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<div class="vector-toc-text"> 1.1 Present value, expectation and utility </div> </a> <ul id="toc-Present_value,_expectation_and_utility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arbitrage-free_pricing_and_equilibrium" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arbitrage-free_pricing_and_equilibrium"> <div class="vector-toc-text"> 1.2 Arbitrage-free pricing and equilibrium </div> </a> <ul id="toc-Arbitrage-free_pricing_and_equilibrium-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-State_prices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#State_prices"> <div class="vector-toc-text"> 1.3 State prices </div> </a> <ul id="toc-State_prices-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Resultant_models" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Resultant_models"> <div class="vector-toc-text"> 2 Resultant models </div> </a> <button aria-controls="toc-Resultant_models-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Resultant models subsection </button> <ul id="toc-Resultant_models-sublist" class="vector-toc-list"> <li id="toc-Certainty" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Certainty"> <div class="vector-toc-text"> 2.1 Certainty </div> </a> <ul id="toc-Certainty-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uncertainty" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uncertainty"> <div class="vector-toc-text"> 2.2 Uncertainty </div> </a> <ul id="toc-Uncertainty-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> 3 Extensions </div> </a> <button aria-controls="toc-Extensions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Extensions subsection </button> <ul id="toc-Extensions-sublist" class="vector-toc-list"> <li id="toc-Portfolio_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Portfolio_theory"> <div class="vector-toc-text"> 3.1 Portfolio theory </div> </a> <ul id="toc-Portfolio_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivative_pricing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivative_pricing"> <div class="vector-toc-text"> 3.2 Derivative pricing </div> </a> <ul id="toc-Derivative_pricing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Corporate_finance_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Corporate_finance_theory"> <div class="vector-toc-text"> 3.3 Corporate finance theory </div> </a> <ul id="toc-Corporate_finance_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Financial_markets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Financial_markets"> <div class="vector-toc-text"> 3.4 Financial markets </div> </a> <ul id="toc-Financial_markets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Challenges_and_criticism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Challenges_and_criticism"> <div class="vector-toc-text"> 4 Challenges and criticism </div> </a> <button aria-controls="toc-Challenges_and_criticism-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Challenges and criticism subsection </button> <ul id="toc-Challenges_and_criticism-sublist" class="vector-toc-list"> <li id="toc-Departures_from_normality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Departures_from_normality"> <div class="vector-toc-text"> 4.1 Departures from normality </div> </a> <ul id="toc-Departures_from_normality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Departures_from_rationality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Departures_from_rationality"> <div class="vector-toc-text"> 4.2 Departures from rationality </div> </a> <ul id="toc-Departures_from_rationality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 5 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical_notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historical_notes"> <div class="vector-toc-text"> 6 Historical notes </div> </a> <ul id="toc-Historical_notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 7 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 8 Bibliography </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 9 External links </div> </a> <ul id="toc-External_links-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 vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Financial economics</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 27 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-27" aria-hidden="true" > 27 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%82%D8%AA%D8%B5%D8%A7%D8%AF_%D9%85%D8%A7%D9%84%D9%8A" title="اقتصاد مالي – Arabic" lang="ar" hreflang="ar" data-title="اقتصاد مالي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Maliyy%C9%99_iqtisadiyyat%C4%B1" title="Maliyyə iqtisadiyyatı – Azerbaijani" lang="az" hreflang="az" data-title="Maliyyə iqtisadiyyatı" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D0%B8%D0%BD%D0%B0%D0%BD%D1%81%D0%BE%D0%B2%D0%B0_%D0%B8%D0%BA%D0%BE%D0%BD%D0%BE%D0%BC%D0%B8%D0%BA%D0%B0" title="Финансова икономика – Bulgarian" lang="bg" hreflang="bg" data-title="Финансова икономика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Finansiel_%C3%B8konomi" title="Finansiel økonomi – Danish" lang="da" hreflang="da" data-title="Finansiel økonomi" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Finanz%C3%B6konomik" title="Finanzökonomik – German" lang="de" hreflang="de" data-title="Finanzökonomik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Finantsmajandus" title="Finantsmajandus – Estonian" lang="et" hreflang="et" data-title="Finantsmajandus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%82%D8%AA%D8%B5%D8%A7%D8%AF_%D9%85%D8%A7%D9%84%DB%8C" title="اقتصاد مالی – Persian" lang="fa" hreflang="fa" data-title="اقتصاد مالی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B8%88%EC%9C%B5%EA%B2%BD%EC%A0%9C%ED%95%99" title="금융경제학 – Korean" lang="ko" hreflang="ko" data-title="금융경제학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%AB%D5%B6%D5%A1%D5%B6%D5%BD%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D5%B6%D5%BF%D5%A5%D5%BD%D5%A1%D5%A3%D5%AB%D5%BF%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ֆինանսական տնտեսագիտություն – Armenian" lang="hy" hreflang="hy" data-title="Ֆինանսական տնտեսագիտություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%85%E0%A4%B0%E0%A5%8D%E0%A4%A5%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="वित्तीय अर्थशास्त्र – Hindi" lang="hi" hreflang="hi" data-title="वित्तीय अर्थशास्त्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Financal_ekonomiko" title="Financal ekonomiko – Ido" lang="io" hreflang="io" data-title="Financal ekonomiko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ekonomika_keuangan" title="Ekonomika keuangan – Indonesian" lang="id" hreflang="id" data-title="Ekonomika keuangan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Economia_finanziaria" title="Economia finanziaria – Italian" lang="it" hreflang="it" data-title="Economia finanziaria" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B9%E0%B2%A3%E0%B2%95%E0%B2%BE%E0%B2%B8%E0%B2%BF%E0%B2%A8_%E0%B2%85%E0%B2%B0%E0%B3%8D%E0%B2%A5%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0" title="ಹಣಕಾಸಿನ ಅರ್ಥಶಾಸ್ತ್ರ – Kannada" lang="kn" hreflang="kn" data-title="ಹಣಕಾಸಿನ ಅರ್ಥಶಾಸ್ತ್ರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Finans%C5%B3_ekonomika" title="Finansų ekonomika – Lithuanian" lang="lt" hreflang="lt" data-title="Finansų ekonomika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%87%91%E8%9E%8D%E7%B5%8C%E6%B8%88%E5%AD%A6" title="金融経済学 – Japanese" lang="ja" hreflang="ja" data-title="金融経済学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Finansiell_%C3%B8konomi" title="Finansiell økonomi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Finansiell økonomi" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%85%D8%A7%D9%84%D9%8A_%D8%A7%D9%82%D8%AA%D8%B5%D8%A7%D8%AF" title="مالي اقتصاد – Pashto" lang="ps" hreflang="ps" data-title="مالي اقتصاد" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ekonomia_finansowa" title="Ekonomia finansowa – Polish" lang="pl" hreflang="pl" data-title="Ekonomia finansowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Economia_financeira" title="Economia financeira – Portuguese" lang="pt" hreflang="pt" data-title="Economia financeira" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B8%E0%B7%96%E0%B6%BD%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B6%B8%E0%B6%BA_%E0%B6%86%E0%B6%BB%E0%B7%8A%E0%B6%AE%E0%B7%92%E0%B6%9A_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80" title="මූල්‍යමය ආර්ථික විද්‍යාව – Sinhala" lang="si" hreflang="si" data-title="මූල්‍යමය ආර්ථික විද්‍යාව" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rahoitustaloustiede" title="Rahoitustaloustiede – Finnish" lang="fi" hreflang="fi" data-title="Rahoitustaloustiede" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Finansiell_ekonomi" title="Finansiell ekonomi – Swedish" lang="sv" hreflang="sv" data-title="Finansiell ekonomi" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Finansal_ekonomi" title="Finansal ekonomi – Turkish" lang="tr" hreflang="tr" data-title="Finansal ekonomi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kinh_t%E1%BA%BF_h%E1%BB%8Dc_t%C3%A0i_ch%C3%ADnh" title="Kinh tế học tài chính – Vietnamese" lang="vi" hreflang="vi" data-title="Kinh tế học tài chính" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%87%91%E8%9E%8D%E7%B6%93%E6%BF%9F%E5%AD%B8" 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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:Economics_sidebar" title="Template:Economics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Economics_sidebar" title="Template talk:Economics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Economics_sidebar" title="Special:EditPage/Template:Economics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Financial economics is the branch of <a href="/wiki/Economics" title="Economics">economics</a> characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on both sides of a trade".<a href="#cite_note-stanford1-1">[1]</a> Its concern is thus the interrelation of financial variables, such as share prices, <a href="/wiki/Interest_rate" title="Interest rate">interest rates</a> and exchange rates, as opposed to those concerning the <a href="/wiki/Real_economy" title="Real economy">real economy</a>. It has two main areas of focus:<a href="#cite_note-Miller-2">[2]</a> <a href="/wiki/Asset_pricing" title="Asset pricing">asset pricing</a> and <a href="/wiki/Corporate_finance" title="Corporate finance">corporate finance</a>; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital. It thus provides the theoretical underpinning for much of <a href="/wiki/Finance" title="Finance">finance</a>. The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment".<a href="#cite_note-3">[3]</a><a href="#cite_note-Fama_and_Miller-4">[4]</a> It therefore centers on decision making under uncertainty in the context of the financial markets, and the resultant <a href="/wiki/Economic_model" title="Economic model">economic</a> and <a href="/wiki/Financial_model" class="mw-redirect" title="Financial model">financial models</a> and principles, and is concerned with deriving testable or policy implications from acceptable assumptions. It thus also includes a formal study of the <a href="/wiki/Financial_markets" class="mw-redirect" title="Financial markets">financial markets</a> themselves, especially <a href="/wiki/Market_microstructure" title="Market microstructure">market microstructure</a> and <a href="/wiki/Financial_regulation" title="Financial regulation">market regulation</a>. It is built on the foundations of <a href="/wiki/Microeconomics" title="Microeconomics">microeconomics</a> and <a href="/wiki/Decision_theory" title="Decision theory">decision theory</a>. <a href="/wiki/Financial_econometrics" title="Financial econometrics">Financial econometrics</a> is the branch of financial economics that uses <a href="/wiki/Econometrics" title="Econometrics">econometric</a> techniques to parameterise the relationships identified. <a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a> is related in that it will derive and extend the mathematical or numerical models suggested by financial economics. Whereas financial economics has a primarily microeconomic focus, <a href="/wiki/Monetary_economics" title="Monetary economics">monetary economics</a> is primarily <a href="/wiki/Macroeconomic" class="mw-redirect" title="Macroeconomic">macroeconomic</a> in nature. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Underlying_economics">Underlying economics</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=1" title="Edit section: Underlying economics">edit</a>]</div> <table class="wikitable floatright" width="250"> <tbody><tr style="text-align:center;"> <td>Fundamental valuation equation <a href="#cite_note-Cochrane_&_Culp-5">[5]</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Price_{j}=\sum _{s}(p_{s}Y_{s}X_{sj})/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>r</mi> <mi>i</mi> <mi>c</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Price_{j}=\sum _{s}(p_{s}Y_{s}X_{sj})/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6ce80cde546ce3a614782b0ee153783b6ebf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.202ex; height:5.509ex;" alt="{\displaystyle Price_{j}=\sum _{s}(p_{s}Y_{s}X_{sj})/r}"> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{s}(q_{s}X_{sj})/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{s}(q_{s}X_{sj})/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5175be8a413209f46ebe3d090855394390033664" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.474ex; height:5.509ex;" alt="{\displaystyle =\sum _{s}(q_{s}X_{sj})/r}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{s}p_{s}X_{sj}{\tilde {m}}_{s}=E[X_{s}{\tilde {m}}_{s}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{s}p_{s}X_{sj}{\tilde {m}}_{s}=E[X_{s}{\tilde {m}}_{s}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772024175a0be2129a6b4d8c2a8ce2273a5d0ebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.156ex; height:5.509ex;" alt="{\displaystyle =\sum _{s}p_{s}X_{sj}{\tilde {m}}_{s}=E[X_{s}{\tilde {m}}_{s}]}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{s}\pi _{s}X_{sj}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{s}\pi _{s}X_{sj}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6452aca5730b854f4dcb42061e5ce11e36ceb1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.129ex; height:5.509ex;" alt="{\displaystyle =\sum _{s}\pi _{s}X_{sj}}"></dd></dl></dd></dl> Four equivalent formulations,<a href="#cite_note-Rubinstein-6">[6]</a> where: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"> is the asset or security</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> are the various states</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> is the risk-free return</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{sj}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{sj}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2e49b71bd333d783c587ccc799b2169b5cac15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.605ex; height:2.843ex;" alt="{\displaystyle X_{sj}}"> dollar payoffs in each state</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8201eb39454219ddec719cdb8b3ffd2b3be2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}"> a subjective, personal probability assigned to the state; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{s}p_{s}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{s}p_{s}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc58c68c8da5ffba2e7012b273558ac72acff627" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.176ex; height:5.509ex;" alt="{\displaystyle \sum _{s}p_{s}=1}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6802614359e5b2a13d3dc8cee4640865ae03d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.354ex; height:2.509ex;" alt="{\displaystyle Y_{s}}"> risk aversion factors by state, normalized s.t. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{s}q_{s}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{s}q_{s}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/755d8fb5a4614c4f946ac0375d411b2ae552aa89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.043ex; height:5.509ex;" alt="{\displaystyle \sum _{s}q_{s}=1}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{s}\equiv p_{s}Y_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{s}\equiv p_{s}Y_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed4ee79969b11f288e17c26dbe3694ecb9e820e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.665ex; height:2.509ex;" alt="{\displaystyle q_{s}\equiv p_{s}Y_{s}}">, risk neutral probabilities</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {m}}\equiv Y/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>≡</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {m}}\equiv Y/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e868a8f26b6721ed80efbc9dfac885120d98f1ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.123ex; height:2.843ex;" alt="{\displaystyle {\tilde {m}}\equiv Y/r}"> the stochastic discount factor</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{s}=q_{s}/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{s}=q_{s}/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c7965e099282e3af6c9fcfb5c9d7f591143245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.678ex; height:2.843ex;" alt="{\displaystyle \pi _{s}=q_{s}/r}"> state prices; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{s}\pi _{s}=1/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{s}\pi _{s}=1/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbe51b4723526db5a789906184d84b0c371d7a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.542ex; height:5.509ex;" alt="{\displaystyle \sum _{s}\pi _{s}=1/r}"></dd></dl> </td></tr></tbody></table> Financial economics studies how <a href="/wiki/Homo_economicus" title="Homo economicus">rational investors</a> would apply <a href="/wiki/Decision_theory" title="Decision theory">decision theory</a> to <a href="/wiki/Investment_management" title="Investment management">investment management</a>. The subject is thus built on the foundations of <a href="/wiki/Microeconomics" title="Microeconomics">microeconomics</a> and derives several key results for the application of <a href="/wiki/Decision_making" class="mw-redirect" title="Decision making">decision making</a> under uncertainty to the <a href="/wiki/Financial_market" title="Financial market">financial markets</a>. The underlying economic logic yields the <a href="/wiki/Fundamental_theorem_of_asset_pricing" title="Fundamental theorem of asset pricing">fundamental theorem of asset pricing</a>, which gives the conditions for <a href="/wiki/Arbitrage" title="Arbitrage">arbitrage</a>-free asset pricing.<a href="#cite_note-Rubinstein-6">[6]</a><a href="#cite_note-Cochrane_&_Culp-5">[5]</a> The various "fundamental" valuation formulae result directly. <div class="mw-heading mw-heading3"><h3 id="Present_value,_expectation_and_utility">Present value, expectation and utility</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=2" title="Edit section: Present value, expectation and utility">edit</a>]</div> Underlying all of financial economics are the concepts of <a href="/wiki/Present_value" title="Present value">present value</a> and <a href="/wiki/Expected_value" title="Expected value">expectation</a>.<a href="#cite_note-Rubinstein-6">[6]</a> Calculating their present value, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{sj}/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{sj}/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8010ad855d7e0cd7bdc5688fa5034ff2072de48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.816ex; height:3.009ex;" alt="{\displaystyle X_{sj}/r}"> in the first formula, allows the decision maker to aggregate the <a href="/wiki/Cashflow" class="mw-redirect" title="Cashflow">cashflows</a> (or other returns) to be produced by the asset in the future to a single value at the date in question, and to thus more readily compare two opportunities; this concept is then the starting point for financial decision making. <a href="#cite_note-10">[note 1]</a> (Note that here, "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">" represents a generic (or arbitrary) <a href="/wiki/Discounted_cash_flow#Discount_rate" title="Discounted cash flow">discount rate</a> applied to the cash flows, whereas in the valuation formulae, the <a href="/wiki/Risk-free_rate" title="Risk-free rate">risk-free rate</a> is applied once these have been "adjusted" for their riskiness; see below.) An immediate extension is to combine probabilities with present value, leading to the <a href="/wiki/Expected_value" title="Expected value">expected value criterion</a> which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1b40229a003b8b719a29ac70c15fe4f704a777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8201eb39454219ddec719cdb8b3ffd2b3be2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}"> respectively. <a href="#cite_note-11">[note 2]</a> This decision method, however, fails to consider <a href="/wiki/Risk_aversion" title="Risk aversion">risk aversion</a>. In other words, since individuals receive greater <a href="/wiki/Utility#Applications" title="Utility">utility</a> from an extra dollar when they are poor and less utility when comparatively rich, the approach is therefore to "adjust" the weight assigned to the various outcomes, i.e. "states", correspondingly: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6802614359e5b2a13d3dc8cee4640865ae03d5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.354ex; height:2.509ex;" alt="{\displaystyle Y_{s}}">. See <a href="/wiki/Indifference_price" title="Indifference price">indifference price</a>. (Some investors may in fact be <a href="/wiki/Risk_seeking" class="mw-redirect" title="Risk seeking">risk seeking</a> as opposed to <a href="/wiki/Risk_aversion" title="Risk aversion">risk averse</a>, but the same logic would apply.) Choice under uncertainty here may then be defined as the maximization of <a href="/wiki/Expected_utility" class="mw-redirect" title="Expected utility">expected utility</a>. More formally, the resulting <a href="/wiki/Expected_utility_hypothesis" title="Expected utility hypothesis">expected utility hypothesis</a> states that, if certain axioms are satisfied, the <a href="/wiki/Subjective_theory_of_value" title="Subjective theory of value">subjective</a> value associated with a gamble by an individual is that individual's <a href="/wiki/Expected_value" title="Expected value">statistical expectation</a> of the valuations of the outcomes of that gamble. The impetus for these ideas arises from various inconsistencies observed under the expected value framework, such as the <a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg paradox</a> and the <a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg paradox</a>. <a href="#cite_note-12">[note 3]</a> <div class="mw-heading mw-heading3"><h3 id="Arbitrage-free_pricing_and_equilibrium">Arbitrage-free pricing and equilibrium</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=3" title="Edit section: Arbitrage-free pricing and equilibrium">edit</a>]</div> <table class="wikitable floatright" width="250"> <tbody><tr style="text-align:center;"> <td>JEL classification codes </td></tr> <tr> <td>In the <a href="/wiki/JEL_classification_codes" class="mw-redirect" title="JEL classification codes">Journal of Economic Literature classification codes</a>, Financial Economics is one of the 19 primary classifications, at JEL: G. It follows <a href="/wiki/Monetary_economics" title="Monetary economics">Monetary</a> and <a href="/wiki/International_economics" title="International economics">International Economics</a> and precedes <a href="/wiki/Public_economics" title="Public economics">Public Economics</a>. For detailed subclassifications see <a href="/wiki/JEL_classification_codes#G._Financial_Economics" class="mw-redirect" title="JEL classification codes">JEL classification codes § G. Financial Economics</a>. <a href="/wiki/The_New_Palgrave_Dictionary_of_Economics" title="The New Palgrave Dictionary of Economics">The New Palgrave Dictionary of Economics</a> (2008, 2nd ed.) also uses the JEL codes to classify its entries in v. 8, Subject Index, including Financial Economics at pp. 863–64. The below have links to entry <a href="/wiki/Abstract_(summary)" title="Abstract (summary)">abstracts</a> of The New Palgrave <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130210185054/http://www.dictionaryofeconomics.com/dictionary">Online</a> for each primary or secondary JEL category (10 or fewer per page, similar to <a href="/wiki/Google" title="Google">Google</a> searches): <dl><dd><a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/search_results?,q=&field=content&edition=all&topicid=G">JEL: G</a> – <a href="/wiki/Financial_Economics" class="mw-redirect" title="Financial Economics">Financial Economics</a></dd> <dd><a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G0">JEL: G0</a> – General</dd> <dd><a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G1">JEL: G1</a> – <a href="/wiki/Financial_market" title="Financial market">General Financial Markets</a></dd> <dd><a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G2">JEL: G2</a> – <a href="/wiki/Financial_institution" title="Financial institution">Financial institutions</a> and <a href="/wiki/Financial_services" title="Financial services">Services</a></dd> <dd><a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G3">JEL: G3</a> – <a href="/wiki/Corporate_finance" title="Corporate finance">Corporate finance</a> and <a href="/wiki/Corporate_governance" title="Corporate governance">Governance</a></dd></dl> Tertiary category entries can also be searched.<a href="#cite_note-13">[10]</a> </td></tr></tbody></table> The concepts of <a href="/wiki/Arbitrage" title="Arbitrage">arbitrage</a>-free, "rational", pricing and equilibrium are then coupled <a href="#cite_note-Varian-14">[11]</a> with the above to derive various of the "classical"<a href="#cite_note-Rubinstein2-15">[12]</a> (or <a href="/wiki/Neoclassical_economics" title="Neoclassical economics">"neo-classical"</a><a href="#cite_note-Derman-16">[13]</a>) financial economics models. <a href="/wiki/Rational_pricing" title="Rational pricing">Rational pricing</a> is the assumption that asset prices (and hence asset pricing models) will reflect the <a href="/wiki/Arbitrage-free" class="mw-redirect" title="Arbitrage-free">arbitrage-free price</a> of the asset, as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments. <a href="/wiki/Economic_equilibrium" title="Economic equilibrium">Economic equilibrium</a> is a state in which economic forces such as supply and demand are balanced, and in the absence of external influences these equilibrium values of economic variables will not change. <a href="/wiki/General_equilibrium_theory" title="General equilibrium theory">General equilibrium</a> deals with the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium. (This is in contrast to partial equilibrium, which only analyzes single markets.) The two concepts are linked as follows: where market prices are <a href="/wiki/Complete_market" title="Complete market">complete</a> and do not allow profitable arbitrage, i.e. they comprise an arbitrage-free market, then these prices are also said to constitute an "arbitrage equilibrium". Intuitively, this may be seen by considering that where an arbitrage opportunity does exist, then prices can be expected to change, and they are therefore not in equilibrium.<a href="#cite_note-Delbaen_Schachermayer-17">[14]</a> An arbitrage equilibrium is thus a precondition for a general economic equilibrium. "Complete" here means that there is a price for every asset in every possible state of the world, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}">, and that the complete set of possible bets on future states-of-the-world can therefore be constructed with existing assets (assuming <a href="/wiki/Frictionless_market" title="Frictionless market">no friction</a>): essentially <a href="/wiki/System_of_linear_equations" title="System of linear equations">solving simultaneously</a> for n (risk-neutral) probabilities, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff98d38af85e0c551a806894448a9b12a5a61cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.04ex; height:2.009ex;" alt="{\displaystyle q_{s}}">, given n prices. For a simplified example see <a href="/wiki/Rational_pricing#Risk_neutral_valuation" title="Rational pricing">Rational pricing § Risk neutral valuation</a>, where the economy has only two possible states – up and down – and where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{up}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{up}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e754006c3908268225b7d4e5afb069f1e4ad365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.036ex; height:2.343ex;" alt="{\displaystyle q_{up}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{down}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>o</mi> <mi>w</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{down}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfea5bf50562a19a9db6744d1ccf9912e97c0451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.089ex; height:2.009ex;" alt="{\displaystyle q_{down}}"> (=<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-q_{up}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-q_{up}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36cbcea84e0b2fd557d7076d359a0d8451b0599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.039ex; height:2.843ex;" alt="{\displaystyle 1-q_{up}}">) are the two corresponding probabilities, and in turn, the derived distribution, or <a href="/wiki/Probability_measure" title="Probability measure">"measure"</a>. The formal derivation will proceed by arbitrage arguments.<a href="#cite_note-Rubinstein-6">[6]</a><a href="#cite_note-Delbaen_Schachermayer-17">[14]</a><a href="#cite_note-Varian-14">[11]</a>The analysis here is often undertaken assuming a <a href="/wiki/Representative_agent" title="Representative agent">representative agent</a>,<a href="#cite_note-Farmer_Geanakoplos-18">[15]</a> essentially treating all market participants, "<a href="/wiki/Agent_(economics)" title="Agent (economics)">agents</a>", as identical (or, at least, assuming that they <a href="/wiki/Heterogeneity_in_economics#Economic_models_with_heterogeneous_agents" title="Heterogeneity in economics">act in such a way that</a> the sum of their choices is equivalent to the decision of one individual) with the effect that <a href="/wiki/Unreasonable_ineffectiveness_of_mathematics#Economics_and_finance" title="Unreasonable ineffectiveness of mathematics">the problems are then</a> mathematically tractable. With this measure in place, the expected, <a href="/wiki/Required_return" class="mw-redirect" title="Required return">i.e. required</a>, return of any security (or portfolio) will then equal the risk-free return, plus an "adjustment for risk",<a href="#cite_note-Rubinstein-6">[6]</a> i.e. a security-specific <a href="/wiki/Risk_premium" title="Risk premium">risk premium</a>, compensating for the extent to which its cashflows are unpredictable. All pricing models are then essentially variants of this, given specific assumptions or conditions.