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</div> </a> <ul id="toc-Mathematical_tools-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivatives_pricing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivatives_pricing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Derivatives pricing</span> </div> </a> <ul id="toc-Derivatives_pricing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Portfolio_modelling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Portfolio_modelling"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Portfolio modelling</span> </div> </a> <ul id="toc-Portfolio_modelling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Other</span> </div> </a> <ul id="toc-Other-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc 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href="https://ar.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA_%D9%85%D8%A7%D9%84%D9%8A%D8%A9" title="رياضيات مالية – Arabic" lang="ar" hreflang="ar" data-title="رياضيات مالية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riyazi_maliyy%C9%99" title="Riyazi maliyyə – Azerbaijani" lang="az" hreflang="az" data-title="Riyazi maliyyə" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8_%D1%84%D0%B8%D0%BD%D0%B0%D0%BD%D1%81%D0%B8" title="Математически финанси – Bulgarian" lang="bg" hreflang="bg" data-title="Математически финанси" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Finansijska_matematika" title="Finansijska matematika – Bosnian" lang="bs" hreflang="bs" data-title="Finansijska matematika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matem%C3%A0tica_financera" title="Matemàtica financera – Catalan" lang="ca" hreflang="ca" data-title="Matemàtica financera" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Finan%C4%8Dn%C3%AD_matematika" title="Finanční matematika – Czech" lang="cs" hreflang="cs" data-title="Finanční matematika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Finanzmathematik" title="Finanzmathematik – German" lang="de" hreflang="de" data-title="Finanzmathematik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matem%C3%A1tica_financiera" title="Matemática financiera – Spanish" lang="es" hreflang="es" data-title="Matemática financiera" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Finantza-matematika" title="Finantza-matematika – Basque" lang="eu" hreflang="eu" data-title="Finantza-matematika" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA_%D9%85%D8%A7%D9%84%DB%8C" title="ریاضیات مالی – Persian" lang="fa" hreflang="fa" data-title="ریاضیات مالی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Math%C3%A9matiques_financi%C3%A8res" title="Mathématiques financières – French" lang="fr" hreflang="fr" data-title="Mathématiques financières" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EB%A6%AC%EA%B8%88%EC%9C%B5%ED%95%99" title="수리금융학 – Korean" lang="ko" hreflang="ko" data-title="수리금융학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B5%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%A4" title="गणितीय वित्त – Hindi" lang="hi" hreflang="hi" data-title="गणितीय वित्त" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Financijska_matematika" title="Financijska matematika – Croatian" lang="hr" hreflang="hr" data-title="Financijska matematika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Financala_matematiko" title="Financala matematiko – Ido" lang="io" hreflang="io" data-title="Financala matematiko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matematika_keuangan" title="Matematika keuangan – Indonesian" lang="id" hreflang="id" data-title="Matematika keuangan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matematica_finanziaria" title="Matematica finanziaria – Italian" lang="it" hreflang="it" data-title="Matematica finanziaria" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94_%D7%A4%D7%99%D7%A0%D7%A0%D7%A1%D7%99%D7%AA" title="מתמטיקה פיננסית – Hebrew" lang="he" hreflang="he" data-title="מתמטיקה פיננסית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Finans%C5%B3_matematika" title="Finansų matematika – Lithuanian" lang="lt" hreflang="lt" data-title="Finansų matematika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Matematik_kewangan" title="Matematik kewangan – Malay" lang="ms" hreflang="ms" data-title="Matematik kewangan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Financi%C3%ABle_wiskunde" title="Financiële wiskunde – Dutch" lang="nl" hreflang="nl" data-title="Financiële wiskunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E7%90%86%E3%83%95%E3%82%A1%E3%82%A4%E3%83%8A%E3%83%B3%E3%82%B9" title="数理ファイナンス – Japanese" lang="ja" hreflang="ja" data-title="数理ファイナンス" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Matematyka_finansowa" title="Matematyka finansowa – Polish" lang="pl" hreflang="pl" data-title="Matematyka finansowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matem%C3%A1tica_financeira" title="Matemática financeira – Portuguese" lang="pt" hreflang="pt" data-title="Matemática financeira" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%B8%D0%BD%D0%B0%D0%BD%D1%81%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Финансовая математика – Russian" lang="ru" hreflang="ru" data-title="Финансовая математика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Finansijska_matematika" title="Finansijska matematika – Serbian" lang="sr" hreflang="sr" data-title="Finansijska matematika" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rahoitusmatematiikka" title="Rahoitusmatematiikka – Finnish" lang="fi" hreflang="fi" data-title="Rahoitusmatematiikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Finansiell_matematik" title="Finansiell matematik – Swedish" lang="sv" hreflang="sv" data-title="Finansiell matematik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%87%E0%B8%B4%E0%B8%99" title="คณิตศาสตร์การเงิน – Thai" lang="th" hreflang="th" data-title="คณิตศาสตร์การเงิน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Finansal_matematik" title="Finansal matematik – Turkish" lang="tr" hreflang="tr" data-title="Finansal matematik" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D1%96%D0%BD%D0%B0%D0%BD%D1%81%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Фінансова математика – Ukrainian" lang="uk" hreflang="uk" data-title="Фінансова математика" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%A7%D9%84%DB%8C%D8%A7%D8%AA%DB%8C_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="مالیاتی ریاضی – Urdu" lang="ur" hreflang="ur" data-title="مالیاتی ریاضی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/To%C3%A1n_t%C3%A0i_ch%C3%ADnh" title="Toán tài chính – Vietnamese" lang="vi" hreflang="vi" data-title="Toán tài chính" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E7%90%86%E9%87%91%E8%9E%8D%E5%AD%B8" title="數理金融學 – Cantonese" lang="yue" hreflang="yue" data-title="數理金融學" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B8%E7%90%86%E9%87%91%E8%9E%8D%E5%AD%B8" title="數理金融學 – Chinese" lang="zh" hreflang="zh" data-title="數理金融學" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Financial_instrument" title="Financial instrument">Instruments</a></div><div class="sidebar-list-content mw-collapsible-content" style="padding-left:2.0em;padding-right:2.0em;"><div class="hlist"> <ul><li><a href="/wiki/Bond_(finance)" title="Bond (finance)">Bond</a></li> <li><a href="/wiki/Cash" title="Cash">Cash</a></li> <li><a href="/wiki/Collateralised_debt_obligation" class="mw-redirect" title="Collateralised debt obligation">Collateralised debt obligation</a></li> <li><a href="/wiki/Credit_default_swap" title="Credit default swap">Credit default swap</a></li> <li><a href="/wiki/Time_deposit" title="Time deposit">Time deposit</a> (<a href="/wiki/Certificate_of_deposit" title="Certificate of deposit">certificate of deposit</a>)</li> <li><a href="/wiki/Line_of_credit" title="Line of credit">Credit line</a></li> <li><a href="/wiki/Deposit_account" title="Deposit account">Deposit</a></li> <li><a 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agreement">Repurchase agreement</a></li> <li><a href="/wiki/Stock" title="Stock">Stock</a></li> <li><a href="/wiki/Security_(finance)" title="Security (finance)">Security</a></li> <li><a href="/wiki/Syndicated_loan" title="Syndicated loan">Syndicated loan</a></li> <li><a href="/wiki/Synthetic_CDO" title="Synthetic CDO">Synthetic CDO</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Corporate_finance" title="Corporate finance">Corporate</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;"> General</th></tr><tr><td class="sidebar-content hlist" style="padding-left:0.5em;padding-right:0.5em;"> <ul><li><a href="/wiki/Accounting" title="Accounting">Accounting</a></li> <li><a href="/wiki/Audit" title="Audit">Audit</a></li> <li><a href="/wiki/Capital_budgeting" title="Capital budgeting">Capital budgeting</a></li> <li><a href="/wiki/Capital_structure" title="Capital structure">Capital structure</a></li> <li><a href="/wiki/Corporate_finance" title="Corporate finance">Corporate finance</a></li> <li><a href="/wiki/Credit_rating_agency" title="Credit rating agency">Credit rating agency</a></li> <li><a href="/wiki/Enterprise_risk_management" title="Enterprise risk management">Enterprise risk management</a></li> <li><a href="/wiki/Enterprise_value" title="Enterprise value">Enterprise value</a></li> <li><a href="/wiki/Financial_risk_management" title="Financial risk management">Risk management</a></li> <li><a href="/wiki/Financial_statement" title="Financial statement">Financial statements</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;"> Transactions</th></tr><tr><td class="sidebar-content hlist" style="padding-left:0.5em;padding-right:0.5em;"> <ul><li><a href="/wiki/Leveraged_buyout" title="Leveraged buyout">Leveraged buyout</a></li> <li><a href="/wiki/Mergers_and_acquisitions" title="Mergers and acquisitions">Mergers and acquisitions</a></li> <li><a href="/wiki/Structured_finance" title="Structured finance">Structured finance</a></li> <li><a href="/wiki/Venture_capital" title="Venture capital">Venture capital</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;"> <a href="/wiki/Corporate_tax" title="Corporate tax">Taxation</a></th></tr><tr><td class="sidebar-content hlist" style="padding-left:0.5em;padding-right:0.5em;"> <ul><li><a href="/wiki/Base_erosion_and_profit_shifting" title="Base erosion and profit shifting">Base erosion and profit shifting</a> (BEPS)</li> <li><a href="/wiki/Corporate_tax_haven" class="mw-redirect" title="Corporate tax haven">Corporate tax haven</a></li> <li><a href="/wiki/Tax_inversion" title="Tax inversion">Tax inversion</a></li> <li><a href="/wiki/Tax_haven" title="Tax haven">Tax haven</a></li> <li><a href="/wiki/Transfer_pricing" title="Transfer pricing">Transfer pricing</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Personal_finance" title="Personal finance">Personal</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Credit_(finance)" class="mw-redirect" title="Credit (finance)">Credit</a> / <a href="/wiki/Consumer_debt" title="Consumer debt">Debt</a></li> <li><a href="/wiki/Employment_contract" title="Employment contract">Employment contract</a></li> <li><a href="/wiki/Financial_planner" title="Financial planner">Financial planning</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Retirement" title="Retirement">Retirement</a></li><li><a href="/wiki/Student_loan" title="Student loan">Student loan</a></li></ul></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Public_finance" title="Public finance">Public</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;;display:block;margin-top:0.15em;"> <a href="/wiki/Government_spending" title="Government spending">Government spending</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Government_final_consumption_expenditure" title="Government final consumption expenditure">Final consumption expenditure</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Government_operations" class="mw-redirect" title="Government operations">Operations</a></li><li><a href="/wiki/Redistribution_of_income_and_wealth" title="Redistribution of income and wealth">Redistribution</a></li></ul></div></li> <li><a href="/wiki/Transfer_payment" title="Transfer payment">Transfer payment</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;"> <a href="/wiki/Government_revenue" title="Government revenue">Government