t}^{j},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7d592333694e71c54ab6c9ad941b816f4727c165)\n\nwhere the correlation between ![{\\\\displaystyle S\\_{t}^{i}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/816d2d0f0589121853483186d52a881bb2f16dfa) and ![{\\\\displaystyle S\\_{t}^{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5aa071c0b70847f867a603e15ef699e1c0b9a593) is now expressed through the ![{\\\\displaystyle \\\\sigma \\_{i,j}=\\\\rho \\_{i,j}\\\\,\\\\sigma \\_{i}\\\\,\\\\sigma \\_{j}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7e2e8fbcde3e6eb651edf66912e0d67881ce52cf) terms.\n\nGeometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-Hull-4)\n\nSome of the arguments for using GBM to model stock prices are:\n\n- The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-Hull-4)\n- A GBM process only assumes positive values, just like real stock prices.\n- A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices.\n- Calculations with GBM processes are relatively easy.\n\nHowever, GBM is not a completely realistic model, in particular it falls short of reality in the following points:\n\n- In real stock prices, volatility changes over time (possibly [stochastically](https:\/\/en.wikipedia.org\/wiki\/Stochastic_volatility \"Stochastic volatility\")), but in GBM, volatility is assumed constant.\n- In real life, stock prices often show jumps caused by unpredictable events or news, but in GBM, the path is continuous (no discontinuity).\n\nApart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-5)\n\nIn an attempt to make GBM more realistic as a model for stock prices, also in relation to the [volatility smile](https:\/\/en.wikipedia.org\/wiki\/Volatility_smile \"Volatility smile\") problem, one can drop the assumption that the volatility (![{\\\\displaystyle \\\\sigma }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)) is constant. If we assume that the volatility is a [deterministic](https:\/\/en.wikipedia.org\/wiki\/Deterministic_system \"Deterministic system\") function of the stock price and time, this is called a [local volatility](https:\/\/en.wikipedia.org\/wiki\/Local_volatility \"Local volatility\") model. A straightforward extension of the Black Scholes GBM is a local volatility SDE whose distribution is a mixture of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-musielarutkowski-3)[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-6)[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-7)[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-8) If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a [stochastic volatility](https:\/\/en.wikipedia.org\/wiki\/Stochastic_volatility \"Stochastic volatility\") model, see for example the [Heston model](https:\/\/en.wikipedia.org\/wiki\/Heston_model \"Heston model\").[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_note-9)\n\n- [Brownian surface](https:\/\/en.wikipedia.org\/wiki\/Brownian_surface \"Brownian surface\")\n- [Feynman–Kac formula](https:\/\/en.wikipedia.org\/wiki\/Feynman%E2%80%93Kac_formula \"Feynman–Kac formula\")\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-1)**\n   Ross, Sheldon M. (2014). [\"Variations on Brownian Motion\"](https:\/\/books.google.com\/books?id=A3YpAgAAQBAJ&pg=PA612). *Introduction to Probability Models* (11th ed.). Amsterdam: Elsevier. pp. 612–14\\. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-12-407948-9](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-12-407948-9 \"Special:BookSources\/978-0-12-407948-9\")\n   \n   .\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-2)**\n   Øksendal, Bernt K. (2002), *Stochastic Differential Equations: An Introduction with Applications*, Springer, p. 326, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [3-540-63720-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/3-540-63720-6 \"Special:BookSources\/3-540-63720-6\")\n3. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-musielarutkowski_3-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-musielarutkowski_3-1) Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.\n4. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-Hull_4-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-Hull_4-1)\n   Hull, John (2009). \"12.3\". *Options, Futures, and other Derivatives* (7 ed.).\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-5)**\n   Rej, A.; Seager, P.; Bouchaud, J.-P. (January 2018). [\"You are in a drawdown. When should you start worrying?\"](https:\/\/onlinelibrary.wiley.com\/doi\/abs\/10.1002\/wilm.10646). *Wilmott*. **2018** (93): 56–59\\. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[1707\\.01457](https:\/\/arxiv.org\/abs\/1707.01457). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/wilm.10646](https:\/\/doi.org\/10.1002%2Fwilm.10646). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [157827746](https:\/\/api.semanticscholar.org\/CorpusID:157827746).\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-6)** Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI <https:\/\/doi.org\/10.1007\/3-540-30591-2>\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-7)**\n   [Brigo, Damiano](https:\/\/en.wikipedia.org\/wiki\/Damiano_Brigo \"Damiano Brigo\"); [Mercurio, Fabio](https:\/\/en.wikipedia.org\/wiki\/Fabio_Mercurio \"Fabio Mercurio\") (2002). \"Lognormal-mixture dynamics and calibration to market volatility smiles\". *International Journal of Theoretical and Applied Finance*. **5** (4): 427–446\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1142\/S0219024902001511](https:\/\/doi.org\/10.1142%2FS0219024902001511).\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-8)** Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183, [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1469-7688](https:\/\/search.worldcat.org\/issn\/1469-7688)\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Geometric_Brownian_motion#cite_ref-9)**\n   [Heston, Steven L.](https:\/\/en.wikipedia.org\/wiki\/Steven_L._Heston \"Steven L. Heston\") (1993). \"A closed-form solution for options with stochastic volatility with applications to bond and currency options\". *Review of Financial Studies*. **6** (2): 327–343\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1093\/rfs\/6.2.327](https:\/\/doi.org\/10.1093%2Frfs%2F6.2.327). [JSTOR](https:\/\/en.wikipedia.org\/wiki\/JSTOR_\\(identifier\\) \"JSTOR (identifier)\") [2962057](https:\/\/www.jstor.org\/stable\/2962057). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [16091300](https:\/\/api.semanticscholar.org\/CorpusID:16091300).\n\n- [Geometric Brownian motion models for stock movement except in rare events.](https:\/\/web.archive.org\/web\/20120130222949\/http:\/\/math.nyu.edu\/financial_mathematics\/content\/02_financial\/02.html)\n- [Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices](https:\/\/web.archive.org\/web\/20170402233858\/http:\/\/excelandfinance.com\/simulation-of-stock-prices\/brownian-motion\/)\n- [\"Interactive Web Application: Stochastic Processes used in Quantitative Finance\"](https:\/\/web.archive.org\/web\/20150920231636\/http:\/\/turingfinance.com\/interactive-stochastic-processes\/). Archived from [the original](http:\/\/turingfinance.com\/interactive-stochastic-processes\/) on 2015-09-20. Retrieved 2015-07-03.","meta_canonical":null}

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Boilerpipe Text	From Wikipedia, the free encyclopedia For the simulation generating the realizations, see below. A geometric Brownian motion ( GBM ), also known as an exponential Brownian motion , is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion with drift . [ 1 ] It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model . Stochastical differential equation [ edit ] A stochastic process S t is said to follow a GBM if it satisfies the following stochastic differential equation (SDE): where is a Wiener process or Brownian motion , and ('the percentage drift') and ('the percentage volatility') are constants. The former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion. For an arbitrary initial value S 0 the above SDE has the analytic solution (under Itô's interpretation ): The derivation requires the use of Itô calculus . Applying Itô's formula leads to where is the quadratic variation of the SDE. When , converges to 0 faster than , since . So the above infinitesimal can be simplified by Plugging the value of in the above equation and simplifying we obtain Taking the exponential and multiplying both sides by gives the solution claimed above. Arithmetic Brownian motion [ edit ] The process for , satisfying the SDE or more generally the process solving the SDE where and are real constants and for an initial condition , is called an Arithmetic Brownian Motion (ABM). This was the model postulated by Louis Bachelier in 1900 for stock prices, in the first published attempt to model Brownian motion, known today as Bachelier model . As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula. Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula. The above solution (for any value of t) is a log-normally distributed random variable with expected value and variance given by [ 2 ] They can be derived using the fact that is a martingale , and that The probability density function of is: Derivation of GBM probability density function To derive the probability density function for GBM, we must use the Fokker–Planck equation to evaluate the time evolution of the PDF: where is the Dirac delta function . To simplify the computation, we may introduce a logarithmic transform , leading to the form of GBM: Then the equivalent Fokker–Planck equation for the evolution of the PDF becomes: Define and . By introducing the new variables and , the derivatives in the Fokker–Planck equation may be transformed as: Leading to the new form of the Fokker–Planck equation: However, this is the canonical form of the heat equation . which has the solution given by the heat kernel : Plugging in the original variables leads to the PDF for GBM: When deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log( S t ). This is an interesting process, because in the Black–Scholes model it is related to the log return of the stock price. Using Itô's lemma with f ( S ) = log( S ) gives It follows that . This result can also be derived by applying the logarithm to the explicit solution of GBM: Taking the expectation yields the same result as above: . Multivariate version [ edit ] GBM can be extended to the case where there are multiple correlated price paths. [ 3 ] Each price path follows the underlying process where the Wiener processes are correlated such that where . For the multivariate case, this implies that A multivariate formulation that maintains the driving Brownian motions independent is where the correlation between and is now expressed through the terms. Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior. [ 4 ] Some of the arguments for using GBM to model stock prices are: The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality. [ 4 ] A GBM process only assumes positive values, just like real stock prices. A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices. Calculations with GBM processes are relatively easy. However, GBM is not a completely realistic model, in particular it falls short of reality in the following points: In real stock prices, volatility changes over time (possibly stochastically ), but in GBM, volatility is assumed constant. In real life, stock prices often show jumps caused by unpredictable events or news, but in GBM, the path is continuous (no discontinuity). Apart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies. [ 5 ] In an attempt to make GBM more realistic as a model for stock prices, also in relation to the volatility smile problem, one can drop the assumption that the volatility ( ) is constant. If we assume that the volatility is a deterministic function of the stock price and time, this is called a local volatility model. A straightforward extension of the Black Scholes GBM is a local volatility SDE whose distribution is a mixture of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options. [ 3 ] [ 6 ] [ 7 ] [ 8 ] If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a stochastic volatility model, see for example the Heston model . [ 9 ] Brownian surface Feynman–Kac formula ^ Ross, Sheldon M. (2014). "Variations on Brownian Motion" . Introduction to Probability Models (11th ed.). Amsterdam: Elsevier. pp. 612– 14. ISBN 978-0-12-407948-9 . ^ Øksendal, Bernt K. (2002), Stochastic Differential Equations: An Introduction with Applications , Springer, p. 326, ISBN 3-540-63720-6 ^ a b Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin. ^ a b Hull, John (2009). "12.3". Options, Futures, and other Derivatives (7 ed.). ^ Rej, A.; Seager, P.; Bouchaud, J.-P. (January 2018). "You are in a drawdown. When should you start worrying?" . Wilmott . 2018 (93): 56– 59. arXiv : 1707.01457 . doi : 10.1002/wilm.10646 . S2CID 157827746 . ^ Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI https://doi.org/10.1007/3-540-30591-2 ^ Brigo, Damiano ; Mercurio, Fabio (2002). "Lognormal-mixture dynamics and calibration to market volatility smiles". International Journal of Theoretical and Applied Finance . 5 (4): 427– 446. doi : 10.1142/S0219024902001511 . ^ Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183, ISSN 1469-7688 ^ Heston, Steven L. (1993). "A closed-form solution for options with stochastic volatility with applications to bond and currency options". Review of Financial Studies . 6 (2): 327– 343. doi : 10.1093/rfs/6.2.327 . JSTOR 2962057 . S2CID 16091300 . Geometric Brownian motion models for stock movement except in rare events. Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices "Interactive Web Application: Stochastic Processes used in Quantitative Finance" . Archived from the original on 2015-09-20 . Retrieved 2015-07-03 .
Markdown	[Jump to content](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#bodyContent) Main menu Main menu move to sidebar hide Navigation - [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]") - [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia") - [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events") - [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]") - [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works") - [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia") Contribute - [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia") - [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia") - [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors") - [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]") - [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia") - [Special pages](https://en.wikipedia.org/wiki/Special:SpecialPages "A list of all special pages [q]") [![](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) [Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]") Appearance - [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en) - [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Geometric+Brownian+motion "You are encouraged to create an account and log in; however, it is not mandatory") - [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Geometric+Brownian+motion "You're encouraged to log in; however, it's not mandatory. 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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Geometric_Brownian_motion) - [1 Stochastical differential equation](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#Stochastical_differential_equation) Toggle Stochastical differential equation subsection - [1\.1 Solution](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#Solution) - [2 Arithmetic Brownian motion](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#Arithmetic_Brownian_motion) - [3 Properties](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#Properties) - [4 Multivariate version](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#Multivariate_version) - [5 Use in finance](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#Use_in_finance) - [6 Extensions](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#Extensions) - [7 See also](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#See_also) - [8 References](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#References) - [9 External links](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#External_links) Toggle the table of contents # Geometric Brownian motion 12 languages - [Català](https://ca.wikipedia.org/wiki/Moviment_browni%C3%A0_geom%C3%A8tric "Moviment brownià geomètric – Catalan") - [Deutsch](https://de.wikipedia.org/wiki/Geometrische_brownsche_Bewegung "Geometrische brownsche Bewegung – German") - [Español](https://es.wikipedia.org/wiki/Movimiento_browniano_geom%C3%A9trico "Movimiento browniano geométrico – Spanish") - [עברית](https://he.wikipedia.org/wiki/%D7%AA%D7%A0%D7%95%D7%A2%D7%94_%D7%91%D7%A8%D7%90%D7%95%D7%A0%D7%99%D7%AA_%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%AA "תנועה בראונית גאומטרית – Hebrew") - [Italiano](https://it.wikipedia.org/wiki/Moto_browniano_geometrico "Moto browniano geometrico – Italian") - [日本語](https://ja.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95%E3%83%96%E3%83%A9%E3%82%A6%E3%83%B3%E9%81%8B%E5%8B%95 "幾何ブラウン運動 – Japanese") - [Português](https://pt.wikipedia.org/wiki/Movimento_browniano_geom%C3%A9trico "Movimento browniano geométrico – Portuguese") - [Русский](https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B1%D1%80%D0%BE%D1%83%D0%BD%D0%BE%D0%B2%D1%81%D0%BA%D0%BE%D0%B5_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5 "Геометрическое броуновское движение – Russian") - [Svenska](https://sv.wikipedia.org/wiki/Geometrisk_brownsk_r%C3%B6relse "Geometrisk brownsk rörelse – Swedish") - [Türkçe](https://tr.wikipedia.org/wiki/Geometrik_Brown_hareketi "Geometrik Brown hareketi – Turkish") - [Українська](https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%B1%D1%80%D0%BE%D1%83%D0%BD%D1%96%D0%B2%D1%81%D1%8C%D0%BA%D0%B8%D0%B9_%D1%80%D1%83%D1%85 "Геометричний броунівський рух – Ukrainian") - [中文](https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%B8%83%E6%9C%97%E8%BF%90%E5%8A%A8 "几何布朗运动 – Chinese") [Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q1503307#sitelinks-wikipedia "Edit interlanguage links") - [Article](https://en.wikipedia.org/wiki/Geometric_Brownian_motion "View the content page [c]") - 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[Wikimedia Commons](https://commons.wikimedia.org/wiki/Category:Brownian_motion) - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q1503307 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Continuous stochastic process [![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/GBM2.png/500px-GBM2.png)](https://en.wikipedia.org/wiki/File:GBM2.png) For the simulation generating the realizations, see below. A geometric Brownian motion (GBM), also known as an exponential Brownian motion, is a continuous-time [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") in which the [logarithm](https://en.wikipedia.org/wiki/Logarithm "Logarithm") of the randomly varying quantity follows a [Brownian motion](https://en.wikipedia.org/wiki/Wiener_process "Wiener process") with [drift](https://en.wikipedia.org/wiki/Stochastic_drift "Stochastic drift").[\[1\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-1) It is an important example of stochastic processes satisfying a [stochastic differential equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation "Stochastic differential equation") (SDE); in particular, it is used in [mathematical finance](https://en.wikipedia.org/wiki/Mathematical_finance "Mathematical finance") to model stock prices in the [Black–Scholes model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model "Black–Scholes model"). ## Stochastical differential equation \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=1 "Edit section: Stochastical differential equation")\] A stochastic process St is said to follow a GBM if it satisfies the following [stochastic differential equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation "Stochastic differential equation") (SDE): d S t \= μ S t d t \+ σ S t d W t {\\displaystyle dS\_{t}=\\mu S\_{t}\\,dt+\\sigma S\_{t}\\,dW\_{t}} ![{\\displaystyle dS\_{t}=\\mu S\_{t}\\,dt+\\sigma S\_{t}\\,dW\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a3c0d1dcb510719effde045c26f1a9d0b9cb2d) where W t {\\displaystyle W\_{t}} ![{\\displaystyle W\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50680c5535c83badfa630dba63b583d2eeaa2977) is a [Wiener process or Brownian motion](https://en.wikipedia.org/wiki/Wiener_process "Wiener process"), and μ {\\displaystyle \\mu } ![{\\displaystyle \\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) ('the percentage drift') and σ {\\displaystyle \\sigma } ![{\\displaystyle \\sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36) ('the percentage volatility') are constants. The former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion. ### Solution \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=2 "Edit section: Solution")\] For an arbitrary initial value S0 the above SDE has the analytic solution (under [Itô's interpretation](https://en.wikipedia.org/wiki/It%C3%B4_calculus "Itô calculus")): S t \= S 0 exp ⁡ ( ( μ − σ 2 2 ) t \+ σ W t ) . {\\displaystyle S\_{t}=S\_{0}\\exp \\left(\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}\\right).} ![{\\displaystyle S\_{t}=S\_{0}\\exp \\left(\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/266807b65fd50635526a766c0c89a2913085d0c2) The derivation requires the use of [Itô calculus](https://en.wikipedia.org/wiki/It%C3%B4_calculus "Itô calculus"). Applying [Itô's formula](https://en.wikipedia.org/wiki/It%C3%B4%27s_formula "Itô's formula") leads to d ( ln ⁡ S t ) \= ( ln ⁡ S t ) ′ d S t \+ 1 2 ( ln ⁡ S t ) ″ d S t d S t \= d S t S t − 1 2 1 S t 2 d S t d S t {\\displaystyle d(\\ln S\_{t})=(\\ln S\_{t})'dS\_{t}+{\\frac {1}{2}}(\\ln S\_{t})''\\,dS\_{t}\\,dS\_{t}={\\frac {dS\_{t}}{S\_{t}}}-{\\frac {1}{2}}\\,{\\frac {1}{S\_{t}^{2}}}\\,dS\_{t}\\,dS\_{t}} ![{\\displaystyle d(\\ln S\_{t})=(\\ln S\_{t})'dS\_{t}+{\\frac {1}{2}}(\\ln S\_{t})''\\,dS\_{t}\\,dS\_{t}={\\frac {dS\_{t}}{S\_{t}}}-{\\frac {1}{2}}\\,{\\frac {1}{S\_{t}^{2}}}\\,dS\_{t}\\,dS\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbef6e7e7776ca5963d9179d9d0e1d198e26fd63) where d S t d S t {\\displaystyle dS\_{t}\\,dS\_{t}} ![{\\displaystyle dS\_{t}\\,dS\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7dd8f0d0a80348c6459b950635d83110c4453ae) is the [quadratic variation](https://en.wikipedia.org/wiki/Quadratic_variation "Quadratic variation") of the SDE. d S t d S t \= σ 2 S t 2 d W t 2 \+ 2 σ S t 2 μ d W t d t \+ μ 2 S t 2 d t 2 {\\displaystyle dS\_{t}\\,dS\_{t}\\,=\\,\\sigma ^{2}\\,S\_{t}^{2}\\,dW\_{t}^{2}+2\\sigma S\_{t}^{2}\\mu \\,dW\_{t}\\,dt+\\mu ^{2}S\_{t}^{2}\\,dt^{2}} ![{\\displaystyle dS\_{t}\\,dS\_{t}\\,=\\,\\sigma ^{2}\\,S\_{t}^{2}\\,dW\_{t}^{2}+2\\sigma S\_{t}^{2}\\mu \\,dW\_{t}\\,dt+\\mu ^{2}S\_{t}^{2}\\,dt^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a2a5473dd3d9151bdc6f88a4eb51ed055c1797) When d t → 0 {\\displaystyle dt\\to 0} ![{\\displaystyle dt\\to 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02f957c6106281a61de6adf0ec77aaf6ea42aba9), d t {\\displaystyle dt} ![{\\displaystyle dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebee76a835701fd1f26047a09855f2ea36bb08fc) converges to 0 faster than d W t {\\displaystyle dW\_{t}} ![{\\displaystyle dW\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc431f447c4bc6c9063431fe612625992c3cca91), since d W t 2 \= O ( d t ) {\\displaystyle dW\_{t}^{2}=O(dt)} ![{\\displaystyle dW\_{t}^{2}=O(dt)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f08e336e1625340ddcf50ada36b7bf8cd083b1). So the above infinitesimal can be simplified by d S t d S t \= σ 2 S t 2 d t {\\displaystyle dS\_{t}\\,dS\_{t}\\,=\\,\\sigma ^{2}\\,S\_{t}^{2}\\,dt} ![{\\displaystyle dS\_{t}\\,dS\_{t}\\,=\\,\\sigma ^{2}\\,S\_{t}^{2}\\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dceb3d38f0dd8adce9525559a5d9c0ee497b137f) Plugging the value of d S t {\\displaystyle dS\_{t}} ![{\\displaystyle dS\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c796c605ee3a300e8b295d4225bb163e899af5ca) in the above equation and simplifying we obtain ln ⁡ S t S 0 \= ( μ − σ 2 2 ) t \+ σ W t . {\\displaystyle \\ln {\\frac {S\_{t}}{S\_{0}}}=\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\,\\right)t+\\sigma W\_{t}\\,.