<a href="#cite_note-Rubinstein-6">[6]</a><a href="#cite_note-Cochrane_&_Culp-5">[5]</a><a href="#cite_note-Backus-19">[16]</a> This approach is consistent with <a href="#Present_value,_expectation_and_utility">the above</a>, but with the expectation based on "the market" (i.e. arbitrage-free, and, per the theorem, therefore in equilibrium) as opposed to individual preferences. Continuing the example, in pricing a <a href="/wiki/Derivative_(finance)" title="Derivative (finance)">derivative instrument</a>, its forecasted cashflows in the abovementioned up- and down-states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{up}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{up}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59c18c06baef2d07e661a92dac36fffe91d550f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.924ex; height:2.843ex;" alt="{\displaystyle X_{up}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{down}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>o</mi> <mi>w</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{down}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2f9d0cd28c547bcf5c3f8675edec9093ca102d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.977ex; height:2.509ex;" alt="{\displaystyle X_{down}}">, are multiplied through by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{up}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{up}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e754006c3908268225b7d4e5afb069f1e4ad365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.036ex; height:2.343ex;" alt="{\displaystyle q_{up}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{down}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>o</mi> <mi>w</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{down}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfea5bf50562a19a9db6744d1ccf9912e97c0451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.089ex; height:2.009ex;" alt="{\displaystyle q_{down}}">, and are then <a href="/wiki/Present_value" title="Present value">discounted</a> at the risk-free interest rate; per the second equation above. In pricing a "fundamental", underlying, instrument (in equilibrium), on the other hand, a risk-appropriate premium over risk-free is required in the discounting, essentially employing the first equation with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> combined. This premium may be derived by the <a href="/wiki/Capital_asset_pricing_model" title="Capital asset pricing model">CAPM</a> (or extensions) as will be seen under <a href="#Uncertainty">§ Uncertainty</a>. The difference is explained as follows: By construction, the value of the derivative will (must) grow at the risk free rate, and, by arbitrage arguments, its value must then be discounted correspondingly; in the case of an option, this is achieved by "manufacturing" the instrument as a combination of the <a href="/wiki/Underlying" class="mw-redirect" title="Underlying">underlying</a> and a risk free "bond"; see <a href="/wiki/Rational_pricing#Delta_hedging" title="Rational pricing">Rational pricing § Delta hedging</a> (and <a href="#Uncertainty">§ Uncertainty</a> below). Where the underlying is itself being priced, such "manufacturing" is of course not possible – the instrument being "fundamental", i.e. as opposed to "derivative" – and a premium is then required for risk. (Correspondingly, mathematical finance separates into <a href="/wiki/Mathematical_finance#History:_Q_versus_P" title="Mathematical finance">two analytic regimes</a>: risk and portfolio management (generally) use physical (or actual or actuarial) probability, denoted by "P"; while derivatives pricing uses risk-neutral probability (or arbitrage-pricing probability), denoted by "Q". In specific applications the lower case is used, as in the above equations.) <div class="mw-heading mw-heading3"><h3 id="State_prices">State prices</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=4" title="Edit section: State prices">edit</a>]</div> With the above relationship established, the further specialized <a href="/wiki/Arrow%E2%80%93Debreu_model" title="Arrow–Debreu model">Arrow–Debreu model</a> may be derived. <a href="#cite_note-23">[note 4]</a> This result suggests that, under certain economic conditions, there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy. The Arrow–Debreu model applies to economies with maximally <a href="/wiki/Complete_market" title="Complete market">complete markets</a>, in which there exists a market for every time period and forward prices for every commodity at all time periods. A direct extension, then, is the concept of a <a href="/wiki/State_price" class="mw-redirect" title="State price">state price</a> security, also called an Arrow–Debreu security, a contract that agrees to pay one unit of a <a href="/wiki/Numeraire" class="mw-redirect" title="Numeraire">numeraire</a> (a currency or a commodity) if a particular state occurs ("up" and "down" in the simplified example above) at a particular time in the future and pays zero numeraire in all the other states. The price of this security is the state price <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcfaac8814f5e62e87bca6bec267ee6e80808db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.328ex; height:2.009ex;" alt="{\displaystyle \pi _{s}}"> of this particular state of the world; also referred to as a "Risk Neutral Density".<a href="#cite_note-Figlewski-24">[20]</a> In the above example, the state prices, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{up}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{up}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e05b6b4b9a8acc051acaf0ed07f6474e4988d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.324ex; height:2.343ex;" alt="{\displaystyle \pi _{up}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{down}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>o</mi> <mi>w</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{down}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589f88df29cd01c2394174a1f7ca61e495bd878b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.009ex;" alt="{\displaystyle \pi _{down}}">would equate to the present values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \$q_{up}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">$</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \$q_{up}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b340e953725a14b584e3509cfc45fd1089a3a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.199ex; height:3.009ex;" alt="{\displaystyle \$q_{up}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \$q_{down}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">$</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>o</mi> <mi>w</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \$q_{down}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1cce03b9394a9155cd5d93f32734202b5b9720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.252ex; height:2.676ex;" alt="{\displaystyle \$q_{down}}">: i.e. what one would pay today, respectively, for the up- and down-state securities; the <a href="/wiki/State_price_vector" class="mw-redirect" title="State price vector">state price vector</a> is the vector of state prices for all states. Applied to derivative valuation, the price today would simply be [<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{up}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{up}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e05b6b4b9a8acc051acaf0ed07f6474e4988d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.324ex; height:2.343ex;" alt="{\displaystyle \pi _{up}}">×<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{up}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{up}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59c18c06baef2d07e661a92dac36fffe91d550f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.924ex; height:2.843ex;" alt="{\displaystyle X_{up}}"> + <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{down}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>o</mi> <mi>w</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{down}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589f88df29cd01c2394174a1f7ca61e495bd878b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.009ex;" alt="{\displaystyle \pi _{down}}">×<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{down}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>o</mi> <mi>w</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{down}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2f9d0cd28c547bcf5c3f8675edec9093ca102d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.977ex; height:2.509ex;" alt="{\displaystyle X_{down}}">]: the fourth formula (see above regarding the absence of a risk premium here). For a <a href="/wiki/Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous random variable</a> indicating a continuum of possible states, the value is found by <a href="/wiki/Integration_(mathematics)" class="mw-redirect" title="Integration (mathematics)">integrating</a> over the state price "density". State prices find immediate application as a conceptual tool ("<a href="/wiki/Contingent_claim_analysis" class="mw-redirect" title="Contingent claim analysis">contingent claim analysis</a>");<a href="#cite_note-Rubinstein-6">[6]</a> but can also be applied to valuation problems.<a href="#cite_note-corp_fin_state_prices-25">[21]</a> Given the pricing mechanism described, one can decompose the derivative value – true in fact for "every security"<a href="#cite_note-Miller-2">[2]</a> – as a linear combination of its state-prices; i.e. back-solve for the state-prices corresponding to observed derivative prices.<a href="#cite_note-Chance2-26">[22]</a><a href="#cite_note-corp_fin_state_prices-25">[21]</a> <a href="#cite_note-Figlewski-24">[20]</a> These recovered state-prices can then be used for valuation of other instruments with exposure to the underlyer, or for other decision making relating to the underlyer itself. Using the related <a href="/wiki/Stochastic_discount_factor" title="Stochastic discount factor">stochastic discount factor</a> - also called the pricing kernel - the asset price is computed by "discounting" the future cash flow by the stochastic factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98556e7119feca9d7c240177d1155ecba5c966b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.176ex;" alt="{\displaystyle {\tilde {m}}}">, and then taking the expectation;<a href="#cite_note-Backus-19">[16]</a> the third equation above. Essentially, this factor divides expected utility at the relevant future period - a function of the possible asset values realized under each state - by the utility due to today's wealth, and is then also referred to as "the intertemporal <a href="/wiki/Marginal_rate_of_substitution" title="Marginal rate of substitution">marginal rate of substitution</a>". <div class="mw-heading mw-heading2"><h2 id="Resultant_models">Resultant models</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=5" title="Edit section: Resultant models">edit</a>]</div> <table class="wikitable floatright" width="250"> <tbody><tr style="text-align:left;"> <td> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{t=1}^{n}{\frac {C}{(1+i)^{t}}}+{\frac {F}{(1+i)^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{t=1}^{n}{\frac {C}{(1+i)^{t}}}+{\frac {F}{(1+i)^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651b48a05c1622326115b50c716083f2c2b45a72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.528ex; height:6.843ex;" alt="{\displaystyle \sum _{t=1}^{n}{\frac {C}{(1+i)^{t}}}+{\frac {F}{(1+i)^{n}}}}"></dd></dl> <a href="/wiki/Bond_valuation#Present_value_approach" title="Bond valuation">Bond valuation formula</a> where Coupons and Face value are discounted at the appropriate rate, "i": typically reflecting a spread over the risk free rate <a href="/wiki/Bond_valuation#Relative_price_approach" title="Bond valuation">as a function of credit risk</a>; often quoted as a "<a href="/wiki/Yield_to_maturity" title="Yield to maturity">yield to maturity</a>". See body for discussion re the relationship with the above pricing formulae. </td></tr></tbody></table> <table class="wikitable floatright" width="250"> <tbody><tr style="text-align:left;"> <td><div style="font-size:85%;"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{t=1}^{n}{\frac {FCFF_{t}}{(1+WACC_{t})^{t}}}+{\frac {\left[{\frac {FCFF_{n+1}}{(WACC_{n+1}-g_{n+1})}}\right]}{(1+WACC_{n})^{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mi>C</mi> <mi>F</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>W</mi> <mi>A</mi> <mi>C</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mi>C</mi> <mi>F</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>W</mi> <mi>A</mi> <mi>C</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>W</mi> <mi>A</mi> <mi>C</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{t=1}^{n}{\frac {FCFF_{t}}{(1+WACC_{t})^{t}}}+{\frac {\left[{\frac {FCFF_{n+1}}{(WACC_{n+1}-g_{n+1})}}\right]}{(1+WACC_{n})^{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22b05672cab39b95519d0347e9a4892559f1531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.417ex; height:8.676ex;" alt="{\displaystyle \sum _{t=1}^{n}{\frac {FCFF_{t}}{(1+WACC_{t})^{t}}}+{\frac {\left[{\frac {FCFF_{n+1}}{(WACC_{n+1}-g_{n+1})}}\right]}{(1+WACC_{n})^{n}}}}"></dd></dl></div> <a href="/wiki/Valuation_using_discounted_cash_flows#Basic_formula_for_firm_valuation_using_DCF_model" title="Valuation using discounted cash flows">DCF valuation formula</a>, where the <a href="/wiki/Business_valuation" title="Business valuation">value of the firm</a>, is its forecasted <a href="/wiki/Free_cash_flow" title="Free cash flow">free cash flows</a> discounted to the present using the <a href="/wiki/Weighted_average_cost_of_capital" title="Weighted average cost of capital">weighted average cost of capital</a>, i.e. <a href="/wiki/Cost_of_equity" title="Cost of equity">cost of equity</a> and <a href="/wiki/Cost_of_debt" class="mw-redirect" title="Cost of debt">cost of debt</a>, with the former (often) derived using the below CAPM. For <a href="/wiki/Stock_valuation" title="Stock valuation">share valuation</a> investors use the related <a href="/wiki/Dividend_discount_model" title="Dividend discount model">dividend discount model</a>. </td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:MM2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/MM2.png/220px-MM2.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/MM2.png/330px-MM2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/74/MM2.png 2x" data-file-width="336" data-file-height="255" /></a><figcaption>Modigliani–Miller Proposition II with risky debt. Even if <a href="/wiki/Leverage_(finance)" title="Leverage (finance)">leverage</a> (<a href="/wiki/Debt_to_equity_ratio" class="mw-redirect" title="Debt to equity ratio">D/E</a>) increases, the <a href="/wiki/Weighted_average_cost_of_capital" title="Weighted average cost of capital">WACC</a> (k0) stays constant.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Markowitz_frontier.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Markowitz_frontier.jpg/220px-Markowitz_frontier.jpg" decoding="async" width="220" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Markowitz_frontier.jpg/330px-Markowitz_frontier.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e1/Markowitz_frontier.jpg 2x" data-file-width="434" data-file-height="239" /></a><figcaption>Efficient Frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet', and its upward sloped portion is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier. The graphic displays the CAL, <a href="/wiki/Capital_allocation_line" title="Capital allocation line">Capital allocation line</a>, formed when the risky asset is a single-asset rather than the market, in which case the line is the CML.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CML-plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/CML-plot.png/220px-CML-plot.png" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/CML-plot.png/330px-CML-plot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/CML-plot.png/440px-CML-plot.png 2x" data-file-width="537" data-file-height="439" /></a><figcaption>The <a href="/wiki/Capital_market_line" title="Capital market line">Capital market line</a> is the tangent line drawn from the point of the risk-free asset to the <a href="/wiki/Feasible_region" title="Feasible region">feasible region</a> for risky assets. The tangency point M represents the <a href="/wiki/Market_portfolio" title="Market portfolio">market portfolio</a>. The CML results from the combination of the market portfolio and the risk-free asset (the point L). Addition of leverage (the point R) creates levered portfolios that are also on the CML.</figcaption></figure> <table class="wikitable floatright" width="250"> <tbody><tr style="text-align:center;"> <td><div style="font-size:85%;">The capital asset pricing model (CAPM): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(R_{i})=R_{f}+\beta _{i}(E(R_{m})-R_{f})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(R_{i})=R_{f}+\beta _{i}(E(R_{m})-R_{f})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2caeda09206938d724047133529f3f56984f7b1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.677ex; height:3.009ex;" alt="{\displaystyle E(R_{i})=R_{f}+\beta _{i}(E(R_{m})-R_{f})}"></dd></dl></div> The <a href="/wiki/Required_rate_of_return" class="mw-redirect" title="Required rate of return">expected return</a> used when discounting cashflows on an asset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">, is the risk-free rate plus the <a href="/wiki/Capital_asset_pricing_model#Formula" title="Capital asset pricing model">market premium</a> multiplied by <a href="/wiki/Beta_(finance)" title="Beta (finance)">beta</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i,m}{\frac {\sigma _{i}}{\sigma _{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i,m}{\frac {\sigma _{i}}{\sigma _{m}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4dfc3f9cca1e664366fde02b89bb64c43b56ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.74ex; height:5.009ex;" alt="{\displaystyle \rho _{i,m}{\frac {\sigma _{i}}{\sigma _{m}}}}">), the asset's correlated volatility relative to the overall market <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">. </td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SML-chart.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/f3/SML-chart.png/220px-SML-chart.png" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f3/SML-chart.png/330px-SML-chart.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f3/SML-chart.png/440px-SML-chart.png 2x" data-file-width="574" data-file-height="469" /></a><figcaption><a href="/wiki/Security_market_line" title="Security market line">Security market line</a>: the representation of the CAPM displaying the expected rate of return of an individual security as a function of its systematic, non-diversifiable risk.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stockpricesimulation.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Stockpricesimulation.jpg/220px-Stockpricesimulation.jpg" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Stockpricesimulation.jpg/330px-Stockpricesimulation.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Stockpricesimulation.jpg/440px-Stockpricesimulation.jpg 2x" data-file-width="1593" data-file-height="1206" /></a><figcaption>Simulated geometric Brownian motions with parameters from market data</figcaption></figure> <table class="wikitable floatright" width="250"> <tbody><tr style="text-align:center;"> <td><div style="font-size:85%;"><a href="/wiki/Black%E2%80%93Scholes_equation" title="Black–Scholes equation">The Black–Scholes equation:</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}=rV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>r</mi> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>r</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}=rV}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c08ad5f350ae6037a16e03d37dde320a67768a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.026ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}=rV}"></dd></dl> <a href="/wiki/Black%E2%80%93Scholes_equation#Financial_interpretation_of_the_Black–Scholes_PDE" title="Black–Scholes equation">Interpretation:</a> by arbitrage arguments, the instantaneous impact of time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> and change in <a href="/wiki/Spot_price" class="mw-redirect" title="Spot price">spot price</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> on an option price <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> will (must) realize as growth at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, the risk free rate, when the option is correctly <a href="/wiki/Rational_pricing#Delta_hedging" title="Rational pricing">"manufactured"</a>.</div> </td></tr></tbody></table> <table class="wikitable floatright" width="250"> <tbody><tr style="text-align:center;"> <td><div style="font-size:85%;"><a href="/wiki/Black%E2%80%93Scholes_model#Black–Scholes_formula" title="Black–Scholes model">The Black–Scholes formula</a> for the value of a call option: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}C(S,t)&=N(d_{1})S-N(d_{2})Ke^{-r(T-t)}\\d_{1}&={\frac {1}{\sigma {\sqrt {T-t}}}}\left[\ln \left({\frac {S}{K}}\right)+\left(r+{\frac {\sigma ^{2}}{2}}\right)(T-t)\right]\\d_{2}&=d_{1}-\sigma {\sqrt {T-t}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>C</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>S</mi> <mo>−</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>K</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>T</mi> <mo>−</mo> <mi>t</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>K</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>T</mi> <mo>−</mo> <mi>t</mi> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}C(S,t)&=N(d_{1})S-N(d_{2})Ke^{-r(T-t)}\\d_{1}&={\frac {1}{\sigma {\sqrt {T-t}}}}\left[\ln \left({\frac {S}{K}}\right)+\left(r+{\frac {\sigma ^{2}}{2}}\right)(T-t)\right]\\d_{2}&=d_{1}-\sigma {\sqrt {T-t}}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ed0aef39b1aee3a602de0faf6224848c506363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:52.205ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}C(S,t)&=N(d_{1})S-N(d_{2})Ke^{-r(T-t)}\\d_{1}&={\frac {1}{\sigma {\sqrt {T-t}}}}\left[\ln \left({\frac {S}{K}}\right)+\left(r+{\frac {\sigma ^{2}}{2}}\right)(T-t)\right]\\d_{2}&=d_{1}-\sigma {\sqrt {T-t}}\\\end{aligned}}}"></dd></dl> <a href="/wiki/Black%E2%80%93Scholes_model#Interpretation" title="Black–Scholes model">Interpretation</a>: The value of a call is the <a href="/wiki/Risk_free_rate" class="mw-redirect" title="Risk free rate">risk free rated</a> present value of its expected <a href="/wiki/In_the_money" class="mw-redirect" title="In the money">in the money</a> value - i.e. a specific formulation of the fundamental valuation result. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(d_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(d_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89cce403c6e8cca0981f18f218be2e7c9a63d301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.136ex; height:2.843ex;" alt="{\displaystyle N(d_{2})}"> is the probability that the call will be exercised; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(d_{1})S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(d_{1})S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfe845e15c502c19ce3d02e871b34274a78f5d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.635ex; height:2.843ex;" alt="{\displaystyle N(d_{1})S}"> is the present value of the expected asset price at expiration, <a href="/wiki/Conditional_probability" title="Conditional probability">given that</a> the asset price at expiration is above the exercise price.</div> </td></tr></tbody></table> Applying the above economic concepts, we may then derive various <a href="/wiki/Economic_model" title="Economic model">economic-</a> and financial models and principles. As above, the two usual areas of focus are Asset Pricing and Corporate Finance, the first being the perspective of providers of capital, the second of users of capital. Here, and for (almost) all other financial economics models, the questions addressed are typically framed in terms of "time, uncertainty, options, and information",<a href="#cite_note-stanford1-1">[1]</a><a href="#cite_note-Farmer_Geanakoplos-18">[15]</a> as will be seen below. <ul><li>Time: money now is traded for money in the future.</li> <li>Uncertainty (or risk): The amount of money to be transferred in the future is uncertain.</li> <li><a href="/wiki/Option_(finance)" title="Option (finance)">Options</a>: one party to the transaction can make a decision at a later time that will affect subsequent transfers of money.</li> <li><a href="/wiki/Perfect_information" title="Perfect information">Information</a>: knowledge of the future can reduce, or possibly eliminate, the uncertainty associated with <a href="/wiki/Future_value" title="Future value">future monetary value</a> (FMV).</li></ul> Applying this framework, with the above concepts, leads to the required models. This derivation begins with the assumption of "no uncertainty" and is then expanded to incorporate the other considerations.<a href="#cite_note-Fama_and_Miller-4">[4]</a> (This division sometimes denoted "<a href="/wiki/Deterministic" class="mw-redirect" title="Deterministic">deterministic</a>" and "random",<a href="#cite_note-Luenberger-27">[23]</a> or "<a href="/wiki/Stochastic" title="Stochastic">stochastic</a>".) <div class="mw-heading mw-heading3"><h3 id="Certainty">Certainty</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=6" title="Edit section: Certainty">edit</a>]</div> The starting point here is "Investment under certainty", and usually framed in the context of a corporation. The <a href="/wiki/Fisher_separation_theorem" title="Fisher separation theorem">Fisher separation theorem</a>, asserts that the objective of the corporation will be the maximization of its present value, regardless of the preferences of its shareholders. Related is the <a href="/wiki/Modigliani%E2%80%93Miller_theorem" title="Modigliani–Miller theorem">Modigliani–Miller theorem</a>, which shows that, under certain conditions, the value of a firm is unaffected by how that firm is financed, and depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt. The proof here proceeds using arbitrage arguments, and acts as a benchmark <a href="#cite_note-Varian-14">[11]</a> for evaluating the effects of factors outside the model that do affect value. <a href="#cite_note-30">[note 5]</a> The mechanism for determining (corporate) value is provided by <a href="#cite_note-New_School-31">[26]</a> <a href="#cite_note-Rubinstein_2-32">[27]</a> <a href="/wiki/John_Burr_Williams" title="John Burr Williams">John Burr Williams</a>' <a href="/wiki/The_Theory_of_Investment_Value" class="mw-redirect" title="The Theory of Investment Value">The Theory of Investment Value</a>, which proposes that the value of an asset should be calculated using "evaluation by the rule of present worth". Thus, for a common stock, the <a href="/wiki/Intrinsic_value_(finance)#Equity" title="Intrinsic value (finance)">"intrinsic"</a>, long-term worth is the present value of its future net cashflows, in the form of <a href="/wiki/Dividend" title="Dividend">dividends</a>. What remains to be determined is the appropriate discount rate. Later developments show that, "rationally", i.e. in the formal sense, the appropriate discount rate here will (should) depend on the asset's riskiness relative to the overall market, as opposed to its owners' preferences; see below. <a href="/wiki/Net_present_value" title="Net present value">Net present value</a> (NPV) is the direct extension of these ideas typically applied to Corporate Finance decisioning. For other results, as well as specific models developed here, see the list of "Equity valuation" topics under <a href="/wiki/Outline_of_finance#Discounted_cash_flow_valuation" title="Outline of finance">Outline of finance § Discounted cash flow valuation</a>. <a href="#cite_note-33">[note 6]</a> <a href="/wiki/Bond_valuation" title="Bond valuation">Bond valuation</a>, in that cashflows (<a href="/wiki/Coupon_(finance)" title="Coupon (finance)">coupons</a> and return of principal, or "<a href="/wiki/Face_value" title="Face value">Face value</a>") are deterministic, may proceed in the same fashion.<a href="#cite_note-Luenberger-27">[23]</a> An immediate extension, <a href="/wiki/Bond_valuation#Arbitrage-free_pricing_approach" title="Bond valuation">Arbitrage-free bond pricing</a>, discounts each cashflow at the market derived rate – i.e. at each coupon's corresponding <a href="/wiki/Zero_rate" class="mw-redirect" title="Zero rate">zero rate</a>, and of equivalent credit worthiness – as opposed to an overall rate. In many treatments bond valuation precedes <a href="/wiki/Equity_valuation" class="mw-redirect" title="Equity valuation">equity valuation</a>, under which cashflows (dividends) are not "known" per se. Williams and onward allow for forecasting as to these – based on <a href="/wiki/Dividend_payout_ratio" title="Dividend payout ratio">historic ratios</a> or published <a href="/wiki/Dividend_policy" title="Dividend policy">dividend policy</a> – and cashflows are then treated as essentially deterministic; see below under <a href="#Corporate_finance_theory">§ Corporate finance theory</a>. For both stocks and bonds, "under certainty, with the focus on cash flows from securities over time," valuation based on a <a href="/wiki/Yield_curve" title="Yield curve">term structure of interest rates</a> is in fact consistent with arbitrage-free pricing.