revenue</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;;padding-left:0.5em;padding-right:0.5em;"> <ul><li><div class="hlist"><ul><li><a href="/wiki/Tax" title="Tax">Taxation</a></li><li><a href="/wiki/Deficit_spending" title="Deficit spending">Deficit spending</a></li></ul></div></li> <li><div class="hlist"><ul><li><a href="/wiki/Government_budget" title="Government budget">Budget</a> (<a href="/wiki/Government_budget_balance" title="Government budget balance">balance</a>)</li><li><a href="/wiki/Government_debt" title="Government debt">Debt</a></li></ul></div></li> <li><a href="/wiki/Non-tax_revenue" title="Non-tax revenue">Non-tax revenue</a></li> <li><a href="/wiki/Warrant_of_payment" title="Warrant of payment">Warrant of payment</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Bank" title="Bank">Banking</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Central_bank" title="Central bank">Central bank</a></li> <li><a href="/wiki/Deposit_account" title="Deposit account">Deposit account</a></li> <li><a href="/wiki/Fractional-reserve_banking" title="Fractional-reserve banking">Fractional-reserve</a></li> <li><a href="/wiki/Full-reserve_banking" title="Full-reserve banking">Full-reserve</a></li> <li><a href="/wiki/Investment_banking" title="Investment banking">Investment banking</a></li> <li><a href="/wiki/Loan" title="Loan">Loan</a></li> <li><a href="/wiki/Money_supply" title="Money supply">Money supply</a></li></ul> </div> <ul><li><a href="/wiki/Lists_of_banks" title="Lists of banks">Lists of banks</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;;display:block;margin-top:0.15em;"> <a href="/wiki/Bank_regulation" class="mw-redirect" title="Bank regulation">Bank regulation</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Banking_license" title="Banking license">Banking license</a></li> <li><a href="/wiki/Basel_Accords" title="Basel Accords">Basel Accords</a></li> <li><a href="/wiki/Bank_for_International_Settlements" title="Bank for International Settlements">Bank for International Settlements</a></li> <li><a href="/wiki/Financial_Stability_Board" title="Financial Stability Board">Financial Stability Board</a></li> <li><a href="/wiki/Deposit_insurance" title="Deposit insurance">Deposit insurance</a></li> <li><a href="/wiki/Separation_of_investment_and_retail_banking" title="Separation of investment and retail banking">Separation of investment and retail banking</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Financial_regulation" title="Financial regulation">Regulation</a> <b>·</b> <a href="/wiki/Financial_law" title="Financial law">Financial law</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/International_Financial_Reporting_Standards" title="International Financial Reporting Standards">International Financial Reporting Standards</a></li> <li><a href="/wiki/ISO_31000" title="ISO 31000">ISO 31000</a></li> <li><a href="/wiki/Professional_certification_in_financial_services" title="Professional certification in financial services">Professional certification</a></li> <li><a href="/wiki/Fund_governance" title="Fund governance">Fund governance</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Economic_history" title="Economic history">Economic history</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_private_equity_and_venture_capital" title="History of private equity and venture capital">Private equity and venture capital</a></li> <li><a href="/wiki/Recession" title="Recession">Recession</a></li> <li><a href="/wiki/Stock_market_bubble" title="Stock market bubble">Stock market bubble</a></li> <li><a href="/wiki/Stock_market_crash" title="Stock market crash">Stock market crash</a></li> <li><a href="/wiki/Accounting_scandals" title="Accounting scandals">Accounting scandals</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><a href="/wiki/Outline_of_finance" title="Outline of finance">Outline</a></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Emblem-money.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/16px-Emblem-money.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/24px-Emblem-money.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/32px-Emblem-money.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span> </span><a href="/wiki/Portal:Business_and_Economics" class="mw-redirect" title="Portal:Business and Economics">Business and Economics portal</a></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Bills_and_coins.svg/16px-Bills_and_coins.svg.png" decoding="async" width="16" height="8" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Bills_and_coins.svg/24px-Bills_and_coins.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Bills_and_coins.svg/32px-Bills_and_coins.svg.png 2x" data-file-width="960" data-file-height="465" /></span></span> </span><a href="/wiki/Portal:Money" title="Portal:Money">Money portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Finance_sidebar" title="Template:Finance sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Finance_sidebar" title="Template talk:Finance sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Finance_sidebar" title="Special:EditPage/Template:Finance sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Application of mathematical and statistical methods in finance</div> <p><b>Mathematical finance</b>, also known as <b>quantitative finance</b> and <b>financial mathematics</b>, is a field of <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>, concerned with mathematical modeling in the financial field. </p><p>In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and <a href="/wiki/Risk_management" title="Risk management">risk</a> and <a href="/wiki/Investment_management#Investment_managers_and_portfolio_structures" title="Investment management">portfolio management</a> on the other.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Mathematical finance overlaps heavily with the fields of <a href="/wiki/Computational_finance" title="Computational finance">computational finance</a> and <a href="/wiki/Financial_engineering" title="Financial engineering">financial engineering</a>. The latter focuses on applications and modeling, often with the help of <a href="/wiki/Stochastic_asset_model" class="mw-redirect" title="Stochastic asset model">stochastic asset models</a>, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is <a href="/wiki/Quantitative_investing" class="mw-redirect" title="Quantitative investing">quantitative investing</a>, which relies on statistical and numerical models (and lately <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>) as opposed to traditional <a href="/wiki/Fundamental_analysis" title="Fundamental analysis">fundamental analysis</a> when <a href="/wiki/Investment_management" title="Investment management">managing portfolios</a>. </p><p>French mathematician <a href="/wiki/Louis_Bachelier" title="Louis Bachelier">Louis Bachelier</a>'s doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical finance. But mathematical finance emerged as a discipline in the 1970s, following the work of <a href="/wiki/Fischer_Black" title="Fischer Black">Fischer Black</a>, <a href="/wiki/Myron_Scholes" title="Myron Scholes">Myron Scholes</a> and <a href="/wiki/Robert_C._Merton" title="Robert C. Merton">Robert Merton</a> on option pricing theory. <a href="/wiki/Quantitative_analysis_(finance)#Quantitative_investment_management)" title="Quantitative analysis (finance)">Mathematical investing</a> originated from the research of mathematician <a href="/wiki/Edward_O._Thorp" title="Edward O. Thorp">Edward Thorp</a> who used statistical methods to first invent <a href="/wiki/Card_counting" title="Card counting">card counting</a> in <a href="/wiki/Blackjack" title="Blackjack">blackjack</a> and then applied its principles to modern systematic investing.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The subject has a close relationship with the discipline of <a href="/wiki/Financial_economics" title="Financial economics">financial economics</a>, which is concerned with much of the underlying theory that is involved in financial mathematics. While trained economists use complex <a href="/wiki/Economic_models" class="mw-redirect" title="Economic models">economic models</a> that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend the <a href="/wiki/Mathematical_model" title="Mathematical model">mathematical</a> or <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical</a> models without necessarily establishing a link to financial theory, taking observed market prices as input. See: <a href="/wiki/Valuation_of_options" title="Valuation of options">Valuation of options</a>; <a href="/wiki/Financial_modeling#Quantitative_finance" title="Financial modeling">Financial modeling</a>; <a href="/wiki/Asset_pricing" title="Asset pricing">Asset pricing</a>. The <a href="/wiki/Fundamental_theorem_of_arbitrage-free_pricing" class="mw-redirect" title="Fundamental theorem of arbitrage-free pricing">fundamental theorem of arbitrage-free pricing</a> is one of the key theorems in mathematical finance, while the <a href="/wiki/Black%E2%80%93Scholes" class="mw-redirect" title="Black–Scholes">Black–Scholes</a> equation and formula are amongst the key results.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Today many universities offer degree and research programs in mathematical finance. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History:_Q_versus_P">History: Q versus P</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=1" title="Edit section: History: Q versus P"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are two separate branches of finance that require advanced quantitative techniques: derivatives pricing, and risk and portfolio management. One of the main differences is that they use different probabilities such as the risk-neutral probability (or arbitrage-pricing probability), denoted by "Q", and the actual (or actuarial) probability, denoted by "P". </p> <div class="mw-heading mw-heading3"><h3 id="Derivatives_pricing:_the_Q_world">Derivatives pricing: the Q world</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=2" title="Edit section: Derivatives pricing: the Q world"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable floatright"> <caption><b>The Q world</b> </caption> <tbody><tr> <td>Goal </td> <td>"extrapolate the present" </td></tr> <tr> <td>Environment </td> <td>risk-neutral probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> </td></tr> <tr> <td>Processes </td> <td>continuous-time martingales </td></tr> <tr> <td>Dimension </td> <td>low </td></tr> <tr> <td>Tools </td> <td>Itō calculus, PDEs </td></tr> <tr> <td>Challenges </td> <td>calibration </td></tr> <tr> <td>Business </td> <td>sell-side </td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Risk-neutral_measure" title="Risk-neutral measure">Risk-neutral measure</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes model</a>, <a href="/wiki/Brownian_model_of_financial_markets" title="Brownian model of financial markets">Brownian model of financial markets</a>, <a href="/wiki/Martingale_pricing" title="Martingale pricing">Martingale pricing</a>, and <a href="/wiki/Quantitative_analysis_(finance)#History" title="Quantitative analysis (finance)">Quantitative analysis (finance) § History</a></div> <p>The goal of derivatives pricing is to determine the fair price of a given security in terms of more <a href="/wiki/Market_liquidity" title="Market liquidity">liquid securities</a> whose price is determined by the law of <a href="/wiki/Supply_and_demand" title="Supply and demand">supply and demand</a>. The meaning of "fair" depends, of course, on whether one considers buying or selling the security. Examples of securities being priced are <a href="/wiki/Option_(finance)" title="Option (finance)">plain vanilla</a> and <a href="/wiki/Exotic_option" title="Exotic option">exotic options</a>, <a href="/wiki/Convertible_bond" title="Convertible bond">convertible bonds</a>, etc. </p><p>Once a fair price has been determined, the sell-side trader can make a market on the security. Therefore, derivatives pricing is a complex "extrapolation" exercise to define the current market value of a security, which is then used by the sell-side community. Quantitative derivatives pricing was initiated by <a href="/wiki/Louis_Bachelier" title="Louis Bachelier">Louis Bachelier</a> in <i>The Theory of Speculation</i> ("Théorie de la spéculation", published 1900), with the introduction of the most basic and most influential of processes, <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>, and its applications to the pricing of options.