} ![{\\displaystyle \\ln {\\frac {S\_{t}}{S\_{0}}}=\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\,\\right)t+\\sigma W\_{t}\\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adee3ba14c6e0d50918fefe1f352991b13631265) Taking the exponential and multiplying both sides by S 0 {\\displaystyle S\_{0}} ![{\\displaystyle S\_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe0ac45a38c4437bd2689a14ec434cd499e7e49) gives the solution claimed above. ## Arithmetic Brownian motion \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=3 "Edit section: Arithmetic Brownian motion")\] The process for X t \= ln ⁡ S t S 0 {\\displaystyle X\_{t}=\\ln {\\frac {S\_{t}}{S\_{0}}}} ![{\\displaystyle X\_{t}=\\ln {\\frac {S\_{t}}{S\_{0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7184af654d06bc2453b2bc139a81ad9458ca897b), satisfying the SDE d X t \= ( μ − σ 2 2 ) d t \+ σ d W t , {\\displaystyle dX\_{t}=\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\,\\right)dt+\\sigma dW\_{t}\\,,} ![{\\displaystyle dX\_{t}=\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\,\\right)dt+\\sigma dW\_{t}\\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/387886bce7774547c64416a186455791dda2c4e5) or more generally the process solving the SDE d X t \= m d t \+ v d W t , {\\displaystyle dX\_{t}=m\\,dt+v\\,dW\_{t}\\,,} ![{\\displaystyle dX\_{t}=m\\,dt+v\\,dW\_{t}\\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f3370b73be582ffd0d49245c6cfc05801e91df) where m {\\displaystyle m} ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) and v \> 0 {\\displaystyle v\>0} ![{\\displaystyle v\>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c314fc908a83c555d34968d25e86c5ae0b76ef6f) are real constants and for an initial condition X 0 {\\displaystyle X\_{0}} ![{\\displaystyle X\_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6381fdad2b9f11954b1fc2db08bbaccf634ededa), is called an Arithmetic Brownian Motion (ABM). This was the model postulated by [Louis Bachelier](https://en.wikipedia.org/wiki/Louis_Bachelier "Louis Bachelier") in 1900 for stock prices, in the first published attempt to model Brownian motion, known today as [Bachelier model](https://en.wikipedia.org/wiki/Bachelier_model "Bachelier model"). As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula. Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula. ## Properties \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=4 "Edit section: Properties")\] The above solution S t {\\displaystyle S\_{t}} ![{\\displaystyle S\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2391e6e796fbf718be3828080775ac2ac3d3d4) (for any value of t) is a [log-normally distributed](https://en.wikipedia.org/wiki/Log-normal_distribution "Log-normal distribution") [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable") with [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value") and [variance](https://en.wikipedia.org/wiki/Variance "Variance") given by[\[2\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-2) E ⁡ ( S t ) \= S 0 e μ t , {\\displaystyle \\operatorname {E} (S\_{t})=S\_{0}e^{\\mu t},} ![{\\displaystyle \\operatorname {E} (S\_{t})=S\_{0}e^{\\mu t},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/029367c196d22b8abeeb9e362cfe5cf10d505b5c) Var ⁡ ( S t ) \= S 0 2 e 2 μ t ( e σ 2 t − 1 ) . {\\displaystyle \\operatorname {Var} (S\_{t})=S\_{0}^{2}e^{2\\mu t}\\left(e^{\\sigma ^{2}t}-1\\right).} ![{\\displaystyle \\operatorname {Var} (S\_{t})=S\_{0}^{2}e^{2\\mu t}\\left(e^{\\sigma ^{2}t}-1\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6035329909c079220525421bada1f1b2d67f8d0) They can be derived using the fact that Z t \= exp ⁡ ( σ W t − 1 2 σ 2 t ) {\\displaystyle Z\_{t}=\\exp \\left(\\sigma W\_{t}-{\\frac {1}{2}}\\sigma ^{2}t\\right)} ![{\\displaystyle Z\_{t}=\\exp \\left(\\sigma W\_{t}-{\\frac {1}{2}}\\sigma ^{2}t\\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c181320a17700bcd9460adcb293ad092f50eccc4) is a [martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)"), and that E ⁡ \[ exp ⁡ ( 2 σ W t − σ 2 t ) ∣ F s \] \= e σ 2 ( t − s ) exp ⁡ ( 2 σ W s − σ 2 s ) , ∀ 0 ≤ s \< t . {\\displaystyle \\operatorname {E} \\left\[\\exp \\left(2\\sigma W\_{t}-\\sigma ^{2}t\\right)\\mid {\\mathcal {F}}\_{s}\\right\]=e^{\\sigma ^{2}(t-s)}\\exp \\left(2\\sigma W\_{s}-\\sigma ^{2}s\\right),\\quad \\forall 0\\leq s\<t.} ![{\\displaystyle \\operatorname {E} \\left\[\\exp \\left(2\\sigma W\_{t}-\\sigma ^{2}t\\right)\\mid {\\mathcal {F}}\_{s}\\right\]=e^{\\sigma ^{2}(t-s)}\\exp \\left(2\\sigma W\_{s}-\\sigma ^{2}s\\right),\\quad \\forall 0\\leq s\<t.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/121f4efbdcac966a20535657392d5de0345d6f9b) The [probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") of S t {\\displaystyle S\_{t}} ![{\\displaystyle S\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2391e6e796fbf718be3828080775ac2ac3d3d4) is: f S t ( s ; μ , σ , t ) \= 1 2 π 1 s σ t exp ⁡ ( − ( ln ⁡ s − ln ⁡ S 0 − ( μ − 1 2 σ 2 ) t ) 2 2 σ 2 t ) . {\\displaystyle f\_{S\_{t}}(s;\\mu ,\\sigma ,t)={\\frac {1}{\\sqrt {2\\pi }}}\\,{\\frac {1}{s\\sigma {\\sqrt {t}}}}\\,\\exp \\left(-{\\frac {\\left(\\ln s-\\ln S\_{0}-\\left(\\mu -{\\frac {1}{2}}\\sigma ^{2}\\right)t\\right)^{2}}{2\\sigma ^{2}t}}\\right).} ![{\\displaystyle f\_{S\_{t}}(s;\\mu ,\\sigma ,t)={\\frac {1}{\\sqrt {2\\pi }}}\\,{\\frac {1}{s\\sigma {\\sqrt {t}}}}\\,\\exp \\left(-{\\frac {\\left(\\ln s-\\ln S\_{0}-\\left(\\mu -{\\frac {1}{2}}\\sigma ^{2}\\right)t\\right)^{2}}{2\\sigma ^{2}t}}\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90a9074c173307646da549284ffc5d19c8b85b07) \| Derivation of GBM probability density function \| \|---\| \| To derive the probability density function for GBM, we must use the [Fokker–Planck equation](https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation "Fokker–Planck equation") to evaluate the time evolution of the PDF: ∂ p ∂ t \+ ∂ ∂ S \[ μ ( t , S ) p ( t , S ) \] \= 1 2 ∂ 2 ∂ S 2 \[ σ 2 ( t , S ) p ( t , S ) \] , p ( 0 , S ) \= δ ( S − S 0 ) {\\displaystyle {\\partial p \\over {\\partial t}}+{\\partial \\over {\\partial S}}\[\\mu (t,S)p(t,S)\]={1 \\over {2}}{\\partial ^{2} \\over {\\partial S^{2}}}\[\\sigma ^{2}(t,S)p(t,S)\],\\quad p(0,S)=\\delta (S-S\_{0})} ![{\\displaystyle {\\partial p \\over {\\partial t}}+{\\partial \\over {\\partial S}}\[\\mu (t,S)p(t,S)\]={1 \\over {2}}{\\partial ^{2} \\over {\\partial S^{2}}}\[\\sigma ^{2}(t,S)p(t,S)\],\\quad p(0,S)=\\delta (S-S\_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6827fa340748de87d1aa6a2b7d99d356ef38e6f) where δ ( S ) {\\displaystyle \\delta (S)} ![{\\displaystyle \\delta (S)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f29eec89c4928b10514c6993b942ea248b3b5a57) is the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function"). To simplify the computation, we may introduce a logarithmic transform x \= log ⁡ ( S / S 0 ) {\\displaystyle x=\\log(S/S\_{0})} ![{\\displaystyle x=\\log(S/S\_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84aae4276b1b7e359e466df00fc19d2786f95717), leading to the form of GBM: d x \= ( μ − 1 2 σ 2 ) d t \+ σ d W {\\displaystyle dx=\\left(\\mu -{1 \\over {2}}\\sigma ^{2}\\right)dt+\\sigma \\,dW} ![{\\displaystyle dx=\\left(\\mu -{1 \\over {2}}\\sigma ^{2}\\right)dt+\\sigma \\,dW}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0f82ac78a1936876587eb26c15fa7b2f32e64a) Then the equivalent Fokker–Planck equation for the evolution of the PDF becomes: ∂ p ∂ t \+ ( μ − 1 2 σ 2 ) ∂ p ∂ x \= 1 2 σ 2 ∂ 2 p ∂ x 2 , p ( 0 , x ) \= δ ( x ) {\\displaystyle {\\partial p \\over {\\partial t}}+\\left(\\mu -{1 \\over {2}}\\sigma ^{2}\\right){\\partial p \\over {\\partial x}}={1 \\over {2}}\\sigma ^{2}{\\partial ^{2}p \\over {\\partial x^{2}}},\\quad p(0,x)=\\delta (x)} ![{\\displaystyle {\\partial p \\over {\\partial t}}+\\left(\\mu -{1 \\over {2}}\\sigma ^{2}\\right){\\partial p \\over {\\partial x}}={1 \\over {2}}\\sigma ^{2}{\\partial ^{2}p \\over {\\partial x^{2}}},\\quad p(0,x)=\\delta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18070926ea29bd1d7df4d204932749ba98c4d6f5) Define V \= μ − σ 2 / 2 {\\displaystyle V=\\mu -\\sigma ^{2}/2} ![{\\displaystyle V=\\mu -\\sigma ^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f993e6cf8ae456b42f51d4bee95123a852990451) and D \= σ 2 / 2 {\\displaystyle D=\\sigma ^{2}/2} ![{\\displaystyle D=\\sigma ^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/657c30593c183d872f0b006925f3266080c180fe). By introducing the new variables ξ \= x − V t {\\displaystyle \\xi =x-Vt} ![{\\displaystyle \\xi =x-Vt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/061ed424493e686baedd2938042ca86a38f6bb62) and τ \= D t {\\displaystyle \\tau =Dt} ![{\\displaystyle \\tau =Dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fad3874220f39bba512b64c56aa1ddb442039787), the derivatives in the Fokker–Planck equation may be transformed as: ∂ t p \= D ∂ τ p − V ∂ ξ p ∂ x p \= ∂ ξ p ∂ x 2 p \= ∂ ξ 2 p {\\displaystyle {\\begin{aligned}\\partial \_{t}p&=D\\partial \_{\\tau }p-V\\partial \_{\\xi }p\\\\\\partial \_{x}p&=\\partial \_{\\xi }p\\\\\\partial \_{x}^{2}p&=\\partial \_{\\xi }^{2}p\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\partial \_{t}p&=D\\partial \_{\\tau }p-V\\partial \_{\\xi }p\\\\\\partial \_{x}p&=\\partial \_{\\xi }p\\\\\\partial \_{x}^{2}p&=\\partial \_{\\xi }^{2}p\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dae8470306338fcfcfcfb5c35b0509309525d195) Leading to the new form of the Fokker–Planck equation: ∂ p ∂ τ \= ∂ 2 p ∂ ξ 2 , p ( 0 , ξ ) \= δ ( ξ ) {\\displaystyle {\\partial p \\over {\\partial \\tau }}={\\partial ^{2}p \\over {\\partial \\xi ^{2}}},\\quad p(0,\\xi )=\\delta (\\xi )} ![{\\displaystyle {\\partial p \\over {\\partial \\tau }}={\\partial ^{2}p \\over {\\partial \\xi ^{2}}},\\quad p(0,\\xi )=\\delta (\\xi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3524160633d67751dbd36d9320a3d0d9cac24350) However, this is the canonical form of the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation"). which has the solution given by the [heat kernel](https://en.wikipedia.org/wiki/Heat_kernel "Heat kernel"): p ( τ , ξ ) \= 1 4 π τ exp ⁡ ( − ξ 2 4 τ ) {\\displaystyle p(\\tau ,\\xi )={1 \\over {\\sqrt {4\\pi \\tau }}}\\exp \\left(-{\\xi ^{2} \\over 4\\tau }\\right)} ![{\\displaystyle p(\\tau ,\\xi )={1 \\over {\\sqrt {4\\pi \\tau }}}\\exp \\left(-{\\xi ^{2} \\over 4\\tau }\\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39476d718fdd67a80bb7626ccb7dcf07e548158f) Plugging in the original variables leads to the PDF for GBM: p ( t , S ) \= 1 S 2 π σ 2 t exp ⁡ { − \[ log ⁡ ( S / S 0 ) − ( μ − 1 2 σ 2 ) t \] 2 2 σ 2 t } {\\displaystyle p(t,S)={1 \\over {S{\\sqrt {2\\pi \\sigma ^{2}t}}}}\\exp \\left\\{-{\\left\[\\log(S/S\_{0})-\\left(\\mu -{1 \\over 2}\\sigma ^{2}\\right)t\\right\]^{2} \\over {2\\sigma ^{2}t}}\\right\\}} ![{\\displaystyle p(t,S)={1 \\over {S{\\sqrt {2\\pi \\sigma ^{2}t}}}}\\exp \\left\\{-{\\left\[\\log(S/S\_{0})-\\left(\\mu -{1 \\over 2}\\sigma ^{2}\\right)t\\right\]^{2} \\over {2\\sigma ^{2}t}}\\right\\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4a7389649e234f4214a50946c1928ea272d0d2b) \| When deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log(St). This is an interesting process, because in the Black–Scholes model it is related to the [log return](https://en.wikipedia.org/wiki/Log_return "Log return") of the stock price. Using [Itô's lemma](https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma "Itô's lemma") with f(S) = log(S) gives d log ⁡ ( S ) \= f ′ ( S ) d S \+ 1 2 f ″ ( S ) S 2 σ 2 d t \= 1 S ( σ S d W t \+ μ S d t ) − 1 2 σ 2 d t \= σ d W t \+ ( μ − σ 2 / 2 ) d t . {\\displaystyle {\\begin{alignedat}{2}d\\log(S)&=f'(S)\\,dS+{\\frac {1}{2}}f''(S)S^{2}\\sigma ^{2}\\,dt\\\\\[6pt\]&={\\frac {1}{S}}\\left(\\sigma S\\,dW\_{t}+\\mu S\\,dt\\right)-{\\frac {1}{2}}\\sigma ^{2}\\,dt\\\\\[6pt\]&=\\sigma \\,dW\_{t}+(\\mu -\\sigma ^{2}/2)\\,dt.