<a href="#cite_note-34">[28]</a> Indeed, a corollary of <a href="#Arbitrage-free_pricing_and_equilibrium">the above</a> is that "<a href="/wiki/Rational_pricing#The_law_of_one_price" title="Rational pricing">the law of one price</a> implies the existence of a discount factor"; <a href="#cite_note-35">[29]</a> correspondingly, as formulated, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{s}\pi _{s}=1/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{s}\pi _{s}=1/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c81a3435d4e176cc0f0ecb31e064e43d4c8dd28e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.645ex; height:3.009ex;" alt="{\textstyle \sum _{s}\pi _{s}=1/r}">. Whereas these "certainty" results are all commonly employed under corporate finance, uncertainty is the focus of "asset pricing models" as follows. <a href="/wiki/Irving_Fisher#Interest_and_capital" title="Irving Fisher">Fisher's formulation</a> of the theory here - developing <a href="/wiki/Intertemporal_choice#Fisher's_model_of_intertemporal_consumption" title="Intertemporal choice">an intertemporal equilibrium model</a> - underpins also <a href="#cite_note-New_School-31">[26]</a> the below applications to uncertainty; <a href="#cite_note-36">[note 7]</a> see <a href="#cite_note-37">[30]</a> for the development. <div class="mw-heading mw-heading3"><h3 id="Uncertainty">Uncertainty</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=7" title="Edit section: Uncertainty">edit</a>]</div> For <a href="/wiki/Decision_theory#Choice_under_uncertainty" title="Decision theory">"choice under uncertainty"</a> the twin assumptions of rationality and <a href="/wiki/Financial_market_efficiency" title="Financial market efficiency">market efficiency</a>, as more closely defined, lead to <a href="/wiki/Modern_portfolio_theory" title="Modern portfolio theory">modern portfolio theory</a> (MPT) with its <a href="/wiki/Capital_asset_pricing_model" title="Capital asset pricing model">capital asset pricing model</a> (CAPM) – an equilibrium-based result – and to the <a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes–Merton theory</a> (BSM; often, simply Black–Scholes) for <a href="/wiki/Valuation_of_options" title="Valuation of options">option pricing</a> – an arbitrage-free result. As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitrage-free with respect to the more fundamental, equilibrium determined, securities prices; see <a href="/wiki/Asset_pricing#Interrelationship" title="Asset pricing">Asset pricing § Interrelationship</a>. Briefly, and intuitively – and consistent with <a href="#Arbitrage-free_pricing_and_equilibrium">§ Arbitrage-free pricing and equilibrium</a> above – the relationship between rationality and efficiency is as follows.<a href="#cite_note-38">[31]</a> Given the ability to profit from <a href="/wiki/Privacy" title="Privacy">private information</a>, self-interested traders are motivated to acquire and act on their private information. In doing so, traders contribute to more and more "correct", i.e. efficient, prices: the <a href="/wiki/Efficient-market_hypothesis" title="Efficient-market hypothesis">efficient-market hypothesis</a>, or EMH. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See <a href="/wiki/Earnings_response_coefficient" title="Earnings response coefficient">earnings response coefficient</a>.) The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information are identical to the best guess of the future: the assumption of <a href="/wiki/Rational_expectations" title="Rational expectations">rational expectations</a>. The EMH does allow that when faced with new information, some investors may overreact and some may underreact, <a href="#cite_note-affirmative_case-39">[32]</a> but what is required, however, is that investors' reactions follow a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> – so that the net effect on market prices cannot be reliably exploited <a href="#cite_note-affirmative_case-39">[32]</a> to make an abnormal profit. In the competitive limit, then, market prices will reflect all available information and prices can only move in response to news: <a href="#cite_note-Shiller-40">[33]</a> the <a href="/wiki/Random_walk_hypothesis" title="Random walk hypothesis">random walk hypothesis</a>. This news, of course, could be "good" or "bad", minor or, less common, major; and these moves are then, correspondingly, normally distributed; with the price therefore following a log-normal distribution. <a href="#cite_note-42">[note 8]</a> Under these conditions, investors can then be assumed to act rationally: their investment decision must be calculated or a loss is sure to follow; <a href="#cite_note-affirmative_case-39">[32]</a> correspondingly, where an arbitrage opportunity presents itself, then arbitrageurs will exploit it, reinforcing this equilibrium. Here, as under the certainty-case above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends, <a href="#cite_note-Cochrane_&_Culp-5">[5]</a> <a href="#cite_note-Shiller-40">[33]</a> <a href="#cite_note-Farmer_Geanakoplos-18">[15]</a> as based on currently available information. What is required though, is a theory for determining the appropriate discount rate, i.e. "required return", given this uncertainty: this is provided by the MPT and its CAPM. Relatedly, rationality – in the sense of arbitrage-exploitation – gives rise to Black–Scholes; option values here ultimately consistent with the CAPM. In general, then, while portfolio theory studies how investors should balance risk and return when investing in many assets or securities, the CAPM is more focused, describing how, in equilibrium, markets set the prices of assets in relation to how risky they are. <a href="#cite_note-44">[note 9]</a> This result will be independent of the investor's level of risk aversion and assumed <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility function</a>, thus providing a readily determined discount rate for corporate finance decision makers <a href="#Certainty">as above</a>,<a href="#cite_note-Jensen&Smith-45">[36]</a> and for other investors. The argument <a href="/wiki/Modern_portfolio_theory#Mathematical_model" title="Modern portfolio theory">proceeds as follows</a>: <a href="#cite_note-Bollerslev-46">[37]</a> If one can construct an <a href="/wiki/Efficient_frontier" title="Efficient frontier">efficient frontier</a> – i.e. each combination of assets offering the best possible expected level of return for its level of risk, see diagram – then mean-variance efficient portfolios can be formed simply as a combination of holdings of the <a href="/wiki/Risk-free_interest_rate" class="mw-redirect" title="Risk-free interest rate">risk-free asset</a> and the "<a href="/wiki/Market_portfolio" title="Market portfolio">market portfolio</a>" (the <a href="/wiki/Mutual_fund_separation_theorem" title="Mutual fund separation theorem">Mutual fund separation theorem</a>), with the combinations here plotting as the <a href="/wiki/Capital_market_line" title="Capital market line">capital market line</a>, or CML. Then, given this CML, the required return on a risky security will be independent of the investor's <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility function</a>, and solely determined by its <a href="/wiki/Covariance" title="Covariance">covariance</a> ("beta") with aggregate, i.e. market, risk. This is because investors here can then maximize utility through leverage as opposed to pricing; see <a href="/wiki/Separation_property_(finance)" title="Separation property (finance)">Separation property (finance)</a>, <a href="/wiki/Markowitz_model#Choosing_the_best_portfolio" title="Markowitz model">Markowitz model § Choosing the best portfolio</a> and CML diagram aside. As can be seen in the formula aside, this result is consistent with <a href="#Arbitrage-free_pricing_and_equilibrium">the preceding</a>, equaling the riskless return plus an adjustment for risk.<a href="#cite_note-Cochrane_&_Culp-5">[5]</a> A more modern, direct, derivation is as described at the bottom of this section; which can be generalized to derive other equilibrium-pricing models. Black–Scholes provides a mathematical model of a financial market containing <a href="/wiki/Derivative_(finance)" title="Derivative (finance)">derivative</a> instruments, and the resultant formula for the price of <a href="/wiki/Option_style" title="Option style">European-styled options</a>. <a href="#cite_note-52">[note 10]</a> The model is expressed as the Black–Scholes equation, a <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a> describing the changing price of the option over time; it is derived assuming log-normal, <a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">geometric Brownian motion</a> (see <a href="/wiki/Brownian_model_of_financial_markets" title="Brownian model of financial markets">Brownian model of financial markets</a>). The key financial insight behind the model is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk", absenting the risk adjustment from the pricing (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}">, the value, or price, of the option, grows at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, the risk-free rate).<a href="#cite_note-Rubinstein-6">[6]</a><a href="#cite_note-Cochrane_&_Culp-5">[5]</a> This hedge, in turn, implies that there is only one right price – in an arbitrage-free sense – for the option. And this price is returned by the Black–Scholes option pricing formula. (The formula, and hence the price, is consistent with the equation, as the formula is the <a href="/wiki/Partial_differential_equation#Analytical_solutions" title="Partial differential equation">solution</a> to the equation.) Since the formula is without reference to the share's expected return, Black–Scholes inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with <a href="#Arbitrage-free_pricing_and_equilibrium">§ Arbitrage-free pricing and equilibrium</a> above. Relatedly, therefore, the pricing formula <a href="/wiki/Black%E2%80%93Scholes_model#Derivations" title="Black–Scholes model">may also be derived</a> directly via risk neutral expectation. <a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a> provides <a href="/wiki/It%C3%B4%27s_lemma#Black–Scholes_formula" title="Itô's lemma">the underlying mathematics</a>, and, with <a href="/wiki/It%C3%B4_calculus" title="Itô calculus">Itô calculus</a> more generally, remains fundamental in quantitative finance. <a href="#cite_note-55">[note 11]</a> As implied by the Fundamental Theorem, <a href="/wiki/Asset_pricing#Interrelationship" title="Asset pricing">the two major results are consistent</a>. Here, the Black Scholes equation can alternatively be derived from the CAPM, and the price obtained from the Black–Scholes model is thus consistent with the assumptions of the CAPM.<a href="#cite_note-Chance1-56">[45]</a><a href="#cite_note-Derman-16">[13]</a> The Black–Scholes theory, although built on Arbitrage-free pricing, is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via state-pricing – essentially, by expanding the above fundamental equations – further explaining, and if required demonstrating, this consistency. <a href="#cite_note-Rubinstein-6">[6]</a> Here, the CAPM is derived by linking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}">, risk aversion, to overall market return, and setting the return on security <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}/Price_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> <mi>r</mi> <mi>i</mi> <mi>c</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}/Price_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a820a1492097361db0fd51bf646ba6231349f7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.593ex; height:3.009ex;" alt="{\displaystyle X_{j}/Price_{j}}">; see <a href="/wiki/Stochastic_discount_factor#Properties" title="Stochastic discount factor">Stochastic discount factor § Properties</a>. The Black–Scholes formula is found, in the limit, by attaching a <a href="/wiki/Binomial_probability" class="mw-redirect" title="Binomial probability">binomial probability</a><a href="#cite_note-Varian-14">[11]</a> to each of numerous possible <a href="/wiki/Spot_price" class="mw-redirect" title="Spot price">spot-prices</a> (i.e. states) and then rearranging for the terms corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(d_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(d_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1deee939132dff6d0e1c017ff6c26eb3b3768be1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.136ex; height:2.843ex;" alt="{\displaystyle N(d_{1})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(d_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(d_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89cce403c6e8cca0981f18f218be2e7c9a63d301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.136ex; height:2.843ex;" alt="{\displaystyle N(d_{2})}">, per the boxed description; see <a href="/wiki/Binomial_options_pricing_model#Relationship_with_Black–Scholes" title="Binomial options pricing model">Binomial options pricing model § Relationship with Black–Scholes</a>. <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=8" title="Edit section: Extensions">edit</a>]</div> More recent work further generalizes and extends these models. As regards <a href="/wiki/Asset_pricing" title="Asset pricing">asset pricing</a>, developments in equilibrium-based pricing are discussed under "Portfolio theory" below, while "Derivative pricing" relates to risk-neutral, i.e. arbitrage-free, pricing. As regards the use of capital, "Corporate finance theory" relates, mainly, to the application of these models. <div class="mw-heading mw-heading3"><h3 id="Portfolio_theory">Portfolio theory</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=9" title="Edit section: Portfolio theory">edit</a>]</div> <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">See also: <a href="/wiki/Post-modern_portfolio_theory" title="Post-modern portfolio theory">Post-modern portfolio theory</a> and <a href="/wiki/Mathematical_finance#Risk_and_portfolio_management:_the_P_world" title="Mathematical finance">Mathematical finance § Risk and portfolio management: the P world</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pareto_Efficient_Frontier_for_the_Markowitz_Portfolio_selection_problem..png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Pareto_Efficient_Frontier_for_the_Markowitz_Portfolio_selection_problem..png/200px-Pareto_Efficient_Frontier_for_the_Markowitz_Portfolio_selection_problem..png" decoding="async" width="200" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Pareto_Efficient_Frontier_for_the_Markowitz_Portfolio_selection_problem..png/300px-Pareto_Efficient_Frontier_for_the_Markowitz_Portfolio_selection_problem..png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Pareto_Efficient_Frontier_for_the_Markowitz_Portfolio_selection_problem..png/400px-Pareto_Efficient_Frontier_for_the_Markowitz_Portfolio_selection_problem..png 2x" data-file-width="1392" data-file-height="936" /></a><figcaption>Plot of two criteria when maximizing return and minimizing risk in financial portfolios (Pareto-optimal points in red)</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Four_Correlations.svg" class="mw-file-description"><img alt="Examples of bivariate copulæ used in finance." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Four_Correlations.svg/220px-Four_Correlations.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Four_Correlations.svg/330px-Four_Correlations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Four_Correlations.svg/440px-Four_Correlations.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>Examples of bivariate copulæ used in finance.</figcaption></figure> The majority of developments here relate to required return, i.e. pricing, extending the basic CAPM. Multi-factor models such as the <a href="/wiki/Fama%E2%80%93French_three-factor_model" title="Fama–French three-factor model">Fama–French three-factor model</a> and the <a href="/wiki/Carhart_four-factor_model" title="Carhart four-factor model">Carhart four-factor model</a>, propose factors other than market return as relevant in pricing. The <a href="/wiki/Intertemporal_CAPM" title="Intertemporal CAPM">intertemporal CAPM</a> and <a href="/wiki/Consumption-based_capital_asset_pricing_model" title="Consumption-based capital asset pricing model">consumption-based CAPM</a> similarly extend the model. With <a href="/wiki/Intertemporal_portfolio_choice" title="Intertemporal portfolio choice">intertemporal portfolio choice</a>, the investor now repeatedly optimizes her portfolio; while the inclusion of <a href="/wiki/Consumption_(economics)" title="Consumption (economics)">consumption (in the economic sense)</a> then incorporates all sources of wealth, and not just market-based investments, into the investor's calculation of required return. Whereas the above extend the CAPM, the <a href="/wiki/Single-index_model" title="Single-index model">single-index model</a> is a more simple model. It assumes, only, a correlation between security and market returns, without (numerous) other economic assumptions. It is useful in that it simplifies the estimation of correlation between securities, significantly reducing the inputs for building the correlation matrix required for portfolio optimization. The <a href="/wiki/Arbitrage_pricing_theory" title="Arbitrage pricing theory">arbitrage pricing theory</a> (APT) similarly differs as regards its assumptions. APT "gives up the notion that there is one right portfolio for everyone in the world, and ...replaces it with an explanatory model of what drives asset returns."<a href="#cite_note-57">[46]</a> It returns the required (expected) return of a financial asset as a linear function of various macro-economic factors, and assumes that arbitrage should bring incorrectly priced assets back into line.<a href="#cite_note-60">[note 12]</a> The linear factor model structure of the APT is used as the basis for many of the commercial risk systems employed by asset managers. As regards <a href="/wiki/Portfolio_optimization" title="Portfolio optimization">portfolio optimization</a>, the <a href="/wiki/Black%E2%80%93Litterman_model" title="Black–Litterman model">Black–Litterman model</a><a href="#cite_note-61">[49]</a> departs from the original <a href="/wiki/Markowitz_model" title="Markowitz model">Markowitz model</a> – i.e. of constructing portfolios via an <a href="/wiki/Efficient_frontier" title="Efficient frontier">efficient frontier</a>. Black–Litterman instead starts with an equilibrium assumption, and is then modified to take into account the 'views' (i.e., the specific opinions about asset returns) of the investor in question to arrive at a bespoke <a href="#cite_note-62">[50]</a> asset allocation. Where factors additional to volatility are considered (kurtosis, skew...) then <a href="/wiki/Multiple-criteria_decision_analysis" title="Multiple-criteria decision analysis">multiple-criteria decision analysis</a> can be applied; here deriving a <a href="/wiki/Pareto_efficient" class="mw-redirect" title="Pareto efficient">Pareto efficient</a> portfolio. The <a href="/wiki/Universal_portfolio_algorithm" title="Universal portfolio algorithm">universal portfolio algorithm</a> applies <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> to asset selection, learning adaptively from historical data. <a href="/wiki/Behavioral_portfolio_theory" title="Behavioral portfolio theory">Behavioral portfolio theory</a> recognizes that investors have varied aims and create an investment portfolio that meets a broad range of goals. Copulas have <a href="/wiki/Copula_(probability_theory)#Quantitative_finance" class="mw-redirect" title="Copula (probability theory)">lately been applied here</a>; recently this is the case also <a href="/wiki/List_of_genetic_algorithm_applications#Finance_and_Economics" title="List of genetic algorithm applications">for genetic algorithms</a> and <a href="/wiki/Machine_learning#Applications" title="Machine learning">Machine learning, more generally</a>. <a href="/wiki/Tail_risk_parity" title="Tail risk parity">(Tail)</a> <a href="/wiki/Risk_parity" title="Risk parity">risk parity</a> focuses on allocation of risk, rather than allocation of capital. <a href="#cite_note-63">[note 13]</a> See <a href="/wiki/Portfolio_optimization#Improving_portfolio_optimization" title="Portfolio optimization">Portfolio optimization § Improving portfolio optimization</a> for other techniques and objectives, and <a href="/wiki/Financial_risk_management#Investment_management" title="Financial risk management">Financial risk management § Investment management</a> for discussion. <div class="mw-heading mw-heading3"><h3 id="Derivative_pricing">Derivative pricing</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=10" title="Edit section: Derivative pricing">edit</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/Mathematical_finance#Derivatives_pricing:_the_Q_world" title="Mathematical finance">Mathematical finance § Derivatives pricing: the Q world</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/Quantitative_analyst#History" class="mw-redirect" title="Quantitative analyst">Quantitative analyst § History</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Arbre_Binomial_Options_Reelles.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Arbre_Binomial_Options_Reelles.png/220px-Arbre_Binomial_Options_Reelles.png" decoding="async" width="220" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Arbre_Binomial_Options_Reelles.png/330px-Arbre_Binomial_Options_Reelles.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Arbre_Binomial_Options_Reelles.png/440px-Arbre_Binomial_Options_Reelles.png 2x" data-file-width="537" data-file-height="314" /></a><figcaption> Binomial Lattice with <a href="/wiki/Binomial_options_pricing_model#STEP_1:_Create_the_binomial_price_tree" title="Binomial options pricing model">CRR formulae</a> </figcaption></figure> <table class="wikitable floatright" width="250"> <tbody><tr style="text-align:center;"> <td><div style="font-size:85%;">PDE for a zero-coupon bond: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\sigma (r)^{2}{\frac {\partial ^{2}P}{\partial r^{2}}}+[a(r)+\sigma (r)+\varphi (r,t)]{\frac {\partial P}{\partial r}}+{\frac {\partial P}{\partial t}}=rP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>r</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\sigma (r)^{2}{\frac {\partial ^{2}P}{\partial r^{2}}}+[a(r)+\sigma (r)+\varphi (r,t)]{\frac {\partial P}{\partial r}}+{\frac {\partial P}{\partial t}}=rP}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fac48f16eaeec661a5c4cc1d966b021ff1b25c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:53.094ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{2}}\sigma (r)^{2}{\frac {\partial ^{2}P}{\partial r^{2}}}+[a(r)+\sigma (r)+\varphi (r,t)]{\frac {\partial P}{\partial r}}+{\frac {\partial P}{\partial t}}=rP}"></dd></dl></div> <a href="/wiki/Bond_valuation#Stochastic_calculus_approach" title="Bond valuation">Interpretation:</a> Analogous to Black–Scholes, <a href="#cite_note-64">[51]</a> arbitrage arguments describe the instantaneous change in the bond price <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> for changes in the (risk-free) short rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">; the analyst selects the specific <a href="/wiki/Short-rate_model" title="Short-rate model">short-rate model</a> to be employed. </td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Volatility_smile.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Volatility_smile.svg/220px-Volatility_smile.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Volatility_smile.svg/330px-Volatility_smile.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Volatility_smile.svg/440px-Volatility_smile.svg.png 2x" data-file-width="150" data-file-height="150" /></a><figcaption>Stylized volatility smile: showing the (implied) volatility by strike-price, for which the <a href="/wiki/Black%E2%80%93Scholes_formula" class="mw-redirect" title="Black–Scholes formula">Black–Scholes formula</a> returns market prices.</figcaption></figure> In pricing derivatives, the <a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">binomial options pricing model</a> provides a discretized version of Black–Scholes, useful for the valuation of <a href="/wiki/American_option" class="mw-redirect" title="American option">American styled options</a>. Discretized models of this type are built – at least implicitly – using state-prices (<a href="#State_prices">as above</a>); relatedly, a large number of researchers <a href="/wiki/Contingent_claim_analysis" class="mw-redirect" title="Contingent claim analysis">have used options</a> to extract state-prices for a variety of other applications in financial economics.<a href="#cite_note-Rubinstein-6">[6]</a><a href="#cite_note-Chance1-56">[45]</a><a href="#cite_note-Chance2-26">[22]</a> For <a href="/wiki/Option_style#Non-vanilla_path-dependent_"exotic"_options" title="Option style">path dependent derivatives</a>, <a href="/wiki/Monte_Carlo_methods_for_option_pricing" title="Monte Carlo methods for option pricing">Monte Carlo methods for option pricing</a> are employed; here the modelling is in continuous time, but similarly uses risk neutral expected value. Various <a href="/wiki/Option_(finance)#Model_implementation" title="Option (finance)">other numeric techniques</a> have also been developed. The theoretical framework too has been extended such that <a href="/wiki/Martingale_pricing" title="Martingale pricing">martingale pricing</a> is now the standard approach. <a href="#cite_note-71">[note 14]</a> Drawing on these techniques, models for various other underlyings and applications have also been developed, all based on the same logic (using "<a href="/wiki/Contingent_claim_analysis" class="mw-redirect" title="Contingent claim analysis">contingent claim analysis</a>"). <a href="/wiki/Real_options_valuation" title="Real options valuation">Real options valuation</a> allows that option holders can influence the option's underlying; models for <a href="/wiki/Employee_stock_option#Valuation" title="Employee stock option">employee stock option valuation</a> explicitly assume non-rationality on the part of option holders; <a href="/wiki/Credit_derivative" title="Credit derivative">Credit derivatives</a> allow that payment obligations or delivery requirements might not be honored. <a href="/wiki/Exotic_derivative" title="Exotic derivative">Exotic derivatives</a> are now routinely valued. Multi-asset underlyers are handled via simulation or <a href="/wiki/Copula_(probability_theory)#Quantitative_finance" class="mw-redirect" title="Copula (probability theory)">copula based analysis</a>. Similarly, the various <a href="/wiki/Short-rate_model" title="Short-rate model">short-rate models</a> allow for an extension of these techniques to <a href="/wiki/Fixed_income#Derivatives" title="Fixed income">fixed income-</a> and <a href="/wiki/Interest_rate_derivative" title="Interest rate derivative">interest rate derivatives</a>. (The <a href="/wiki/Vasicek_model" title="Vasicek model">Vasicek</a> and <a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">CIR</a> models are equilibrium-based, while <a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a> and subsequent models are based on arbitrage-free pricing.) The more general <a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">HJM Framework</a> describes the dynamics of the full <a href="/wiki/Forward_rate" title="Forward rate">forward-rate</a> curve – as opposed to working with short rates – and is then more widely applied. The valuation of the underlying instrument – additional to its derivatives – is relatedly extended, particularly for <a href="/wiki/Hybrid_security" title="Hybrid security">hybrid securities</a>, where credit risk is combined with uncertainty re future rates; see <a href="/wiki/Bond_valuation#Stochastic_calculus_approach" title="Bond valuation">Bond valuation § Stochastic calculus approach</a> and <a href="/wiki/Lattice_model_(finance)#Hybrid_securities" title="Lattice model (finance)">Lattice model (finance) § Hybrid securities</a>. <a href="#cite_note-74">[note 15]</a> Following the <a href="/wiki/Black_Monday_(1987)" title="Black Monday (1987)">Crash of 1987</a>, equity options traded in American markets began to exhibit what is known as a "<a href="/wiki/Volatility_smile" title="Volatility smile">volatility smile</a>"; that is, for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices, and thus <a href="/wiki/Implied_volatility" title="Implied volatility">implied volatilities</a>, than what is suggested by BSM. (The pattern differs across various markets.) Modelling the volatility smile is an active area of research, and developments here – as well as implications re the standard theory – are discussed <a href="#Departures_from_normality">in the next section</a>. After the <a href="/wiki/2007%E2%80%932008_financial_crisis" class="mw-redirect" title="2007–2008 financial crisis">2007–2008 financial crisis</a>, a further development:<a href="#cite_note-Youmbi-75">[60]</a> as outlined, (<a href="/wiki/Over-the-counter_(finance)" title="Over-the-counter (finance)">over the counter</a>) derivative pricing had relied on the BSM risk neutral pricing framework, under the assumptions of funding at the risk free rate and the ability to perfectly replicate cashflows so as to fully hedge. This, in turn, is built on the assumption of a credit-risk-free environment – called into question during the crisis. Addressing this, therefore, issues such as <a href="/wiki/Counterparty_credit_risk" class="mw-redirect" title="Counterparty credit risk">counterparty credit risk</a>, funding costs and costs of capital are now additionally considered when pricing,<a href="#cite_note-76">[61]</a> and a <a href="/wiki/Credit_valuation_adjustment" title="Credit valuation adjustment">credit valuation adjustment</a>, or CVA – and potentially other valuation adjustments, collectively <a href="/wiki/XVA" title="XVA">xVA</a> – is generally added to the risk-neutral derivative value. The standard economic arguments can be extended to incorporate these various adjustments.<a href="#cite_note-Hull_White_2-77">[62]</a> A related, and perhaps more fundamental change, is that discounting is now on the <a href="/wiki/Overnight_index_swap" class="mw-redirect" title="Overnight index swap">Overnight Index Swap</a> (OIS) curve, as opposed to <a href="/wiki/LIBOR" class="mw-redirect" title="LIBOR">LIBOR</a> as used previously.<a href="#cite_note-Youmbi-75">[60]</a> This is because post-crisis, the <a href="/wiki/Overnight_rate" title="Overnight rate">overnight rate</a> is considered a better proxy for the "risk-free rate".<a href="#cite_note-78">[63]</a> (Also, practically, the interest paid on cash <a href="/wiki/Collateral_(finance)" title="Collateral (finance)">collateral</a> is usually the overnight rate; OIS discounting is then, sometimes, referred to as "<a href="/wiki/Credit_Support_Annex" title="Credit Support Annex">CSA</a> discounting".) <a href="/wiki/Swap_(finance)#Valuation" title="Swap (finance)">Swap pricing</a> – and, therefore, <a href="/wiki/Yield_curve" title="Yield curve">yield curve</a> construction – is further modified: previously, swaps were valued off a single "self discounting" interest rate curve; whereas post crisis, to accommodate OIS discounting, valuation is now under a "<a href="/wiki/Multi-curve_framework" class="mw-redirect" title="Multi-curve framework">multi-curve framework</a>" where "forecast curves" are constructed for each floating-leg <a href="/wiki/Libor#Maturities" title="Libor">LIBOR tenor</a>, with discounting on the common OIS curve. <div class="mw-heading mw-heading3"><h3 id="Corporate_finance_theory">Corporate finance theory</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=11" title="Edit section: Corporate finance theory">edit</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/Outline_of_corporate_finance#Theory" title="Outline of corporate finance">Outline of corporate finance § Theory</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Manual_decision_tree.