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Brownian motion is derived using the <a href="/wiki/Langevin_equation" title="Langevin equation">Langevin equation</a> and the discrete <a href="/wiki/Random_walk" title="Random walk">random walk</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Bachelier modeled the <a href="/wiki/Time_series" title="Time series">time series</a> of changes in the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of stock prices as a <a href="/wiki/Random_walk" title="Random walk">random walk</a> in which the short-term changes had a finite <a href="/wiki/Variance" title="Variance">variance</a>. This causes longer-term changes to follow a <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distribution</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The theory remained dormant until <a href="/wiki/Fischer_Black" title="Fischer Black">Fischer Black</a> and <a href="/wiki/Myron_Scholes" title="Myron Scholes">Myron Scholes</a>, along with fundamental contributions by <a href="/wiki/Robert_C._Merton" title="Robert C. Merton">Robert C. Merton</a>, applied the second most influential process, the <a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">geometric Brownian motion</a>, to <a href="/wiki/Option_pricing" class="mw-redirect" title="Option pricing">option pricing</a>. For this M. Scholes and R. Merton were awarded the 1997 <a href="/wiki/Nobel_Memorial_Prize_in_Economic_Sciences" title="Nobel Memorial Prize in Economic Sciences">Nobel Memorial Prize in Economic Sciences</a>. Black was ineligible for the prize because he died in 1995.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The next important step was the <a href="/wiki/Fundamental_theorem_of_asset_pricing" title="Fundamental theorem of asset pricing">fundamental theorem of asset pricing</a> by Harrison and Pliska (1981), according to which the suitably normalized current price <i>P<sub>0</sub></i> of security is arbitrage-free, and thus truly fair only if there exists a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> <i>P<sub>t</sub></i> with constant <a href="/wiki/Expected_value" title="Expected value">expected value</a> which describes its future evolution:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}=\mathbf {E} _{0}(P_{t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}=\mathbf {E} _{0}(P_{t})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0242237ec2a7bef16d0c916237de9a2623d341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.584ex; height:2.843ex;" alt="{\displaystyle P_{0}=\mathbf {E} _{0}(P_{t})}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>A process satisfying (<b><a href="#math_1">1</a></b>) is called a "<a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a>". A martingale does not reward risk. Thus the probability of the normalized security price process is called "risk-neutral" and is typically denoted by the <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard font</a> letter "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>". </p><p>The relationship (<b><a href="#math_1">1</a></b>) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time. </p><p>The <a href="/wiki/Quantitative_analyst" class="mw-redirect" title="Quantitative analyst">quants</a> who operate in the Q world of derivatives pricing are specialists with deep knowledge of the specific products they model. </p><p>Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature. Calibration is one of the main challenges of the Q world: once a continuous-time parametric process has been calibrated to a set of traded securities through a relationship such as (<b><a href="#math_1">1</a></b>), a similar relationship is used to define the price of new derivatives. </p><p>The main quantitative tools necessary to handle continuous-time Q-processes are <a href="/wiki/It%C3%B4_calculus" title="Itô calculus">Itô's stochastic calculus</a>, <a href="/wiki/Monte_Carlo_methods_in_finance" title="Monte Carlo methods in finance">simulation</a> and <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> (PDEs).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Risk_and_portfolio_management:_the_P_world">Risk and portfolio management: the P world</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=3" title="Edit section: Risk and portfolio management: the P world"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable floatright"> <caption><b>The P world</b> </caption> <tbody><tr> <td>Goal </td> <td>"model the future" </td></tr> <tr> <td>Environment </td> <td>real-world probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"></span> </td></tr> <tr> <td>Processes </td> <td>discrete-time series </td></tr> <tr> <td>Dimension </td> <td>large </td></tr> <tr> <td>Tools </td> <td>multivariate statistics </td></tr> <tr> <td>Challenges </td> <td>estimation </td></tr> <tr> <td>Business </td> <td>buy-side </td></tr></tbody></table> <p>Risk and portfolio management aims to model the statistically derived probability distribution of the market prices of all the securities at a given future investment horizon. This "real" probability distribution of the market prices is typically denoted by the blackboard font letter "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"></span>", as opposed to the "risk-neutral" probability "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>" used in derivatives pricing. Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss profile of their positions considered as a portfolio. Increasingly, elements of this process are automated; see <a href="/wiki/Outline_of_finance#Quantitative_investing" title="Outline of finance">Outline of finance § Quantitative investing</a> for a listing of relevant articles. </p><p>For their pioneering work, <a href="/wiki/Harry_Markowitz" title="Harry Markowitz">Markowitz</a> and <a href="/wiki/William_F._Sharpe" title="William F. Sharpe">Sharpe</a>, along with <a href="/wiki/Merton_Miller" title="Merton Miller">Merton Miller</a>, shared the 1990 <a href="/wiki/Nobel_Memorial_Prize_in_Economic_Sciences" title="Nobel Memorial Prize in Economic Sciences">Nobel Memorial Prize in Economic Sciences</a>, for the first time ever awarded for a work in finance. </p><p>The portfolio-selection work of Markowitz and Sharpe introduced mathematics to <a href="/wiki/Investment_management" title="Investment management">investment management</a>. With time, the mathematics has become more sophisticated. Thanks to Robert Merton and Paul Samuelson, one-period models were replaced by continuous time, <a href="/wiki/Brownian_Model_of_Financial_Markets" class="mw-redirect" title="Brownian Model of Financial Markets">Brownian-motion models</a>, and the quadratic utility function implicit in mean–variance optimization was replaced by more general increasing, concave utility functions.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Furthermore, in recent years the focus shifted toward estimation risk, i.e., the dangers of incorrectly assuming that advanced time series analysis alone can provide completely accurate estimates of the market parameters.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> See <a href="/wiki/Financial_risk_management#Investment_management" title="Financial risk management">Financial risk management § Investment management</a>. </p><p>Much effort has gone into the study of financial markets and how prices vary with time. <a href="/wiki/Charles_Dow" title="Charles Dow">Charles Dow</a>, one of the founders of <a href="/wiki/Dow_Jones_%26_Company" title="Dow Jones & Company">Dow Jones & Company</a> and <a href="/wiki/The_Wall_Street_Journal" title="The Wall Street Journal">The Wall Street Journal</a>, enunciated a set of ideas on the subject which are now called <a href="/wiki/Dow_Theory" class="mw-redirect" title="Dow Theory">Dow Theory</a>. This is the basis of the so-called <a href="/wiki/Technical_analysis" title="Technical analysis">technical analysis</a> method of attempting to predict future changes. One of the tenets of "technical analysis" is that <a href="/wiki/Market_trend" title="Market trend">market trends</a> give an indication of the future, at least in the short term. The claims of the technical analysts are disputed by many academics.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2021)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Criticism">Criticism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=4" title="Edit section: Criticism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Financial_economics#Challenges_and_criticism" title="Financial economics">Financial economics § Challenges and criticism</a>, and <a href="/wiki/Financial_engineering#Criticisms" title="Financial engineering">Financial engineering § Criticisms</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Financial_models_with_long-tailed_distributions_and_volatility_clustering" title="Financial models with long-tailed distributions and volatility clustering">Financial models with long-tailed distributions and volatility clustering</a></div> <p>Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the <a href="/wiki/Financial_crisis_of_2007%E2%80%932010" class="mw-redirect" title="Financial crisis of 2007–2010">financial crisis of 2007–2010</a>. Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by <a href="/wiki/Paul_Wilmott" title="Paul Wilmott">Paul Wilmott</a>, and by <a href="/wiki/Nassim_Nicholas_Taleb" title="Nassim Nicholas Taleb">Nassim Nicholas Taleb</a>, in his book <a href="/wiki/The_Black_Swan_(Taleb_book)" class="mw-redirect" title="The Black Swan (Taleb book)"><i>The Black Swan</i></a>.<sup id="cite_ref-Black_Swan_13-0" class="reference"><a href="#cite_note-Black_Swan-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Taleb claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading. Wilmott and <a href="/wiki/Emanuel_Derman" title="Emanuel Derman">Emanuel Derman</a> published the <i><a href="/wiki/Financial_Modelers%27_Manifesto" title="Financial Modelers' Manifesto">Financial Modelers' Manifesto</a></i> in January 2009<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> which addresses some of the most serious concerns. Bodies such as the <a href="/wiki/Institute_for_New_Economic_Thinking" title="Institute for New Economic Thinking">Institute for New Economic Thinking</a> are now attempting to develop new theories and methods.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>In general, modeling the changes by distributions with finite variance is, increasingly, said to be inappropriate.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In the 1960s it was discovered by <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a> that changes in prices do not follow a <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distribution</a>, but are rather modeled better by Lévy alpha-<a href="/wiki/Stable_distribution" title="Stable distribution">stable distributions</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The scale of change, or volatility, depends on the length of the time interval to a <a href="/wiki/Power_law" title="Power law">power</a> a bit more than 1/2. Large changes up or down are more likely than what one would calculate using a Gaussian distribution with an estimated <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>. But the problem is that it does not solve the problem as it makes parametrization much harder and risk control less reliable.