\\end{alignedat}}} ![{\\displaystyle {\\begin{alignedat}{2}d\\log(S)&=f'(S)\\,dS+{\\frac {1}{2}}f''(S)S^{2}\\sigma ^{2}\\,dt\\\\\[6pt\]&={\\frac {1}{S}}\\left(\\sigma S\\,dW\_{t}+\\mu S\\,dt\\right)-{\\frac {1}{2}}\\sigma ^{2}\\,dt\\\\\[6pt\]&=\\sigma \\,dW\_{t}+(\\mu -\\sigma ^{2}/2)\\,dt.\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b72bd76b7255e8dbd8e7c224df8ec2ac15ba9d) It follows that E ⁡ log ⁡ ( S t ) \= log ⁡ ( S 0 ) \+ ( μ − σ 2 / 2 ) t {\\displaystyle \\operatorname {E} \\log(S\_{t})=\\log(S\_{0})+(\\mu -\\sigma ^{2}/2)t} ![{\\displaystyle \\operatorname {E} \\log(S\_{t})=\\log(S\_{0})+(\\mu -\\sigma ^{2}/2)t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16ac3c7e020b0fbbea10e3ae8ced60fe9212604f). This result can also be derived by applying the logarithm to the explicit solution of GBM: log ⁡ ( S t ) \= log ⁡ ( S 0 exp ⁡ ( ( μ − σ 2 2 ) t \+ σ W t ) ) \= log ⁡ ( S 0 ) \+ ( μ − σ 2 2 ) t \+ σ W t . {\\displaystyle {\\begin{alignedat}{2}\\log(S\_{t})&=\\log \\left(S\_{0}\\exp \\left(\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}\\right)\\right)\\\\\[6pt\]&=\\log(S\_{0})+\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}.\\end{alignedat}}} ![{\\displaystyle {\\begin{alignedat}{2}\\log(S\_{t})&=\\log \\left(S\_{0}\\exp \\left(\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}\\right)\\right)\\\\\[6pt\]&=\\log(S\_{0})+\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}.\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/592321ca2ac7a8f230e2854bef22804a8c3d1064) Taking the expectation yields the same result as above: E ⁡ log ⁡ ( S t ) \= log ⁡ ( S 0 ) \+ ( μ − σ 2 / 2 ) t {\\displaystyle \\operatorname {E} \\log(S\_{t})=\\log(S\_{0})+(\\mu -\\sigma ^{2}/2)t} ![{\\displaystyle \\operatorname {E} \\log(S\_{t})=\\log(S\_{0})+(\\mu -\\sigma ^{2}/2)t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16ac3c7e020b0fbbea10e3ae8ced60fe9212604f). ## Multivariate version \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=5 "Edit section: Multivariate version")\] GBM can be extended to the case where there are multiple correlated price paths.[\[3\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-musielarutkowski-3) Each price path follows the underlying process d S t i \= μ i S t i d t \+ σ i S t i d W t i , {\\displaystyle dS\_{t}^{i}=\\mu \_{i}S\_{t}^{i}\\,dt+\\sigma \_{i}S\_{t}^{i}\\,dW\_{t}^{i},} ![{\\displaystyle dS\_{t}^{i}=\\mu \_{i}S\_{t}^{i}\\,dt+\\sigma \_{i}S\_{t}^{i}\\,dW\_{t}^{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41c7438fc68d06af0978e28caeb7f4f2ea4588cc) where the Wiener processes are correlated such that E ⁡ ( d W t i d W t j ) \= ρ i , j d t {\\displaystyle \\operatorname {E} (dW\_{t}^{i}\\,dW\_{t}^{j})=\\rho \_{i,j}\\,dt} ![{\\displaystyle \\operatorname {E} (dW\_{t}^{i}\\,dW\_{t}^{j})=\\rho \_{i,j}\\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d59a9358bd9a5d6d00a1f1610f58f3244be0761c) where ρ i , i \= 1 {\\displaystyle \\rho \_{i,i}=1} ![{\\displaystyle \\rho \_{i,i}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05c5293f972f2fb115ca086541bbeed11a7f2c3d). For the multivariate case, this implies that Cov ⁡ ( S t i , S t j ) \= S 0 i S 0 j e ( μ i \+ μ j ) t ( e ρ i , j σ i σ j t − 1 ) . {\\displaystyle \\operatorname {Cov} (S\_{t}^{i},S\_{t}^{j})=S\_{0}^{i}S\_{0}^{j}e^{(\\mu \_{i}+\\mu \_{j})t}\\left(e^{\\rho \_{i,j}\\sigma \_{i}\\sigma \_{j}t}-1\\right).} ![{\\displaystyle \\operatorname {Cov} (S\_{t}^{i},S\_{t}^{j})=S\_{0}^{i}S\_{0}^{j}e^{(\\mu \_{i}+\\mu \_{j})t}\\left(e^{\\rho \_{i,j}\\sigma \_{i}\\sigma \_{j}t}-1\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8f47f21d96baa1dbcd2ac72458f427304aec24) A multivariate formulation that maintains the driving Brownian motions W t i {\\displaystyle W\_{t}^{i}} ![{\\displaystyle W\_{t}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3514f7cf4ee08e76eed5be82f080bdc9768727a0) independent is d S t i \= μ i S t i d t \+ ∑ j \= 1 d σ i , j S t i d W t j , {\\displaystyle dS\_{t}^{i}=\\mu \_{i}S\_{t}^{i}\\,dt+\\sum \_{j=1}^{d}\\sigma \_{i,j}S\_{t}^{i}\\,dW\_{t}^{j},} ![{\\displaystyle dS\_{t}^{i}=\\mu \_{i}S\_{t}^{i}\\,dt+\\sum \_{j=1}^{d}\\sigma \_{i,j}S\_{t}^{i}\\,dW\_{t}^{j},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d592333694e71c54ab6c9ad941b816f4727c165) where the correlation between S t i {\\displaystyle S\_{t}^{i}} ![{\\displaystyle S\_{t}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/816d2d0f0589121853483186d52a881bb2f16dfa) and S t j {\\displaystyle S\_{t}^{j}} ![{\\displaystyle S\_{t}^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa071c0b70847f867a603e15ef699e1c0b9a593) is now expressed through the σ i , j \= ρ i , j σ i σ j {\\displaystyle \\sigma \_{i,j}=\\rho \_{i,j}\\,\\sigma \_{i}\\,\\sigma \_{j}} ![{\\displaystyle \\sigma \_{i,j}=\\rho \_{i,j}\\,\\sigma \_{i}\\,\\sigma \_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2e8fbcde3e6eb651edf66912e0d67881ce52cf) terms. ## Use in finance \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=6 "Edit section: Use in finance")\] Main article: [Black–Scholes model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model "Black–Scholes model") Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.[\[4\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-Hull-4) Some of the arguments for using GBM to model stock prices are: - The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality.[\[4\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-Hull-4) - A GBM process only assumes positive values, just like real stock prices. - A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices. - Calculations with GBM processes are relatively easy. However, GBM is not a completely realistic model, in particular it falls short of reality in the following points: - In real stock prices, volatility changes over time (possibly [stochastically](https://en.wikipedia.org/wiki/Stochastic_volatility "Stochastic volatility")), but in GBM, volatility is assumed constant. - In real life, stock prices often show jumps caused by unpredictable events or news, but in GBM, the path is continuous (no discontinuity). Apart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies.[\[5\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-5) ## Extensions \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=7 "Edit section: Extensions")\] In an attempt to make GBM more realistic as a model for stock prices, also in relation to the [volatility smile](https://en.wikipedia.org/wiki/Volatility_smile "Volatility smile") problem, one can drop the assumption that the volatility (σ {\\displaystyle \\sigma } ![{\\displaystyle \\sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)) is constant. If we assume that the volatility is a [deterministic](https://en.wikipedia.org/wiki/Deterministic_system "Deterministic system") function of the stock price and time, this is called a [local volatility](https://en.wikipedia.org/wiki/Local_volatility "Local volatility") model. A straightforward extension of the Black Scholes GBM is a local volatility SDE whose distribution is a mixture of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options.[\[3\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-musielarutkowski-3)[\[6\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-6)[\[7\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-7)[\[8\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-8) If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a [stochastic volatility](https://en.wikipedia.org/wiki/Stochastic_volatility "Stochastic volatility") model, see for example the [Heston model](https://en.wikipedia.org/wiki/Heston_model "Heston model").[\[9\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-9) ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=8 "Edit section: See also")\] - [Brownian surface](https://en.wikipedia.org/wiki/Brownian_surface "Brownian surface") - [Feynman–Kac formula](https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula "Feynman–Kac formula") ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=9 "Edit section: References")\] 1. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-1) Ross, Sheldon M. (2014). ["Variations on Brownian Motion"](https://books.google.com/books?id=A3YpAgAAQBAJ&pg=PA612). Introduction to Probability Models (11th ed.). Amsterdam: Elsevier. pp. 612–14\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-12-407948-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-407948-9 "Special:BookSources/978-0-12-407948-9") . 2. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-2) Øksendal, Bernt K. (2002), Stochastic Differential Equations: An Introduction with Applications, Springer, p. 326, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [3-540-63720-6](https://en.wikipedia.org/wiki/Special:BookSources/3-540-63720-6 "Special:BookSources/3-540-63720-6") 3. ^ [*a](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-musielarutkowski_3-0) [b](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-musielarutkowski_3-1) Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin. 4. ^ [a](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-Hull_4-0) [b](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-Hull_4-1) Hull, John (2009). "12.3". Options, Futures, and other Derivatives* (7 ed.). 5. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-5) Rej, A.; Seager, P.; Bouchaud, J.-P. (January 2018). ["You are in a drawdown. When should you start worrying?"](https://onlinelibrary.wiley.com/doi/abs/10.1002/wilm.10646). Wilmott. 2018 (93): 56–59\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1707\.01457](https://arxiv.org/abs/1707.01457). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/wilm.10646](https://doi.org/10.1002%2Fwilm.10646). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [157827746](https://api.semanticscholar.org/CorpusID:157827746). 6. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-6) Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI <https://doi.org/10.1007/3-540-30591-2> 7. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-7) [Brigo, Damiano](https://en.wikipedia.org/wiki/Damiano_Brigo "Damiano Brigo"); [Mercurio, Fabio](https://en.wikipedia.org/wiki/Fabio_Mercurio "Fabio Mercurio") (2002). "Lognormal-mixture dynamics and calibration to market volatility smiles". International Journal of Theoretical and Applied Finance. 5 (4): 427–446\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1142/S0219024902001511](https://doi.org/10.1142%2FS0219024902001511). 8. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-8) Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183, [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1469-7688](https://search.worldcat.org/issn/1469-7688) 9. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-9) [Heston, Steven L.](https://en.wikipedia.org/wiki/Steven_L._Heston "Steven L. Heston") (1993). "A closed-form solution for options with stochastic volatility with applications to bond and currency options". Review of Financial Studies. 6 (2): 327–343\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1093/rfs/6.2.327](https://doi.org/10.1093%2Frfs%2F6.2.327). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2962057](https://www.jstor.org/stable/2962057). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [16091300](https://api.semanticscholar.org/CorpusID:16091300). ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=10 "Edit section: External links")\] - [Geometric Brownian motion models for stock movement except in rare events.](https://web.archive.org/web/20120130222949/http://math.nyu.edu/financial_mathematics/content/02_financial/02.html) - [Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices](https://web.archive.org/web/20170402233858/http://excelandfinance.com/simulation-of-stock-prices/brownian-motion/) - ["Interactive Web Application: Stochastic Processes used in Quantitative Finance"](https://web.archive.org/web/20150920231636/http://turingfinance.com/interactive-stochastic-processes/). Archived from [the original](http://turingfinance.com/interactive-stochastic-processes/) on 2015-09-20. Retrieved 2015-07-03. \| [v](https://en.wikipedia.org/wiki/Template:Stochastic_processes "Template:Stochastic processes") [t](https://en.wikipedia.org/wiki/Template_talk:Stochastic_processes "Template talk:Stochastic processes") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Stochastic_processes "Special:EditPage/Template:Stochastic processes")[Stochastic processes](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") \| \| \|---\|---\| \| [Discrete time](https://en.wikipedia.org/wiki/Discrete-time_stochastic_process "Discrete-time stochastic process") \| [Bernoulli process](https://en.wikipedia.org/wiki/Bernoulli_process "Bernoulli process") [Branching process](https://en.wikipedia.org/wiki/Branching_process "Branching process") [Chinese restaurant process](https://en.wikipedia.org/wiki/Chinese_restaurant_process "Chinese restaurant process") [Galton–Watson process](https://en.wikipedia.org/wiki/Galton%E2%80%93Watson_process "Galton–Watson process") [Independent and identically distributed random variables](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables "Independent and identically distributed random variables") [Markov chain](https://en.wikipedia.org/wiki/Markov_chain "Markov chain") [Moran process](https://en.wikipedia.org/wiki/Moran_process "Moran process") [Random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk") [Loop-erased](https://en.wikipedia.org/wiki/Loop-erased_random_walk "Loop-erased random walk") [Self-avoiding](https://en.wikipedia.org/wiki/Self-avoiding_walk "Self-avoiding walk") [Biased](https://en.wikipedia.org/wiki/Biased_random_walk_on_a_graph "Biased random walk on a graph") [Maximal entropy](https://en.wikipedia.org/wiki/Maximal_entropy_random_walk "Maximal entropy random walk") \| \| [Continuous time](https://en.wikipedia.org/wiki/Continuous-time_stochastic_process "Continuous-time stochastic process") \| [Additive process](https://en.wikipedia.org/wiki/Additive_process "Additive process") [Airy process](https://en.wikipedia.org/wiki/Airy_process "Airy process") [Bessel process](https://en.wikipedia.org/wiki/Bessel_process "Bessel process") [Birth–death process](https://en.wikipedia.org/wiki/Birth%E2%80%93death_process "Birth–death process") [pure birth](https://en.wikipedia.org/wiki/Birth_process "Birth process") [Brownian motion](https://en.wikipedia.org/wiki/Wiener_process "Wiener process") [Bridge](https://en.wikipedia.org/wiki/Brownian_bridge "Brownian bridge") [Dyson](https://en.wikipedia.org/wiki/Dyson_Brownian_motion "Dyson Brownian motion") [Excursion](https://en.wikipedia.org/wiki/Brownian_excursion "Brownian excursion") [Fractional](https://en.wikipedia.org/wiki/Fractional_Brownian_motion "Fractional Brownian motion") [Geometric]() [Meander](https://en.wikipedia.org/wiki/Brownian_meander "Brownian meander") [Cauchy process](https://en.wikipedia.org/wiki/Cauchy_process "Cauchy process") [Contact process](https://en.wikipedia.org/wiki/Contact_process_\(mathematics\) "Contact process (mathematics)") [Continuous-time random walk](https://en.wikipedia.org/wiki/Continuous-time_random_walk "Continuous-time random walk") [Cox process](https://en.wikipedia.org/wiki/Cox_process "Cox process") [Diffusion process](https://en.wikipedia.org/wiki/Diffusion_process "Diffusion process") [Empirical process](https://en.wikipedia.org/wiki/Empirical_process "Empirical process") [Feller process](https://en.wikipedia.org/wiki/Feller_process "Feller process") [Fleming–Viot process](https://en.wikipedia.org/wiki/Fleming%E2%80%93Viot_process "Fleming–Viot process") [Gamma process](https://en.wikipedia.org/wiki/Gamma_process "Gamma process") [Geometric process](https://en.wikipedia.org/wiki/Geometric_process "Geometric process") [Hawkes process](https://en.wikipedia.org/wiki/Hawkes_process "Hawkes process") [Hunt process](https://en.wikipedia.org/wiki/Hunt_process "Hunt process") [Interacting particle systems](https://en.wikipedia.org/wiki/Interacting_particle_system "Interacting particle system") [Itô diffusion](https://en.wikipedia.org/wiki/It%C3%B4_diffusion "Itô diffusion") [Itô process](https://en.wikipedia.org/wiki/It%C3%B4_process "Itô process") [Jump diffusion](https://en.wikipedia.org/wiki/Jump_diffusion "Jump diffusion") [Jump process](https://en.wikipedia.org/wiki/Jump_process "Jump process") [Lévy process](https://en.wikipedia.org/wiki/L%C3%A9vy_process "Lévy process") [Local time](https://en.wikipedia.org/wiki/Local_time_\(mathematics\) "Local time (mathematics)") [Markov additive process](https://en.wikipedia.org/wiki/Markov_additive_process "Markov additive process") [McKean–Vlasov process](https://en.wikipedia.org/wiki/McKean%E2%80%93Vlasov_process "McKean–Vlasov process") [Ornstein–Uhlenbeck process](https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process "Ornstein–Uhlenbeck process") [Poisson process](https://en.wikipedia.org/wiki/Poisson_point_process "Poisson point process") [Compound](https://en.wikipedia.org/wiki/Compound_Poisson_process "Compound Poisson process") [Non-homogeneous](https://en.wikipedia.org/wiki/Non-homogeneous_Poisson_process "Non-homogeneous Poisson process") [Quasimartingale](https://en.wikipedia.org/wiki/Quasimartingale "Quasimartingale") [Schramm–Loewner evolution](https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution "Schramm–Loewner evolution") [Semimartingale](https://en.wikipedia.org/wiki/Semimartingale "Semimartingale") [Sigma-martingale](https://en.wikipedia.org/wiki/Sigma-martingale "Sigma-martingale") [Stable process](https://en.wikipedia.org/wiki/Stable_process "Stable process") [Superprocess](https://en.wikipedia.org/wiki/Superprocess "Superprocess") [Telegraph process](https://en.wikipedia.org/wiki/Telegraph_process "Telegraph process") [Variance gamma process](https://en.wikipedia.org/wiki/Variance_gamma_process "Variance gamma process") [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process") [Wiener sausage](https://en.wikipedia.org/wiki/Wiener_sausage "Wiener sausage") \| \| Both \| [Branching process](https://en.wikipedia.org/wiki/Branching_process "Branching process") [Gaussian process](https://en.wikipedia.org/wiki/Gaussian_process "Gaussian process") [Hidden Markov model (HMM)](https://en.wikipedia.org/wiki/Hidden_Markov_model "Hidden Markov model") [Markov process](https://en.wikipedia.org/wiki/Markov_process "Markov process") [Martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)") [Differences](https://en.wikipedia.org/wiki/Martingale_difference_sequence "Martingale difference sequence") [Local](https://en.wikipedia.org/wiki/Local_martingale "Local martingale") [Sub-](https://en.wikipedia.org/wiki/Submartingale "Submartingale") [Super-](https://en.wikipedia.org/wiki/Supermartingale "Supermartingale") [Random dynamical system](https://en.wikipedia.org/wiki/Random_dynamical_system "Random dynamical system") [Regenerative process](https://en.wikipedia.org/wiki/Regenerative_process "Regenerative process") [Renewal process](https://en.wikipedia.org/wiki/Renewal_process "Renewal process") [Stochastic chains with memory of variable length](https://en.wikipedia.org/wiki/Stochastic_chains_with_memory_of_variable_length "Stochastic chains with memory of variable length") [White noise](https://en.wikipedia.org/wiki/White_noise "White noise") \| \| Fields and other \| [Dirichlet process](https://en.wikipedia.org/wiki/Dirichlet_process "Dirichlet process") [Gaussian random field](https://en.wikipedia.org/wiki/Gaussian_random_field "Gaussian random field") [Gibbs measure](https://en.wikipedia.org/wiki/Gibbs_measure "Gibbs measure") [Hopfield model](https://en.wikipedia.org/wiki/Hopfield_model "Hopfield model") [Ising model](https://en.wikipedia.org/wiki/Ising_model "Ising model") [Potts model](https://en.wikipedia.org/wiki/Potts_model "Potts model") [Boolean network](https://en.wikipedia.org/wiki/Boolean_network "Boolean network") [Markov random field](https://en.wikipedia.org/wiki/Markov_random_field "Markov random field") [Percolation](https://en.wikipedia.org/wiki/Percolation_theory "Percolation theory") [Pitman–Yor process](https://en.wikipedia.org/wiki/Pitman%E2%80%93Yor_process "Pitman–Yor process") [Point process](https://en.wikipedia.org/wiki/Point_process "Point process") [Cox](https://en.wikipedia.org/wiki/Point_process#Cox_point_process "Point process") [Determinantal](https://en.wikipedia.org/wiki/Determinantal_point_process "Determinantal point process") [Poisson](https://en.wikipedia.org/wiki/Poisson_point_process "Poisson point process") [Random field](https://en.wikipedia.org/wiki/Random_field "Random field") [Random graph](https://en.wikipedia.org/wiki/Random_graph "Random graph") \| \| [Time series models](https://en.wikipedia.org/wiki/Time_series "Time series") \| [Autoregressive conditional heteroskedasticity (ARCH) model](https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity "Autoregressive conditional heteroskedasticity") [Autoregressive integrated moving average (ARIMA) model](https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average "Autoregressive integrated moving average") [Autoregressive (AR) model](https://en.wikipedia.org/wiki/Autoregressive_model "Autoregressive model") [Autoregressive moving-average (ARMA) model](https://en.wikipedia.org/wiki/Autoregressive_moving-average_model "Autoregressive moving-average model") [Generalized autoregressive conditional heteroskedasticity (GARCH) model](https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity "Autoregressive conditional heteroskedasticity") [Moving-average (MA) model](https://en.wikipedia.org/wiki/Moving-average_model "Moving-average model") \| \| [Financial models](https://en.wikipedia.org/wiki/Asset_pricing_model "Asset pricing model") \| [Binomial options