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Manual_decision_tree.jpg/220px-Manual_decision_tree.jpg" decoding="async" width="220" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Manual_decision_tree.jpg/330px-Manual_decision_tree.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c6/Manual_decision_tree.jpg 2x" data-file-width="400" data-file-height="473" /></a><figcaption>Project valuation via decision tree.</figcaption></figure> Mirroring the <a href="#Certainty">above</a> developments, corporate finance valuations and decisioning no longer need assume "certainty". <a href="/wiki/Monte_Carlo_methods_in_finance" title="Monte Carlo methods in finance">Monte Carlo methods in finance</a> allow financial analysts to construct "<a href="/wiki/Stochastic" title="Stochastic">stochastic</a>" or <a href="/wiki/Probabilistic" class="mw-redirect" title="Probabilistic">probabilistic</a> corporate finance models, as opposed to the traditional static and <a href="/wiki/Deterministic" class="mw-redirect" title="Deterministic">deterministic</a> models;<a href="#cite_note-Damodaran_Risk-79">[64]</a> see <a href="/wiki/Corporate_finance#Quantifying_uncertainty" title="Corporate finance">Corporate finance § Quantifying uncertainty</a>. Relatedly, <a href="/wiki/Real_options" class="mw-redirect" title="Real options">Real Options theory</a> allows for owner – i.e. managerial – actions that impact underlying value: by incorporating option pricing logic, these actions are then applied to a distribution of future outcomes, changing with time, which then determine the "project's" valuation today.<a href="#cite_note-Damodaran-80">[65]</a> More traditionally, <a href="/wiki/Decision_tree" title="Decision tree">decision trees</a> – which are complementary – have been used to evaluate projects, by incorporating in the valuation (all) <a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">possible events</a> (or states) and consequent <a href="/wiki/Decision_making#Decision_making_in_business_and_management" class="mw-redirect" title="Decision making">management decisions</a>;<a href="#cite_note-81">[66]</a><a href="#cite_note-Damodaran_Risk-79">[64]</a> the correct discount rate here reflecting each decision-point's "non-diversifiable risk looking forward."<a href="#cite_note-Damodaran_Risk-79">[64]</a> <a href="#cite_note-83">[note 16]</a> Related to this, is the treatment of forecasted cashflows in <a href="/wiki/Equity_valuation" class="mw-redirect" title="Equity valuation">equity valuation</a>. In many cases, following Williams <a href="#Certainty">above</a>, the average (or most likely) cash-flows were discounted,<a href="#cite_note-Markowitz_interview-84">[68]</a> as opposed to a theoretically correct state-by-state treatment under uncertainty; see comments under <a href="/wiki/Financial_modeling#Accounting" title="Financial modeling">Financial modeling § Accounting</a>. In more modern treatments, then, it is the expected cashflows (in the <a href="/wiki/Expected_value" title="Expected value">mathematical sense</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{s}p_{s}X_{sj}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{s}p_{s}X_{sj}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef87829830fbf92ab4702deed0a5ddbb08e21a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.622ex; height:3.009ex;" alt="{\textstyle \sum _{s}p_{s}X_{sj}}">) combined into an overall value per forecast period which are discounted. <a href="#cite_note-Kruschwitz_and_Löffler-85">[69]</a> <a href="#cite_note-welch-86">[70]</a> <a href="#cite_note-87">[71]</a> <a href="#cite_note-Damodaran_Risk-79">[64]</a> And using the CAPM – or extensions – the discounting here is at the risk-free rate plus a premium linked to the uncertainty of the entity or project cash flows <a href="#cite_note-Damodaran_Risk-79">[64]</a> (essentially, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> combined). Other developments here include<a href="#cite_note-88">[72]</a> <a href="/wiki/Agency_theory" class="mw-redirect" title="Agency theory">agency theory</a>, which analyses the difficulties in motivating corporate management (the "agent"; in a different sense to the above) to act in the best interests of shareholders (the "principal"), rather than in their own interests; here emphasizing the issues interrelated with capital structure. <a href="#cite_note-89">[73]</a> <a href="/wiki/Clean_surplus_accounting" title="Clean surplus accounting">Clean surplus accounting</a> and the related <a href="/wiki/Residual_income_valuation" title="Residual income valuation">residual income valuation</a> provide a model that returns price as a function of earnings, expected returns, and change in <a href="/wiki/Book_value" title="Book value">book value</a>, as opposed to dividends. This approach, to some extent, arises due to the implicit contradiction of seeing value as a function of dividends, while also holding that dividend policy cannot influence value per Modigliani and Miller's "<a href="/wiki/Irrelevance_principle" class="mw-redirect" title="Irrelevance principle">Irrelevance principle</a>"; see <a href="/wiki/Dividend_policy#Relevance_of_dividend_policy" title="Dividend policy">Dividend policy § Relevance of dividend policy</a>. "Corporate finance" as a discipline more generally, building on Fisher <a href="#Certainty">above</a>, relates to the long term objective of maximizing the <a href="/wiki/Enterprise_value" title="Enterprise value">value of the firm</a> - and its <a href="/wiki/Total_shareholder_return" title="Total shareholder return">return to shareholders</a> - and thus also incorporates the areas of <a href="/wiki/Capital_structure" title="Capital structure">capital structure</a> and <a href="/wiki/Dividend_policy" title="Dividend policy">dividend policy</a>. <a href="#cite_note-90">[74]</a> Extensions of the theory here then also consider these latter, as follows: (i) <a href="/wiki/Corporate_finance#Capitalization_structure" title="Corporate finance">optimization re capitalization structure</a>, and theories here as to corporate choices and behavior: <a href="/wiki/Capital_structure_substitution_theory" title="Capital structure substitution theory">Capital structure substitution theory</a>, <a href="/wiki/Pecking_order_theory" title="Pecking order theory">Pecking order theory</a>, <a href="/wiki/Market_timing_hypothesis" title="Market timing hypothesis">Market timing hypothesis</a>, <a href="/wiki/Trade-off_theory_of_capital_structure" title="Trade-off theory of capital structure">Trade-off theory</a>; (ii) <a href="/wiki/Corporate_finance#Dividend_policy" title="Corporate finance">considerations and analysis re dividend policy</a>, additional to - and sometimes contrasting with - Modigliani-Miller, include: the <a href="/wiki/Dividend_policy#Walter's_model" title="Dividend policy">Walter model</a>, <a href="/wiki/John_Lintner#Lintner's_dividend_policy_model" title="John Lintner">Lintner model</a>, <a href="/wiki/Dividend_policy#Residuals_theory_of_dividends" title="Dividend policy">Residuals theory</a> and <a href="/wiki/Dividend_policy#Dividend_signaling_hypothesis" title="Dividend policy">signaling hypothesis</a>, as well as discussion re the observed <a href="/wiki/Clientele_effect" title="Clientele effect">clientele effect</a> and <a href="/wiki/Dividend_puzzle" title="Dividend puzzle">dividend puzzle</a>. As described, the typical application of real options is to <a href="/wiki/Capital_budgeting" title="Capital budgeting">capital budgeting</a> type problems. However, here, they are <a href="/wiki/Corporate_finance#Corporate_governance" title="Corporate finance">also applied</a> to problems of capital structure and dividend policy, and to the related design of corporate securities; <a href="#cite_note-Garbade-91">[75]</a> and since stockholder and bondholders have different objective functions, in the analysis of the <a href="/wiki/Corporate_finance#Corporate_governance" title="Corporate finance">related agency problems</a>. <a href="#cite_note-Damodaran-80">[65]</a> In all of these cases, state-prices can provide the market-implied information relating to the corporate, <a href="#State_prices">as above</a>, which is then applied to the analysis. For example, <a href="/wiki/Convertible_bond" title="Convertible bond">convertible bonds</a> can (must) be priced consistent with the (recovered) state-prices of the corporate's equity.<a href="#cite_note-corp_fin_state_prices-25">[21]</a><a href="#cite_note-Kruschwitz_and_Löffler-85">[69]</a> <div class="mw-heading mw-heading3"><h3 id="Financial_markets">Financial markets</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=12" title="Edit section: Financial markets">edit</a>]</div> The discipline, as outlined, also includes a formal study of <a href="/wiki/Financial_market" title="Financial market">financial markets</a>. Of interest especially are market regulation and <a href="/wiki/Market_microstructure" title="Market microstructure">market microstructure</a>, and their relationship to <a href="/wiki/Financial_market_efficiency" title="Financial market efficiency">price efficiency</a>. <a href="/wiki/Regulatory_economics" title="Regulatory economics">Regulatory economics</a> studies, in general, the economics of regulation. In the context of finance, it will address the impact of <a href="/wiki/Financial_regulation" title="Financial regulation">financial regulation</a> on the functioning of markets and the efficiency of prices, while also weighing the corresponding increases in market confidence and <a href="/wiki/Financial_stability" title="Financial stability">financial stability</a>. Research here considers how, and to what extent, regulations relating to disclosure (<a href="/wiki/Earnings_guidance" title="Earnings guidance">earnings guidance</a>, <a href="/wiki/Annual_report" title="Annual report">annual reports</a>), <a href="/wiki/Insider_trading" title="Insider trading">insider trading</a>, and <a href="/wiki/Short_(finance)#Regulations" title="Short (finance)">short-selling</a> will impact price efficiency, the <a href="/wiki/Cost_of_equity" title="Cost of equity">cost of equity</a>, and <a href="/wiki/Market_liquidity" title="Market liquidity">market liquidity</a>.<a href="#cite_note-92">[76]</a> Market microstructure is concerned with the details of how exchange occurs in markets (with <a href="/wiki/Walrasian_auction" title="Walrasian auction">Walrasian-</a>, <a href="/wiki/Matching_market" class="mw-redirect" title="Matching market">matching-</a>, <a href="/wiki/Fisher_market" title="Fisher market">Fisher-</a>, and <a href="/wiki/Arrow%E2%80%93Debreu_model#Intuitive_description_of_the_Arrow–Debreu_model" title="Arrow–Debreu model">Arrow-Debreu markets</a> as prototypes), and "analyzes how specific trading mechanisms affect the price formation process",<a href="#cite_note-93">[77]</a> examining the ways in which the processes of a market affect determinants of <a href="/wiki/Transaction_costs" class="mw-redirect" title="Transaction costs">transaction costs</a>, prices, quotes, volume, and trading behavior. It has been used, for example, in providing explanations for <a href="/wiki/Real_exchange-rate_puzzles" title="Real exchange-rate puzzles">long-standing exchange rate puzzles</a>,<a href="#cite_note-94">[78]</a> and for the <a href="/wiki/Equity_premium_puzzle" title="Equity premium puzzle">equity premium puzzle</a>.<a href="#cite_note-95">[79]</a> In contrast to the above classical approach, models here explicitly allow for (testing the impact of) <a href="/wiki/Frictionless_market" title="Frictionless market">market frictions</a> and other <a href="/wiki/Perfect_market" class="mw-redirect" title="Perfect market">imperfections</a>; see also <a href="/wiki/Market_design" title="Market design">market design</a>. For both regulation <a href="#cite_note-96">[80]</a> and microstructure,<a href="#cite_note-97">[81]</a> and generally,<a href="#cite_note-LeBaron-98">[82]</a> <a href="/wiki/Agent-based_model#In_economics_and_social_sciences" title="Agent-based model">agent-based models</a> can be developed <a href="#cite_note-ERIM-99">[83]</a> to <a href="/wiki/Agent-based_computational_economics#Example:_finance" title="Agent-based computational economics">examine any impact</a> due to a change in structure or policy - or <a href="/wiki/Artificial_economics#Method" class="mw-redirect" title="Artificial economics">to make inferences</a> re market dynamics - <a href="/wiki/Computer_experiment" title="Computer experiment">by testing these</a> in an artificial financial market, or AFM. <a href="#cite_note-101">[note 17]</a> This approach, essentially <a href="/wiki/Discrete-event_simulation" title="Discrete-event simulation">simulated</a> trade between numerous <a href="/wiki/Agent_(economics)" title="Agent (economics)">agents</a>, "typically uses <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a> technologies [often <a href="/wiki/Genetic_algorithms" class="mw-redirect" title="Genetic algorithms">genetic algorithms</a> and <a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">neural nets</a>] to represent the <a href="/wiki/Adaptive_market_hypothesis" title="Adaptive market hypothesis">adaptive behaviour</a> of market participants".<a href="#cite_note-ERIM-99">[83]</a> These <a href="/wiki/Microfoundations" title="Microfoundations">'bottom-up' models</a> "start from first principals of agent behavior",<a href="#cite_note-LeBaron2-100">[84]</a> with participants modifying their trading strategies having learned over time, and "are able to describe macro features [i.e. <a href="/wiki/Stylized_fact" title="Stylized fact">stylized facts</a>] <a href="/wiki/Emergence#Economics" title="Emergence">emerging</a> from a soup of individual interacting strategies".<a href="#cite_note-LeBaron2-100">[84]</a> Agent-based models depart further from the classical approach — the <a href="/wiki/Representative_agent" title="Representative agent">representative agent</a>, as outlined — in that they introduce <a href="/wiki/Heterogeneity_in_economics" title="Heterogeneity in economics">heterogeneity</a> into the environment (thereby addressing, also, the <a href="/wiki/Aggregation_problem" title="Aggregation problem">aggregation problem</a>). <div class="mw-heading mw-heading2"><h2 id="Challenges_and_criticism">Challenges and criticism</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=13" title="Edit section: Challenges and criticism">edit</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/Neoclassical_economics#Criticisms" title="Neoclassical economics">Neoclassical economics § Criticisms</a>, <a href="/wiki/Financial_mathematics#Criticism" class="mw-redirect" title="Financial mathematics">Financial mathematics § Criticism</a>, and <a href="/wiki/Financial_engineering#Criticisms" title="Financial engineering">Financial engineering § Criticisms</a></div> As above, there is a very close link between: the <a href="/wiki/Random_walk_hypothesis" title="Random walk hypothesis">random walk hypothesis</a>, with the associated belief that price changes should follow a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, on the one hand; and market efficiency and <a href="/wiki/Rational_expectations" title="Rational expectations">rational expectations</a>, on the other. Wide departures from these are commonly observed, and there are thus, respectively, two main sets of challenges. <div class="mw-heading mw-heading3"><h3 id="Departures_from_normality">Departures from normality</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=14" title="Edit section: Departures from normality">edit</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/Capital_asset_pricing_model#Problems" title="Capital asset pricing model">Capital asset pricing model § Problems</a>, and <a href="/wiki/Black%E2%80%93Scholes_model#Criticism_and_comments" title="Black–Scholes model">Black–Scholes model § Criticism and comments</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ivsrf.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Ivsrf.gif/220px-Ivsrf.gif" decoding="async" width="220" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Ivsrf.gif/330px-Ivsrf.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b1/Ivsrf.gif 2x" data-file-width="395" data-file-height="361" /></a><figcaption>Implied volatility surface. The Z-axis represents implied volatility in percent, and X and Y axes represent the <a href="/wiki/Greeks_(finance)#Delta" title="Greeks (finance)">option delta</a>, and the days to maturity.</figcaption></figure> As discussed, the assumptions that market prices follow a <a href="/wiki/Random_walk" title="Random walk">random walk</a> and that asset returns are normally distributed are fundamental. Empirical evidence, however, suggests that these assumptions may not hold, and that in practice, traders, analysts <a href="/wiki/Financial_risk_management#Banking" title="Financial risk management">and risk managers</a> frequently modify the "standard models" (see <a href="/wiki/Kurtosis_risk" title="Kurtosis risk">Kurtosis risk</a>, <a href="/wiki/Skewness_risk" title="Skewness risk">Skewness risk</a>, <a href="/wiki/Long_tail" title="Long tail">Long tail</a>, <a href="/wiki/Model_risk" title="Model risk">Model risk</a>). In fact, <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a> had discovered already in the 1960s <a href="#cite_note-102">[85]</a> that changes in financial prices do not follow a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, the basis for much option pricing theory, although this observation was slow to find its way into mainstream financial economics. <a href="#cite_note-Taleb_Mandelbrot-103">[86]</a> <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> have been introduced to overcome problems with the realism of the above "classical" financial models; while <a href="/wiki/Jump_diffusion#In_economics_and_finance" title="Jump diffusion">jump diffusion models</a> allow for (option) pricing incorporating <a href="/wiki/Jump_process" title="Jump process">"jumps"</a> in the <a href="/wiki/Spot_price" class="mw-redirect" title="Spot price">spot price</a>.<a href="#cite_note-holes-104">[87]</a> Risk managers, similarly, complement (or substitute) the standard <a href="/wiki/Value_at_risk" title="Value at risk">value at risk</a> models with <a href="/wiki/Historical_simulation_(finance)" title="Historical simulation (finance)">historical simulations</a>, <a href="/wiki/Mixture_model#A_financial_model" title="Mixture model">mixture models</a>, <a href="/wiki/Principal_component_analysis" title="Principal component analysis">principal component analysis</a>, <a href="/wiki/Extreme_value_theory" title="Extreme value theory">extreme value theory</a>, as well as models for <a href="/wiki/Volatility_clustering" title="Volatility clustering">volatility clustering</a>.<a href="#cite_note-105">[88]</a> For further discussion see <a href="/wiki/Fat-tailed_distribution#Applications_in_economics" title="Fat-tailed distribution">Fat-tailed distribution § Applications in economics</a>, and <a href="/wiki/Value_at_risk#Criticism" title="Value at risk">Value at risk § Criticism</a>. Portfolio managers, likewise, have modified their optimization criteria and algorithms; see <a href="#Portfolio_theory">§ Portfolio theory</a> above. Closely related is the <a href="/wiki/Volatility_smile" title="Volatility smile">volatility smile</a>, where, as above, <a href="/wiki/Implied_volatility" title="Implied volatility">implied volatility</a> – the volatility corresponding to the BSM price – is observed to differ as a function of <a href="/wiki/Strike_price" title="Strike price">strike price</a> (i.e. <a href="/wiki/Moneyness" title="Moneyness">moneyness</a>), true only if the price-change distribution is non-normal, unlike that assumed by BSM. The term structure of volatility describes how (implied) volatility differs for related options with different maturities. An implied volatility surface is then a three-dimensional surface plot of volatility smile and term structure. These empirical phenomena negate the assumption of constant volatility – and <a href="/wiki/Log-normal" class="mw-redirect" title="Log-normal">log-normality</a> – upon which Black–Scholes is built.<a href="#cite_note-Haug_Taleb-49">[40]</a><a href="#cite_note-holes-104">[87]</a> Within institutions, the function of Black–Scholes is now, largely, to communicate prices via implied volatilities, much like bond prices are communicated via <a href="/wiki/Yield_to_maturity" title="Yield to maturity">YTM</a>; see <a href="/wiki/Black%E2%80%93Scholes_model#The_volatility_smile" title="Black–Scholes model">Black–Scholes model § The volatility smile</a>. In consequence traders (<a href="/wiki/Financial_risk_management#Banking" title="Financial risk management">and risk managers</a>) now, instead, use "smile-consistent" models, firstly, when valuing derivatives not directly mapped to the surface, facilitating the pricing of other, i.e. non-quoted, strike/maturity combinations, or of non-European derivatives, and generally for hedging purposes. The two main approaches are <a href="/wiki/Local_volatility" title="Local volatility">local volatility</a> and <a href="/wiki/Stochastic_volatility" title="Stochastic volatility">stochastic volatility</a>. The first returns the volatility which is "local" to each spot-time point of the <a href="/wiki/Finite_difference_methods_for_option_pricing" title="Finite difference methods for option pricing">finite difference-</a> or <a href="/wiki/Monte_Carlo_methods_for_option_pricing" title="Monte Carlo methods for option pricing">simulation-based valuation</a>; i.e. as opposed to implied volatility, which holds overall. In this way calculated prices – and numeric structures – are market-consistent in an arbitrage-free sense. The second approach assumes that the volatility of the underlying price is a stochastic process rather than a constant. Models here are first <a href="/wiki/Stochastic_volatility#Calibration_and_estimation" title="Stochastic volatility">calibrated to observed prices</a>, and are then applied to the valuation or hedging in question; the most common are <a href="/wiki/Heston_model" title="Heston model">Heston</a>, <a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR</a> and <a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">CEV</a>. This approach addresses certain problems identified with hedging under local volatility.<a href="#cite_note-106">[89]</a> Related to local volatility are the <a href="/wiki/Lattice_model_(finance)" title="Lattice model (finance)">lattice</a>-based <a href="/wiki/Implied_binomial_tree" class="mw-redirect" title="Implied binomial tree">implied-binomial</a> and <a href="/wiki/Implied_trinomial_tree" class="mw-redirect" title="Implied trinomial tree">-trinomial trees</a> – essentially a discretization of the approach – which are similarly, but less commonly,<a href="#cite_note-Figlewski-24">[20]</a> used for pricing; these are built on state-prices recovered from the surface. <a href="/wiki/Edgeworth_binomial_tree" class="mw-redirect" title="Edgeworth binomial tree">Edgeworth binomial trees</a> allow for a specified (i.e. non-Gaussian) <a href="/wiki/Skewness" title="Skewness">skew</a> and <a href="/wiki/Kurtosis" title="Kurtosis">kurtosis</a> in the spot price; priced here, options with differing strikes will return differing implied volatilities, and the tree can be calibrated to the smile as required.<a href="#cite_note-107">[90]</a> Similarly purposed (and derived) <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form models</a> were also developed. <a href="#cite_note-108">[91]</a> As discussed, additional to assuming log-normality in returns, "classical" BSM-type models also (implicitly) assume the existence of a credit-risk-free environment, where one can perfectly replicate cashflows so as to fully hedge, and then discount at "the" risk-free-rate. And therefore, post crisis, the various x-value adjustments must be employed, effectively correcting the risk-neutral value for <a href="/wiki/Counterparty_credit_risk" class="mw-redirect" title="Counterparty credit risk">counterparty-</a> and <a href="/wiki/XVA#Valuation_adjustments" title="XVA">funding-related</a> risk. These xVA are additional to any smile or surface effect. This is valid as the surface is built on price data relating to fully collateralized positions, and there is therefore no "<a href="/wiki/Double_counting_(accounting)" title="Double counting (accounting)">double counting</a>" of credit risk (etc.) when appending xVA. (Were this not the case, then each counterparty would have its own surface...) As mentioned at top, mathematical finance (and particularly <a href="/wiki/Financial_engineering" title="Financial engineering">financial engineering</a>) is more concerned with mathematical consistency (and market realities) than compatibility with economic theory, and the above "extreme event" approaches, smile-consistent modeling, and valuation adjustments should then be seen in this light. Recognizing this, critics of financial economics - especially vocal since the <a href="/wiki/2007%E2%80%932008_financial_crisis" class="mw-redirect" title="2007–2008 financial crisis">2007–2008 financial crisis</a> - suggest that instead, the theory needs revisiting almost entirely: <a href="#cite_note-109">[note 18]</a> <dl><dd>"The current system, based on the idea that risk is distributed in the shape of a bell curve, is flawed... The problem is [that economists and practitioners] never abandon the bell curve. They are like medieval astronomers who believe the sun revolves around the earth and are <a href="/wiki/Geocentric_model#Ptolemaic_system" title="Geocentric model">furiously tweaking their geo-centric math</a> in the face of contrary evidence. They will never get this right; <a href="/wiki/Copernican_Revolution" title="Copernican Revolution">they need their Copernicus</a>."<a href="#cite_note-110">[92]</a></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Departures_from_rationality">Departures from rationality</h3>[<a href="/w/index.php?title=Financial_economics&action=edit&section=15" title="Edit section: Departures from rationality">edit</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/Efficient-market_hypothesis#Criticism" title="Efficient-market hypothesis">Efficient-market hypothesis § Criticism</a>, and <a href="/wiki/Rational_expectations#Criticism" title="Rational expectations">Rational expectations § Criticism</a></div> <table class="wikitable floatright" width="200"> <tbody><tr align="center"> <td colspan="1"><a href="/wiki/Market_anomaly" title="Market anomaly">Market anomalies</a> and <a href="/wiki/Economic_puzzle" title="Economic puzzle">economic puzzles</a> </td></tr> <tr> <td rowspan="2"> <ul><li><a href="/wiki/Calendar_effect" title="Calendar effect">Calendar effect</a> <ul><li><a href="/wiki/January_effect" title="January effect">January effect</a></li> <li><a href="/wiki/Sell_in_May" title="Sell in May">Sell in May</a></li> <li><a href="/wiki/Mark_Twain_effect" title="Mark Twain effect">Mark Twain effect</a></li> <li><a href="/wiki/Santa_Claus_rally" title="Santa Claus rally">Santa Claus rally</a></li></ul></li> <li><a href="/wiki/Closed-end_fund_puzzle" class="mw-redirect" title="Closed-end fund puzzle">Closed-end fund puzzle</a></li> <li><a href="/wiki/Dividend_puzzle" title="Dividend puzzle">Dividend puzzle</a></li> <li><a href="/wiki/Equity_home_bias_puzzle" title="Equity home bias puzzle">Equity home bias puzzle</a></li> <li><a href="/wiki/Equity_premium_puzzle" title="Equity premium puzzle">Equity premium puzzle</a></li> <li><a href="/wiki/Forward_premium_anomaly" title="Forward premium anomaly">Forward premium anomaly</a></li> <li><a href="/wiki/Low-volatility_anomaly" title="Low-volatility anomaly">Low-volatility anomaly</a></li> <li><a href="/wiki/Momentum_(finance)" title="Momentum (finance)">Momentum anomaly</a></li> <li><a href="/wiki/Neglected_firm_effect" title="Neglected firm effect">Neglected firm effect</a></li> <li><a href="/wiki/Post-earnings-announcement_drift" class="mw-redirect" title="Post-earnings-announcement drift">Post-earnings-announcement drift</a></li> <li><a href="/wiki/Real_exchange-rate_puzzles" title="Real exchange-rate puzzles">Real exchange-rate puzzles</a></li></ul> </td></tr> </tbody></table> As seen, a common assumption is that financial decision makers act rationally; see <a href="/wiki/Homo_economicus" title="Homo economicus">Homo economicus</a>. Recently, however, researchers in <a href="/wiki/Experimental_economics" title="Experimental economics">experimental economics</a> and <a href="/wiki/Experimental_finance" title="Experimental finance">experimental finance</a> have challenged this assumption <a href="/wiki/Empirical_evidence" title="Empirical evidence">empirically</a>. These assumptions are also challenged <a href="/wiki/Theory" title="Theory">theoretically</a>, by <a href="/wiki/Behavioral_finance" class="mw-redirect" title="Behavioral finance">behavioral finance</a>, a discipline primarily concerned with the limits to rationality of economic agents. <a href="#cite_note-111">[note 19]</a> For related criticisms re corporate finance theory vs its practice see:.