<sup id="cite_ref-Black_Swan_13-1" class="reference"><a href="#cite_note-Black_Swan-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Perhaps more fundamental: though mathematical finance models may generate a profit in the short-run, this type of modeling is often in conflict with a central tenet of modern macroeconomics, the <a href="/wiki/Lucas_critique" title="Lucas critique">Lucas critique</a> - or <a href="/wiki/Rational_expectations" title="Rational expectations">rational expectations</a> - which states that observed relationships may not be structural in nature and thus may not be possible to exploit for public policy or for profit unless we have identified relationships using <a href="/wiki/Causal_analysis" title="Causal analysis">causal analysis</a> and <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Mathematical finance models do not, therefore, incorporate complex elements of human psychology that are critical to modeling modern macroeconomic movements such as the <a href="/wiki/Self-fulfilling_prophecy" title="Self-fulfilling prophecy">self-fulfilling</a> panic that motivates <a href="/wiki/Bank_runs" class="mw-redirect" title="Bank runs">bank runs</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Outline_of_finance#Financial_mathematics" title="Outline of finance">Outline of finance § Financial mathematics</a>, <a href="/wiki/Outline_of_finance#Mathematical_tools" title="Outline of finance">Outline of finance § Mathematical tools</a>, <a href="/wiki/Outline_of_finance#Derivatives_pricing" title="Outline of finance">Outline of finance § Derivatives pricing</a>, and <a href="/wiki/Outline_of_corporate_finance" title="Outline of corporate finance">Outline of corporate finance</a></div> <div class="mw-heading mw-heading3"><h3 id="Mathematical_tools">Mathematical tools</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=6" title="Edit section: Mathematical tools"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">Asymptotic analysis</a></li> <li><a href="/wiki/Backward_stochastic_differential_equation" title="Backward stochastic differential equation">Backward stochastic differential equation</a></li> <li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Copula_(probability_theory)" class="mw-redirect" title="Copula (probability theory)">Copulas</a>, including Gaussian</li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Finance#Quantitative_finance" title="Finance">Finance § Quantitative finance</a></li> <li><a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Mathematical_model" title="Mathematical model">Mathematical models</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Mathematical optimization</a> <ul><li><a href="/wiki/Linear_programming" title="Linear programming">Linear programming</a></li> <li><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a></li> <li><a href="/wiki/Quadratic_programming" title="Quadratic programming">Quadratic programming</a></li></ul></li> <li><a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> <ul><li><a href="/wiki/Gaussian_quadrature" title="Gaussian quadrature">Gaussian quadrature</a></li></ul></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equations</a> <ul><li><a href="/wiki/Heat_equation" title="Heat equation">Heat equation</a></li> <li><a href="/wiki/Numerical_partial_differential_equations" class="mw-redirect" title="Numerical partial differential equations">Numerical partial differential equations</a> <ul><li><a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson method</a></li> <li><a href="/wiki/Finite_difference#Numerical_analysis" title="Finite difference">Finite difference method</a></li></ul></li></ul></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> <ul><li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's SU-distribution</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal distribution</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t-distribution</a></li></ul></li> <li><a href="/wiki/Quantile_function" title="Quantile function">Quantile functions</a></li> <li><a href="/wiki/Radon%E2%80%93Nikodym_derivative" class="mw-redirect" title="Radon–Nikodym derivative">Radon–Nikodym derivative</a></li> <li><a href="/wiki/Risk-neutral_measure" title="Risk-neutral measure">Risk-neutral measure</a></li> <li><a href="/wiki/Scenario_optimization" title="Scenario optimization">Scenario optimization</a></li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic calculus</a> <ul><li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li></ul></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li> <li><a href="/wiki/Stochastic_optimization" title="Stochastic optimization">Stochastic optimization</a></li> <li><a href="/wiki/Stochastic_volatility" title="Stochastic volatility">Stochastic volatility</a></li> <li><a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></li> <li><a href="/wiki/Value_at_risk" title="Value at risk">Value at risk</a></li> <li><a href="/wiki/Volatility_(finance)" title="Volatility (finance)">Volatility</a> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">ARCH model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">GARCH model</a></li></ul></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Derivatives_pricing">Derivatives pricing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=7" title="Edit section: Derivatives pricing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 20em;"> <ul><li>The <a href="/wiki/Brownian_model_of_financial_markets" title="Brownian model of financial markets">Brownian model of financial markets</a></li> <li><a href="/wiki/Rational_pricing" title="Rational pricing">Rational pricing</a> assumptions <ul><li><a href="/wiki/Risk-neutral_measure" title="Risk-neutral measure">Risk neutral valuation</a></li> <li><a href="/wiki/Arbitrage" title="Arbitrage">Arbitrage</a>-free pricing</li></ul></li> <li>Valuation adjustments <ul><li><a href="/wiki/Credit_valuation_adjustment" title="Credit valuation adjustment">Credit valuation adjustment</a></li> <li><a href="/wiki/XVA" title="XVA">XVA</a></li></ul></li> <li><a href="/wiki/Yield_curve" title="Yield curve">Yield curve</a> modelling <ul><li><a href="/wiki/Multi-curve_framework" class="mw-redirect" title="Multi-curve framework">Multi-curve framework</a></li> <li><a href="/wiki/Bootstrapping_(finance)" title="Bootstrapping (finance)">Bootstrapping</a></li> <li><a href="/wiki/Yield_curve#Construction_of_the_full_yield_curve_from_market_data" title="Yield curve">Construction from market data</a></li> <li><a href="/wiki/Fixed-income_attribution#Modeling_the_yield_curve" title="Fixed-income attribution">Fixed-income attribution</a></li> <li><a href="/wiki/Nelson-Siegel" class="mw-redirect" title="Nelson-Siegel">Nelson-Siegel</a></li> <li><a href="/wiki/Principal_component_analysis#Quantitative_finance" title="Principal component analysis">Principal component analysis</a></li></ul></li> <li><a href="/wiki/Forward_price#Forward_Price_Formula" title="Forward price">Forward Price Formula</a></li> <li><a href="/wiki/Futures_contract#Pricing" title="Futures contract">Futures contract pricing</a></li> <li><a href="/wiki/Swap_(finance)#Valuation_and_Pricing" title="Swap (finance)">Swap valuation</a> <ul><li><a href="/wiki/Currency_swap#Valuation_and_Pricing" title="Currency swap">Currency swap#Valuation and Pricing</a></li> <li><a href="/wiki/Interest_rate_swap#Valuation_and_pricing" title="Interest rate swap">Interest rate swap#Valuation and pricing</a> <ul><li><a href="/wiki/Multi-curve_framework" class="mw-redirect" title="Multi-curve framework">Multi-curve framework</a></li></ul></li> <li><a href="/wiki/Variance_swap#Pricing_and_valuation" title="Variance swap">Variance swap#Pricing and valuation</a></li> <li><a href="/wiki/Asset_swap#Computing_the_asset_swap_spread" title="Asset swap">Asset swap #Computing the asset swap spread</a></li> <li><a href="/wiki/Credit_default_swap#Pricing_and_valuation" title="Credit default swap">Credit default swap #Pricing and valuation</a></li></ul></li> <li>Options <ul><li><a href="/wiki/Put%E2%80%93call_parity" title="Put–call parity">Put–call parity</a> (Arbitrage relationships for options)</li> <li><a href="/wiki/Intrinsic_value_(finance)" title="Intrinsic value (finance)">Intrinsic value</a>, <a href="/wiki/Option_time_value" title="Option time value">Time value</a></li> <li><a href="/wiki/Moneyness" title="Moneyness">Moneyness</a></li> <li>Pricing <a href="/wiki/Mathematical_model" title="Mathematical model">models</a> <ul><li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes model</a></li> <li><a href="/wiki/Black_model" title="Black model">Black model</a></li> <li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options model</a> <ul><li><a href="/wiki/Implied_binomial_tree" class="mw-redirect" title="Implied binomial tree">Implied binomial tree</a></li> <li><a href="/wiki/Edgeworth_binomial_tree" class="mw-redirect" title="Edgeworth binomial tree">Edgeworth binomial tree</a></li></ul></li> <li><a href="/wiki/Monte_Carlo_option_model" class="mw-redirect" title="Monte Carlo option model">Monte Carlo option model</a></li> <li><a href="/wiki/Implied_volatility" title="Implied volatility">Implied volatility</a>, <a href="/wiki/Volatility_smile" title="Volatility smile">Volatility smile</a></li> <li><a href="/wiki/Local_volatility" title="Local volatility">Local volatility</a></li> <li><a href="/wiki/Stochastic_volatility" title="Stochastic volatility">Stochastic volatility</a> <ul><li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance model</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston model</a> <ul><li><a href="/wiki/Stochastic_volatility_jump" title="Stochastic volatility jump">Stochastic volatility jump</a></li></ul></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility model</a></li></ul></li> <li><a href="/wiki/Markov_switching_multifractal" title="Markov switching multifractal">Markov switching multifractal</a></li> <li><a href="/wiki/Greeks_(finance)" title="Greeks (finance)">The Greeks</a></li> <li><a href="/wiki/Finite_difference_methods_for_option_pricing" title="Finite difference methods for option pricing">Finite difference methods for option pricing</a></li> <li><a href="/wiki/Vanna%E2%80%93Volga_pricing" title="Vanna–Volga pricing">Vanna–Volga pricing</a></li> <li><a href="/wiki/Trinomial_tree" title="Trinomial tree">Trinomial tree</a> <ul><li><a href="/wiki/Implied_trinomial_tree" class="mw-redirect" title="Implied trinomial tree">Implied trinomial tree</a></li></ul></li> <li><a href="/wiki/Garman-Kohlhagen_model" class="mw-redirect" title="Garman-Kohlhagen model">Garman-Kohlhagen model</a></li> <li><a href="/wiki/Lattice_model_(finance)" title="Lattice model (finance)">Lattice model (finance)</a></li> <li><a href="/wiki/Margrabe%27s_formula" title="Margrabe's formula">Margrabe's formula</a></li> <li><a href="/wiki/Carr%E2%80%93Madan_formula" title="Carr–Madan formula">Carr–Madan formula</a></li></ul></li> <li>Pricing of American options <ul><li><a href="/wiki/Barone-Adesi_and_Whaley" class="mw-redirect" title="Barone-Adesi and Whaley">Barone-Adesi and Whaley</a></li> <li><a href="/wiki/Bjerksund_and_Stensland" class="mw-redirect" title="Bjerksund and Stensland">Bjerksund and Stensland</a></li> <li><a href="/wiki/Black%27s_approximation" title="Black's approximation">Black's approximation</a></li> <li><a href="/wiki/Monte_Carlo_methods_for_option_pricing#Least_Square_Monte_Carlo" title="Monte Carlo methods for option pricing">Least Square Monte Carlo</a></li> <li><a href="/wiki/Optimal_stopping" title="Optimal stopping">Optimal stopping</a></li> <li><a href="/wiki/Roll-Geske-Whaley" class="mw-redirect" title="Roll-Geske-Whaley">Roll-Geske-Whaley</a></li></ul></li></ul></li> <li><a href="/wiki/Interest_rate_derivative" title="Interest rate derivative">Interest rate derivatives</a> <ul><li><a href="/wiki/Black_model" title="Black model">Black model</a> <ul><li><a href="/wiki/Interest_rate_cap_and_floor#Black_model" title="Interest rate cap and floor">caps and floors</a></li> <li><a href="/wiki/Swaption#Valuation" title="Swaption">swaptions</a></li> <li><a href="/wiki/Bond_option#Valuation" title="Bond option">Bond options</a></li></ul></li> <li><a href="/wiki/Short-rate_model" title="Short-rate model">Short-rate models</a> <ul><li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter model</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vasicek model</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee model</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White model</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross model</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy model</a></li> <li><a href="/wiki/Kalotay%E2%80%93Williams%E2%80%93Fabozzi_model" class="mw-redirect" title="Kalotay–Williams–Fabozzi model">Kalotay–Williams–Fabozzi model</a></li> <li><a href="/wiki/Longstaff%E2%80%93Schwartz_model" class="mw-redirect" title="Longstaff–Schwartz