pricing model](https://en.wikipedia.org/wiki/Binomial_options_pricing_model "Binomial options pricing model") [Black–Derman–Toy](https://en.wikipedia.org/wiki/Black%E2%80%93Derman%E2%80%93Toy_model "Black–Derman–Toy model") [Black–Karasinski](https://en.wikipedia.org/wiki/Black%E2%80%93Karasinski_model "Black–Karasinski model") [Black–Scholes](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model "Black–Scholes model") [Chan–Karolyi–Longstaff–Sanders (CKLS)](https://en.wikipedia.org/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process "Chan–Karolyi–Longstaff–Sanders process") [Chen](https://en.wikipedia.org/wiki/Chen_model "Chen model") [Constant elasticity of variance (CEV)](https://en.wikipedia.org/wiki/Constant_elasticity_of_variance_model "Constant elasticity of variance model") [Cox–Ingersoll–Ross (CIR)](https://en.wikipedia.org/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model "Cox–Ingersoll–Ross model") [Garman–Kohlhagen](https://en.wikipedia.org/wiki/Garman%E2%80%93Kohlhagen_model "Garman–Kohlhagen model") [Heath–Jarrow–Morton (HJM)](https://en.wikipedia.org/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework "Heath–Jarrow–Morton framework") [Heston](https://en.wikipedia.org/wiki/Heston_model "Heston model") [Ho–Lee](https://en.wikipedia.org/wiki/Ho%E2%80%93Lee_model "Ho–Lee model") [Hull–White](https://en.wikipedia.org/wiki/Hull%E2%80%93White_model "Hull–White model") [Korn-Kreer-Lenssen](https://en.wikipedia.org/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model "Korn–Kreer–Lenssen model") [LIBOR market](https://en.wikipedia.org/wiki/LIBOR_market_model "LIBOR market model") [Rendleman–Bartter](https://en.wikipedia.org/wiki/Rendleman%E2%80%93Bartter_model "Rendleman–Bartter model") [SABR volatility](https://en.wikipedia.org/wiki/SABR_volatility_model "SABR volatility model") [Vašíček](https://en.wikipedia.org/wiki/Vasicek_model "Vasicek model") [Wilkie](https://en.wikipedia.org/wiki/Wilkie_investment_model "Wilkie investment model") \| \| [Actuarial models](https://en.wikipedia.org/wiki/Actuarial_mathematics "Actuarial mathematics") \| [Bühlmann](https://en.wikipedia.org/wiki/B%C3%BChlmann_model "Bühlmann model") [Cramér–Lundberg](https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Lundberg_model "Cramér–Lundberg model") [Risk process](https://en.wikipedia.org/wiki/Risk_process "Risk process") [Sparre–Anderson](https://en.wikipedia.org/wiki/Sparre%E2%80%93Anderson_model "Sparre–Anderson model") \| \| [Queueing models](https://en.wikipedia.org/wiki/Queueing_model "Queueing model") \| [Bulk](https://en.wikipedia.org/wiki/Bulk_queue "Bulk queue") [Fluid](https://en.wikipedia.org/wiki/Fluid_queue "Fluid queue") [Generalized queueing network](https://en.wikipedia.org/wiki/G-network "G-network") [M/G/1](https://en.wikipedia.org/wiki/M/G/1_queue "M/G/1 queue") [M/M/1](https://en.wikipedia.org/wiki/M/M/1_queue "M/M/1 queue") [M/M/c](https://en.wikipedia.org/wiki/M/M/c_queue "M/M/c queue") \| \| Properties \| [Càdlàg paths](https://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g "Càdlàg") [Continuous](https://en.wikipedia.org/wiki/Continuous_stochastic_process "Continuous stochastic process") [Continuous paths](https://en.wikipedia.org/wiki/Sample-continuous_process "Sample-continuous process") [Ergodic](https://en.wikipedia.org/wiki/Ergodicity "Ergodicity") [Exchangeable](https://en.wikipedia.org/wiki/Exchangeable_random_variables "Exchangeable random variables") [Feller-continuous](https://en.wikipedia.org/wiki/Feller-continuous_process "Feller-continuous process") [Gauss–Markov](https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_process "Gauss–Markov process") [Markov](https://en.wikipedia.org/wiki/Markov_property "Markov property") [Mixing](https://en.wikipedia.org/wiki/Mixing_\(mathematics\) "Mixing (mathematics)") [Piecewise-deterministic](https://en.wikipedia.org/wiki/Piecewise-deterministic_Markov_process "Piecewise-deterministic Markov process") [Predictable](https://en.wikipedia.org/wiki/Predictable_process "Predictable process") [Progressively measurable](https://en.wikipedia.org/wiki/Progressively_measurable_process "Progressively measurable process") [Self-similar](https://en.wikipedia.org/wiki/Self-similar_process "Self-similar process") [Stationary](https://en.wikipedia.org/wiki/Stationary_process "Stationary process") [Time-reversible](https://en.wikipedia.org/wiki/Time_reversibility "Time reversibility") \| \| Limit theorems \| [Central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem "Central limit theorem") [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem") [Doob's martingale convergence theorems](https://en.wikipedia.org/wiki/Doob%27s_martingale_convergence_theorems "Doob's martingale convergence theorems") [Ergodic theorem](https://en.wikipedia.org/wiki/Ergodic_theory "Ergodic theory") [Fisher–Tippett–Gnedenko theorem](https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem "Fisher–Tippett–Gnedenko theorem") [Large deviation principle](https://en.wikipedia.org/wiki/Large_deviation_principle "Large deviation principle") [Law of large numbers (weak/strong)](https://en.wikipedia.org/wiki/Law_of_large_numbers "Law of large numbers") [Law of the iterated logarithm](https://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm "Law of the iterated logarithm") [Maximal ergodic theorem](https://en.wikipedia.org/wiki/Maximal_ergodic_theorem "Maximal ergodic theorem") [Sanov's theorem](https://en.wikipedia.org/wiki/Sanov%27s_theorem "Sanov's theorem") [Zero–one laws](https://en.wikipedia.org/wiki/Zero%E2%80%93one_law "Zero–one law") ([Blumenthal](https://en.wikipedia.org/wiki/Blumenthal%27s_zero%E2%80%93one_law "Blumenthal's zero–one law"), [Borel–Cantelli](https://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma "Borel–Cantelli lemma"), [Engelbert–Schmidt](https://en.wikipedia.org/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law "Engelbert–Schmidt zero–one law"), [Hewitt–Savage](https://en.wikipedia.org/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law "Hewitt–Savage zero–one law"), [Kolmogorov](https://en.wikipedia.org/wiki/Kolmogorov%27s_zero%E2%80%93one_law "Kolmogorov's zero–one law"), [Lévy](https://en.wikipedia.org/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law "Lévy's zero–one law")) \| \| [Inequalities](https://en.wikipedia.org/wiki/List_of_inequalities#Probability_theory_and_statistics "List of inequalities") \| [Burkholder–Davis–Gundy](https://en.wikipedia.org/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities "Burkholder–Davis–Gundy inequalities") [Doob's martingale](https://en.wikipedia.org/wiki/Doob%27s_martingale_inequality "Doob's martingale inequality") [Doob's upcrossing](https://en.wikipedia.org/wiki/Doob%27s_upcrossing_inequality "Doob's upcrossing inequality") [Kunita–Watanabe](https://en.wikipedia.org/wiki/Kunita%E2%80%93Watanabe_inequality "Kunita–Watanabe inequality") [Marcinkiewicz–Zygmund](https://en.wikipedia.org/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality "Marcinkiewicz–Zygmund inequality") \| \| Tools \| [Cameron–Martin theorem](https://en.wikipedia.org/wiki/Cameron%E2%80%93Martin_theorem "Cameron–Martin theorem") [Convergence of random variables](https://en.wikipedia.org/wiki/Convergence_of_random_variables "Convergence of random variables") [Doléans-Dade exponential](https://en.wikipedia.org/wiki/Dol%C3%A9ans-Dade_exponential "Doléans-Dade exponential") [Doob decomposition theorem](https://en.wikipedia.org/wiki/Doob_decomposition_theorem "Doob decomposition theorem") [Doob–Meyer decomposition theorem](https://en.wikipedia.org/wiki/Doob%E2%80%93Meyer_decomposition_theorem "Doob–Meyer decomposition theorem") [Doob's optional stopping theorem](https://en.wikipedia.org/wiki/Doob%27s_optional_stopping_theorem "Doob's optional stopping theorem") [Dynkin's formula](https://en.wikipedia.org/wiki/Dynkin%27s_formula "Dynkin's formula") [Feynman–Kac formula](https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula "Feynman–Kac formula") [Filtration](https://en.wikipedia.org/wiki/Filtration_\(probability_theory\) "Filtration (probability theory)") [Girsanov theorem](https://en.wikipedia.org/wiki/Girsanov_theorem "Girsanov theorem") [Infinitesimal generator](https://en.wikipedia.org/wiki/Infinitesimal_generator_\(stochastic_processes\) "Infinitesimal generator (stochastic processes)") [Itô integral](https://en.wikipedia.org/wiki/It%C3%B4_integral "Itô integral") [Itô's lemma](https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma "Itô's lemma") [Kolmogorov continuity theorem](https://en.wikipedia.org/wiki/Kolmogorov_continuity_theorem "Kolmogorov continuity theorem") [Kolmogorov extension theorem](https://en.wikipedia.org/wiki/Kolmogorov_extension_theorem "Kolmogorov extension theorem") [Kosambi–Karhunen–Loève theorem](https://en.wikipedia.org/wiki/Kosambi%E2%80%93Karhunen%E2%80%93Lo%C3%A8ve_theorem "Kosambi–Karhunen–Loève theorem") [Lévy–Prokhorov metric](https://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric "Lévy–Prokhorov metric") [Malliavin calculus](https://en.wikipedia.org/wiki/Malliavin_calculus "Malliavin calculus") [Martingale representation theorem](https://en.wikipedia.org/wiki/Martingale_representation_theorem "Martingale representation theorem") [Optional stopping theorem](https://en.wikipedia.org/wiki/Optional_stopping_theorem "Optional stopping theorem") [Prokhorov's theorem](https://en.wikipedia.org/wiki/Prokhorov%27s_theorem "Prokhorov's theorem") [Quadratic variation](https://en.wikipedia.org/wiki/Quadratic_variation "Quadratic variation") [Reflection principle](https://en.wikipedia.org/wiki/Reflection_principle_\(Wiener_process\) "Reflection principle (Wiener process)") [Skorokhod integral](https://en.wikipedia.org/wiki/Skorokhod_integral "Skorokhod integral") [Skorokhod's representation theorem](https://en.wikipedia.org/wiki/Skorokhod%27s_representation_theorem "Skorokhod's representation theorem") [Skorokhod space](https://en.wikipedia.org/wiki/Skorokhod_space "Skorokhod space") [Snell envelope](https://en.wikipedia.org/wiki/Snell_envelope "Snell envelope") [Stochastic differential equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation "Stochastic differential equation") [Tanaka](https://en.wikipedia.org/wiki/Tanaka_equation "Tanaka equation") [Stopping time](https://en.wikipedia.org/wiki/Stopping_time "Stopping time") [Stratonovich integral](https://en.wikipedia.org/wiki/Stratonovich_integral "Stratonovich integral") [Uniform integrability](https://en.wikipedia.org/wiki/Uniform_integrability "Uniform integrability") [Usual hypotheses](https://en.wikipedia.org/wiki/Usual_hypotheses "Usual hypotheses") Wiener space [Classical](https://en.wikipedia.org/wiki/Classical_Wiener_space "Classical Wiener space") [Abstract](https://en.wikipedia.org/wiki/Abstract_Wiener_space "Abstract Wiener space") \| \| Disciplines \| [Actuarial mathematics](https://en.wikipedia.org/wiki/Actuarial_mathematics "Actuarial mathematics") [Control theory](https://en.wikipedia.org/wiki/Stochastic_control "Stochastic control") [Econometrics](https://en.wikipedia.org/wiki/Econometrics "Econometrics") [Ergodic theory](https://en.wikipedia.org/wiki/Ergodic_theory "Ergodic theory") [Extreme value theory (EVT)](https://en.wikipedia.org/wiki/Extreme_value_theory "Extreme value theory") [Large deviations theory](https://en.wikipedia.org/wiki/Large_deviations_theory "Large deviations theory") [Mathematical finance](https://en.wikipedia.org/wiki/Mathematical_finance "Mathematical finance") [Mathematical statistics](https://en.wikipedia.org/wiki/Mathematical_statistics "Mathematical statistics") [Probability theory](https://en.wikipedia.org/wiki/Probability_theory "Probability theory") [Queueing theory](https://en.wikipedia.org/wiki/Queueing_theory "Queueing theory") [Renewal theory](https://en.wikipedia.org/wiki/Renewal_theory "Renewal theory") [Ruin theory](https://en.wikipedia.org/wiki/Ruin_theory "Ruin theory") [Signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") [Statistics](https://en.wikipedia.org/wiki/Statistics "Statistics") [Stochastic analysis](https://en.wikipedia.org/wiki/Stochastic_analysis "Stochastic analysis") [Time series analysis](https://en.wikipedia.org/wiki/Time_series_analysis "Time series analysis") [Machine learning](https://en.wikipedia.org/wiki/Machine_learning "Machine learning") \| \| [List of topics](https://en.wikipedia.org/wiki/List_of_stochastic_processes_topics "List of stochastic processes topics") [Category](https://en.wikipedia.org/wiki/Category:Stochastic_processes "Category:Stochastic processes") \| \| 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Readable Markdown	From Wikipedia, the free encyclopedia [![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/GBM2.png/500px-GBM2.png)](https://en.wikipedia.org/wiki/File:GBM2.png) For the simulation generating the realizations, see below. A geometric Brownian motion (GBM), also known as an exponential Brownian motion, is a continuous-time [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") in which the [logarithm](https://en.wikipedia.org/wiki/Logarithm "Logarithm") of the randomly varying quantity follows a [Brownian motion](https://en.wikipedia.org/wiki/Wiener_process "Wiener process") with [drift](https://en.wikipedia.org/wiki/Stochastic_drift "Stochastic drift").