<a href="#cite_note-112">[93]</a> Various persistent <a href="/wiki/Market_anomaly" title="Market anomaly">market anomalies</a> have also been documented as consistent with and complementary to price or return distortions – e.g. <a href="/wiki/Size_premium" title="Size premium">size premiums</a> – which appear to contradict the <a href="/wiki/Efficient-market_hypothesis" title="Efficient-market hypothesis">efficient-market hypothesis</a>. Within these market anomalies, <a href="/wiki/Calendar_effect" title="Calendar effect">calendar effects</a> are the most commonly referenced group. Related to these are various of the <a href="/wiki/Economic_puzzle" title="Economic puzzle">economic puzzles</a>, concerning phenomena similarly contradicting the theory. The <a href="/wiki/Equity_premium_puzzle" title="Equity premium puzzle">equity premium puzzle</a>, as one example, arises in that the difference between the observed returns on stocks as compared to government bonds is consistently higher than the <a href="/wiki/Risk_premium" title="Risk premium">risk premium</a> rational equity investors should demand, an "<a href="/wiki/Abnormal_return" title="Abnormal return">abnormal return</a>". For further context see <a href="/wiki/Random_walk_hypothesis#A_non-random_walk_hypothesis" title="Random walk hypothesis">Random walk hypothesis § A non-random walk hypothesis</a>, and sidebar for specific instances. More generally, and, again, particularly following the <a href="/wiki/2007%E2%80%932008_financial_crisis" class="mw-redirect" title="2007–2008 financial crisis">2007–2008 financial crisis</a>, financial economics (and <a href="/wiki/Mathematical_finance" title="Mathematical finance">mathematical finance</a>) has been subjected to deeper criticism. Notable here is <a href="/wiki/Nassim_Nicholas_Taleb" title="Nassim Nicholas Taleb">Nassim Nicholas Taleb</a>, whose critique overlaps the above, but extends also to the institutional <a href="#cite_note-113">[94]</a> aspects of (academic) finance. His <a href="/wiki/Black_swan_theory" title="Black swan theory">Black swan theory</a> emphasizes that although events of large magnitude and consequence play a major role in finance, since these are (statistically) unexpected, they are "ignored" by economists and traders. Thus, although a "<a href="/wiki/Taleb_distribution" title="Taleb distribution">Taleb distribution</a>" - which normally provides a payoff of small positive returns, while carrying a small but significant risk of catastrophic losses - more realistically describes markets than current models, the latter continue to be preferred (even with <a href="/wiki/Risk_manager" class="mw-redirect" title="Risk manager">professionals here</a> acknowledging that it only "generally works" or only "works on average"). <a href="#cite_note-114">[95]</a> Here, <a href="/wiki/Financial_crises" class="mw-redirect" title="Financial crises">financial crises</a> have been a topic of interest <a href="#cite_note-115">[96]</a> and, in particular, <a href="/wiki/2008_financial_crisis#Prediction_by_economists" title="2008 financial crisis">the failure</a> of (financial) economics to model (and predict) these. See <a href="/wiki/Financial_crisis#Theories" title="Financial crisis">Financial crisis § Theories</a>. The related problem of <a href="/wiki/Systemic_risk" title="Systemic risk">systemic risk</a>, has also received attention. Where companies hold securities in each other, then this interconnectedness may entail a "valuation chain" – and the performance of one company, or security, here will impact all, a phenomenon not easily modeled, regardless of whether the individual models are correct. See: <a href="/wiki/Systemic_risk#Inadequacy_of_classic_valuation_models" title="Systemic risk">Systemic risk § Inadequacy of classic valuation models</a>; <a href="/wiki/Cascades_in_financial_networks" title="Cascades in financial networks">Cascades in financial networks</a>; <a href="/wiki/Flight-to-quality" title="Flight-to-quality">Flight-to-quality</a>. Areas of research attempting to explain (or at least model) these phenomena, and crises, include <a href="#cite_note-Farmer_Geanakoplos-18">[15]</a> <a href="/wiki/Market_microstructure" title="Market microstructure">market microstructure</a> and <a href="/wiki/Heterogeneous_agent_model" class="mw-redirect" title="Heterogeneous agent model">Heterogeneous agent models</a>, as above. The latter is extended to <a href="/wiki/Agent-based_computational_economics" title="Agent-based computational economics">agent-based computational models</a>; here, <a href="#cite_note-LeBaron-98">[82]</a> as mentioned, price is treated as an <a href="/wiki/Emergent_phenomenon" class="mw-redirect" title="Emergent phenomenon">emergent phenomenon</a>, resulting from the interaction of the various market participants (agents). The <a href="/wiki/Noisy_market_hypothesis" title="Noisy market hypothesis">noisy market hypothesis</a> argues that prices can be influenced by speculators and <a href="/wiki/Momentum_trader" class="mw-redirect" title="Momentum trader">momentum traders</a>, as well as by <a href="/wiki/Insider_trading" title="Insider trading">insiders</a> and institutions that often buy and sell stocks for reasons unrelated to <a href="/wiki/Fundamental_value" class="mw-redirect" title="Fundamental value">fundamental value</a>; see <a href="/wiki/Noise_(economic)" title="Noise (economic)">Noise (economic)</a> and <a href="/wiki/Noise_trader" title="Noise trader">Noise trader</a>. The <a href="/wiki/Adaptive_market_hypothesis" title="Adaptive market hypothesis">adaptive market hypothesis</a> is an attempt to reconcile the efficient market hypothesis with behavioral economics, by applying the principles of <a href="/wiki/Evolution" title="Evolution">evolution</a> to financial interactions. An <a href="/wiki/Information_cascade" title="Information cascade">information cascade</a>, alternatively, shows market participants engaging in the same acts as others ("<a href="/wiki/Herd_behavior" title="Herd behavior">herd behavior</a>"), despite contradictions with their private information. <a href="/wiki/Copula_(probability_theory)#Quantitative_finance" class="mw-redirect" title="Copula (probability theory)">Copula-based modelling</a> has similarly been applied. See also <a href="/wiki/Hyman_Minsky" title="Hyman Minsky">Hyman Minsky</a>'s <a href="/wiki/Hyman_Minsky#Minsky's_financial_instability-hypothesis" title="Hyman Minsky">"financial instability hypothesis"</a>, as well as <a href="/wiki/George_Soros#Reflexivity,_financial_markets,_and_economic_theory" title="George Soros">George Soros' application</a> of <a href="/wiki/Reflexivity_(social_theory)#In_economics" title="Reflexivity (social theory)">"reflexivity"</a>. However, studies show that despite inefficiencies, asset prices generally follow a random walk, making it difficult to consistently outperform market averages and achieve <a href="/wiki/Alpha_(investment)" class="mw-redirect" title="Alpha (investment)">"alpha"</a>.<a href="#cite_note-116">[97]</a> As an explanation for these inefficiencies, institutional <a href="/wiki/Limits_to_arbitrage" title="Limits to arbitrage">limits to arbitrage</a> are sometimes referenced (as opposed to factors directly contradictory to the theory). The practical implication is that <a href="/wiki/Passive_investing" class="mw-redirect" title="Passive investing">passive investing</a> (e.g. via low-cost <a href="/wiki/Index_fund" title="Index fund">index funds</a>) should, on average, serve better than any other <a href="/wiki/Active_investing" class="mw-redirect" title="Active investing">active strategy</a>.<a href="#cite_note-two-117">[98]</a> <a href="#cite_note-118">[note 20]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=16" title="Edit section: See also">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><a href="/wiki/Category:Finance_theories" title="Category:Finance theories">Category:Finance theories</a></li> <li><a href="/wiki/Category:Financial_models" title="Category:Financial models">Category:Financial models</a></li> <li><a href="/wiki/Deutsche_Bank_Prize_in_Financial_Economics" title="Deutsche Bank Prize in Financial Economics">Deutsche Bank Prize in Financial Economics</a> – scholarly awardPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Finance#Financial_theory" title="Finance">Finance § Financial theory</a></li> <li><a href="/wiki/Fischer_Black_Prize" title="Fischer Black Prize">Fischer Black Prize</a> – economics awardPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/List_of_financial_economics_articles" class="mw-redirect" title="List of financial economics articles">List of financial economics articles</a> – Overview of finance and finance-related topicsPages displaying short descriptions of redirect targets</li> <li><a href="/wiki/List_of_financial_economists" class="mw-redirect" title="List of financial economists">List of financial economists</a></li> <li><a href="/wiki/List_of_unsolved_problems_in_economics#Financial_economics" title="List of unsolved problems in economics">List of unsolved problems in economics § Financial economics</a></li> <li><a href="/wiki/Master_of_Financial_Economics" title="Master of Financial Economics">Master of Financial Economics</a> – graduate degree with a major in financial economicsPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Monetary_economics" title="Monetary economics">Monetary economics</a> – Branch of economics covering theories of money</li> <li><a href="/wiki/Outline_of_economics" title="Outline of economics">Outline of economics</a></li> <li><a href="/wiki/Outline_of_corporate_finance" title="Outline of corporate finance">Outline of corporate finance</a> – Overview of corporate finance and corporate finance-related topics</li> <li><a href="/wiki/Outline_of_finance" title="Outline of finance">Outline of finance</a> – Overview of finance and finance-related topics</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Historical_notes">Historical notes</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=17" title="Edit section: Historical notes">edit</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/Quantitative_analyst#Seminal_publications" class="mw-redirect" title="Quantitative analyst">Quantitative analyst § Seminal publications</a>, and <a href="/wiki/List_of_quantitative_analysts" title="List of quantitative analysts">List of quantitative analysts</a></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Its history is correspondingly early: <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a> developed the concept of present value already in 1202 in his <a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a>. <a href="/wiki/Compound_interest" title="Compound interest">Compound interest</a> was discussed in depth by <a href="/wiki/Richard_Witt" class="mw-redirect" title="Richard Witt">Richard Witt</a> in 1613, in his Arithmeticall Questions,<a href="#cite_note-7">[7]</a> and was further developed by <a href="/wiki/Johan_de_Witt" title="Johan de Witt">Johan de Witt</a> in 1671 <a href="#cite_note-8">[8]</a> and by <a href="/wiki/Edmond_Halley" title="Edmond Halley">Edmond Halley</a> in 1705.<a href="#cite_note-9">[9]</a> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> These ideas originate with <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> and <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> in 1654. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> The development here is originally due to <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> in 1738; it was later formalized by <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> and <a href="/wiki/Oskar_Morgenstern" title="Oskar Morgenstern">Oskar Morgenstern</a> in 1947. </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> State prices originate with <a href="/wiki/Kenneth_Arrow" title="Kenneth Arrow">Kenneth Arrow</a> and <a href="/wiki/G%C3%A9rard_Debreu" title="Gérard Debreu">Gérard Debreu</a> in 1954.<a href="#cite_note-20">[17]</a> <a href="/wiki/Lionel_W._McKenzie" title="Lionel W. McKenzie">Lionel W. McKenzie</a> is also cited for his independent proof of equilibrium existence in 1954.<a href="#cite_note-21">[18]</a> <a href="/wiki/Douglas_Breeden" title="Douglas Breeden">Breeden</a> and <a href="/wiki/Robert_Litzenberger" title="Robert Litzenberger">Litzenberger's</a> work in 1978<a href="#cite_note-22">[19]</a> established the use of state prices in financial economics. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> The theorem of <a href="/wiki/Franco_Modigliani" title="Franco Modigliani">Franco Modigliani</a> and <a href="/wiki/Merton_Miller" title="Merton Miller">Merton Miller</a> is often called the "capital structure irrelevance principle"; it is presented in two key papers of 1958,<a href="#cite_note-MM1-28">[24]</a> and 1963.<a href="#cite_note-MM2-29">[25]</a> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <a href="/wiki/John_Burr_Williams" title="John Burr Williams">John Burr Williams</a> published his "Theory" in 1938; NPV was recommended to corporate managers by <a href="/wiki/Joel_Dean_(economist)" title="Joel Dean (economist)">Joel Dean</a> in 1951. </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> In fact, "Fisher (1930, [The Theory of Interest]) is the seminal work for most of the financial theory of investments during the twentieth century… Fisher develops the first formal equilibrium model of an economy with both intertemporal exchange and production. In so doing, at one swoop, he not only derives present value calculations as a natural economic outcome in calculating wealth, he also justifies the maximization of present value as the goal of production and derives determinants of the interest rates that are used to calculate present value."<a href="#cite_note-Rubinstein2-15">[12]</a>: 55  </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> The EMH was presented by <a href="/wiki/Eugene_Fama" title="Eugene Fama">Eugene Fama</a> in a 1970 review paper,<a href="#cite_note-41">[34]</a> consolidating previous works re random walks in stock prices: <a href="/wiki/Jules_Regnault" title="Jules Regnault">Jules Regnault</a> (1863); <a href="/wiki/Louis_Bachelier" title="Louis Bachelier">Louis Bachelier</a> (1900); <a href="/wiki/Maurice_Kendall" title="Maurice Kendall">Maurice Kendall</a> (1953); <a href="/wiki/Paul_Cootner" title="Paul Cootner">Paul Cootner</a> (1964); and <a href="/wiki/Paul_Samuelson" title="Paul Samuelson">Paul Samuelson</a> (1965), among others. </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> The efficient frontier was introduced by <a href="/wiki/Harry_Markowitz" title="Harry Markowitz">Harry Markowitz</a> in 1952. The CAPM was derived by <a href="/wiki/Jack_L._Treynor" title="Jack L. Treynor">Jack Treynor</a> (1961, 1962), <a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William F. Sharpe</a> (1964), <a href="/wiki/John_Lintner" title="John Lintner">John Lintner</a> (1965), and <a href="/wiki/Jan_Mossin" title="Jan Mossin">Jan Mossin</a> (1966) independently. Already in 1940, <a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a><a href="#cite_note-43">[35]</a> had described the mean-variance method, in the context of <a href="/wiki/Reinsurance" title="Reinsurance">reinsurance</a>. </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> "BSM" – two seminal 1973 papers by <a href="/wiki/Fischer_Black" title="Fischer Black">Fischer Black</a> and <a href="/wiki/Myron_Scholes" title="Myron Scholes">Myron Scholes</a>,<a href="#cite_note-BlackScholes_paper-47">[38]</a> and <a href="/wiki/Robert_C._Merton" title="Robert C. Merton">Robert C. Merton</a><a href="#cite_note-Merton_paper-48">[39]</a> – is consistent with "previous versions of the formula" of <a href="/wiki/Louis_Bachelier" title="Louis Bachelier">Louis Bachelier</a> (1900) and <a href="/wiki/Edward_O._Thorp" title="Edward O. Thorp">Edward O. Thorp</a> (1967);<a href="#cite_note-Haug_Taleb-49">[40]</a> although these were more "actuarial" in flavor, and had not established risk-neutral discounting.<a href="#cite_note-Derman-16">[13]</a> <a href="/wiki/Vinzenz_Bronzin" title="Vinzenz Bronzin">Vinzenz Bronzin</a> (1908) produced very early results, also. <a href="/wiki/Case_Sprenkle" title="Case Sprenkle">Case Sprenkle</a> (1961) <a href="#cite_note-50">[41]</a> had published a formula for the price of a call-option which, with adjustments, satisfied the BSM partial differential equation.<a href="#cite_note-51">[42]</a> </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <a href="/wiki/Kiyosi_It%C3%B4" title="Kiyosi Itô">Kiyosi Itô</a> published his Lemma in 1944. <a href="/wiki/Paul_Samuelson" title="Paul Samuelson">Paul Samuelson</a><a href="#cite_note-53">[43]</a> introduced this area of mathematics into finance in 1965; Robert Merton promoted continuous <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">stochastic calculus</a> and continuous-time processes from 1969. <a href="#cite_note-54">[44]</a> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> The single-index model was developed by William Sharpe in 1963. <a href="#cite_note-58">[47]</a> APT was developed by <a href="/wiki/Stephen_Ross_(economist)" title="Stephen Ross (economist)">Stephen Ross</a> in 1976. <a href="#cite_note-59">[48]</a> </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> The universal portfolio algorithm was published by <a href="/wiki/Thomas_M._Cover" title="Thomas M. Cover">Thomas M. Cover</a> in 1991. The Black–Litterman model was developed in 1990 at <a href="/wiki/Goldman_Sachs" title="Goldman Sachs">Goldman Sachs</a> by Fischer Black and <a href="/wiki/Robert_Litterman" title="Robert Litterman">Robert Litterman</a>, and published in 1991. </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> The binomial model was first proposed by <a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William Sharpe</a> in the 1978 edition of Investments (<style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/013504605X" title="Special:BookSources/013504605X">013504605X</a>), and in 1979 formalized by <a href="/wiki/John_Carrington_Cox" title="John Carrington Cox">Cox</a>, <a href="/wiki/Stephen_Ross_(economist)" title="Stephen Ross (economist)">Ross</a> and <a href="/wiki/Mark_Rubinstein" title="Mark Rubinstein">Rubinstein</a> <a href="#cite_note-65">[52]</a> and by Rendleman and Bartter. <a href="#cite_note-66">[53]</a> <a href="/wiki/Finite_difference_methods_for_option_pricing" title="Finite difference methods for option pricing">Finite difference methods for option pricing</a> were due to <a href="/wiki/Eduardo_Schwartz" title="Eduardo Schwartz">Eduardo Schwartz</a> in 1977.<a href="#cite_note-Schwartz-67">[54]</a> <a href="/wiki/Monte_Carlo_methods_for_option_pricing" title="Monte Carlo methods for option pricing">Monte Carlo methods for option pricing</a> were originated by <a href="/wiki/Phelim_Boyle" title="Phelim Boyle">Phelim Boyle</a> in 1977; <a href="#cite_note-68">[55]</a> In 1996, methods were developed for <a href="/wiki/American_option" class="mw-redirect" title="American option">American</a> <a href="#cite_note-69">[56]</a> and <a href="/wiki/Asian_option" title="Asian option">Asian options</a>. <a href="#cite_note-70">[57]</a> </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> <a href="/wiki/Oldrich_Vasicek" class="mw-redirect" title="Oldrich Vasicek">Oldrich Vasicek</a> developed his pioneering short-rate model in 1977. <a href="#cite_note-72">[58]</a> The HJM framework originates from the work of <a href="/wiki/David_Heath_(probabilist)" title="David Heath (probabilist)">David Heath</a>, <a href="/wiki/Robert_A._Jarrow" title="Robert A. Jarrow">Robert A. Jarrow</a>, and Andrew Morton in 1987. <a href="#cite_note-73">[59]</a> </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> Simulation was first applied to (corporate) finance by <a href="/wiki/David_B._Hertz" title="David B. Hertz">David B. Hertz</a> in 1964. Decision trees, a standard <a href="/wiki/Operations_research" title="Operations research">operations research</a> tool, were applied to corporate finance also in the 1960s.<a href="#cite_note-82">[67]</a> Real options in corporate finance were first discussed by <a href="/wiki/Stewart_Myers" title="Stewart Myers">Stewart Myers</a> in 1977. </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> The Benchmark here is the pioneering AFM of the <a href="/wiki/Santa_Fe_Institute" title="Santa Fe Institute">Santa Fe Institute</a> developed in the early 1990s. See <a href="#cite_note-LeBaron2-100">[84]</a> for discussion of other early models. </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> This quote, from banker and author <a href="/wiki/James_Rickards" title="James Rickards">James Rickards</a>, is representative. Prominent and earlier criticism <a href="#cite_note-Taleb_Mandelbrot-103">[86]</a> is from <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a> <a href="/wiki/Emanuel_Derman" title="Emanuel Derman">Emanuel Derman</a>, <a href="/wiki/Paul_Wilmott" title="Paul Wilmott">Paul Wilmott</a>, <a href="/wiki/Nassim_Taleb" class="mw-redirect" title="Nassim Taleb">Nassim Taleb</a>, <a href="/wiki/Financial_engineering#Criticisms" title="Financial engineering">and others</a>. Well known popularizations include Taleb's <a href="/wiki/Fooled_by_Randomness" title="Fooled by Randomness">Fooled by Randomness</a> and <a href="/wiki/The_Black_Swan:_The_Impact_of_the_Highly_Improbable" title="The Black Swan: The Impact of the Highly Improbable">The Black Swan</a>, Mandelbrot's <a href="/wiki/Benoit_Mandelbrot#Bibliography" title="Benoit Mandelbrot">The Misbehavior of Markets</a>, and Derman's <a href="/wiki/Emanuel_Derman#Models.Behaving.Badly" title="Emanuel Derman">Models.Behaving.Badly</a> and, with Wimott, the <a href="/wiki/Financial_Modelers%27_Manifesto" title="Financial Modelers' Manifesto">Financial Modelers' Manifesto</a>. </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> An early anecdotal treatment is <a href="/wiki/Benjamin_Graham" title="Benjamin Graham">Benjamin Graham</a>'s "<a href="/wiki/Mr._Market" title="Mr. Market">Mr. Market</a>", discussed in his <a href="/wiki/The_Intelligent_Investor" title="The Intelligent Investor">The Intelligent Investor</a> in 1949. See also <a href="/wiki/John_Maynard_Keynes" title="John Maynard Keynes">John Maynard Keynes</a>' 1936 discussion of <a href="/wiki/Animal_spirits_(Keynes)" title="Animal spirits (Keynes)">"Animal spirits"</a>, and the related <a href="/wiki/Keynesian_beauty_contest" title="Keynesian beauty contest">Keynesian beauty contest</a>, in his <a href="/wiki/The_General_Theory_of_Employment,_Interest_and_Money#Chapter_12:_Animal_spirits" title="The General Theory of Employment, Interest and Money">General Theory, Ch. 12.</a> <a href="/wiki/Extraordinary_Popular_Delusions_and_the_Madness_of_Crowds" title="Extraordinary Popular Delusions and the Madness of Crowds">Extraordinary Popular Delusions and the Madness of Crowds</a> is a study of <a href="/wiki/Crowd_psychology" title="Crowd psychology">crowd psychology</a> by Scottish journalist <a href="/wiki/Charles_Mackay_(author)" title="Charles Mackay (author)">Charles Mackay</a>, first published in 1841, with Volume I discussing <a href="/wiki/Economic_bubbles" class="mw-redirect" title="Economic bubbles">economic bubbles</a>. </li> <li id="cite_note-118"><a href="#cite_ref-118">^</a> <a href="/wiki/Burton_Malkiel" title="Burton Malkiel">Burton Malkiel</a>'s <a href="/wiki/A_Random_Walk_Down_Wall_Street" title="A Random Walk Down Wall Street">A Random Walk Down Wall Street</a> – first published in 1973, and in its 13th edition as of 2023 – is a widely read popularization of these arguments. See also <a href="/wiki/John_C._Bogle" title="John C. Bogle">John C. Bogle</a>'s <a href="/wiki/Common_Sense_on_Mutual_Funds" title="Common Sense on Mutual Funds">Common Sense on Mutual Funds</a>; but compare <a href="/wiki/Warren_Buffett" title="Warren Buffett">Warren Buffett</a>'s <a href="/wiki/The_Superinvestors_of_Graham-and-Doddsville" title="The Superinvestors of Graham-and-Doddsville">The Superinvestors of Graham-and-Doddsville</a>. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=18" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 20em;"> <ol class="references"> <li id="cite_note-stanford1-1">^ <a href="#cite_ref-stanford1_1-0">a</a> <a href="#cite_ref-stanford1_1-1">b</a> <a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William F. Sharpe</a>, <a rel="nofollow" class="external text" href="http://www.stanford.edu/~wfsharpe/mia/int/mia_int2.htm">"Financial Economics"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040604105441/http://www.stanford.edu/~wfsharpe/mia/int/mia_int2.htm">Archived</a> 2004-06-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, in <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.stanford.edu/~wfsharpe/mia/MIA.HTM">"Macro-Investment Analysis"</a>. Stanford University (manuscript). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140714034144/https://web.stanford.edu/~wfsharpe/mia/mia.htm">Archived</a> from the original on 2014-07-14. Retrieved 2009-08-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Macro-Investment+Analysis&rft.pub=Stanford+University+%28manuscript%29&rft_id=https%3A%2F%2Fweb.stanford.edu%2F~wfsharpe%2Fmia%2FMIA.HTM&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Miller-2">^ <a href="#cite_ref-Miller_2-0">a</a> <a href="#cite_ref-Miller_2-1">b</a> <a href="/wiki/Merton_H._Miller" class="mw-redirect" title="Merton H. Miller">Merton H. Miller</a>, (1999). The History of Finance: An Eyewitness Account, Journal of Portfolio Management. Summer 1999. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="/wiki/Robert_C._Merton" title="Robert C. Merton">Robert C. Merton</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://nobelprize.org/nobel_prizes/economics/laureates/1997/merton-lecture.pdf">"Nobel Lecture"</a> (PDF). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090319202149/http://nobelprize.org/nobel_prizes/economics/laureates/1997/merton-lecture.pdf">Archived</a> (PDF) from the original on 2009-03-19. Retrieved 2009-08-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Nobel+Lecture&rft_id=http%3A%2F%2Fnobelprize.org%2Fnobel_prizes%2Feconomics%2Flaureates%2F1997%2Fmerton-lecture.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Fama_and_Miller-4">^ <a href="#cite_ref-Fama_and_Miller_4-0">a</a> <a href="#cite_ref-Fama_and_Miller_4-1">b</a> See Fama and Miller (1972), The Theory of Finance, in Bibliography. </li> <li id="cite_note-Cochrane_&_Culp-5">^ <a href="#cite_ref-Cochrane_&_Culp_5-0">a</a> <a href="#cite_ref-Cochrane_&_Culp_5-1">b</a> <a href="#cite_ref-Cochrane_&_Culp_5-2">c</a> <a href="#cite_ref-Cochrane_&_Culp_5-3">d</a> <a href="#cite_ref-Cochrane_&_Culp_5-4">e</a> <a href="#cite_ref-Cochrane_&_Culp_5-5">f</a> Christopher L. Culp and <a href="/wiki/John_H._Cochrane" title="John H. Cochrane">John H. Cochrane</a>. (2003). "<a rel="nofollow" class="external text" href="http://faculty.chicagobooth.edu/john.cochrane/research/Papers/cochrane-culp%20asset%20pricing.pdf">"Equilibrium Asset Pricing and Discount Factors: Overview and Implications for Derivatives Valuation and Risk Management"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304190225/http://faculty.chicagobooth.edu/john.cochrane/research/Papers/cochrane-culp%20asset%20pricing.pdf">Archived</a> 2016-03-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, in Modern Risk Management: A History. Peter Field, ed. London: Risk Books, 2003. <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/1904339050" title="Special:BookSources/1904339050">1904339050</a> </li> <li id="cite_note-Rubinstein-6">^ <a href="#cite_ref-Rubinstein_6-0">a</a> <a href="#cite_ref-Rubinstein_6-1">b</a> <a href="#cite_ref-Rubinstein_6-2">c</a> <a href="#cite_ref-Rubinstein_6-3">d</a> <a href="#cite_ref-Rubinstein_6-4">e</a> <a href="#cite_ref-Rubinstein_6-5">f</a> <a href="#cite_ref-Rubinstein_6-6">g</a> <a href="#cite_ref-Rubinstein_6-7">h</a> <a href="#cite_ref-Rubinstein_6-8">i</a> <a href="#cite_ref-Rubinstein_6-9">j</a> <a href="/wiki/Mark_Rubinstein" title="Mark Rubinstein">Rubinstein, Mark</a>. (2005). "Great Moments in Financial Economics: IV. The Fundamental Theorem (Part I)", Journal of Investment Management, Vol. 3, No. 4, Fourth Quarter 2005; ~ (2006). Part II, Vol. 4, No. 1, First Quarter 2006. (See under "External links".) </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> C. Lewin (1970). <a rel="nofollow" class="external text" href="https://www.actuaries.org.uk/system/files/documents/pdf/0121-0132.pdf">An early book on compound interest</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161221163926/https://www.actuaries.org.uk/system/files/documents/pdf/0121-0132.pdf">Archived</a> 2016-12-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Institute and Faculty of Actuaries </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> James E. Ciecka. 2008. <a rel="nofollow" class="external text" href="https://fac.comtech.depaul.edu/jciecka/deWitt.pdf">"The First Mathematically Correct Life Annuity"</a>. Journal of Legal Economics 15(1): pp. 59-63 </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> James E. Ciecka (2008). <a rel="nofollow" class="external text" href="https://fac.comtech.depaul.edu/jciecka/Halley.pdf">"Edmond Halley’s Life Table and Its Uses"</a>. Journal of Legal Economics 15(1): pp. 65-74. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> For example, <a rel="nofollow" class="external free" href="http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G00">http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G00</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130529074942/http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G00">Archived</a> 2013-05-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. </li> <li id="cite_note-Varian-14">^ <a href="#cite_ref-Varian_14-0">a</a> <a href="#cite_ref-Varian_14-1">b</a> <a href="#cite_ref-Varian_14-2">c</a> <a href="#cite_ref-Varian_14-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVarian1987" class="citation journal cs1"><a href="/wiki/Hal_Varian" title="Hal Varian">Varian, Hal R.</a> (1987). <a rel="nofollow" class="external text" href="https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.1.2.55">"The Arbitrage Principle in Financial Economics"</a>. Economic Perspectives. 1 (2): 55–72. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1257%2Fjep.1.2.55">10.1257/jep.1.2.55</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1942981">1942981</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Economic+Perspectives&rft.atitle=The+Arbitrage+Principle+in+Financial+Economics&rft.volume=1&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E55-%3C%2Fspan%3E72&rft.date=1987&rft_id=info%3Adoi%2F10.1257%2Fjep.1.2.55&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1942981%23id-name%3DJSTOR&rft.aulast=Varian&rft.aufirst=Hal+R.&rft_id=https%3A%2F%2Fpubs.aeaweb.org%2Fdoi%2Fpdfplus%2F10.1257%2Fjep.1.2.55&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Rubinstein2-15">^ <a href="#cite_ref-Rubinstein2_15-0">a</a> <a href="#cite_ref-Rubinstein2_15-1">b</a> See Rubinstein (2006), under "Bibliography". </li> <li id="cite_note-Derman-16">^ <a href="#cite_ref-Derman_16-0">a</a> <a href="#cite_ref-Derman_16-1">b</a> <a href="#cite_ref-Derman_16-2">c</a> Emanuel Derman, <a rel="nofollow" class="external text" href="http://www.emanuelderman.com/media/Scientific_Approach_to_Finance.pdf">A Scientific Approach to CAPM and Options Valuation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160330002200/http://www.emanuelderman.com/media/Scientific_Approach_to_Finance.pdf">Archived</a> 2016-03-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-Delbaen_Schachermayer-17">^ <a href="#cite_ref-Delbaen_Schachermayer_17-0">a</a> <a href="#cite_ref-Delbaen_Schachermayer_17-1">b</a> Freddy Delbaen and Walter Schachermayer. (2004). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200405/what-is.pdf">"What is... a Free Lunch?"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304061252/http://www.ams.org/notices/200405/what-is.pdf">Archived</a> 2016-03-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (pdf). Notices of the AMS 51 (5): 526–528 </li> <li id="cite_note-Farmer_Geanakoplos-18">^ <a href="#cite_ref-Farmer_Geanakoplos_18-0">a</a> <a href="#cite_ref-Farmer_Geanakoplos_18-1">b</a> <a href="#cite_ref-Farmer_Geanakoplos_18-2">c</a> <a href="#cite_ref-Farmer_Geanakoplos_18-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarmer_J._Doyne,_Geanakoplos_John2009" class="citation journal cs1">Farmer J. Doyne, Geanakoplos John (2009). <a rel="nofollow" class="external text" href="https://campuspress.yale.edu/johngeanakoplos/files/2017/07/63.-The-Virtues-and-Vices-of-Equilbrium-and-the-Future-of-Financial-Economics-2009-26baz0x.pdf">"The virtues and vices of equilibrium and the future of financial economics"</a> (PDF). Complexity. 14 (3): 11–38. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0803.2996">0803.2996</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009Cmplx..14c..11F">2009Cmplx..14c..11F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcplx.20261">10.1002/cplx.20261</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4506630">4506630</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Complexity&rft.atitle=The+virtues+and+vices+of+equilibrium+and+the+future+of+financial+economics&rft.volume=14&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E11-%3C%2Fspan%3E38&rft.date=2009&rft_id=info%3Aarxiv%2F0803.2996&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4506630%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2Fcplx.20261&rft_id=info%3Abibcode%2F2009Cmplx..14c..11F&rft.au=Farmer+J.+Doyne%2C+Geanakoplos+John&rft_id=https%3A%2F%2Fcampuspress.yale.edu%2Fjohngeanakoplos%2Ffiles%2F2017%2F07%2F63.-The-Virtues-and-Vices-of-Equilbrium-and-the-Future-of-Financial-Economics-2009-26baz0x.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Backus-19">^ <a href="#cite_ref-Backus_19-0">a</a> <a href="#cite_ref-Backus_19-1">b</a> See: <a href="/wiki/David_K._Backus" title="David K. Backus">David K. Backus</a> (2015). <a rel="nofollow" class="external text" href="http://pages.stern.nyu.edu/~dbackus/233/notes_econ_assetpricing.pdf">Fundamentals of Asset Pricing</a>, Stern NYU </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArrowDebreu1954" class="citation journal cs1">Arrow, K. J.; Debreu, G. (1954). "Existence of an equilibrium for a competitive economy". Econometrica. 22 (3): 265–290. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1907353">10.2307/1907353</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1907353">1907353</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=Existence+of+an+equilibrium+for+a+competitive+economy&rft.volume=22&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E265-%3C%2Fspan%3E290&rft.date=1954&rft_id=info%3Adoi%2F10.2307%2F1907353&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1907353%23id-name%3DJSTOR&rft.aulast=Arrow&rft.aufirst=K.+J.&rft.au=Debreu%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcKenzie1954" class="citation journal cs1">McKenzie, Lionel W. (1954). "On Equilibrium in Graham's Model of World Trade and Other Competitive Systems". 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"Prices of State-Contingent Claims Implicit in Option Prices". <a href="/wiki/Journal_of_Business" class="mw-redirect" title="Journal of Business">Journal of Business</a>. 51 (4): 621–651. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F296025">10.1086/296025</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2352653">2352653</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:153841737">153841737</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Business&rft.atitle=Prices+of+State-Contingent+Claims+Implicit+in+Option+Prices&rft.volume=51&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E621-%3C%2Fspan%3E651&rft.date=1978&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A153841737%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2352653%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1086%2F296025&rft.aulast=Breeden&rft.aufirst=Douglas+T.&rft.au=Litzenberger%2C+Robert+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Figlewski-24">^ <a href="#cite_ref-Figlewski_24-0">a</a> <a href="#cite_ref-Figlewski_24-1">b</a> <a href="#cite_ref-Figlewski_24-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFiglewski2018" class="citation journal cs1">Figlewski, Stephen (2018). <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3120028">"Risk-Neutral Densities: A Review Annual Review of Financial Economics"</a>. <a href="/wiki/Annual_Review_of_Financial_Economics" title="Annual Review of Financial Economics">Annual Review of Financial Economics</a>. 10: 329–359. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev-financial-110217-022944">10.1146/annurev-financial-110217-022944</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:158075926">158075926</a>. <a href="/wiki/SSRN_(identifier)" class="mw-redirect" title="SSRN (identifier)">SSRN</a> <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3120028">3120028</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Financial+Economics&rft.atitle=Risk-Neutral+Densities%3A+A+Review+Annual+Review+of+Financial+Economics&rft.volume=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E329-%3C%2Fspan%3E359&rft.date=2018&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A158075926%23id-name%3DS2CID&rft_id=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D3120028%23id-name%3DSSRN&rft_id=info%3Adoi%2F10.1146%2Fannurev-financial-110217-022944&rft.aulast=Figlewski&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D3120028&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-corp_fin_state_prices-25">^ <a href="#cite_ref-corp_fin_state_prices_25-0">a</a> <a href="#cite_ref-corp_fin_state_prices_25-1">b</a> <a href="#cite_ref-corp_fin_state_prices_25-2">c</a> See de Matos, as well as Bossaerts and Ødegaard, under bibliography. </li> <li id="cite_note-Chance2-26">^ <a href="#cite_ref-Chance2_26-0">a</a> <a href="#cite_ref-Chance2_26-1">b</a> Don M. Chance (2008). <a rel="nofollow" class="external text" href="http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN97-13.pdf">"Option Prices and State Prices"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120209215717/http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN97-13.pdf">Archived</a> 2012-02-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-Luenberger-27">^ <a href="#cite_ref-Luenberger_27-0">a</a> <a href="#cite_ref-Luenberger_27-1">b</a> See Luenberger's Investment Science, under Bibliography. </li> <li id="cite_note-MM1-28"><a href="#cite_ref-MM1_28-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFModiglianiMiller,_M.1958" class="citation journal cs1">Modigliani, F.; Miller, M. (1958). "The Cost of Capital, Corporation Finance and the Theory of Investment". 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(1963). "Corporate income taxes and the cost of capital: a correction". American Economic Review. 53 (3): 433–443. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1809167">1809167</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Economic+Review&rft.atitle=Corporate+income+taxes+and+the+cost+of+capital%3A+a+correction&rft.volume=53&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E433-%3C%2Fspan%3E443&rft.date=1963&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1809167%23id-name%3DJSTOR&rft.aulast=Modigliani&rft.aufirst=F.&rft.au=Miller%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-New_School-31">^ <a href="#cite_ref-New_School_31-0">a</a> <a href="#cite_ref-New_School_31-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThe_New_School" class="citation web cs1"><a href="/wiki/The_New_School" title="The New School">The New School</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060702212228/http://cepa.newschool.edu/het/schools/finance.htm">"Finance Theory"</a>. Archived from <a rel="nofollow" class="external text" href="http://cepa.newschool.edu/het/schools/finance.htm">the original</a> on 2006-07-02. Retrieved 2006-06-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Finance+Theory&rft.au=The+New+School&rft_id=http%3A%2F%2Fcepa.newschool.edu%2Fhet%2Fschools%2Ffinance.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Rubinstein_2-32"><a href="#cite_ref-Rubinstein_2_32-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Rubinstein2002" class="citation web cs1"><a href="/wiki/Mark_Rubinstein" title="Mark Rubinstein">Mark Rubinstein</a> (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070713043745/http://www.in-the-money.com/artandpap/I%20Present%20Value.doc">"Great Moments in Financial Economics: I. Present Value"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.in-the-money.com/artandpap/I%20Present%20Value.doc">the original</a> on 2007-07-13. Retrieved 2007-06-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Great+Moments+in+Financial+Economics%3A+I.+Present+Value&rft.date=2002&rft.au=Mark+Rubinstein&rft_id=http%3A%2F%2Fwww.in-the-money.com%2Fartandpap%2FI%2520Present%2520Value.doc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> See footnote 3 under Rubinstein (2005). "The Fundamental Theorem (Part I)", refenced below. </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> § 4.1 "Law of one price and existence of a discount factor" in Cochrane (2005). </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> Gonçalo L. Fonseca (N.D.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080429203224/http://cepa.newschool.edu/het/essays/capital/fisherinvest.htm">Irving Fisher's Theory of Investment</a>. History of Economic Thought series, <a href="/wiki/The_New_School" title="The New School">The New School</a>. </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> For a more formal treatment, see, for example: Eugene F. Fama. (1965). <a rel="nofollow" class="external text" href="http://www.cfapubs.org/toc/faj/1965/21/5">"Random Walks in Stock Market Prices"</a>. <a href="/wiki/Financial_Analysts_Journal" title="Financial Analysts Journal">Financial Analysts Journal</a>, September/October 1965, Vol. 21, No. 5: 55–59. </li> <li id="cite_note-affirmative_case-39">^ <a href="#cite_ref-affirmative_case_39-0">a</a> <a href="#cite_ref-affirmative_case_39-1">b</a> <a href="#cite_ref-affirmative_case_39-2">c</a> <a href="/wiki/Mark_Rubinstein" title="Mark Rubinstein">Mark Rubinstein</a> (2001). <a rel="nofollow" class="external text" href="https://escholarship.org/content/qt22q318mh/qt22q318mh_noSplash_0a514e78eb5c8e16a9ead63a4f4a628e.pdf?t=krnd5a">"Rational Markets: Yes or No? 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Insurance: Mathematics and Economics. 19: 19–30. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0167-6687%2896%2900004-2">10.1016/S0167-6687(96)00004-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Insurance%3A+Mathematics+and+Economics&rft.atitle=Valuation+of+the+early-exercise+price+for+options+using+simulations+and+nonparametric+regression&rft.volume=19&rft.pages=%3Cspan+class%3D%22nowrap%22%3E19-%3C%2Fspan%3E30&rft.date=1996&rft_id=info%3Adoi%2F10.1016%2FS0167-6687%2896%2900004-2&rft.aulast=Carriere&rft.aufirst=Jacques&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBroadieGlasserman1996" class="citation journal cs1">Broadie, M.; Glasserman, P. (1996). <a rel="nofollow" class="external text" href="http://www.columbia.edu/~mnb2/broadie/Assets/bg_ms_1996.pdf">"Estimating Security Price Derivatives Using Simulation"</a> (PDF). Management Science. 42 (2): 269–285. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.196.1128">10.1.1.196.1128</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmnsc.42.2.269">10.1287/mnsc.42.2.269</a>. Retrieved June 28, 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Management+Science&rft.atitle=Estimating+Security+Price+Derivatives+Using+Simulation&rft.volume=42&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E269-%3C%2Fspan%3E285&rft.date=1996&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.196.1128%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1287%2Fmnsc.42.2.269&rft.aulast=Broadie&rft.aufirst=M.&rft.au=Glasserman%2C+P.&rft_id=http%3A%2F%2Fwww.columbia.edu%2F~mnb2%2Fbroadie%2FAssets%2Fbg_ms_1996.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVasicek1977" class="citation journal cs1">Vasicek, O. (1977). "An equilibrium characterization of the term structure". <a href="/wiki/Journal_of_Financial_Economics" title="Journal of Financial Economics">Journal of Financial Economics</a>. 5 (2): 177–188. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.164.447">10.1.1.164.447</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-405X%2877%2990016-2">10.1016/0304-405X(77)90016-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Financial+Economics&rft.atitle=An+equilibrium+characterization+of+the+term+structure&rft.volume=5&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E177-%3C%2Fspan%3E188&rft.date=1977&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.164.447%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1016%2F0304-405X%2877%2990016-2&rft.aulast=Vasicek&rft.aufirst=O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> David Heath, Robert A. Jarrow, and Andrew Morton (1987). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2951677">Bond pricing and the term structure of interest rates: a new methodology</a> – working paper, Cornell University </li> <li id="cite_note-Youmbi-75">^ <a href="#cite_ref-Youmbi_75-0">a</a> <a href="#cite_ref-Youmbi_75-1">b</a> Didier Kouokap Youmbi (2017). "<a rel="nofollow" class="external text" href="https://ssrn.com/abstract=2511585">Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach</a>". <a href="/wiki/Bank_of_England" title="Bank of England">Bank of England</a> – <a href="/wiki/Prudential_Regulation_Authority_(United_Kingdom)" title="Prudential Regulation Authority (United Kingdom)">Prudential Regulation Authority</a> </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <a rel="nofollow" class="external text" href="http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf">"Post-Crisis Pricing of Swaps using xVAs"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160917015231/http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf">Archived</a> 2016-09-17 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Christian Kjølhede & Anders Bech, Master thesis, <a href="/wiki/Aarhus_University" title="Aarhus University">Aarhus University</a> </li> <li id="cite_note-Hull_White_2-77"><a href="#cite_ref-Hull_White_2_77-0">^</a> John C. Hull and Alan White (2014). <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2212953">Collateral and Credit Issues in Derivatives Pricing</a>. Rotman School of Management Working Paper No. 2212953 </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHullWhite2013" class="citation journal cs1">Hull, John; White, Alan (2013). "LIBOR vs. OIS: The Derivatives Discounting Dilemma". <a href="/wiki/Journal_of_Investment_Management" title="Journal of Investment Management">Journal of Investment Management</a>. 11 (3): 14–27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Investment+Management&rft.atitle=LIBOR+vs.+OIS%3A+The+Derivatives+Discounting+Dilemma&rft.volume=11&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E14-%3C%2Fspan%3E27&rft.date=2013&rft.aulast=Hull&rft.aufirst=John&rft.au=White%2C+Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Damodaran_Risk-79">^ <a href="#cite_ref-Damodaran_Risk_79-0">a</a> <a href="#cite_ref-Damodaran_Risk_79-1">b</a> <a href="#cite_ref-Damodaran_Risk_79-2">c</a> <a href="#cite_ref-Damodaran_Risk_79-3">d</a> <a href="#cite_ref-Damodaran_Risk_79-4">e</a> <a href="/wiki/Aswath_Damodaran" title="Aswath Damodaran">Aswath Damodaran</a> (2007). <a rel="nofollow" class="external text" href="http://www.stern.nyu.edu/~adamodar/pdfiles/papers/probabilistic.pdf">"Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations"</a>. In Strategic Risk Taking: A Framework for Risk Management. Prentice Hall. <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/0137043775" title="Special:BookSources/0137043775">0137043775</a> </li> <li id="cite_note-Damodaran-80">^ <a href="#cite_ref-Damodaran_80-0">a</a> <a href="#cite_ref-Damodaran_80-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDamodaran2005" class="citation journal cs1"><a href="/wiki/Aswath_Damodaran" title="Aswath Damodaran">Damodaran, Aswath</a> (2005). <a rel="nofollow" class="external text" href="http://stern.nyu.edu/~adamodar/pdfiles/papers/realopt.pdf">"The Promise and Peril of Real Options"</a> (PDF). NYU Working Paper (S-DRP-05-02). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20010613082802/http://www.stern.nyu.edu/~adamodar/pdfiles/papers/realopt.pdf">Archived</a> (PDF) from the original on 2001-06-13. Retrieved 2016-12-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=NYU+Working+Paper&rft.atitle=The+Promise+and+Peril+of+Real+Options&rft.issue=S-DRP-05-02&rft.date=2005&rft.aulast=Damodaran&rft.aufirst=Aswath&rft_id=http%3A%2F%2Fstern.nyu.edu%2F~adamodar%2Fpdfiles%2Fpapers%2Frealopt.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithNau1995" class="citation journal cs1">Smith, James E.; Nau, Robert F. (1995). <a rel="nofollow" class="external text" href="https://faculty.fuqua.duke.edu/~jes9/bio/Valuing_Risky_Projects.pdf">"Valuing Risky Projects: Option Pricing Theory and Decision Analysis"</a> (PDF). Management Science. 41 (5): 795–816. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmnsc.41.5.795">10.1287/mnsc.41.5.795</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100612170613/http://faculty.fuqua.duke.edu/%7Ejes9/bio/Valuing_Risky_Projects.pdf">Archived</a> (PDF) from the original on 2010-06-12. Retrieved 2017-08-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Management+Science&rft.atitle=Valuing+Risky+Projects%3A+Option+Pricing+Theory+and+Decision+Analysis&rft.volume=41&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E795-%3C%2Fspan%3E816&rft.date=1995&rft_id=info%3Adoi%2F10.1287%2Fmnsc.41.5.795&rft.aulast=Smith&rft.aufirst=James+E.&rft.au=Nau%2C+Robert+F.&rft_id=https%3A%2F%2Ffaculty.fuqua.duke.edu%2F~jes9%2Fbio%2FValuing_Risky_Projects.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> See for example: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMagee1964" class="citation journal cs1 cs1-prop-long-vol">Magee, John F. (1964). <a rel="nofollow" class="external text" href="https://hbr.org/1964/07/decision-trees-for-decision-making">"Decision Trees for Decision Making"</a>. <a href="/wiki/Harvard_Business_Review" title="Harvard Business Review">Harvard Business Review</a>. July 1964: 795–816. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170816192517/https://hbr.org/1964/07/decision-trees-for-decision-making">Archived</a> from the original on 2017-08-16. Retrieved 2017-08-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Harvard+Business+Review&rft.atitle=Decision+Trees+for+Decision+Making&rft.volume=July+1964&rft.pages=%3Cspan+class%3D%22nowrap%22%3E795-%3C%2Fspan%3E816&rft.date=1964&rft.aulast=Magee&rft.aufirst=John+F.&rft_id=https%3A%2F%2Fhbr.org%2F1964%2F07%2Fdecision-trees-for-decision-making&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Markowitz_interview-84"><a href="#cite_ref-Markowitz_interview_84-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKritzman2017" class="citation journal cs1">Kritzman, Mark (2017). "An Interview with Nobel Laureate Harry M. Markowitz". Financial Analysts Journal. 73 (4): 16–21. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2469%2Ffaj.v73.n4.3">10.2469/faj.v73.n4.3</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:158093964">158093964</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Financial+Analysts+Journal&rft.atitle=An+Interview+with+Nobel+Laureate+Harry+M.+Markowitz&rft.volume=73&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E16-%3C%2Fspan%3E21&rft.date=2017&rft_id=info%3Adoi%2F10.2469%2Ffaj.v73.n4.3&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A158093964%23id-name%3DS2CID&rft.aulast=Kritzman&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Kruschwitz_and_Löffler-85">^ <a href="#cite_ref-Kruschwitz_and_Löffler_85-0">a</a> <a href="#cite_ref-Kruschwitz_and_Löffler_85-1">b</a> See Kruschwitz and Löffler under Bibliography. </li> <li id="cite_note-welch-86"><a href="#cite_ref-welch_86-0">^</a> <a rel="nofollow" class="external text" href="http://book.ivo-welch.info/read/chap13.pdf">"Capital Budgeting Applications and Pitfalls"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170815234404/http://book.ivo-welch.info/read/chap13.pdf">Archived</a> 2017-08-15 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Ch 13 in <a href="/wiki/Ivo_Welch" title="Ivo Welch">Ivo Welch</a> (2017). Corporate Finance: 4th Edition </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> George Chacko and Carolyn Evans (2014). Valuation: Methods and Models in Applied Corporate Finance. FT Press. <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/0132905221" title="Special:BookSources/0132905221">0132905221</a> </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> See Jensen and Smith under "External links", as well as Rubinstein under "Bibliography". </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJensenMeckling1976" class="citation journal cs1">Jensen, Michael; Meckling, William (1976). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-405X%2876%2990026-X">"Theory of the firm: Managerial behavior, agency costs and ownership structure"</a>. Journal of Financial Economics. 3 (4): 305–360. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-405X%2876%2990026-X">10.1016/0304-405X(76)90026-X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Financial+Economics&rft.atitle=Theory+of+the+firm%3A+Managerial+behavior%2C+agency+costs+and+ownership+structure&rft.volume=3&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E305-%3C%2Fspan%3E360&rft.date=1976&rft_id=info%3Adoi%2F10.1016%2F0304-405X%2876%2990026-X&rft.aulast=Jensen&rft.aufirst=Michael&rft.au=Meckling%2C+William&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0304-405X%252876%252990026-X&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> <a rel="nofollow" class="external text" href="http://pages.stern.nyu.edu/~adamodar/New_Home_Page/AppldCF/other/Image2.gif">Corporate Finance: First Principles</a>, from <a href="/wiki/Aswath_Damodaran" title="Aswath Damodaran">Aswath Damodaran</a> (2022). Applied Corporate Finance: A User's Manual. Wiley. <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/978-1118808931" title="Special:BookSources/978-1118808931">978-1118808931</a> </li> <li id="cite_note-Garbade-91"><a href="#cite_ref-Garbade_91-0">^</a> Kenneth D. Garbade (2001). Pricing Corporate Securities as Contingent Claims. <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>. <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/9780262072236" title="Special:BookSources/9780262072236">9780262072236</a> </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> See for example: Hazem Daouk, Charles M.C. Lee, David Ng. (2006). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/S0929119905000507">"Capital Market Governance: How Do Security Laws Affect Market Performance?"</a>. Journal of Corporate Finance, Volume 12, Issue 3; Emilios Avgouleas (2010). <a rel="nofollow" class="external text" href="https://core.ac.uk/download/pdf/6631008.pdf">"The Regulation of Short Sales and its Reform"</a> <a rel="nofollow" class="external text" href="https://dice.ifo.de/">DICE Report</a>, Vol. 8, Iss. 1. </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> <a href="/wiki/Maureen_O%27Hara_(financial_economist)" title="Maureen O'Hara (financial economist)">O'Hara, Maureen</a>, Market Microstructure Theory, Blackwell, Oxford, 1995, <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/1-55786-443-8" title="Special:BookSources/1-55786-443-8">1-55786-443-8</a>, p.1. </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> King, Michael, Osler, Carol and Rime, Dagfinn (2013). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/S0261560613000594">"The market microstructure approach to foreign exchange: Looking back and looking forward"</a>, Journal of International Money and Finance. Volume 38, November 2013, Pages 95-119 </li> <li id="cite_note-95"><a href="#cite_ref-95">^</a> Randi Næs, Johannes Skjeltorp (2006). <a rel="nofollow" class="external text" href="https://www.norges-bank.no/globalassets/upload/english/publications/economic-bulletin/2006-03/naes.pdf">"Is the market microstructure of stock markets important?"</a>. <a href="/wiki/Norges_Bank" title="Norges Bank">Norges Bank</a> Economic Bulletin 3/06 (Vol. 77) </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> See, e.g., Westerhoff, Frank H. (2008). <a rel="nofollow" class="external text" href="https://ideas.repec.org/a/jns/jbstat/v228y2008i2-3p195-227.html">"The Use of Agent-Based Financial Market Models to Test the Effectiveness of Regulatory Policies"</a>, Journal of Economics and Statistics </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> See, e.g., Mizuta, Takanobu (2019). <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/1906.06000.pdf">"An agent-based model for designing a financial market that works well"</a>. 2020 IEEE Symposium Series on Computational Intelligence (SSCI). </li> <li id="cite_note-LeBaron-98">^ <a href="#cite_ref-LeBaron_98-0">a</a> <a href="#cite_ref-LeBaron_98-1">b</a> For a survey see: LeBaron, Blake (2006). <a rel="nofollow" class="external text" href="https://peeps.unet.brandeis.edu/~blebaron/wps/hbook.pdf">"Agent-based Computational Finance"</a>. <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/handbook/handbook-of-computational-economics">Handbook of Computational Economics</a>. Elsevier </li> <li id="cite_note-ERIM-99">^ <a href="#cite_ref-ERIM_99-0">a</a> <a href="#cite_ref-ERIM_99-1">b</a> Katalin Boer, Arie De Bruin, Uzay Kaymak (2005). <a rel="nofollow" class="external text" href="https://core.ac.uk/download/pdf/18517225.pdf">"On the Design of Artificial Stock Markets"</a>. Research In Management <a href="/wiki/Erasmus_Research_Institute_of_Management" title="Erasmus Research Institute of Management">ERIM</a> Report Series </li> <li id="cite_note-LeBaron2-100">^ <a href="#cite_ref-LeBaron2_100-0">a</a> <a href="#cite_ref-LeBaron2_100-1">b</a> <a href="#cite_ref-LeBaron2_100-2">c</a> LeBaron, B. (2002). <a rel="nofollow" class="external text" href="https://www2.econ.iastate.edu/tesfatsi/blake.sfisum.pdf">"Building the Santa Fe artificial stock market"</a>. <a href="/wiki/Physica_(journal)" title="Physica (journal)">Physica A</a>, 1, 20. </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMandelbrot1963" class="citation journal cs1"><a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Mandelbrot, Benoit</a> (1963). <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> (PDF). <a href="/wiki/The_Journal_of_Business" title="The Journal of Business">The Journal of Business</a>. 36 (Oct): 394–419. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F294632">10.1086/294632</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Business&rft.atitle=The+Variation+of+Certain+Speculative+Prices&rft.volume=36&rft.issue=Oct&rft.pages=%3Cspan+class%3D%22nowrap%22%3E394-%3C%2Fspan%3E419&rft.date=1963&rft_id=info%3Adoi%2F10.1086%2F294632&rft.aulast=Mandelbrot&rft.aufirst=Benoit&rft_id=http%3A%2F%2Fweb.williams.edu%2FMathematics%2Fsjmiller%2Fpublic_html%2F341Fa09%2Fecon%2FMandelbroit_VariationCertainSpeculativePrices.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-Taleb_Mandelbrot-103">^ <a href="#cite_ref-Taleb_Mandelbrot_103-0">a</a> <a href="#cite_ref-Taleb_Mandelbrot_103-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNassim_Taleb_and_Benoit_Mandelbrot" class="citation web cs1">Nassim Taleb and Benoit Mandelbrot. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101207045925/http://www.fooledbyrandomness.com/fortune.pdf">"How the Finance Gurus Get Risk All Wrong"</a> (PDF). Archived from <a rel="nofollow" class="external text" href="http://www.fooledbyrandomness.com/fortune.pdf">the original</a> (PDF) on 2010-12-07. Retrieved 2010-06-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=How+the+Finance+Gurus+Get+Risk+All+Wrong&rft.au=Nassim+Taleb+and+Benoit+Mandelbrot&rft_id=http%3A%2F%2Fwww.fooledbyrandomness.com%2Ffortune.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-holes-104">^ <a href="#cite_ref-holes_104-0">a</a> <a href="#cite_ref-holes_104-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlack1989" class="citation journal cs1"><a href="/wiki/Fischer_Black" title="Fischer Black">Black, Fischer</a> (1989). "How to use the holes in Black–Scholes". <a href="/wiki/Journal_of_Applied_Corporate_Finance" title="Journal of Applied Corporate Finance">Journal of Applied Corporate Finance</a>. 1 (Jan): 67–73. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1745-6622.1989.tb00175.x">10.1111/j.1745-6622.1989.tb00175.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Corporate+Finance&rft.atitle=How+to+use+the+holes+in+Black%E2%80%93Scholes&rft.volume=1&rft.issue=Jan&rft.pages=%3Cspan+class%3D%22nowrap%22%3E67-%3C%2Fspan%3E73&rft.date=1989&rft_id=info%3Adoi%2F10.1111%2Fj.1745-6622.1989.tb00175.x&rft.aulast=Black&rft.aufirst=Fischer&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> See for example III.A.3, in Carol Alexander, ed. (January 2005). The Professional Risk Managers' Handbook. PRMIA Publications. <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/978-0976609704" title="Special:BookSources/978-0976609704">978-0976609704</a> </li> <li id="cite_note-106"><a href="#cite_ref-106">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHagan2002" class="citation journal cs1">Hagan, Patrick; et al. (2002). "Managing smile risk". <a href="/wiki/Wilmott_Magazine" class="mw-redirect" title="Wilmott Magazine">Wilmott Magazine</a> (Sep): 84–108.