model">Longstaff–Schwartz model</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen model</a></li></ul></li> <li><a href="/wiki/Forward_rate" title="Forward rate">Forward rate</a>-based models <ul><li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market model</a> (Brace–Gatarek–Musiela Model, BGM)</li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton Model</a> (HJM)</li></ul></li></ul></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Portfolio_modelling">Portfolio modelling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=8" title="Edit section: Portfolio modelling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Outline_of_finance#Portfolio_theory" title="Outline of finance">Outline of finance § Portfolio theory</a>, <a href="/wiki/Outline_of_finance#Quantitative_investing" title="Outline of finance">Outline of finance § Quantitative investing</a>, and <a href="/wiki/Outline_of_finance#Portfolio_mathematics" title="Outline of finance">Outline of finance § Portfolio mathematics</a></div> <div class="mw-heading mw-heading3"><h3 id="Other">Other</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=9" title="Edit section: Other"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Computational_finance" title="Computational finance">Computational finance</a></li> <li><a href="/wiki/Derivative_(finance)" title="Derivative (finance)">Derivative (finance)</a>, <a href="/wiki/Outline_of_finance#Derivatives_market" title="Outline of finance">list of derivatives topics</a></li> <li><a href="/wiki/Economic_model" title="Economic model">Economic model</a></li> <li><a href="/wiki/Econophysics" title="Econophysics">Econophysics</a></li> <li><a href="/wiki/Financial_economics" title="Financial economics">Financial economics</a></li> <li><a href="/wiki/Financial_engineering" title="Financial engineering">Financial engineering</a></li> <li><a href="/wiki/Financial_modeling#Quantitative_finance" title="Financial modeling">Financial modeling § Quantitative finance</a></li> <li><a href="/wiki/International_Association_for_Quantitative_Finance" title="International Association for Quantitative Finance">International Association for Quantitative Finance</a></li> <li><a href="/wiki/International_Swaps_and_Derivatives_Association" title="International Swaps and Derivatives Association">International Swaps and Derivatives Association</a></li> <li><a href="/wiki/Index_of_accounting_articles" title="Index of accounting articles">Index of accounting articles</a></li> <li><a href="/wiki/List_of_economists" title="List of economists">List of economists</a></li> <li><a href="/wiki/Master_of_Quantitative_Finance" title="Master of Quantitative Finance">Master of Quantitative Finance</a></li> <li><a href="/wiki/Outline_of_economics" title="Outline of economics">Outline of economics</a></li> <li><a href="/wiki/Outline_of_finance" title="Outline of finance">Outline of finance</a></li> <li><a href="/wiki/Physics_of_financial_markets" title="Physics of financial markets">Physics of financial markets</a></li> <li><a href="/wiki/Quantitative_behavioral_finance" title="Quantitative behavioral finance">Quantitative behavioral finance</a></li> <li><a href="/wiki/Statistical_finance" title="Statistical finance">Statistical finance</a></li> <li><a href="/wiki/Technical_analysis" title="Technical analysis">Technical analysis</a></li> <li><a href="/wiki/XVA" title="XVA">XVA</a></li> <li><a href="/wiki/Quantum_finance" title="Quantum finance">Quantum finance</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=10" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output 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.citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://finance.laws.com/quantitative-finance">"Quantitative Finance"</a>. About.com<span class="reference-accessdate">. Retrieved <span class="nowrap">28 March</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Quantitative+Finance&rft.pub=About.com&rft_id=http%3A%2F%2Ffinance.laws.com%2Fquantitative-finance&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLam" class="citation web cs1">Lam, Leslie P. Norton and Dan. <a rel="nofollow" class="external text" href="https://www.barrons.com/articles/why-edward-thorp-only-owns-berkshire-hathaway-1521547200">"Why Edward Thorp Owns Only Berkshire Hathaway"</a>. <i>www.barrons.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-06-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.barrons.com&rft.atitle=Why+Edward+Thorp+Owns+Only+Berkshire+Hathaway&rft.aulast=Lam&rft.aufirst=Leslie+P.+Norton+and+Dan&rft_id=https%3A%2F%2Fwww.barrons.com%2Farticles%2Fwhy-edward-thorp-only-owns-berkshire-hathaway-1521547200&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson2009" class="citation web cs1">Johnson, Tim (1 September 2009). <a rel="nofollow" class="external text" href="https://plus.maths.org/content/what-financial-mathematics">"What is financial mathematics?"</a>. <i>+Plus Magazine</i><span class="reference-accessdate">. Retrieved <span class="nowrap">1 March</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=%2BPlus+Magazine&rft.atitle=What+is+financial+mathematics%3F&rft.date=2009-09-01&rft.aulast=Johnson&rft.aufirst=Tim&rft_id=https%3A%2F%2Fplus.maths.org%2Fcontent%2Fwhat-financial-mathematics&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE.2004" class="citation book cs1">E., Shreve, Steven (2004). <i>Stochastic calculus for finance</i>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387401003" title="Special:BookSources/9780387401003"><bdi>9780387401003</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/53289874">53289874</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+calculus+for+finance&rft.place=New+York&rft.pub=Springer&rft.date=2004&rft_id=info%3Aoclcnum%2F53289874&rft.isbn=9780387401003&rft.aulast=E.&rft.aufirst=Shreve%2C+Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen.2013" class="citation book cs1">Stephen., Blyth (2013). <i>Introduction to Quantitative Finance</i>. Oxford University Press, USA. p. 157. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780199666591" title="Special:BookSources/9780199666591"><bdi>9780199666591</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/868286679">868286679</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantitative+Finance.&rft.pages=157&rft.pub=Oxford+University+Press%2C+USA&rft.date=2013&rft_id=info%3Aoclcnum%2F868286679&rft.isbn=9780199666591&rft.aulast=Stephen.&rft.aufirst=Blyth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFB.2005" class="citation book cs1">B., Schmidt, Anatoly (2005). <i>Quantitative finance for physicists : an introduction</i>. San Diego, Calif.: Elsevier Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780080492209" title="Special:BookSources/9780080492209"><bdi>9780080492209</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/57743436">57743436</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantitative+finance+for+physicists+%3A+an+introduction&rft.place=San+Diego%2C+Calif.&rft.pub=Elsevier+Academic+Press&rft.date=2005&rft_id=info%3Aoclcnum%2F57743436&rft.isbn=9780080492209&rft.aulast=B.&rft.aufirst=Schmidt%2C+Anatoly&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBachelir" class="citation web cs1">Bachelir, Louis. <a rel="nofollow" class="external text" href="https://docs.google.com/file/d/0B5LLDy7-d3SKNGI0M2E0NGItYzFlMS00NGU2LWE2ZDAtODc3MDY3MzdiNmY0/edit?hl=en_GB&pli=1">"The Theory of Speculation"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">28 March</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Theory+of+Speculation&rft.aulast=Bachelir&rft.aufirst=Louis&rft_id=https%3A%2F%2Fdocs.google.com%2Ffile%2Fd%2F0B5LLDy7-d3SKNGI0M2E0NGItYzFlMS00NGU2LWE2ZDAtODc3MDY3MzdiNmY0%2Fedit%3Fhl%3Den_GB%26pli%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLindbeck" class="citation web cs1">Lindbeck, Assar. <a rel="nofollow" class="external text" href="https://www.nobelprize.org/nobel_prizes/themes/economic-sciences/lindbeck/index.html">"The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1969-2007"</a>. Nobel Prize<span class="reference-accessdate">. Retrieved <span class="nowrap">28 March</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Sveriges+Riksbank+Prize+in+Economic+Sciences+in+Memory+of+Alfred+Nobel+1969-2007&rft.pub=Nobel+Prize&rft.aulast=Lindbeck&rft.aufirst=Assar&rft_id=https%3A%2F%2Fwww.nobelprize.org%2Fnobel_prizes%2Fthemes%2Feconomic-sciences%2Flindbeck%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown2008" class="citation news cs1">Brown, Angus (1 Dec 2008). <a rel="nofollow" class="external text" href="http://plus.maths.org/content/risky-business-how-price-derivatives">"A risky business: How to price derivatives"</a>. <i>Price+ Magazine</i><span class="reference-accessdate">. Retrieved <span class="nowrap">28 March</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Price%2B+Magazine&rft.atitle=A+risky+business%3A+How+to+price+derivatives&rft.date=2008-12-01&rft.aulast=Brown&rft.aufirst=Angus&rft_id=http%3A%2F%2Fplus.maths.org%2Fcontent%2Frisky-business-how-price-derivatives&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">For a survey, see <a rel="nofollow" class="external text" href="https://catalogimages.wiley.com/images/db/pdf/9781118487716.excerpt.pdf">"Financial Models"</a>, from Michael Mastro (2013). <i>Financial Derivative and Energy Market Valuation</i>, John Wiley & Sons. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1118487716" title="Special:BookSources/978-1118487716">978-1118487716</a>.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaratzasShreve1998" class="citation book cs1">Karatzas, Ioannis; Shreve, Steve (1998). <i>Methods of Mathematical Finance</i>. Secaucus, New Jersey, US: Springer-Verlag New York, Incorporated. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387948393" title="Special:BookSources/9780387948393"><bdi>9780387948393</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Mathematical+Finance&rft.place=Secaucus%2C+New+Jersey%2C+US&rft.pub=Springer-Verlag+New+York%2C+Incorporated&rft.date=1998&rft.isbn=9780387948393&rft.aulast=Karatzas&rft.aufirst=Ioannis&rft.au=Shreve%2C+Steve&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeucci2005" class="citation book cs1"><a href="/wiki/Attilio_Meucci" title="Attilio Meucci">Meucci, Attilio</a> (2005). <i>Risk and Asset Allocation</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783642009648" title="Special:BookSources/9783642009648"><bdi>9783642009648</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Risk+and+Asset+Allocation&rft.pub=Springer&rft.date=2005&rft.isbn=9783642009648&rft.aulast=Meucci&rft.aufirst=Attilio&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-Black_Swan-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Black_Swan_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Black_Swan_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaleb2007" class="citation book cs1"><a href="/wiki/Nassim_Nicholas_Taleb" title="Nassim Nicholas Taleb">Taleb, Nassim Nicholas</a> (2007). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/blackswanimpacto00tale"><i>The Black Swan: The Impact of the Highly Improbable</i></a></span>. Random House Trade. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4000-6351-2" title="Special:BookSources/978-1-4000-6351-2"><bdi>978-1-4000-6351-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Black+Swan%3A+The+Impact+of+the+Highly+Improbable&rft.pub=Random+House+Trade&rft.date=2007&rft.isbn=978-1-4000-6351-2&rft.aulast=Taleb&rft.aufirst=Nassim+Nicholas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fblackswanimpacto00tale&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140908100545/http://www.wilmott.com/blogs/paul/index.cfm/2009/1/8/Financial-Modelers-Manifesto">"Financial Modelers' Manifesto"</a>. Paul Wilmott's Blog. January 8, 2009. Archived from <a rel="nofollow" class="external text" href="http://www.wilmott.com/blogs/paul/index.cfm/2009/1/8/Financial-Modelers-Manifesto">the original</a> on September 8, 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">June 1,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Financial+Modelers%27+Manifesto&rft.pub=Paul+Wilmott%27s+Blog&rft.date=2009-01-08&rft_id=http%3A%2F%2Fwww.wilmott.com%2Fblogs%2Fpaul%2Findex.cfm%2F2009%2F1%2F8%2FFinancial-Modelers-Manifesto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGillian_Tett2010" class="citation news cs1">Gillian Tett (April 15, 2010). <a rel="nofollow" class="external text" href="https://www.ft.com/cms/s/0/cfb9c43a-48b7-11df-8af4-00144feab49a.html">"Mathematicians must get out of their ivory towers"</a>. <i><a href="/wiki/Financial_Times" title="Financial Times">Financial Times</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Financial+Times&rft.atitle=Mathematicians+must+get+out+of+their+ivory+towers&rft.date=2010-04-15&rft.au=Gillian+Tett&rft_id=http%3A%2F%2Fwww.ft.com%2Fcms%2Fs%2F0%2Fcfb9c43a-48b7-11df-8af4-00144feab49a.