[\[1\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-1) It is an important example of stochastic processes satisfying a [stochastic differential equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation "Stochastic differential equation") (SDE); in particular, it is used in [mathematical finance](https://en.wikipedia.org/wiki/Mathematical_finance "Mathematical finance") to model stock prices in the [Black–Scholes model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model "Black–Scholes model"). ## Stochastical differential equation \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=1 "Edit section: Stochastical differential equation")\] A stochastic process St is said to follow a GBM if it satisfies the following [stochastic differential equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation "Stochastic differential equation") (SDE): ![{\\displaystyle dS\_{t}=\\mu S\_{t}\\,dt+\\sigma S\_{t}\\,dW\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a3c0d1dcb510719effde045c26f1a9d0b9cb2d) where ![{\\displaystyle W\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50680c5535c83badfa630dba63b583d2eeaa2977) is a [Wiener process or Brownian motion](https://en.wikipedia.org/wiki/Wiener_process "Wiener process"), and ![{\\displaystyle \\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) ('the percentage drift') and ![{\\displaystyle \\sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36) ('the percentage volatility') are constants. The former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion. For an arbitrary initial value S0 the above SDE has the analytic solution (under [Itô's interpretation](https://en.wikipedia.org/wiki/It%C3%B4_calculus "Itô calculus")): ![{\\displaystyle S\_{t}=S\_{0}\\exp \\left(\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/266807b65fd50635526a766c0c89a2913085d0c2) The derivation requires the use of [Itô calculus](https://en.wikipedia.org/wiki/It%C3%B4_calculus "Itô calculus"). Applying [Itô's formula](https://en.wikipedia.org/wiki/It%C3%B4%27s_formula "Itô's formula") leads to ![{\\displaystyle d(\\ln S\_{t})=(\\ln S\_{t})'dS\_{t}+{\\frac {1}{2}}(\\ln S\_{t})''\\,dS\_{t}\\,dS\_{t}={\\frac {dS\_{t}}{S\_{t}}}-{\\frac {1}{2}}\\,{\\frac {1}{S\_{t}^{2}}}\\,dS\_{t}\\,dS\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbef6e7e7776ca5963d9179d9d0e1d198e26fd63) where ![{\\displaystyle dS\_{t}\\,dS\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7dd8f0d0a80348c6459b950635d83110c4453ae) is the [quadratic variation](https://en.wikipedia.org/wiki/Quadratic_variation "Quadratic variation") of the SDE. ![{\\displaystyle dS\_{t}\\,dS\_{t}\\,=\\,\\sigma ^{2}\\,S\_{t}^{2}\\,dW\_{t}^{2}+2\\sigma S\_{t}^{2}\\mu \\,dW\_{t}\\,dt+\\mu ^{2}S\_{t}^{2}\\,dt^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a2a5473dd3d9151bdc6f88a4eb51ed055c1797) When ![{\\displaystyle dt\\to 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02f957c6106281a61de6adf0ec77aaf6ea42aba9), ![{\\displaystyle dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebee76a835701fd1f26047a09855f2ea36bb08fc) converges to 0 faster than ![{\\displaystyle dW\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc431f447c4bc6c9063431fe612625992c3cca91), since ![{\\displaystyle dW\_{t}^{2}=O(dt)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f08e336e1625340ddcf50ada36b7bf8cd083b1). So the above infinitesimal can be simplified by ![{\\displaystyle dS\_{t}\\,dS\_{t}\\,=\\,\\sigma ^{2}\\,S\_{t}^{2}\\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dceb3d38f0dd8adce9525559a5d9c0ee497b137f) Plugging the value of ![{\\displaystyle dS\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c796c605ee3a300e8b295d4225bb163e899af5ca) in the above equation and simplifying we obtain ![{\\displaystyle \\ln {\\frac {S\_{t}}{S\_{0}}}=\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\,\\right)t+\\sigma W\_{t}\\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adee3ba14c6e0d50918fefe1f352991b13631265) Taking the exponential and multiplying both sides by ![{\\displaystyle S\_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe0ac45a38c4437bd2689a14ec434cd499e7e49) gives the solution claimed above. ## Arithmetic Brownian motion \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=3 "Edit section: Arithmetic Brownian motion")\] The process for ![{\\displaystyle X\_{t}=\\ln {\\frac {S\_{t}}{S\_{0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7184af654d06bc2453b2bc139a81ad9458ca897b), satisfying the SDE ![{\\displaystyle dX\_{t}=\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\,\\right)dt+\\sigma dW\_{t}\\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/387886bce7774547c64416a186455791dda2c4e5) or more generally the process solving the SDE ![{\\displaystyle dX\_{t}=m\\,dt+v\\,dW\_{t}\\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f3370b73be582ffd0d49245c6cfc05801e91df) where ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) and ![{\\displaystyle v\>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c314fc908a83c555d34968d25e86c5ae0b76ef6f) are real constants and for an initial condition ![{\\displaystyle X\_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6381fdad2b9f11954b1fc2db08bbaccf634ededa), is called an Arithmetic Brownian Motion (ABM). This was the model postulated by [Louis Bachelier](https://en.wikipedia.org/wiki/Louis_Bachelier "Louis Bachelier") in 1900 for stock prices, in the first published attempt to model Brownian motion, known today as [Bachelier model](https://en.wikipedia.org/wiki/Bachelier_model "Bachelier model"). As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula. Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula. The above solution ![{\\displaystyle S\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2391e6e796fbf718be3828080775ac2ac3d3d4) (for any value of t) is a [log-normally distributed](https://en.wikipedia.org/wiki/Log-normal_distribution "Log-normal distribution") [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable") with [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value") and [variance](https://en.wikipedia.org/wiki/Variance "Variance") given by[\[2\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-2) ![{\\displaystyle \\operatorname {E} (S\_{t})=S\_{0}e^{\\mu t},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/029367c196d22b8abeeb9e362cfe5cf10d505b5c) ![{\\displaystyle \\operatorname {Var} (S\_{t})=S\_{0}^{2}e^{2\\mu t}\\left(e^{\\sigma ^{2}t}-1\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6035329909c079220525421bada1f1b2d67f8d0) They can be derived using the fact that ![{\\displaystyle Z\_{t}=\\exp \\left(\\sigma W\_{t}-{\\frac {1}{2}}\\sigma ^{2}t\\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c181320a17700bcd9460adcb293ad092f50eccc4) is a [martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)"), and that ![{\\displaystyle \\operatorname {E} \\left\[\\exp \\left(2\\sigma W\_{t}-\\sigma ^{2}t\\right)\\mid {\\mathcal {F}}\_{s}\\right\]=e^{\\sigma ^{2}(t-s)}\\exp \\left(2\\sigma W\_{s}-\\sigma ^{2}s\\right),\\quad \\forall 0\\leq s\<t.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/121f4efbdcac966a20535657392d5de0345d6f9b) The [probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") of ![{\\displaystyle S\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2391e6e796fbf718be3828080775ac2ac3d3d4) is: ![{\\displaystyle f\_{S\_{t}}(s;\\mu ,\\sigma ,t)={\\frac {1}{\\sqrt {2\\pi }}}\\,{\\frac {1}{s\\sigma {\\sqrt {t}}}}\\,\\exp \\left(-{\\frac {\\left(\\ln s-\\ln S\_{0}-\\left(\\mu -{\\frac {1}{2}}\\sigma ^{2}\\right)t\\right)^{2}}{2\\sigma ^{2}t}}\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90a9074c173307646da549284ffc5d19c8b85b07) \| Derivation of GBM probability density function \| \|---\| \| To derive the probability density function for GBM, we must use the [Fokker–Planck equation](https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation "Fokker–Planck equation") to evaluate the time evolution of the PDF: ![{\\displaystyle {\\partial p \\over {\\partial t}}+{\\partial \\over {\\partial S}}\[\\mu (t,S)p(t,S)\]={1 \\over {2}}{\\partial ^{2} \\over {\\partial S^{2}}}\[\\sigma ^{2}(t,S)p(t,S)\],\\quad p(0,S)=\\delta (S-S\_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6827fa340748de87d1aa6a2b7d99d356ef38e6f) where ![{\\displaystyle \\delta (S)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f29eec89c4928b10514c6993b942ea248b3b5a57) is the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function"). To simplify the computation, we may introduce a logarithmic transform ![{\\displaystyle x=\\log(S/S\_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84aae4276b1b7e359e466df00fc19d2786f95717), leading to the form of GBM: ![{\\displaystyle dx=\\left(\\mu -{1 \\over {2}}\\sigma ^{2}\\right)dt+\\sigma \\,dW}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0f82ac78a1936876587eb26c15fa7b2f32e64a) Then the equivalent Fokker–Planck equation for the evolution of the PDF becomes: ![{\\displaystyle {\\partial p \\over {\\partial t}}+\\left(\\mu -{1 \\over {2}}\\sigma ^{2}\\right){\\partial p \\over {\\partial x}}={1 \\over {2}}\\sigma ^{2}{\\partial ^{2}p \\over {\\partial x^{2}}},\\quad p(0,x)=\\delta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18070926ea29bd1d7df4d204932749ba98c4d6f5) Define ![{\\displaystyle V=\\mu -\\sigma ^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f993e6cf8ae456b42f51d4bee95123a852990451) and ![{\\displaystyle D=\\sigma ^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/657c30593c183d872f0b006925f3266080c180fe). By introducing the new variables ![{\\displaystyle \\xi =x-Vt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/061ed424493e686baedd2938042ca86a38f6bb62) and ![{\\displaystyle \\tau =Dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fad3874220f39bba512b64c56aa1ddb442039787), the derivatives in the Fokker–Planck equation may be transformed as: ![{\\displaystyle {\\begin{aligned}\\partial \_{t}p&=D\\partial \_{\\tau }p-V\\partial \_{\\xi }p\\\\\\partial \_{x}p&=\\partial \_{\\xi }p\\\\\\partial \_{x}^{2}p&=\\partial \_{\\xi }^{2}p\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dae8470306338fcfcfcfb5c35b0509309525d195) Leading to the new form of the Fokker–Planck equation: ![{\\displaystyle {\\partial p \\over {\\partial \\tau }}={\\partial ^{2}p \\over {\\partial \\xi ^{2}}},\\quad p(0,\\xi )=\\delta (\\xi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3524160633d67751dbd36d9320a3d0d9cac24350) However, this is the canonical form of the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation"). which has the solution given by the [heat kernel](https://en.wikipedia.org/wiki/Heat_kernel "Heat kernel"): ![{\\displaystyle p(\\tau ,\\xi )={1 \\over {\\sqrt {4\\pi \\tau }}}\\exp \\left(-{\\xi ^{2} \\over 4\\tau }\\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39476d718fdd67a80bb7626ccb7dcf07e548158f) Plugging in the original variables leads to the PDF for GBM: ![{\\displaystyle p(t,S)={1 \\over {S{\\sqrt {2\\pi \\sigma ^{2}t}}}}\\exp \\left\\{-{\\left\[\\log(S/S\_{0})-\\left(\\mu -{1 \\over 2}\\sigma ^{2}\\right)t\\right\]^{2} \\over {2\\sigma ^{2}t}}\\right\\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4a7389649e234f4214a50946c1928ea272d0d2b) \| When deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log(St). This is an interesting process, because in the Black–Scholes model it is related to the [log return](https://en.wikipedia.org/wiki/Log_return "Log return") of the stock price. Using [Itô's lemma](https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma "Itô's lemma") with f(S) = log(S) gives ![{\\displaystyle {\\begin{alignedat}{2}d\\log(S)&=f'(S)\\,dS+{\\frac {1}{2}}f''(S)S^{2}\\sigma ^{2}\\,dt\\\\\[6pt\]&={\\frac {1}{S}}\\left(\\sigma S\\,dW\_{t}+\\mu S\\,dt\\right)-{\\frac {1}{2}}\\sigma ^{2}\\,dt\\\\\[6pt\]&=\\sigma \\,dW\_{t}+(\\mu -\\sigma ^{2}/2)\\,dt.\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b72bd76b7255e8dbd8e7c224df8ec2ac15ba9d) It follows that ![{\\displaystyle \\operatorname {E} \\log(S\_{t})=\\log(S\_{0})+(\\mu -\\sigma ^{2}/2)t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16ac3c7e020b0fbbea10e3ae8ced60fe9212604f). This result can also be derived by applying the logarithm to the explicit solution of GBM: ![{\\displaystyle {\\begin{alignedat}{2}\\log(S\_{t})&=\\log \\left(S\_{0}\\exp \\left(\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}\\right)\\right)\\\\\[6pt\]&=\\log(S\_{0})+\\left(\\mu -{\\frac {\\sigma ^{2}}{2}}\\right)t+\\sigma W\_{t}.\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/592321ca2ac7a8f230e2854bef22804a8c3d1064) Taking the expectation yields the same result as above: ![{\\displaystyle \\operatorname {E} \\log(S\_{t})=\\log(S\_{0})+(\\mu -\\sigma ^{2}/2)t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16ac3c7e020b0fbbea10e3ae8ced60fe9212604f). ## Multivariate version \[[edit](https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&action=edit&section=5 "Edit section: Multivariate version")\] GBM can be extended to the case where there are multiple correlated price paths.[\[3\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-musielarutkowski-3) Each price path follows the underlying process ![{\\displaystyle dS\_{t}^{i}=\\mu \_{i}S\_{t}^{i}\\,dt+\\sigma \_{i}S\_{t}^{i}\\,dW\_{t}^{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41c7438fc68d06af0978e28caeb7f4f2ea4588cc) where the Wiener processes are correlated such that ![{\\displaystyle \\operatorname {E} (dW\_{t}^{i}\\,dW\_{t}^{j})=\\rho \_{i,j}\\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d59a9358bd9a5d6d00a1f1610f58f3244be0761c) where ![{\\displaystyle \\rho \_{i,i}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05c5293f972f2fb115ca086541bbeed11a7f2c3d). For the multivariate case, this implies that ![{\\displaystyle \\operatorname {Cov} (S\_{t}^{i},S\_{t}^{j})=S\_{0}^{i}S\_{0}^{j}e^{(\\mu \_{i}+\\mu \_{j})t}\\left(e^{\\rho \_{i,j}\\sigma \_{i}\\sigma \_{j}t}-1\\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8f47f21d96baa1dbcd2ac72458f427304aec24) A multivariate formulation that maintains the driving Brownian motions ![{\\displaystyle W\_{t}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3514f7cf4ee08e76eed5be82f080bdc9768727a0) independent is ![{\\displaystyle dS\_{t}^{i}=\\mu \_{i}S\_{t}^{i}\\,dt+\\sum \_{j=1}^{d}\\sigma \_{i,j}S\_{t}^{i}\\,dW\_{t}^{j},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d592333694e71c54ab6c9ad941b816f4727c165) where the correlation between ![{\\displaystyle S\_{t}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/816d2d0f0589121853483186d52a881bb2f16dfa) and ![{\\displaystyle S\_{t}^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa071c0b70847f867a603e15ef699e1c0b9a593) is now expressed through the ![{\\displaystyle \\sigma \_{i,j}=\\rho \_{i,j}\\,\\sigma \_{i}\\,\\sigma \_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2e8fbcde3e6eb651edf66912e0d67881ce52cf) terms. Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.[\[4\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-Hull-4) Some of the arguments for using GBM to model stock prices are: - The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality.[\[4\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-Hull-4) - A GBM process only assumes positive values, just like real stock prices. - A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices. - Calculations with GBM processes are relatively easy. However, GBM is not a completely realistic model, in particular it falls short of reality in the following points: - In real stock prices, volatility changes over time (possibly [stochastically](https://en.wikipedia.org/wiki/Stochastic_volatility "Stochastic volatility")), but in GBM, volatility is assumed constant. - In real life, stock prices often show jumps caused by unpredictable events or news, but in GBM, the path is continuous (no discontinuity). Apart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies.[\[5\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-5) In an attempt to make GBM more realistic as a model for stock prices, also in relation to the [volatility smile](https://en.wikipedia.org/wiki/Volatility_smile "Volatility smile") problem, one can drop the assumption that the volatility (![{\\displaystyle \\sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)) is constant. If we assume that the volatility is a [deterministic](https://en.wikipedia.org/wiki/Deterministic_system "Deterministic system") function of the stock price and time, this is called a [local volatility](https://en.wikipedia.org/wiki/Local_volatility "Local volatility") model. A straightforward extension of the Black Scholes GBM is a local volatility SDE whose distribution is a mixture of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options.[\[3\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-musielarutkowski-3)[\[6\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-6)[\[7\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-7)[\[8\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-8) If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a [stochastic volatility](https://en.wikipedia.org/wiki/Stochastic_volatility "Stochastic volatility") model, see for example the [Heston model](https://en.wikipedia.org/wiki/Heston_model "Heston model").[\[9\]](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_note-9) - [Brownian surface](https://en.wikipedia.org/wiki/Brownian_surface "Brownian surface") - [Feynman–Kac formula](https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula "Feynman–Kac formula") 1. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-1) Ross, Sheldon M. (2014). ["Variations on Brownian Motion"](https://books.google.com/books?id=A3YpAgAAQBAJ&pg=PA612). Introduction to Probability Models (11th ed.). Amsterdam: Elsevier. pp. 612–14\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-12-407948-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-407948-9 "Special:BookSources/978-0-12-407948-9") . 2. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-2) Øksendal, Bernt K. (2002), Stochastic Differential Equations: An Introduction with Applications, Springer, p. 326, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [3-540-63720-6](https://en.wikipedia.org/wiki/Special:BookSources/3-540-63720-6 "Special:BookSources/3-540-63720-6") 3. ^ [*a](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-musielarutkowski_3-0) [b](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-musielarutkowski_3-1) Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin. 4. ^ [a](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-Hull_4-0) [b](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-Hull_4-1) Hull, John (2009). "12.3". Options, Futures, and other Derivatives* (7 ed.). 5. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-5) Rej, A.; Seager, P.; Bouchaud, J.-P. (January 2018). ["You are in a drawdown. When should you start worrying?"](https://onlinelibrary.wiley.com/doi/abs/10.1002/wilm.10646). Wilmott. 2018 (93): 56–59\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1707\.01457](https://arxiv.org/abs/1707.01457). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/wilm.10646](https://doi.org/10.1002%2Fwilm.10646). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [157827746](https://api.semanticscholar.org/CorpusID:157827746). 6. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-6) Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI <https://doi.org/10.1007/3-540-30591-2> 7. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-7) [Brigo, Damiano](https://en.wikipedia.org/wiki/Damiano_Brigo "Damiano Brigo"); [Mercurio, Fabio](https://en.wikipedia.org/wiki/Fabio_Mercurio "Fabio Mercurio") (2002). "Lognormal-mixture dynamics and calibration to market volatility smiles". International Journal of Theoretical and Applied Finance. 5 (4): 427–446\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1142/S0219024902001511](https://doi.org/10.1142%2FS0219024902001511). 8. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-8) Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183, [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1469-7688](https://search.worldcat.org/issn/1469-7688) 9. [^](https://en.wikipedia.org/wiki/Geometric_Brownian_motion#cite_ref-9) [Heston, Steven L.](https://en.wikipedia.org/wiki/Steven_L._Heston "Steven L. Heston") (1993). "A closed-form solution for options with stochastic volatility with applications to bond and currency options". Review of Financial Studies. 6 (2): 327–343\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1093/rfs/6.2.327](https://doi.org/10.1093%2Frfs%2F6.2.327). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2962057](https://www.jstor.org/stable/2962057). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [16091300](https://api.semanticscholar.org/CorpusID:16091300). - [Geometric Brownian motion models for stock movement except in rare events.](https://web.archive.org/web/20120130222949/http://math.nyu.edu/financial_mathematics/content/02_financial/02.html) - [Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices](https://web.archive.org/web/20170402233858/http://excelandfinance.com/simulation-of-stock-prices/brownian-motion/) - ["Interactive Web Application: Stochastic Processes used in Quantitative Finance"](https://web.archive.org/web/20150920231636/http://turingfinance.com/interactive-stochastic-processes/). Archived from [the original](http://turingfinance.com/interactive-stochastic-processes/) on 2015-09-20. Retrieved 2015-07-03.
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