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wilmott+Magazine&rft.atitle=Managing+smile+risk&rft.issue=Sep&rft.pages=%3Cspan+class%3D%22nowrap%22%3E84-%3C%2Fspan%3E108&rft.date=2002&rft.aulast=Hagan&rft.aufirst=Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> See for example Pg 217 of: Jackson, Mary; Mike Staunton (2001). Advanced modelling in finance using Excel and VBA. New Jersey: Wiley. <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/0-471-49922-6" title="Special:BookSources/0-471-49922-6">0-471-49922-6</a>. </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> These include: <a href="/wiki/Robert_A._Jarrow" title="Robert A. Jarrow">Jarrow</a> and Rudd (1982); Corrado and Su (1996); Brown and Robinson (2002); <a href="/wiki/David_K._Backus" title="David K. Backus">Backus</a>, Foresi, and Wu (2004). See, e.g.: E. Jurczenko, B. Maillet, and B. Negrea (2002). <a rel="nofollow" class="external text" href="http://eprints.lse.ac.uk/24950/1/dp430.pdf">"Revisited multi-moment approximate option pricing models: a general comparison (Part 1)"</a>. Working paper, <a href="/wiki/London_School_of_Economics_and_Political_Science" class="mw-redirect" title="London School of Economics and Political Science">London School of Economics and Political Science</a>. </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> <a rel="nofollow" class="external text" href="https://www.govinfo.gov/content/pkg/CHRG-111hhrg51925/pdf/CHRG-111hhrg51925.pdf">The Risks of Financial Modeling: VAR and the Economic Meltdown</a>, Hearing before the <a href="/wiki/United_States_House_Science_Subcommittee_on_Investigations_and_Oversight" title="United States House Science Subcommittee on Investigations and Oversight">Subcommittee on Investigations and Oversight</a>, <a href="/wiki/United_States_House_Committee_on_Science,_Space,_and_Technology" title="United States House Committee on Science, Space, and Technology">Committee on Science and Technology</a>, <a href="/wiki/United_States_House_of_Representatives" title="United States House of Representatives">House of Representatives</a>, One Hundred Eleventh Congress, first session, September 10, 2009 </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPablo_Fernandez2019" class="citation ssrn cs1">Pablo Fernandez (2019). "Common Sense and Illogical Models: Finance and Financial Economics". <a href="/wiki/SSRN_(identifier)" class="mw-redirect" title="SSRN (identifier)">SSRN</a> <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2906887">2906887</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=Social+Science+Research+Network&rft.atitle=Common+Sense+and+Illogical+Models%3A+Finance+and+Financial+Economics&rft.date=2019&rft_id=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D2906887%23id-name%3DSSRN&rft.au=Pablo+Fernandez&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> See, e.g., this <a href="/wiki/Opinion_piece" title="Opinion piece">opinion piece</a>: <a rel="nofollow" class="external text" href="https://today.thefinancialexpress.com.bd/print/the-pseudo-science-hurting-markets">"The pseudo-science hurting markets"</a> (<a href="/wiki/Financial_Times" title="Financial Times">Financial Times</a>, November 2007). </li> <li id="cite_note-114"><a href="#cite_ref-114">^</a> <a rel="nofollow" class="external text" href="https://www.fooledbyrandomness.com/jorion.html">Against Value-at-Risk: Nassim Taleb Replies to Philippe Jorion</a>, fooledbyrandomness.com </li> <li id="cite_note-115"><a href="#cite_ref-115">^</a> From <a href="/wiki/The_New_Palgrave_Dictionary_of_Economics" title="The New Palgrave Dictionary of Economics">The New Palgrave Dictionary of Economics</a>, Online Editions, 2011, 2012, with abstract links:    • <a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2012_F000330&edition=1">"regulatory responses to the financial crisis: an interim assessment"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130529101109/http://www.dictionaryofeconomics.com/article?id=pde2012_F000330&edition=1">Archived</a> 2013-05-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> by <a href="/wiki/Howard_Davies_(economist)" title="Howard Davies (economist)">Howard Davies</a>    • <a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2011_C000621&edition=">"Credit Crunch Chronology: April 2007–September 2009"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130529092712/http://www.dictionaryofeconomics.com/article?id=pde2011_C000621&edition=">Archived</a> 2013-05-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> by The Statesman's Yearbook team    • <a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2011_M000430&edition=current&q=">"Minsky crisis"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130529172102/http://www.dictionaryofeconomics.com/article?id=pde2011_M000430&edition=current&q=">Archived</a> 2013-05-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> by <a href="/wiki/L._Randall_Wray" title="L. Randall Wray">L. Randall Wray</a>    • <a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2011_E000326&edition=current&q=">"euro zone crisis 2010"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130529092726/http://www.dictionaryofeconomics.com/article?id=pde2011_E000326&edition=current&q=">Archived</a> 2013-05-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> by <a href="/wiki/Daniel_Gros" title="Daniel Gros">Daniel Gros</a> and Cinzia Alcidi.    • <a href="/wiki/Carmen_M._Reinhart" class="mw-redirect" title="Carmen M. Reinhart">Carmen M. Reinhart</a> and <a href="/wiki/Kenneth_S._Rogoff" class="mw-redirect" title="Kenneth S. Rogoff">Kenneth S. Rogoff</a>, 2009. This Time Is Different: Eight Centuries of Financial Folly, Princeton. <a rel="nofollow" class="external text" href="http://press.princeton.edu/titles/8973.html">Description</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130118213207/http://press.princeton.edu/titles/8973.html">Archived</a> 2013-01-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, ch. 1 ("Varieties of Crises and their Dates". pp. <a rel="nofollow" class="external text" href="http://press.princeton.edu/chapters/s8973.pdf">3-20)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120925065855/http://press.princeton.edu/chapters/s8973.pdf">Archived</a> 2012-09-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, and chapter-preview <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ak5fLB24ircC&pg=PR7gbs_atb">links.</a> </li> <li id="cite_note-116"><a href="#cite_ref-116">^</a> <a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William F. Sharpe</a> (1991). <a rel="nofollow" class="external text" href="http://www.stanford.edu/~wfsharpe/art/active/active.htm">"The Arithmetic of Active Management"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131113071513/http://www.stanford.edu/~wfsharpe/art/active/active.htm">Archived</a> 2013-11-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Financial Analysts Journal Vol. 47, No. 1, January/February </li> <li id="cite_note-two-117"><a href="#cite_ref-two_117-0">^</a> <a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William F. Sharpe</a> (2002). <a rel="nofollow" class="external text" href="http://www.stanford.edu/~wfsharpe/art/talks/indexed_investing.htm">Indexed Investing: A Prosaic Way to Beat the Average Investor</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131114160728/http://www.stanford.edu/~wfsharpe/art/talks/indexed_investing.htm">Archived</a> 2013-11-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Presentation: <a href="/wiki/Monterey_Institute_of_International_Studies" class="mw-redirect" title="Monterey Institute of International Studies">Monterey Institute of International Studies</a>. Retrieved May 20, 2010. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=19" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> Financial economics <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoy_E._Bailey2005" class="citation book cs1">Roy E. Bailey (2005). The Economics of Financial Markets. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521612807" title="Special:BookSources/978-0521612807"><bdi>978-0521612807</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+Economics+of+Financial+Markets&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0521612807&rft.au=Roy+E.+Bailey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarcelo_Bianconi2013" class="citation book cs1">Marcelo Bianconi (2013). Financial Economics, Risk and Information (2nd ed.). <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9814355131" title="Special:BookSources/978-9814355131"><bdi>978-9814355131</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Economics%2C+Risk+and+Information&rft.edition=2nd&rft.pub=World+Scientific&rft.date=2013&rft.isbn=978-9814355131&rft.au=Marcelo+Bianconi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZvi_Bodie,_Robert_C._Merton_and_David_Cleeton2008" class="citation book cs1"><a href="/wiki/Zvi_Bodie" title="Zvi Bodie">Zvi Bodie</a>, <a href="/wiki/Robert_C._Merton" title="Robert C. Merton">Robert C. Merton</a> and David Cleeton (2008). Financial Economics (2nd ed.). <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0131856158" title="Special:BookSources/978-0131856158"><bdi>978-0131856158</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Economics&rft.edition=2nd&rft.pub=Prentice+Hall&rft.date=2008&rft.isbn=978-0131856158&rft.au=Zvi+Bodie%2C+Robert+C.+Merton+and+David+Cleeton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_Bradfield2007" class="citation book cs1">James Bradfield (2007). Introduction to the Economics of Financial Markets. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-531063-4" title="Special:BookSources/978-0-19-531063-4"><bdi>978-0-19-531063-4</bdi></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+the+Economics+of+Financial+Markets&rft.pub=Oxford+University+Press&rft.date=2007&rft.isbn=978-0-19-531063-4&rft.au=James+Bradfield&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSatya_R._Chakravarty2014" class="citation book cs1">Satya R. Chakravarty (2014). An Outline of Financial Economics. Anthem Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1783083367" title="Special:BookSources/978-1783083367"><bdi>978-1783083367</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Outline+of+Financial+Economics&rft.pub=Anthem+Press&rft.date=2014&rft.isbn=978-1783083367&rft.au=Satya+R.+Chakravarty&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJakša_Cvitanić_and_Fernando_Zapatero2004" class="citation book cs1"><a href="/wiki/Jak%C5%A1a_Cvitani%C4%87" title="Jakša Cvitanić">Jakša Cvitanić</a> and Fernando Zapatero (2004). Introduction to the Economics and Mathematics of Financial Markets. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0262033206" title="Special:BookSources/978-0262033206"><bdi>978-0262033206</bdi></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+the+Economics+and+Mathematics+of+Financial+Markets&rft.pub=MIT+Press&rft.date=2004&rft.isbn=978-0262033206&rft.au=Jak%C5%A1a+Cvitani%C4%87+and+Fernando+Zapatero&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_M._ConstantinidesMilton_HarrisRené_M._Stulz2003" class="citation book cs1"><a href="/wiki/George_Constantinides" title="George Constantinides">George M. Constantinides</a>; Milton Harris; <a href="/wiki/Ren%C3%A9_M._Stulz" title="René M. Stulz">René M. Stulz</a>, eds. (2003). <a rel="nofollow" class="external text" href="http://econpapers.repec.org/bookchap/eeefinchp/">Handbook of the Economics of Finance</a>. <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0444513632" title="Special:BookSources/978-0444513632"><bdi>978-0444513632</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+the+Economics+of+Finance&rft.pub=Elsevier&rft.date=2003&rft.isbn=978-0444513632&rft_id=http%3A%2F%2Feconpapers.repec.org%2Fbookchap%2Feeefinchp%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeith_CuthbertsonDirk_Nitzsche2004" class="citation book cs1">Keith Cuthbertson; Dirk Nitzsche (2004). Quantitative Financial Economics: Stocks, Bonds and Foreign Exchange. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470091715" title="Special:BookSources/978-0470091715"><bdi>978-0470091715</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantitative+Financial+Economics%3A+Stocks%2C+Bonds+and+Foreign+Exchange&rft.pub=Wiley&rft.date=2004&rft.isbn=978-0470091715&rft.au=Keith+Cuthbertson&rft.au=Dirk+Nitzsche&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJean-Pierre_Danthine,_John_B._Donaldson2005" class="citation book cs1"><a href="/wiki/Jean-Pierre_Danthine" title="Jean-Pierre Danthine">Jean-Pierre Danthine</a>, <a href="/wiki/John_Donaldson_(economist)" title="John Donaldson (economist)">John B. Donaldson</a> (2005). Intermediate Financial Theory (2nd ed.). <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0123693808" title="Special:BookSources/978-0123693808"><bdi>978-0123693808</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Intermediate+Financial+Theory&rft.edition=2nd&rft.pub=Academic+Press&rft.date=2005&rft.isbn=978-0123693808&rft.au=Jean-Pierre+Danthine%2C+John+B.+Donaldson&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLouis_EeckhoudtChristian_Gollier,_Harris_Schlesinger2005" class="citation book cs1">Louis Eeckhoudt; Christian Gollier, <a href="/wiki/American_Risk_and_Insurance_Association#Presidents" title="American Risk and Insurance Association">Harris Schlesinger</a> (2005). Economic and Financial Decisions Under Risk. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12215-1" title="Special:BookSources/978-0-691-12215-1"><bdi>978-0-691-12215-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Economic+and+Financial+Decisions+Under+Risk&rft.pub=Princeton+University+Press&rft.date=2005&rft.isbn=978-0-691-12215-1&rft.au=Louis+Eeckhoudt&rft.au=Christian+Gollier%2C+Harris+Schlesinger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJürgen_EichbergerIan_R._Harper1997" class="citation book cs1">Jürgen Eichberger; <a href="/wiki/Ian_Harper" title="Ian Harper">Ian R. Harper</a> (1997). Financial Economics. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0198775409" title="Special:BookSources/978-0198775409"><bdi>978-0198775409</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Economics&rft.pub=Oxford+University+Press&rft.date=1997&rft.isbn=978-0198775409&rft.au=J%C3%BCrgen+Eichberger&rft.au=Ian+R.+Harper&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIgor_EvstigneevThorsten_HensKlaus_Reiner_Schenk-Hoppé2015" class="citation book cs1">Igor Evstigneev; Thorsten Hens; Klaus Reiner Schenk-Hoppé (2015). Mathematical Financial Economics: A Basic Introduction. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3319165707" title="Special:BookSources/978-3319165707"><bdi>978-3319165707</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Financial+Economics%3A+A+Basic+Introduction&rft.pub=Springer&rft.date=2015&rft.isbn=978-3319165707&rft.au=Igor+Evstigneev&rft.au=Thorsten+Hens&rft.au=Klaus+Reiner+Schenk-Hopp%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrank_J._Fabozzi,_Edwin_H._Neave_and_Guofu_Zhou2011" class="citation book cs1"><a href="/wiki/Frank_J._Fabozzi" title="Frank J. Fabozzi">Frank J. Fabozzi</a>, Edwin H. Neave and Guofu Zhou (2011). Financial Economics. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470596203" title="Special:BookSources/978-0470596203"><bdi>978-0470596203</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Economics&rft.pub=Wiley&rft.date=2011&rft.isbn=978-0470596203&rft.au=Frank+J.+Fabozzi%2C+Edwin+H.+Neave+and+Guofu+Zhou&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEugene_F._Fama_and_Merton_H._Miller1972" class="citation book cs1">Eugene F. Fama and Merton H. Miller (1972). <a rel="nofollow" class="external text" href="https://faculty.chicagobooth.edu/eugene.fama/research/Theory%20of%20Finance/The%20Theory%20of%20Finance%20Preface%20and%20Table%20of%20Contents.pdf">The Theory of Finance</a> (PDF). <a href="/wiki/Holt,_Rinehart_and_Winston" class="mw-redirect" title="Holt, Rinehart and Winston">Holt, Rinehart and Winston</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0030867320" title="Special:BookSources/0030867320"><bdi>0030867320</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+Theory+of+Finance&rft.pub=Holt%2C+Rinehart+and+Winston&rft.date=1972&rft.isbn=0030867320&rft.au=Eugene+F.+Fama+and+Merton+H.+Miller&rft_id=https%3A%2F%2Ffaculty.chicagobooth.edu%2Feugene.fama%2Fresearch%2FTheory%2520of%2520Finance%2FThe%2520Theory%2520of%2520Finance%2520Preface%2520and%2520Table%2520of%2520Contents.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristian_Gollier2004" class="citation book cs1">Christian Gollier (2004). The Economics of Risk and Time (2nd ed.). <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-57224-8" title="Special:BookSources/978-0-262-57224-8"><bdi>978-0-262-57224-8</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+Economics+of+Risk+and+Time&rft.edition=2nd&rft.pub=MIT+Press&rft.date=2004&rft.isbn=978-0-262-57224-8&rft.au=Christian+Gollier&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThorsten_Hens_and_Marc_Oliver_Rieger2010" class="citation book cs1"><a href="/wiki/Thorsten_Hens" title="Thorsten Hens">Thorsten Hens</a> and Marc Oliver Rieger (2010). Financial Economics: A Concise Introduction to Classical and Behavioral Finance. <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3540361466" title="Special:BookSources/978-3540361466"><bdi>978-3540361466</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Economics%3A+A+Concise+Introduction+to+Classical+and+Behavioral+Finance&rft.pub=Springer&rft.date=2010&rft.isbn=978-3540361466&rft.au=Thorsten+Hens+and+Marc+Oliver+Rieger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChi-fu_Huang_and_Robert_H._Litzenberger1998" class="citation book cs1"><a href="/wiki/Chi-fu_Huang" title="Chi-fu Huang">Chi-fu Huang</a> and <a href="/wiki/Robert_Litzenberger" title="Robert Litzenberger">Robert H. Litzenberger</a> (1998). Foundations for Financial Economics. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0135006535" title="Special:BookSources/978-0135006535"><bdi>978-0135006535</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+for+Financial+Economics&rft.pub=Prentice+Hall&rft.date=1998&rft.isbn=978-0135006535&rft.au=Chi-fu+Huang+and+Robert+H.+Litzenberger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJonathan_E._Ingersoll1987" class="citation book cs1"><a href="/wiki/Jonathan_E._Ingersoll" title="Jonathan E. Ingersoll">Jonathan E. Ingersoll</a> (1987). <a rel="nofollow" class="external text" href="https://archive.org/details/theoryoffinancia1987inge">Theory of Financial Decision Making</a>. Rowman & Littlefield. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0847673599" title="Special:BookSources/978-0847673599"><bdi>978-0847673599</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Financial+Decision+Making&rft.pub=Rowman+%26+Littlefield&rft.date=1987&rft.isbn=978-0847673599&rft.au=Jonathan+E.+Ingersoll&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryoffinancia1987inge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_A._Jarrow1988" class="citation book cs1"><a href="/wiki/Robert_A._Jarrow" title="Robert A. Jarrow">Robert A. Jarrow</a> (1988). Finance theory. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0133148657" title="Special:BookSources/978-0133148657"><bdi>978-0133148657</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finance+theory&rft.pub=Prentice+Hall&rft.date=1988&rft.isbn=978-0133148657&rft.au=Robert+A.+Jarrow&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChris_Jones2008" class="citation book cs1">Chris Jones (2008). Financial Economics. <a href="/wiki/Routledge" title="Routledge">Routledge</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0415375856" title="Special:BookSources/978-0415375856"><bdi>978-0415375856</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Economics&rft.pub=Routledge&rft.date=2008&rft.isbn=978-0415375856&rft.au=Chris+Jones&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrian_Kettell2002" class="citation book cs1">Brian Kettell (2002). Economics for Financial Markets. <a href="/wiki/Butterworth-Heinemann" title="Butterworth-Heinemann">Butterworth-Heinemann</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-5384-8" title="Special:BookSources/978-0-7506-5384-8"><bdi>978-0-7506-5384-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Economics+for+Financial+Markets&rft.pub=Butterworth-Heinemann&rft.date=2002&rft.isbn=978-0-7506-5384-8&rft.au=Brian+Kettell&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYvan_Lengwiler2006" class="citation book cs1">Yvan Lengwiler (2006). Microfoundations of Financial Economics: An Introduction to General Equilibrium Asset Pricing. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0691126319" title="Special:BookSources/978-0691126319"><bdi>978-0691126319</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Microfoundations+of+Financial+Economics%3A+An+Introduction+to+General+Equilibrium+Asset+Pricing&rft.pub=Princeton+University+Press&rft.date=2006&rft.isbn=978-0691126319&rft.au=Yvan+Lengwiler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_F._LeRoyJan_Werner2000" class="citation book cs1">Stephen F. LeRoy; Jan Werner (2000). Principles of Financial Economics. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521586054" title="Special:BookSources/978-0521586054"><bdi>978-0521586054</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Financial+Economics&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0521586054&rft.au=Stephen+F.+LeRoy&rft.au=Jan+Werner&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeonard_C._MacLeanWilliam_T._Ziemba2013" class="citation book cs1">Leonard C. MacLean; William T. Ziemba (2013). Handbook of the Fundamentals of Financial Decision Making. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9814417341" title="Special:BookSources/978-9814417341"><bdi>978-9814417341</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+the+Fundamentals+of+Financial+Decision+Making&rft.pub=World+Scientific&rft.date=2013&rft.isbn=978-9814417341&rft.au=Leonard+C.+MacLean&rft.au=William+T.+Ziemba&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntonio_Mele2022" class="citation book cs1"><a href="/wiki/Antonio_Mele" title="Antonio Mele">Antonio Mele</a> (2022). Financial Economics. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262046848" title="Special:BookSources/9780262046848"><bdi>9780262046848</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Economics&rft.pub=MIT+Press&rft.date=2022&rft.isbn=9780262046848&rft.au=Antonio+Mele&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_C._Merton1992" class="citation book cs1">Robert C. Merton (1992). Continuous-Time Finance. <a href="/wiki/Wiley-Blackwell" title="Wiley-Blackwell">Blackwell</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0631185086" title="Special:BookSources/978-0631185086"><bdi>978-0631185086</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Continuous-Time+Finance&rft.pub=Blackwell&rft.date=1992&rft.isbn=978-0631185086&rft.au=Robert+C.+Merton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrederic_S._Mishkin2012" class="citation book cs1"><a href="/wiki/Frederic_S._Mishkin" class="mw-redirect" title="Frederic S. Mishkin">Frederic S. Mishkin</a> (2012). The Economics of Money, Banking, and Financial Markets (3rd ed.). <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0132961974" title="Special:BookSources/978-0132961974"><bdi>978-0132961974</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+Economics+of+Money%2C+Banking%2C+and+Financial+Markets&rft.edition=3rd&rft.pub=Prentice+Hall&rft.date=2012&rft.isbn=978-0132961974&rft.au=Frederic+S.+Mishkin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarry_H._Panjer1998" class="citation book cs1"><a href="/wiki/Harry_Panjer" class="mw-redirect" title="Harry Panjer">Harry H. Panjer</a>, ed. (1998). Financial Economics with Applications. Actuarial Foundation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0938959489" title="Special:BookSources/978-0938959489"><bdi>978-0938959489</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Economics+with+Applications&rft.pub=Actuarial+Foundation&rft.date=1998&rft.isbn=978-0938959489&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeoffrey_Poitras2007" class="citation book cs1">Geoffrey Poitras, ed. (2007). Pioneers of Financial Economics. <a href="/wiki/Edward_Elgar_Publishing" title="Edward Elgar Publishing">Edward Elgar Publishing</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pioneers+of+Financial+Economics&rft.pub=Edward+Elgar+Publishing&rft.date=2007&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"> Volume I <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/978-1845423810" title="Special:BookSources/978-1845423810">978-1845423810</a>; Volume II <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/978-1845423827" title="Special:BookSources/978-1845423827">978-1845423827</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Roll2006" class="citation book cs1"><a href="/wiki/Richard_Roll" title="Richard Roll">Richard Roll</a>, ed. (2006). <a rel="nofollow" class="external text" href="https://www.e-elgar.com/shop/gbp/book-series/economics-and-finance/the-international-library-of-critical-writings-in-financial-economics-series.html">The International Library of Critical Writings in Financial Economics</a>. <a href="/wiki/Cheltenham" title="Cheltenham">Cheltenham</a>: <a href="/wiki/Edward_Elgar_Publishing" title="Edward Elgar Publishing">Edward Elgar Publishing</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+International+Library+of+Critical+Writings+in+Financial+Economics&rft.place=Cheltenham&rft.pub=Edward+Elgar+Publishing&rft.date=2006&rft_id=https%3A%2F%2Fwww.e-elgar.com%2Fshop%2Fgbp%2Fbook-series%2Feconomics-and-finance%2Fthe-international-library-of-critical-writings-in-financial-economics-series.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li></ul> Asset pricing <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKerry_E._Back2010" class="citation book cs1">Kerry E. Back (2010). Asset Pricing and Portfolio Choice Theory. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0195380613" title="Special:BookSources/978-0195380613"><bdi>978-0195380613</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Asset+Pricing+and+Portfolio+Choice+Theory&rft.pub=Oxford+University+Press&rft.date=2010&rft.isbn=978-0195380613&rft.au=Kerry+E.+Back&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTomas_Björk2009" class="citation book cs1">Tomas Björk (2009). Arbitrage Theory in Continuous Time (3rd ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0199574742" title="Special:BookSources/978-0199574742"><bdi>978-0199574742</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arbitrage+Theory+in+Continuous+Time&rft.edition=3rd&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0199574742&rft.au=Tomas+Bj%C3%B6rk&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_H._Cochrane2005" class="citation book cs1"><a href="/wiki/John_H._Cochrane" title="John H. Cochrane">John H. Cochrane</a> (2005). Asset Pricing. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0691121376" title="Special:BookSources/978-0691121376"><bdi>978-0691121376</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Asset+Pricing&rft.pub=Princeton+University+Press&rft.date=2005&rft.isbn=978-0691121376&rft.au=John+H.+Cochrane&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDarrell_Duffie2001" class="citation book cs1"><a href="/wiki/Darrell_Duffie" title="Darrell Duffie">Darrell Duffie</a> (2001). Dynamic Asset Pricing Theory (3rd ed.). Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0691090221" title="Special:BookSources/978-0691090221"><bdi>978-0691090221</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamic+Asset+Pricing+Theory&rft.edition=3rd&rft.pub=Princeton+University+Press&rft.date=2001&rft.isbn=978-0691090221&rft.au=Darrell+Duffie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwin_J._EltonMartin_J._GruberStephen_J._BrownWilliam_N._Goetzmann2014" class="citation book cs1"><a href="/wiki/Edwin_Elton" class="mw-redirect" title="Edwin Elton">Edwin J. Elton</a>; <a href="/wiki/Martin_J._Gruber" title="Martin J. Gruber">Martin J. Gruber</a>; Stephen J. Brown; <a href="/wiki/William_N._Goetzmann" title="William N. Goetzmann">William N. Goetzmann</a> (2014). Modern Portfolio Theory and Investment Analysis (9th ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1118469941" title="Special:BookSources/978-1118469941"><bdi>978-1118469941</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Portfolio+Theory+and+Investment+Analysis&rft.edition=9th&rft.pub=Wiley&rft.date=2014&rft.isbn=978-1118469941&rft.au=Edwin+J.+Elton&rft.au=Martin+J.+Gruber&rft.au=Stephen+J.+Brown&rft.au=William+N.+Goetzmann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_A._Haugen2000" class="citation book cs1"><a href="/wiki/Robert_Haugen" title="Robert Haugen">Robert A. Haugen</a> (2000). Modern Investment Theory (5th ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0130191700" title="Special:BookSources/978-0130191700"><bdi>978-0130191700</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Investment+Theory&rft.edition=5th&rft.pub=Prentice+Hall&rft.date=2000&rft.isbn=978-0130191700&rft.au=Robert+A.+Haugen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_S._Joshi,_Jane_M._Paterson2013" class="citation book cs1"><a href="/wiki/Mark_S._Joshi" title="Mark S. Joshi">Mark S. Joshi</a>, Jane M. Paterson (2013). Introduction to Mathematical Portfolio Theory. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1107042315" title="Special:BookSources/978-1107042315"><bdi>978-1107042315</bdi></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+Mathematical+Portfolio+Theory&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-1107042315&rft.au=Mark+S.+Joshi%2C+Jane+M.+Paterson&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLutz_Kruschwitz,_Andreas_Loeffler2005" class="citation book cs1">Lutz Kruschwitz, Andreas Loeffler (2005). <a rel="nofollow" class="external text" href="https://archive.org/details/discountedcashfl00krus">Discounted Cash Flow: A Theory of the Valuation of Firms</a>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470870440" title="Special:BookSources/978-0470870440"><bdi>978-0470870440</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discounted+Cash+Flow%3A+A+Theory+of+the+Valuation+of+Firms&rft.pub=Wiley&rft.date=2005&rft.isbn=978-0470870440&rft.au=Lutz+Kruschwitz%2C+Andreas+Loeffler&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiscountedcashfl00krus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_G._Luenberger2013" class="citation book cs1"><a href="/wiki/David_Luenberger" title="David Luenberger">David G. Luenberger</a> (2013). Investment Science (2nd ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0199740086" title="Special:BookSources/978-0199740086"><bdi>978-0199740086</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Investment+Science&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=2013&rft.isbn=978-0199740086&rft.au=David+G.+Luenberger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarry_M._Markowitz1991" class="citation book cs1"><a href="/wiki/Harry_M._Markowitz" class="mw-redirect" title="Harry M. Markowitz">Harry M. Markowitz</a> (1991). <a rel="nofollow" class="external text" href="https://cowles.yale.edu/sites/default/files/2022-09/m16-all.pdf">Portfolio Selection: Efficient Diversification of Investments</a> (PDF) (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1557861085" title="Special:BookSources/978-1557861085"><bdi>978-1557861085</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Portfolio+Selection%3A+Efficient+Diversification+of+Investments&rft.edition=2nd&rft.pub=Wiley&rft.date=1991&rft.isbn=978-1557861085&rft.au=Harry+M.