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSvetlozar_T._RachevFrank_J._FabozziChristian_Menn2005" class="citation book cs1">Svetlozar T. Rachev; <a href="/wiki/Frank_J._Fabozzi" title="Frank J. Fabozzi">Frank J. Fabozzi</a>; Christian Menn (2005). <i>Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing</i>. <a href="/wiki/John_Wiley_and_Sons" class="mw-redirect" title="John Wiley and Sons">John Wiley and Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471718864" title="Special:BookSources/978-0471718864"><bdi>978-0471718864</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fat-Tailed+and+Skewed+Asset+Return+Distributions%3A+Implications+for+Risk+Management%2C+Portfolio+Selection%2C+and+Option+Pricing&rft.pub=John+Wiley+and+Sons&rft.date=2005&rft.isbn=978-0471718864&rft.au=Svetlozar+T.+Rachev&rft.au=Frank+J.+Fabozzi&rft.au=Christian+Menn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">B. Mandelbrot</a>, <a rel="nofollow" class="external text" href="http://web.williams.edu/Mathematics/sjmiller/public_html/341Fa09/econ/Mandelbroit_VariationCertainSpeculativePrices.pdf">"The variation of certain Speculative Prices"</a>, <i>The Journal of Business</i> 1963</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLucas" class="citation web cs1">Lucas, Bob. <a rel="nofollow" class="external text" href="https://web.sgh.waw.pl/~atoroj/makroekonomia_zaawansowana/lucas76.pdf">"ECONOMETRIC POEICY EVALUATION: A CRITIQUE"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-08-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ECONOMETRIC+POEICY+EVALUATION%3A+A+CRITIQUE&rft.aulast=Lucas&rft.aufirst=Bob&rft_id=https%3A%2F%2Fweb.sgh.waw.pl%2F~atoroj%2Fmakroekonomia_zaawansowana%2Flucas76.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+finance" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_finance&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Nicole_El_Karoui" title="Nicole El Karoui">Nicole El Karoui</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170101051554/http://www.paristechreview.com/2013/09/06/future-financial-mathematics/">"The future of financial mathematics"</a>, <i><a href="/wiki/ParisTech" title="ParisTech">ParisTech</a> Review</i>, 6 September 2013</li> <li><a href="/wiki/Harold_Markowitz" class="mw-redirect" title="Harold Markowitz">Harold Markowitz</a>, "Portfolio Selection", <i><a href="/wiki/The_Journal_of_Finance" title="The Journal of Finance">The Journal of Finance</a></i>, 7, 1952, pp. 77–91</li> <li><a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William F. Sharpe</a>, <i>Investments</i>, Prentice-Hall, 1985</li> <li>Pierre Henry Labordere (2017). “Model-Free Hedging A Martingale Optimal Transport Viewpoint”. Chapman & Hall/ CRC.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete 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title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, 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angel</a></li> <li><a href="/wiki/Asset_(economics)" title="Asset (economics)">Asset (economics)</a></li> <li><a href="/wiki/Asset_allocation" title="Asset allocation">Asset allocation</a></li> <li><a href="/wiki/Bad_debt" title="Bad debt">Bad debt</a></li> <li><a href="/wiki/Bond_(finance)" title="Bond (finance)">Bond (finance)</a></li> <li><a href="/wiki/Bull_(stock_market_speculator)" title="Bull (stock market speculator)">Bull (stock market speculator)</a></li> <li><a href="/wiki/Capital_appreciation" title="Capital appreciation">Asset growth</a></li> <li><a href="/wiki/Capital_asset" title="Capital asset">Capital asset</a></li> <li><a href="/wiki/Capital_management" title="Capital management">Capital management</a></li> <li><a href="/wiki/Capital_structure" title="Capital structure">Capital structure</a></li> <li><a href="/wiki/Climate_finance" title="Climate finance">Climate finance</a></li> <li><a href="/wiki/Computational_finance" title="Computational finance">Computational finance</a></li> <li><a href="/wiki/Corporate_finance" title="Corporate finance">Corporate finance</a></li> <li><a href="/wiki/Cost_of_capital" title="Cost of capital">Cost of capital</a></li> <li><a href="/wiki/Disinvestment" title="Disinvestment">Disinvestment</a></li> <li><a href="/wiki/Diversification_(finance)" title="Diversification (finance)">Diversification (finance)</a></li> <li><a href="/wiki/Divestment" title="Divestment">Divestment</a></li> <li><a href="/wiki/Eco-investing" title="Eco-investing">Eco-investing</a></li> <li><a href="/wiki/Enterprise_risk_management" title="Enterprise risk management">Enterprise risk management</a></li> <li><a href="/wiki/Environmental_finance" title="Environmental finance">Environmental finance</a></li> <li><a href="/wiki/Equity_(finance)" title="Equity (finance)">Equity (finance)</a></li> <li><a href="/wiki/Environmental,_social,_and_governance" title="Environmental, social, and governance">ESG</a></li> <li><a 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title="Financial institution">institutions</a></li> <li><a href="/wiki/Financial_management" title="Financial management">management</a></li> <li><a href="/wiki/Financial_market" title="Financial market">market</a></li> <li><a href="/wiki/Financial_plan" title="Financial plan">plan</a></li> <li><a href="/wiki/Financial_planner" title="Financial planner">planner</a></li> <li><a href="/wiki/Financial_regulation" title="Financial regulation">regulation</a></li> <li><a href="/wiki/Financial_risk" title="Financial risk">risk</a></li> <li><a href="/wiki/Financial_services" title="Financial services">services</a></li> <li><a href="/wiki/Financial_social_work" title="Financial social work">social work</a></li> <li><a href="/wiki/Financial_system" title="Financial system">system</a></li></ul></li> <li><a href="/wiki/Financial_technology" class="mw-redirect" title="Financial technology">Financial technology</a> (Fintech)</li> <li><a href="/wiki/Fundamental_analysis" title="Fundamental analysis">Fundamental analysis</a></li> <li><a href="/wiki/Government_bond" title="Government bond">Government bond</a></li> <li><a href="/wiki/Greater_fool_theory" title="Greater fool theory">Greater fool theory</a></li> <li><a href="/wiki/Growth_investing" title="Growth investing">Growth investing</a></li> <li><a href="/wiki/Growth_stock" title="Growth stock">Growth stock</a></li> <li><a href="/wiki/Hedge_(finance)" title="Hedge (finance)">Hedge (finance)</a></li> <li><a href="/wiki/History_of_banking" title="History of banking">History of banking</a></li> <li><a href="/wiki/History_of_money" title="History of money">History of money</a></li> <li><a href="/wiki/Impact_investing" title="Impact investing">Impact investing</a></li> <li><a href="/wiki/International_finance" title="International finance">International finance</a></li> <li><a href="/wiki/Investment_advisory" title="Investment advisory">Investment advisory</a></li> <li><a href="/wiki/Investment_banking" title="Investment banking">Investment banking</a></li> <li><a href="/wiki/Investment_management" title="Investment management">Investment management</a></li> <li><a href="/wiki/Investment_performance" title="Investment performance">Investment performance</a></li> <li><a href="/wiki/Investor_profile" title="Investor profile">Investor profile</a></li> <li><a href="/wiki/Market_risk" title="Market risk">Market risk</a></li> <li><a class="mw-selflink selflink">Mathematical finance</a></li> <li><a href="/wiki/Mutual_fund" title="Mutual fund">Mutual fund</a></li> <li><a href="/wiki/Over-the-counter_(finance)" title="Over-the-counter (finance)">Over-the-counter</a></li> <li><a href="/wiki/Pension_fund" title="Pension fund">Pension fund</a></li> <li><a href="/wiki/Personal_finance" title="Personal finance">Personal finance</a></li> <li><a href="/wiki/Position_of_trust" title="Position of trust">Position of trust</a></li> <li><a href="/wiki/Public_finance" title="Public finance">Public finance</a></li> <li><a href="/wiki/Quantitative_behavioral_finance" title="Quantitative behavioral finance">Quantitative behavioral finance</a></li> <li><a href="/wiki/Quantum_finance" title="Quantum finance">Quantum finance</a></li> <li><a href="/wiki/Risk-return_spectrum" class="mw-redirect" title="Risk-return spectrum">Risk-return spectrum</a></li> <li><a href="/wiki/Social_finance" title="Social finance">Social finance</a></li> <li><a href="/wiki/Speculation" title="Speculation">Speculation</a></li> <li><a href="/wiki/Statistical_finance" title="Statistical finance">Statistical finance</a></li> <li><a href="/wiki/Stock_exchange" title="Stock exchange">Stock exchange</a></li> <li><a href="/wiki/Stockbroker" title="Stockbroker">Stockbroker</a></li> <li><a href="/wiki/Strategic_financial_management" title="Strategic financial management">Strategic financial management</a></li> <li><a href="/wiki/Statistical_finance" title="Statistical finance">Statistical finance</a></li> <li><a href="/wiki/Stock" title="Stock">Stock</a></li> <li><a href="/wiki/Structured_finance" title="Structured finance">Structured finance</a></li> <li><a href="/wiki/Structured_product" title="Structured product">Structured product</a></li> <li><a href="/wiki/Sustainability" title="Sustainability">Sustainability</a></li> <li><a href="/wiki/Sustainable_Development_Goals" title="Sustainable Development Goals">Sustainable Development Goals</a></li> <li><a href="/wiki/Sustainable_finance" title="Sustainable finance">Sustainable finance</a></li> <li><a href="/wiki/Swap_(finance)" title="Swap (finance)">Swap (finance)</a></li> <li><a href="/wiki/Systematic_risk" title="Systematic risk">Systematic risk</a></li> <li><a href="/wiki/Too_big_to_fail" title="Too big to fail">Too big to fail</a></li> <li><a href="/wiki/Toxic_asset" title="Toxic asset">Toxic asset</a></li> <li><a href="/wiki/Valuation_using_discounted_cash_flows" title="Valuation using discounted cash flows">Valuation using discounted cash flows</a></li> <li><a href="/wiki/Watered_stock" title="Watered stock">Watered stock</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Financial_risk_and_financial_risk_management" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Financial_risk" title="Template:Financial risk"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Financial_risk" title="Template talk:Financial risk"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Financial_risk" title="Special:EditPage/Template:Financial risk"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Financial_risk_and_financial_risk_management" style="font-size:114%;margin:0 4em"><a href="/wiki/Financial_risk" title="Financial risk">Financial risk</a> and <a href="/wiki/Financial_risk_management" title="Financial risk management">financial risk management</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Categories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Credit_risk" title="Credit risk">Credit risk</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Consumer_credit_risk" title="Consumer credit risk">Consumer credit risk</a></li> <li><a href="/wiki/Sovereign_credit_risk" title="Sovereign credit risk">Sovereign credit risk</a></li> <li><a href="/wiki/Settlement_risk" title="Settlement risk">Settlement risk</a></li> <li><a href="/wiki/Default_(finance)" title="Default (finance)">Default risk</a></li> <li><a href="/wiki/Concentration_risk" title="Concentration risk">Concentration risk</a></li> <li><a href="/wiki/Credit_derivative" title="Credit derivative">Credit derivative</a></li> <li><a href="/wiki/Securitization" title="Securitization">Securitization</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Market_risk" title="Market risk">Market risk</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Commodity_risk" title="Commodity risk">Commodity risk</a> (e.g. <a href="/wiki/Volume_risk" title="Volume risk">Volume risk</a>, <a href="/wiki/Basis_risk" title="Basis risk">Basis risk</a>, <a href="/wiki/Shape_risk" title="Shape risk">Shape risk</a>, <a href="/wiki/Holding_period_risk" title="Holding period risk">Holding period risk</a>, <a href="/wiki/Price_area_risk" class="mw-redirect" title="Price area risk">Price area risk</a>)</li> <li><a href="/wiki/Equity_risk" title="Equity risk">Equity risk</a></li> <li><a href="/wiki/Valuation_risk" title="Valuation risk">Valuation risk</a></li> <li><a href="/wiki/Foreign_exchange_risk" title="Foreign exchange risk">FX risk</a></li> <li><a href="/wiki/Margining_risk" title="Margining risk">Margining risk</a></li> <li><a href="/wiki/Interest_rate_risk" title="Interest rate risk">Interest rate risk</a></li> <li><a href="/wiki/Inflation_risk" class="mw-redirect" title="Inflation risk">Inflation risk</a></li> <li><a href="/wiki/Volatility_risk" title="Volatility risk">Volatility risk</a></li> <li><a href="/wiki/Liquidity_risk" title="Liquidity risk">Liquidity risk</a> (e.g. <a href="/wiki/Refinancing_risk" title="Refinancing risk">Refinancing risk</a>, <a href="/wiki/Deposit_risk" title="Deposit risk">Deposit risk</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operational_risk" title="Operational risk">Operational risk</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Operational_risk_management" title="Operational risk management">Operational risk management</a></li> <li><a href="/wiki/Business_risk" class="mw-redirect" title="Business risk">Business risk</a></li> <li><a href="/wiki/Model_risk" title="Model risk">Model risk</a></li> <li><a href="/wiki/Reputational_risk" class="mw-redirect" title="Reputational risk">Reputational risk</a></li> <li><a href="/wiki/Country_risk" title="Country risk">Country risk</a></li> <li><a href="/wiki/Political_risk" title="Political risk">Political risk</a></li> <li><a href="/wiki/Legal_risk" title="Legal risk">Legal risk</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Execution_risk" class="mw-redirect" title="Execution risk">Execution risk</a></li> <li><a href="/wiki/Profit_risk" title="Profit risk">Profit risk</a></li> <li><a href="/wiki/Systemic_risk" title="Systemic risk">Systemic risk</a></li> <li><a href="/wiki/Non-financial_risk" title="Non-financial risk">Non-financial risk</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Financial_risk_modeling" title="Financial risk modeling">Modeling</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arbitrage_pricing_theory" title="Arbitrage pricing theory">Arbitrage pricing theory</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes model</a></li> <li><a href="/wiki/Replicating_portfolio" title="Replicating portfolio">Replicating portfolio</a></li> <li><a href="/wiki/Cashflow_matching" title="Cashflow matching">Cash flow matching</a></li> <li><a href="/wiki/Expected_shortfall" title="Expected shortfall">Conditional Value-at-Risk (CVaR)</a></li> <li><a href="/wiki/Copula_(probability_theory)" class="mw-redirect" title="Copula (probability theory)">Copula</a></li> <li><a href="/wiki/Drawdown_(economics)" title="Drawdown (economics)">Drawdown</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First-hitting-time model</a></li> <li><a href="/wiki/Immunization_(finance)" title="Immunization (finance)">Interest rate immunization</a></li> <li><a href="/wiki/Market_portfolio" title="Market portfolio">Market portfolio</a></li> <li><a href="/wiki/Modern_portfolio_theory" title="Modern portfolio theory">Modern portfolio theory</a></li> <li><a href="/wiki/Omega_ratio" title="Omega ratio">Omega ratio</a></li> <li><a href="/wiki/Risk-adjusted_return_on_capital" title="Risk-adjusted return on capital">RAROC</a></li> <li><a href="/wiki/Risk-free_interest_rate" class="mw-redirect" title="Risk-free interest rate">Risk-free rate</a></li> <li><a href="/wiki/Risk_parity" title="Risk parity">Risk parity</a></li> <li><a href="/wiki/Sharpe_ratio" title="Sharpe ratio">Sharpe ratio</a></li> <li><a href="/wiki/Sortino_ratio" title="Sortino ratio">Sortino ratio</a></li> <li><a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a> (<a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards model</a>)</li> <li><a href="/wiki/Tracking_error" title="Tracking error">Tracking error</a></li> <li><a href="/wiki/Value_at_risk" title="Value at risk">Value-at-Risk (VaR)</a> and extensions (<a href="/wiki/Profit_at_risk" title="Profit at risk">Profit at risk</a>, <a href="/wiki/Margin_at_risk" title="Margin at risk">Margin at risk</a>, <a href="/wiki/Liquidity_at_risk" title="Liquidity at risk">Liquidity at risk</a>, <a href="/wiki/Cash_flow_at_risk" class="mw-redirect" title="Cash flow at risk">Cash flow at risk</a>, <a href="/wiki/Earnings_at_risk" title="Earnings at risk">Earnings at risk</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asset_allocation" title="Asset allocation">Asset allocation</a></li> <li><a href="/wiki/Asset_and_liability_management" title="Asset and liability management">Asset and liability management</a></li> <li><a href="/wiki/Asset_pricing" title="Asset pricing">Asset pricing</a></li> <li><a href="/wiki/Bad_debt" title="Bad debt">Bad debt</a></li> <li><a href="/wiki/Capital_asset" title="Capital asset">Capital asset</a></li> <li><a href="/wiki/Capital_structure" title="Capital structure">Capital structure</a></li> <li><a href="/wiki/Corporate_finance" title="Corporate finance">Corporate finance</a></li> <li><a href="/wiki/Cost_of_capital" title="Cost of capital">Cost of capital</a></li> <li><a href="/wiki/Diversification_(finance)" title="Diversification (finance)">Diversification</a></li> <li><a href="/wiki/Economic_bubble" title="Economic bubble">Economic bubble</a></li> <li><a href="/wiki/Enterprise_value" title="Enterprise value">Enterprise value</a></li> <li><a href="/wiki/Environmental,_social,_and_governance" title="Environmental, social, and governance">ESG</a></li> <li><a href="/wiki/Exchange_traded_fund" class="mw-redirect" title="Exchange traded fund">Exchange traded fund</a></li> <li><a href="/wiki/Expected_return" title="Expected return">Expected return</a></li> <li><a href="/wiki/Finance" title="Finance">Financial</a> <ul><li><a href="/wiki/Financial_adviser" title="Financial adviser">adviser</a></li> <li><a href="/wiki/Financial_analysis" title="Financial analysis">analysis</a></li> <li><a href="/wiki/Financial_analyst" title="Financial analyst">analyst</a></li> <li><a href="/wiki/Financial_asset" title="Financial asset">asset</a></li> <li><a href="/wiki/Financial_betting" title="Financial betting">betting</a></li> <li><a href="/wiki/Financial_crime" title="Financial crime">crime</a></li> <li><a href="/wiki/Financial_engineering" title="Financial engineering">engineering</a></li> <li><a href="/wiki/Financial_law" title="Financial law">law</a></li> <li><a href="/wiki/Financial_risk" title="Financial risk">risk</a></li> <li><a href="/wiki/Financial_social_work" title="Financial social work">social work</a></li></ul></li> <li><a href="/wiki/Fundamental_analysis" title="Fundamental analysis">Fundamental analysis</a></li> <li><a href="/wiki/Growth_investing" title="Growth investing">Growth investing</a></li> <li><a href="/wiki/Hazard" title="Hazard">Hazard</a></li> <li><a href="/wiki/Hedge_(finance)" title="Hedge (finance)">Hedge</a></li> <li><a href="/wiki/Investment_management" title="Investment management">Investment management</a></li> <li><a href="/wiki/Risk" title="Risk">Risk</a></li> <li><a href="/wiki/Risk_pool" title="Risk pool">Risk pool</a></li> <li><a href="/wiki/Risk_of_ruin" title="Risk of ruin">Risk of ruin</a></li> <li><a href="/wiki/Systematic_risk" title="Systematic risk">Systematic risk</a></li> <li><a class="mw-selflink selflink">Mathematical finance</a></li> <li><a href="/wiki/Moral_hazard" title="Moral hazard">Moral hazard</a></li> <li><a href="/wiki/Risk-return_spectrum" class="mw-redirect" title="Risk-return spectrum">Risk-return spectrum</a></li> <li><a href="/wiki/Speculation" title="Speculation">Speculation</a></li> <li><a href="/wiki/Speculative_attack" title="Speculative attack">Speculative attack</a></li> <li><a href="/wiki/Statistical_finance" title="Statistical finance">Statistical finance</a></li> <li><a href="/wiki/Strategic_financial_management" title="Strategic financial management">Strategic financial management</a></li> <li><a href="/wiki/Stress_test_(financial)" title="Stress test (financial)">Stress test (financial)</a></li> <li><a href="/wiki/Structured_finance" title="Structured finance">Structured finance</a></li> <li><a href="/wiki/Structured_product" title="Structured product">Structured product</a></li> <li><a href="/wiki/Systemic_risk" title="Systemic risk">Systemic risk</a></li> <li><a href="/wiki/Toxic_asset" title="Toxic asset">Toxic asset</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Financial_economics" title="Financial economics">Financial economics</a></li> <li><a href="/wiki/Investment_management" title="Investment management">Investment management</a></li> <li><a class="mw-selflink selflink">Mathematical finance</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"></div><div role="navigation" class="navbox" aria-labelledby="Economics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Economics" title="Template:Economics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Economics" title="Template talk:Economics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Economics" title="Special:EditPage/Template:Economics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Economics" style="font-size:114%;margin:0 4em"><a href="/wiki/Economics" title="Economics">Economics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Economics#Theoretical_research" title="Economics">Theoretical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Microeconomics" title="Microeconomics">Microeconomics</a> <ul><li><a href="/wiki/Decision_theory" title="Decision theory">Decision theory</a></li> <li><a href="/wiki/Price_theory" class="mw-redirect" title="Price theory">Price theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Contract_theory" title="Contract theory">Contract theory</a></li> <li><a href="/wiki/Mechanism_design" title="Mechanism design">Mechanism design</a></li></ul></li> <li><a href="/wiki/Macroeconomics" title="Macroeconomics">Macroeconomics</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Complexity_economics" title="Complexity economics">Complexity economics</a></li> <li><a href="/wiki/Computational_economics" title="Computational economics">Computational economics</a> <ul><li><a href="/wiki/Agent-based_computational_economics" title="Agent-based computational economics">Agent-based computational economics</a></li></ul></li> <li><a href="/wiki/Behavioral_economics" title="Behavioral economics">Behavioral economics</a></li> <li><a href="/wiki/Pluralism_in_economics" title="Pluralism in economics">Pluralism in economics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Economics#Empirical_research" title="Economics">Empirical</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a> <ul><li><a href="/wiki/Economic_statistics" title="Economic statistics">Economic statistics</a></li></ul></li> <li><a href="/wiki/Experimental_economics" title="Experimental economics">Experimental economics</a></li> <li><a href="/wiki/Economic_history" title="Economic history">Economic history</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_economics" title="Applied economics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <div class="excerpt-block"><div class="excerpt"> <ul><li><a href="/wiki/Agricultural_economics" title="Agricultural economics">Agricultural</a></li> <li><a href="/wiki/Behavioral_economics" title="Behavioral economics">Behavioral</a></li> <li><a href="/wiki/Business_economics" title="Business economics">Business</a></li> <li><a href="/wiki/Cultural_economics" title="Cultural economics">Cultural</a></li> <li><a href="/wiki/Democracy_and_economic_growth" title="Democracy and economic growth">Democracy</a></li> <li><a href="/wiki/Demographic_economics" title="Demographic economics">Demographic</a></li> <li><a href="/wiki/Development_economics" title="Development economics">Development</a></li> <li><a