+Markowitz&rft_id=https%3A%2F%2Fcowles.yale.edu%2Fsites%2Fdefault%2Ffiles%2F2022-09%2Fm16-all.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrank_Milne2003" class="citation book cs1"><a href="/wiki/Frank_Milne" title="Frank Milne">Frank Milne</a> (2003). Finance Theory and Asset Pricing (2nd ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0199261079" title="Special:BookSources/978-0199261079"><bdi>978-0199261079</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finance+Theory+and+Asset+Pricing&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=978-0199261079&rft.au=Frank+Milne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_Pennacchi2007" class="citation book cs1"><a href="/wiki/George_Pennacchi" title="George Pennacchi">George Pennacchi</a> (2007). Theory of Asset Pricing. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0321127204" title="Special:BookSources/978-0321127204"><bdi>978-0321127204</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Asset+Pricing&rft.pub=Prentice+Hall&rft.date=2007&rft.isbn=978-0321127204&rft.au=George+Pennacchi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Rubinstein2006" class="citation book cs1"><a href="/wiki/Mark_Rubinstein" title="Mark Rubinstein">Mark Rubinstein</a> (2006). A History of the Theory of Investments. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471770565" title="Special:BookSources/978-0471770565"><bdi>978-0471770565</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Theory+of+Investments&rft.pub=Wiley&rft.date=2006&rft.isbn=978-0471770565&rft.au=Mark+Rubinstein&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_F._Sharpe1999" class="citation book cs1"><a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William F. Sharpe</a> (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/portfoliotheoryc0000shar/page/n5/mode/2up">Portfolio Theory and Capital Markets: The Original Edition</a>. <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0071353205" title="Special:BookSources/978-0071353205"><bdi>978-0071353205</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Portfolio+Theory+and+Capital+Markets%3A+The+Original+Edition&rft.pub=McGraw-Hill&rft.date=1999&rft.isbn=978-0071353205&rft.au=William+F.+Sharpe&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fportfoliotheoryc0000shar%2Fpage%2Fn5%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li></ul> Corporate finance <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJonathan_BerkPeter_DeMarzo2013" class="citation book cs1">Jonathan Berk; Peter DeMarzo (2013). Corporate Finance (3rd ed.). <a href="/wiki/Pearson_Education" title="Pearson Education">Pearson</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0132992473" title="Special:BookSources/978-0132992473"><bdi>978-0132992473</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Corporate+Finance&rft.edition=3rd&rft.pub=Pearson&rft.date=2013&rft.isbn=978-0132992473&rft.au=Jonathan+Berk&rft.au=Peter+DeMarzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_BossaertsBernt_Arne_Ødegaard2006" class="citation book cs1"><a href="/wiki/Peter_Bossaerts" title="Peter Bossaerts">Peter Bossaerts</a>; Bernt Arne Ødegaard (2006). Lectures on Corporate Finance (Second ed.). World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-899-1" title="Special:BookSources/978-981-256-899-1"><bdi>978-981-256-899-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Corporate+Finance&rft.edition=Second&rft.pub=World+Scientific&rft.date=2006&rft.isbn=978-981-256-899-1&rft.au=Peter+Bossaerts&rft.au=Bernt+Arne+%C3%98degaard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_BrealeyStewart_MyersFranklin_Allen2013" class="citation book cs1"><a href="/wiki/Richard_Brealey" class="mw-redirect" title="Richard Brealey">Richard Brealey</a>; <a href="/wiki/Stewart_Myers" title="Stewart Myers">Stewart Myers</a>; <a href="/wiki/Franklin_Allen" title="Franklin Allen">Franklin Allen</a> (2013). <a href="/wiki/Principles_of_Corporate_Finance" title="Principles of Corporate Finance">Principles of Corporate Finance</a>. Mcgraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0078034763" title="Special:BookSources/978-0078034763"><bdi>978-0078034763</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Corporate+Finance&rft.pub=Mcgraw-Hill&rft.date=2013&rft.isbn=978-0078034763&rft.au=Richard+Brealey&rft.au=Stewart+Myers&rft.au=Franklin+Allen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCFA_Institute2022" class="citation book cs1"><a href="/wiki/CFA_Institute" title="CFA Institute">CFA Institute</a> (2022). Corporate Finance: Economic Foundations and Financial Modeling (3rd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1119743767" title="Special:BookSources/978-1119743767"><bdi>978-1119743767</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Corporate+Finance%3A+Economic+Foundations+and+Financial+Modeling&rft.edition=3rd&rft.pub=Wiley&rft.date=2022&rft.isbn=978-1119743767&rft.au=CFA+Institute&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_E._CopelandJ._Fred_WestonKuldeep_Shastri2004" class="citation book cs1">Thomas E. Copeland; J. Fred Weston; Kuldeep Shastri (2004). Financial Theory and Corporate Policy (4th ed.). Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0321127211" title="Special:BookSources/978-0321127211"><bdi>978-0321127211</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Theory+and+Corporate+Policy&rft.edition=4th&rft.pub=Pearson&rft.date=2004&rft.isbn=978-0321127211&rft.au=Thomas+E.+Copeland&rft.au=J.+Fred+Weston&rft.au=Kuldeep+Shastri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJulie_Dahlquist,_Rainford_Knight,_Alan_S._Adams2022" class="citation book cs1">Julie Dahlquist, Rainford Knight, Alan S. Adams (2022). <a rel="nofollow" class="external text" href="https://open.umn.edu/opentextbooks/textbooks/principles-of-finance">Principles of Finance</a>. OpenStax, Rice University. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781951693541" title="Special:BookSources/9781951693541"><bdi>9781951693541</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Finance&rft.pub=OpenStax%2C+Rice+University&rft.date=2022&rft.isbn=9781951693541&rft.au=Julie+Dahlquist%2C+Rainford+Knight%2C+Alan+S.+Adams&rft_id=https%3A%2F%2Fopen.umn.edu%2Fopentextbooks%2Ftextbooks%2Fprinciples-of-finance&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"><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>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAswath_Damodaran1996" class="citation book cs1"><a href="/wiki/Aswath_Damodaran" title="Aswath Damodaran">Aswath Damodaran</a> (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/corporatefinance0000damo">Corporate Finance: Theory and Practice</a>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471076803" title="Special:BookSources/978-0471076803"><bdi>978-0471076803</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Corporate+Finance%3A+Theory+and+Practice&rft.pub=Wiley&rft.date=1996&rft.isbn=978-0471076803&rft.au=Aswath+Damodaran&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcorporatefinance0000damo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoão_Amaro_de_Matos2001" class="citation book cs1">João Amaro de Matos (2001). Theoretical Foundations of Corporate Finance. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691087948" title="Special:BookSources/9780691087948"><bdi>9780691087948</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoretical+Foundations+of+Corporate+Finance&rft.pub=Princeton+University+Press&rft.date=2001&rft.isbn=9780691087948&rft.au=Jo%C3%A3o+Amaro+de+Matos&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC._KrishnamurtiS._R._Vishwanath2010" class="citation book cs1">C. Krishnamurti; S. R. Vishwanath (2010). <a rel="nofollow" class="external text" href="https://www.phindia.com/Books/BookDetail/9788120336117/advanced-corporate-finance-vishwanath-krishnamurti">Advanced Corporate Finance</a>. MediaMatics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-8120336117" title="Special:BookSources/978-8120336117"><bdi>978-8120336117</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Corporate+Finance&rft.pub=MediaMatics&rft.date=2010&rft.isbn=978-8120336117&rft.au=C.+Krishnamurti&rft.au=S.+R.+Vishwanath&rft_id=https%3A%2F%2Fwww.phindia.com%2FBooks%2FBookDetail%2F9788120336117%2Fadvanced-corporate-finance-vishwanath-krishnamurti&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_OgdenFrank_C._JenPhilip_F._O'Connor2002" class="citation book cs1">Joseph Ogden; Frank C. Jen; Philip F. O'Connor (2002). Advanced Corporate Finance. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0130915689" title="Special:BookSources/978-0130915689"><bdi>978-0130915689</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Corporate+Finance&rft.pub=Prentice+Hall&rft.date=2002&rft.isbn=978-0130915689&rft.au=Joseph+Ogden&rft.au=Frank+C.+Jen&rft.au=Philip+F.+O%27Connor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPascal_QuiryYann_Le_FurAntonio_SalviMaurizio_Dallochio2011" class="citation book cs1">Pascal Quiry; Yann Le Fur; Antonio Salvi; Maurizio Dallochio; <a href="/wiki/Pierre_Vernimmen" title="Pierre Vernimmen">Pierre Vernimmen</a> (2011). Corporate Finance: Theory and Practice (3rd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1119975588" title="Special:BookSources/978-1119975588"><bdi>978-1119975588</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Corporate+Finance%3A+Theory+and+Practice&rft.edition=3rd&rft.pub=Wiley&rft.date=2011&rft.isbn=978-1119975588&rft.au=Pascal+Quiry&rft.au=Yann+Le+Fur&rft.au=Antonio+Salvi&rft.au=Maurizio+Dallochio&rft.au=Pierre+Vernimmen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephen_RossRandolph_WesterfieldJeffrey_Jaffe2012" class="citation book cs1"><a href="/wiki/Stephen_Ross_(economist)" title="Stephen Ross (economist)">Stephen Ross</a>; Randolph Westerfield; Jeffrey Jaffe (2012). Corporate Finance (10th ed.). <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0078034770" title="Special:BookSources/978-0078034770"><bdi>978-0078034770</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Corporate+Finance&rft.edition=10th&rft.pub=McGraw-Hill&rft.date=2012&rft.isbn=978-0078034770&rft.au=Stephen+Ross&rft.au=Randolph+Westerfield&rft.au=Jeffrey+Jaffe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoel_M._Stern2003" class="citation book cs1"><a href="/wiki/Joel_Stern" title="Joel Stern">Joel M. Stern</a>, ed. (2003). The Revolution in Corporate Finance (4th ed.). <a href="/wiki/Wiley-Blackwell" title="Wiley-Blackwell">Wiley-Blackwell</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781405107815" title="Special:BookSources/9781405107815"><bdi>9781405107815</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+Revolution+in+Corporate+Finance&rft.edition=4th&rft.pub=Wiley-Blackwell&rft.date=2003&rft.isbn=9781405107815&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJean_Tirole2006" class="citation book cs1"><a href="/wiki/Jean_Tirole" title="Jean Tirole">Jean Tirole</a> (2006). The Theory of Corporate Finance. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0691125565" title="Special:BookSources/978-0691125565"><bdi>978-0691125565</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+Theory+of+Corporate+Finance&rft.pub=Princeton+University+Press&rft.date=2006&rft.isbn=978-0691125565&rft.au=Jean+Tirole&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIvo_Welch2017" class="citation book cs1"><a href="/wiki/Ivo_Welch" title="Ivo Welch">Ivo Welch</a> (2017). Corporate Finance (4th ed.). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9840049-2-8" title="Special:BookSources/978-0-9840049-2-8"><bdi>978-0-9840049-2-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Corporate+Finance&rft.edition=4th&rft.date=2017&rft.isbn=978-0-9840049-2-8&rft.au=Ivo+Welch&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Financial_economics&action=edit&section=20" title="Edit section: External links">edit</a>]</div> <table> <tbody><tr> <td valign="top"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <dl><dt>Surveys</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller_Merton_H2000" class="citation journal cs1"><a href="/wiki/Merton_H._Miller" class="mw-redirect" title="Merton H. Miller">Miller Merton H</a> (2000). "The History of Finance: An Eyewitness Account". Journal of Applied Corporate Finance. 13 (2): 8–14. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1745-6622.2000.tb00050.x">10.1111/j.1745-6622.2000.tb00050.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Corporate+Finance&rft.atitle=The+History+of+Finance%3A+An+Eyewitness+Account&rft.volume=13&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E8-%3C%2Fspan%3E14&rft.date=2000&rft_id=info%3Adoi%2F10.1111%2Fj.1745-6622.2000.tb00050.x&rft.au=Miller+Merton+H&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/I%20Present%20Value.doc">Great Moments in Financial Economics I</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/II%20Modigliani-Miller%20Theorem.doc">II</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/III%20Short-Sales%20and%20Stock%20Prices.doc">III</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/IV%20Fundamental%20Theorem%20-%20Part%20I.doc">IVa</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/IV%20Fundamental%20Theorem%20-%20Part%20II.doc">IVb</a> (<a href="/wiki/Internet_Archive" title="Internet Archive">archived</a>, 2007-06-27). <a href="/wiki/Mark_Rubinstein" title="Mark Rubinstein">Mark Rubinstein</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20030204203936/http://www.finance-and-physics.org/Library/Articles3/scienceandfinance/science.htm">The Scientific Evolution of Finance</a> (<a href="/wiki/Internet_Archive" title="Internet Archive">archived</a>, 2003-04-03). Don Chance and Pamela Peterson</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180615164831/http://www.econ.boun.edu.tr/content/2017/spring/EC-42701/Lecture%20Note-Markowitz%20(1999)-04-05-2017.pdf">The Early History of Portfolio Theory: 1600-1960</a>, Harry M. Markowitz. Financial Analysts Journal, Vol. 55, No. 4 (Jul. – Aug., 1999), pp. 5–16</li> <li><a rel="nofollow" class="external text" href="https://ssrn.com/abstract=244161">The Theory of Corporate Finance: A Historical Overview</a>, <a href="/wiki/Michael_C._Jensen" title="Michael C. Jensen">Michael C. Jensen</a> and Clifford W. Smith.</li> <li><a rel="nofollow" class="external text" href="http://emanuelderman.com/a-stylized-history-of-quantitative-finance/">A Stylized History of Quantitative Finance</a>, Emanuel Derman</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120326213849/http://www.thederivativesbook.com/Chapters/10Chap.pdf">Financial Engineering: Some Tools of the Trade</a> (discusses historical context of derivative pricing). Ch 10 in <a href="/wiki/Phelim_Boyle" title="Phelim Boyle">Phelim Boyle</a> and Feidhlim Boyle (2001). "Derivatives: The Tools That Changed Finance". Risk Books (June 2001). <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/189933288X" title="Special:BookSources/189933288X">189933288X</a></li> <li><a rel="nofollow" class="external text" href="https://www.inkling.com/read/principles-of-corporate-finance-brealey-10th/chapter-34/section-34-1">What We Do Know: The Seven Most Important Ideas in Finance</a>; <a rel="nofollow" class="external text" href="https://www.inkling.com/read/principles-of-corporate-finance-brealey-10th/chapter-34/section-34-2">What We Do Not Know: 10 Unsolved Problems in Finance</a>, <a href="/wiki/Richard_A._Brealey" title="Richard A. Brealey">Richard A. Brealey</a>, <a href="/wiki/Stewart_Myers" title="Stewart Myers">Stewart Myers</a> and <a href="/wiki/Franklin_Allen" title="Franklin Allen">Franklin Allen</a>.</li> <li><a rel="nofollow" class="external text" href="https://dspace.mit.edu/handle/1721.1/48732">An Overview of Modern Financial Economics</a> (<a href="/wiki/MIT" class="mw-redirect" title="MIT">MIT</a> <a href="/wiki/Working_paper" title="Working paper">Working paper</a>). <a href="/wiki/Chi-fu_Huang" title="Chi-fu Huang">Chi-fu Huang</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080429203224/http://cepa.newschool.edu/het/essays/capital/fisherinvest.htm">Irving Fisher's Theory of Investment</a> Gonçalo L. Fonseca, <a href="/wiki/The_New_School" title="The New School">The New School</a></li> <li><a rel="nofollow" class="external text" href="https://sites.bu.edu/perry/files/2019/04/Financial-Economics-Mehrling.pdf">Financial Economics</a>, <a href="/wiki/Perry_Mehrling" title="Perry Mehrling">Perry Mehrling</a>. (From "Handbook of the History of Economic Analysis", 2016.)</li></ul> </div> </td> <td valign="top"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> Course material <ul><li><a rel="nofollow" class="external text" href="http://pages.stern.nyu.edu/~dbackus/233/notes_econ_assetpricing.pdf">Fundamentals of Asset Pricing</a>, <a href="/wiki/David_K._Backus" title="David K. Backus">David K. Backus</a>, <a href="/wiki/New_York_University_Stern_School_of_Business" title="New York University Stern School of Business">NYU, Stern</a></li> <li><a rel="nofollow" class="external text" href="http://antoniomele.org/financial_economics">Financial Economics: Classics and Contemporary</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190423203032/http://antoniomele.org/financial_economics/">Archived</a> 2019-04-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/wiki/Antonio_Mele" title="Antonio Mele">Antonio Mele</a>, <a href="/wiki/Universit%C3%A0_della_Svizzera_Italiana" class="mw-redirect" title="Università della Svizzera Italiana">Università della Svizzera Italiana</a></li> <li><a rel="nofollow" class="external text" href="http://www.ulb.ac.be/cours/solvay/farber/PhD.htm">Microfoundations of Financial Economics</a> André Farber, <a href="/wiki/Solvay_Business_School" class="mw-redirect" title="Solvay Business School">Solvay Business School</a></li> <li><a rel="nofollow" class="external text" href="http://arquivo.pt/wayback/20160516071307/http%3A//viking.som.yale.edu/will/web_pages/will/finman540/classnotes/notes.html">An introduction to investment theory</a>, William Goetzmann, <a href="/wiki/Yale_School_of_Management" title="Yale School of Management">Yale School of Management</a></li> <li><a rel="nofollow" class="external text" href="http://www.stanford.edu/~wfsharpe/mia/MIA.HTM">Macro-Investment Analysis</a>. <a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William F. Sharpe</a>, <a href="/wiki/Stanford_Graduate_School_of_Business" title="Stanford Graduate School of Business">Stanford Graduate School of Business</a></li> <li><a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/sloan-school-of-management/15-401-finance-theory-i-fall-2008/">Finance Theory</a> (<a href="/wiki/MIT_OpenCourseWare" title="MIT OpenCourseWare">MIT OpenCourseWare</a>). <a href="/wiki/Andrew_Lo" title="Andrew Lo">Andrew Lo</a>, <a href="/wiki/MIT" class="mw-redirect" title="MIT">MIT</a>.</li> <li><a rel="nofollow" class="external text" href="http://oyc.yale.edu/economics/econ-251">Financial Theory</a> (<a href="/wiki/Open_Yale_Courses" title="Open Yale Courses">Open Yale Courses</a>). <a href="/wiki/John_Geanakoplos" title="John Geanakoplos">John Geanakoplos</a>, <a href="/wiki/Yale_University" title="Yale University">Yale University</a>.</li> <li><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/20120621123950/http://homepage.newschool.edu/~het/essays/capital/invest.htm">"The Theory of Investment"</a>. Archived from <a rel="nofollow" class="external text" href="http://homepage.newschool.edu/~het/essays/capital/invest.htm">the original</a> on June 21, 2012. Retrieved March 3, 2014.</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+Investment&rft_id=http%3A%2F%2Fhomepage.newschool.edu%2F~het%2Fessays%2Fcapital%2Finvest.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinancial+economics" class="Z3988">. G.L. Fonseca, <a href="/wiki/New_School_for_Social_Research" class="mw-redirect" title="New School for Social Research">New School for Social Research</a></li> <li><a rel="nofollow" class="external text" href="https://www.ma.utexas.edu/rtgs/applied/school2009/rtg09_trans.pdf">Introduction to Financial Economics</a>. Gordan Zitkovi, <a href="/wiki/University_of_Texas_at_Austin" title="University of Texas at Austin">University of Texas at Austin</a></li> <li><a rel="nofollow" class="external text" href="http://jhqian.org/apt/apbook.pdf">An Introduction to Asset Pricing Theory</a>, Junhui Qian, <a href="/wiki/Shanghai_Jiao_Tong_University" title="Shanghai Jiao Tong University">Shanghai Jiao Tong University</a></li></ul> </div> </td> <td valign="top"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> Links and portals <ul><li><a rel="nofollow" class="external text" href="http://www.aeaweb.org/jel/guide/jel.php?class=G">JEL Classification Codes Guide</a></li> <li><a rel="nofollow" class="external text" href="http://rfe.org/showCat.php?cat_id=56">Financial Economics Links on AEA's RFE</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171104233339/http://rfe.org/showCat.php?cat_id=56">Archived</a> 2017-11-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.ssrn.com/en/index.cfm/fen/">SSRN Financial Economics Network</a></li> <li><a rel="nofollow" class="external text" href="http://www.economicsnetwork.ac.uk/subjects/Financial%20Economics">Financial Economics listings on economicsnetwork.ac.uk</a></li> <li><a rel="nofollow" class="external text" href="https://fic.wharton.upenn.edu/policy-briefs/financial-economists-roundtable-fer/">Financial Economists Roundtable</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171107025720/https://fic.wharton.upenn.edu/policy-briefs/financial-economists-roundtable-fer/">Archived</a> 2017-11-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.nber.org/jel/G_index.html">NBER Working Papers in Financial Economics</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140313133821/http://www.qfinance.com/information-sources?section=financial-economics">Financial Economics Resources on QFINANCE</a> (<a href="/wiki/Internet_Archive" title="Internet Archive">archived</a> 2014-03-13)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160324132025/http://www.helsinki.fi/WebEc/webecg.html">Financial Economics Links on WebEc</a> (<a href="/wiki/Internet_Archive" title="Internet Archive">archived</a> 2016-03-24)</li></ul> Actuarial resources <ul><li><a rel="nofollow" class="external text" href="http://www.soa.org/education/exam-req/edu-exam-mfe-detail.aspx">"Models for Financial Economics (MFE)"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220118190929/https://www.soa.org/education/exam-req/edu-exam-mfe-detail.aspx">Archived</a> 2022-01-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/wiki/Society_of_Actuaries" title="Society of Actuaries">Society of Actuaries</a></li> <li><a rel="nofollow" class="external text" href="http://www.actuaries.org.uk/students/pages/ct8-financial-economics">"Financial Economics (CT8)"</a>, <a href="/wiki/Institute_and_Faculty_of_Actuaries" title="Institute and Faculty of Actuaries">Institute and Faculty of Actuaries</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140326125149/http://66.216.104.121/files/sections/prime.pdf">"A Primer In Financial Economics"</a>, S. F. Whelan, D. C. Bowie and A. J. Hibbert. <a href="/wiki/British_Actuarial_Journal" class="mw-redirect" title="British Actuarial Journal">British Actuarial Journal</a>, Volume 8, Issue 1, April 2002, pp. 27–65.</li> <li><a rel="nofollow" class="external text" href="http://www.soa.org/files/sections/actuary-journal-final.pdf">"Pension Actuary's Guide to Financial Economics"</a>. Gordon Enderle, Jeremy Gold, Gordon Latter and Michael Peskin. <a href="/wiki/Society_of_Actuaries" title="Society of Actuaries">Society of Actuaries</a> and <a href="/wiki/American_Academy_of_Actuaries" title="American Academy of Actuaries">American Academy of Actuaries</a>.</li></ul> </div> </td></tr></tbody></table> <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 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class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nobel_Memorial_Prize_in_Economic_Sciences" title="Nobel Memorial Prize in Economic Sciences">Nobel Memorial Prize in Economic Sciences</a></li> <li><a href="/wiki/Germ%C3%A1n_Bern%C3%A1cer_Prize" title="Germán Bernácer Prize">Germán Bernácer Prize</a></li> <li><a href="/wiki/Fischer_Black_Prize" title="Fischer Black Prize">Fischer Black Prize</a></li> <li><a href="/wiki/American_Finance_Association#Morgan_Stanley-AFA_Award_for_Excellence_in_Finance" title="American Finance Association">Morgan Stanley-American Finance Association Award</a></li> <li><a href="/wiki/Deutsche_Bank_Prize_in_Financial_Economics" title="Deutsche Bank Prize in Financial Economics">Deutsche Bank Prize in Financial Economics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Research awards</th><td class="navbox-list-with-group 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href="/wiki/Economic_history" title="Economic history">History</a></li> <li><a href="/wiki/Information_economics" title="Information economics">Information</a></li> <li><a href="/wiki/Institutional_economics" title="Institutional economics">Institutions</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">Management</a></li> <li><a href="/wiki/Organizational_economics" title="Organizational economics">Organization</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 sector</a></li> <li><a href="/wiki/Public_choice" title="Public choice">Public choice</a></li> <li><a href="/wiki/Social_choice" class="mw-redirect" title="Social choice">Social choice</a></li> <li><a href="/wiki/Regional_economics" title="Regional economics">Regional</a></li> <li><a href="/wiki/Natural_resource_economics" title="Natural resource economics">Resources</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/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> (<a href="/wiki/History_of_economic_thought" title="History of economic thought">history</a>) </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/Attention_economy" title="Attention economy">Attention</a></li> <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> </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. Buchanan">Buchanan</a></li> <li><a href="/wiki/Kenneth_Arrow" title="Kenneth Arrow">Arrow</a></li> <li><a href="/wiki/William_Baumol" title="William Baumol">Baumol</a></li> <li><a href="/wiki/Robert_Solow" title="Robert Solow">Solow</a></li> <li><a href="/wiki/Murray_Rothbard" title="Murray Rothbard">Rothbard</a></li> <li><a href="/wiki/Alan_Greenspan" title="Alan Greenspan">Greenspan</a></li> <li><a href="/wiki/Thomas_Sowell" title="Thomas Sowell">Sowell</a></li> <li><a href="/wiki/Gary_Becker" title="Gary Becker">Becker</a></li> <li><a href="/wiki/Elinor_Ostrom" title="Elinor Ostrom">Ostrom</a></li> <li><a href="/wiki/Amartya_Sen" title="Amartya Sen">Sen</a></li> <li><a href="/wiki/Robert_Lucas_Jr." title="Robert Lucas Jr.">Lucas</a></li> <li><a href="/wiki/Joseph_Stiglitz" title="Joseph Stiglitz">Stiglitz</a></li> <li><a href="/wiki/Richard_Thaler" title="Richard Thaler">Thaler</a></li> <li><a href="/wiki/Hans-Hermann_Hoppe" title="Hans-Hermann Hoppe">Hoppe</a></li> <li><a href="/wiki/Paul_Krugman" title="Paul Krugman">Krugman</a></li> <li><a href="/wiki/Thomas_Piketty" title="Thomas Piketty">Piketty</a></li> <li><a href="/wiki/Category:Economists" title="Category:Economists">more</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Economics_lists" title="Category:Economics lists">Lists</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/Glossary_of_economics" title="Glossary of economics">Glossary</a></li> <li><a href="/wiki/List_of_economists" title="List of economists">Economists</a></li> <li><a href="/wiki/List_of_important_publications_in_economics" title="List of important publications in economics">Publications</a> (<a href="/wiki/List_of_economics_journals" title="List of economics journals">journals</a>)</li> <li><a href="/wiki/Schools_of_economic_thought" title="Schools of economic thought">Schools</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Category:Economics" title="Category:Economics">Category</a></li> <li><a href="/wiki/Index_of_economics_articles" title="Index of economics articles">Index</a></li> <li><a href="/wiki/Category:Economics_lists" title="Category:Economics lists">Lists</a></li> <li><a href="/wiki/Outline_of_economics" title="Outline of economics">Outline</a></li> <li><a href="/wiki/List_of_important_publications_in_economics" title="List of important publications in economics">Publications</a></li> <li><a href="/wiki/Portal:Business" title="Portal:Business">Business portal</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="General_areas_of_finance28" style="padding:3px"><table class="nowraplinks 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:Finance" title="Template:Finance"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Finance" title="Template talk:Finance"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Finance" title="Special:EditPage/Template:Finance"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="General_areas_of_finance28" style="font-size:114%;margin:0 4em">General areas of <a href="/wiki/Finance" title="Finance">finance</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternative_investment" title="Alternative investment">Alternative investment</a></li> <li><a href="/wiki/Angel_investor" title="Angel investor">Angel investor</a></li> <li><a href="/wiki/Super_angel" title="Super angel">Super 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 href="/wiki/Exchange_traded_fund" class="mw-redirect" title="Exchange traded fund">Exchange traded fund</a></li> <li><a href="/wiki/Experimental_finance" title="Experimental finance">Experimental finance</a></li> <li><a href="/wiki/Finance" title="Finance">Financial</a> <ul><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_crime" title="Financial crime">crime</a></li> <li><a href="/wiki/Financial_deepening" title="Financial deepening">deepening</a></li> <li><a class="mw-selflink selflink">economics</a></li> <li><a href="/wiki/Financial_engineering" title="Financial engineering">engineering</a></li> <li><a href="/wiki/Financial_inclusion" title="Financial inclusion">inclusion</a></li> <li><a href="/wiki/Financial_institution" 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 href="/wiki/Mathematical_finance" title="Mathematical finance">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 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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_management192" 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> 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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 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