href="/wiki/Economics_of_digitization" title="Economics of digitization">Digitization</a></li> <li><a href="/wiki/Ecological_economics" title="Ecological economics">Ecological</a></li> <li><a href="/wiki/Education_economics" title="Education economics">Education</a></li> <li><a href="/wiki/Engineering_economics" title="Engineering economics">Engineering</a></li> <li><a href="/wiki/Environmental_economics" title="Environmental economics">Environmental</a></li> <li><a href="/wiki/Evolutionary_economics" title="Evolutionary economics">Evolutionary</a></li> <li><a href="/wiki/Expeditionary_economics" title="Expeditionary economics">Expeditionary</a></li> <li><a href="/wiki/Feminist_economics" title="Feminist economics">Feminist</a></li> <li><a href="/wiki/Financial_economics" title="Financial economics">Financial</a></li> <li><a href="/wiki/Economic_geography" title="Economic geography">Geographical</a></li> <li><a href="/wiki/Happiness_economics" title="Happiness economics">Happiness</a></li> <li><a href="/wiki/Health_economics" title="Health economics">Health</a></li> <li><a href="/wiki/Economic_history" title="Economic history">Historical</a></li> <li><a href="/wiki/Humanistic_economics" title="Humanistic economics">Humanistic</a></li> <li><a href="/wiki/Industrial_organization" title="Industrial organization">Industrial organization</a></li> <li><a href="/wiki/Information_economics" title="Information economics">Information</a></li> <li><a href="/wiki/Institutional_economics" title="Institutional economics">Institutional</a></li> <li><a href="/wiki/Knowledge_economy" title="Knowledge economy">Knowledge</a></li> <li><a href="/wiki/Labour_economics" title="Labour economics">Labour</a></li> <li><a href="/wiki/Law_and_economics" title="Law and economics">Law</a></li> <li><a href="/wiki/Managerial_economics" title="Managerial economics">Managerial</a></li> <li><a href="/wiki/Monetary_economics" title="Monetary economics">Monetary</a></li> <li><a href="/wiki/Natural_resource_economics" title="Natural resource economics">Natural resource</a></li> <li><a href="/wiki/Organizational_economics" title="Organizational economics">Organizational</a></li> <li><a href="/wiki/Economics_of_participation" title="Economics of participation">Participation</a></li> <li><a href="/wiki/Personnel_economics" title="Personnel economics">Personnel</a></li> <li><a href="/wiki/Economic_planning" title="Economic planning">Planning</a></li> <li><a href="/wiki/Economic_policy" title="Economic policy">Policy</a></li> <li><a href="/wiki/Public_economics" title="Public economics">Public</a></li> <li><a href="/wiki/Public_choice" title="Public choice">Public choice</a> / <a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice theory</a></li> <li><a href="/wiki/Regional_economics" title="Regional economics">Regional</a></li> <li><a href="/wiki/Rural_economics" title="Rural economics">Rural</a></li> <li><a href="/wiki/Service_economy" title="Service economy">Service</a></li> <li><a href="/wiki/Socioeconomics" class="mw-redirect" title="Socioeconomics">Socio</a></li> <li><a href="/wiki/Economic_sociology" title="Economic sociology">Sociological</a></li> <li><a href="/wiki/Solidarity_economy" title="Solidarity economy">Solidarity</a></li> <li><a href="/wiki/Economic_statistics" title="Economic statistics">Statistics</a></li> <li><a href="/wiki/Urban_economics" title="Urban economics">Urban</a></li> <li><a href="/wiki/Welfare_economics" title="Welfare economics">Welfare</a></li></ul></div></div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Schools_of_economic_thought" title="Schools of economic thought">Schools</a><br />(<a href="/wiki/History_of_economic_thought" title="History of economic thought">history</a>)<br /></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mainstream_economics" title="Mainstream economics">Mainstream</a></li> <li><a href="/wiki/Heterodox_economics" title="Heterodox economics">Heterodox</a></li> <li><a href="/wiki/American_School_(economics)" title="American School (economics)">American (National)</a></li> <li><a href="/wiki/Ancient_economic_thought" title="Ancient economic thought">Ancient thought</a></li> <li><a href="/wiki/Anarchist_economics" class="mw-redirect" title="Anarchist economics">Anarchist</a> <ul><li><a href="/wiki/Mutualism_(economic_theory)" title="Mutualism (economic theory)">Mutualism</a></li></ul></li> <li><a href="/wiki/Austrian_School" class="mw-redirect" title="Austrian School">Austrian</a></li> <li><a href="/wiki/Behavioral_economics" title="Behavioral economics">Behavioral</a></li> <li><a href="/wiki/Buddhist_economics" title="Buddhist economics">Buddhist</a></li> <li><a href="/wiki/Chartalism" title="Chartalism">Chartalism</a> <ul><li><a href="/wiki/Modern_monetary_theory" title="Modern monetary theory">Modern monetary theory</a></li></ul></li> <li><a href="/wiki/Chicago_school_of_economics" title="Chicago school of economics">Chicago</a></li> <li><a href="/wiki/Classical_economics" title="Classical economics">Classical</a></li> <li><a href="/wiki/Critique_of_political_economy" title="Critique of political economy">Critique of political economy</a></li> <li><a href="/wiki/Economic_democracy" title="Economic democracy">Democratic</a></li> <li><a href="/wiki/Disequilibrium_macroeconomics" title="Disequilibrium macroeconomics">Disequilibrium</a></li> <li><a href="/wiki/Ecological_economics" title="Ecological economics">Ecological</a></li> <li><a href="/wiki/Evolutionary_economics" title="Evolutionary economics">Evolutionary</a></li> <li><a href="/wiki/Feminist_economics" title="Feminist economics">Feminist</a></li> <li><a href="/wiki/Georgism" title="Georgism">Georgism</a></li> <li><a href="/wiki/Happiness_economics" title="Happiness economics">Happiness</a></li> <li><a href="/wiki/Historical_school_of_economics" title="Historical school of economics">Historical</a></li> <li><a href="/wiki/Humanistic_economics" title="Humanistic economics">Humanistic</a></li> <li><a href="/wiki/Institutional_economics" title="Institutional economics">Institutional</a></li> <li><a href="/wiki/Keynesian_economics" title="Keynesian economics">Keynesian</a> <ul><li><a href="/wiki/Neo-Keynesian_economics" class="mw-redirect" title="Neo-Keynesian economics">Neo-</a> (<a href="/wiki/Neoclassical_synthesis" title="Neoclassical synthesis">neoclassical–Keynesian synthesis</a>)</li> <li><a href="/wiki/New_Keynesian_economics" title="New Keynesian economics">New</a></li> <li><a href="/wiki/Post-Keynesian_economics" title="Post-Keynesian economics">Post-</a> <ul><li><a href="/wiki/Monetary_circuit_theory" title="Monetary circuit theory">Circuitism</a></li></ul></li></ul></li> <li><a href="/wiki/Malthusianism" title="Malthusianism">Malthusianism</a></li> <li><a href="/wiki/Marginalism" title="Marginalism">Marginalism</a></li> <li><a href="/wiki/Marxian_economics" title="Marxian economics">Marxian</a> <ul><li><a href="/wiki/Neo-Marxian_economics" class="mw-redirect" title="Neo-Marxian economics">Neo-</a></li></ul></li> <li><a href="/wiki/Mercantilism" title="Mercantilism">Mercantilism</a></li> <li><a href="/wiki/Mixed_economy" title="Mixed economy">Mixed</a></li> <li><a href="/wiki/Neoclassical_economics" title="Neoclassical economics">Neoclassical</a> <ul><li><a href="/wiki/Lausanne_School" title="Lausanne School">Lausanne</a></li></ul></li> <li><a href="/wiki/New_classical_macroeconomics" title="New classical macroeconomics">New classical</a> <ul><li><a href="/wiki/Real_business-cycle_theory" title="Real business-cycle theory">Real business-cycle theory</a></li></ul></li> <li><a href="/wiki/New_institutional_economics" title="New institutional economics">New institutional</a></li> <li><a href="/wiki/Physiocracy" title="Physiocracy">Physiocracy</a></li> <li><a href="/wiki/Socialist_economics" title="Socialist economics">Socialist</a></li> <li><a href="/wiki/Stockholm_School_(economics)" title="Stockholm School (economics)">Stockholm</a></li> <li><a href="/wiki/Supply-side_economics" title="Supply-side economics">Supply-side</a></li> <li><a href="/wiki/Thermoeconomics" title="Thermoeconomics">Thermo</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Economist" title="Economist">Economists</a><br /></div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernard_de_Mandeville" class="mw-redirect" title="Bernard de Mandeville">de Mandeville</a></li> <li><a href="/wiki/Fran%C3%A7ois_Quesnay" title="François Quesnay">Quesnay</a></li> <li><a href="/wiki/Adam_Smith" title="Adam Smith">Smith</a></li> <li><a href="/wiki/Thomas_Robert_Malthus" title="Thomas Robert Malthus">Malthus</a></li> <li><a href="/wiki/Jean-Baptiste_Say" title="Jean-Baptiste Say">Say</a></li> <li><a href="/wiki/David_Ricardo" title="David Ricardo">Ricardo</a></li> <li><a href="/wiki/Johann_Heinrich_von_Th%C3%BCnen" title="Johann Heinrich von Thünen">von Thünen</a></li> <li><a href="/wiki/Friedrich_List" title="Friedrich List">List</a></li> <li><a href="/wiki/Fr%C3%A9d%C3%A9ric_Bastiat" title="Frédéric Bastiat">Bastiat</a></li> <li><a href="/wiki/Antoine_Augustin_Cournot" title="Antoine Augustin Cournot">Cournot</a></li> <li><a href="/wiki/John_Stuart_Mill" title="John Stuart Mill">Mill</a></li> <li><a href="/wiki/Hermann_Heinrich_Gossen" title="Hermann Heinrich Gossen">Gossen</a></li> <li><a href="/wiki/Karl_Marx" title="Karl Marx">Marx</a></li> <li><a href="/wiki/L%C3%A9on_Walras" title="Léon Walras">Walras</a></li> <li><a href="/wiki/William_Stanley_Jevons" title="William Stanley Jevons">Jevons</a></li> <li><a href="/wiki/Henry_George" title="Henry George">George</a></li> <li><a href="/wiki/Carl_Menger" title="Carl Menger">Menger</a></li> <li><a href="/wiki/Alfred_Marshall" title="Alfred Marshall">Marshall</a></li> <li><a href="/wiki/Francis_Ysidro_Edgeworth" title="Francis Ysidro Edgeworth">Edgeworth</a></li> <li><a href="/wiki/John_Bates_Clark" title="John Bates Clark">Clark</a></li> <li><a href="/wiki/Vilfredo_Pareto" title="Vilfredo Pareto">Pareto</a></li> <li><a href="/wiki/Eugen_von_B%C3%B6hm-Bawerk" title="Eugen von Böhm-Bawerk">von Böhm-Bawerk</a></li> <li><a href="/wiki/Friedrich_von_Wieser" title="Friedrich von Wieser">von Wieser</a></li> <li><a href="/wiki/Thorstein_Veblen" title="Thorstein Veblen">Veblen</a></li> <li><a href="/wiki/Irving_Fisher" title="Irving Fisher">Fisher</a></li> <li><a href="/wiki/Arthur_Cecil_Pigou" title="Arthur Cecil Pigou">Pigou</a></li> <li><a href="/wiki/Eli_Heckscher" title="Eli Heckscher">Heckscher</a></li> <li><a href="/wiki/Ludwig_von_Mises" title="Ludwig von Mises">von Mises</a></li> <li><a href="/wiki/Joseph_Schumpeter" title="Joseph Schumpeter">Schumpeter</a></li> <li><a href="/wiki/John_Maynard_Keynes" title="John Maynard Keynes">Keynes</a></li> <li><a href="/wiki/Frank_Knight" title="Frank Knight">Knight</a></li> <li><a href="/wiki/Karl_Polanyi" title="Karl Polanyi">Polanyi</a></li> <li><a href="/wiki/Ragnar_Frisch" title="Ragnar Frisch">Frisch</a></li> <li><a href="/wiki/Piero_Sraffa" title="Piero Sraffa">Sraffa</a></li> <li><a href="/wiki/Gunnar_Myrdal" title="Gunnar Myrdal">Myrdal</a></li> <li><a href="/wiki/Friedrich_Hayek" title="Friedrich Hayek">Hayek</a></li> <li><a href="/wiki/Micha%C5%82_Kalecki" title="Michał Kalecki">Kalecki</a></li> <li><a href="/wiki/Wilhelm_R%C3%B6pke" title="Wilhelm Röpke">Röpke</a></li> <li><a href="/wiki/Simon_Kuznets" title="Simon Kuznets">Kuznets</a></li> <li><a href="/wiki/Jan_Tinbergen" title="Jan Tinbergen">Tinbergen</a></li> <li><a href="/wiki/Joan_Robinson" title="Joan Robinson">Robinson</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/John_Hicks" title="John Hicks">Hicks</a></li> <li><a href="/wiki/Oskar_R._Lange" title="Oskar R. Lange">Lange</a></li> <li><a href="/wiki/Wassily_Leontief" title="Wassily Leontief">Leontief</a></li> <li><a href="/wiki/John_Kenneth_Galbraith" title="John Kenneth Galbraith">Galbraith</a></li> <li><a href="/wiki/Tjalling_Koopmans" title="Tjalling Koopmans">Koopmans</a></li> <li><a href="/wiki/E._F._Schumacher" title="E. F. Schumacher">Schumacher</a></li> <li><a href="/wiki/Milton_Friedman" title="Milton Friedman">Friedman</a></li> <li><a href="/wiki/Paul_Samuelson" title="Paul Samuelson">Samuelson</a></li> <li><a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Simon</a></li> <li><a href="/wiki/James_M._Buchanan" title="James M. 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