{"id":25574,"date":"2025-03-09T11:57:05","date_gmt":"2025-03-09T10:57:05","guid":{"rendered":"https:\/\/qooant.com\/?p=25574"},"modified":"2026-07-11T15:50:17","modified_gmt":"2026-07-11T13:50:17","slug":"el-perque-de-lus-de-la-distribucio-normal","status":"publish","type":"post","link":"https:\/\/qooant.com\/ca\/el-perque-de-lus-de-la-distribucio-normal\/","title":{"rendered":"El perqu\u00e8 de l&#8217;us de la distribuci\u00f3 normal"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"25574\" class=\"elementor elementor-25574\">\n\t\t\t\t<div class=\"elementor-element elementor-element-97879ec e-flex e-con-boxed e-con e-parent\" data-id=\"97879ec\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-955cf24 elementor-widget elementor-widget-text-editor\" data-id=\"955cf24\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>L&#8217;an\u00e0lisi quantitativa dels mercats financers sovint assumeix que els preus o els retorns dels actius segueixen una distribuci\u00f3 normal. Aquesta aproximaci\u00f3 no \u00e9s perfecta, per\u00f2 resulta \u00fatil en molts contextos. A continuaci\u00f3, expliquem per qu\u00e8 la distribuci\u00f3 normal s&#8217;ha convertit en un dels models b\u00e0sics per entendre el moviment dels mercats.<\/p><h5>1. <strong>El Teorema Central del L\u00edmit i la Normalitat dels Retorns<\/strong><\/h5><p>Un dels motius principals per utilitzar la distribuci\u00f3 normal \u00e9s el <strong>Teorema Central del L\u00edmit (TCL)<\/strong>. Aquest teorema estableix que la suma o la mitjana d&#8217;un nombre suficientment gran de variables aleat\u00f2ries independents tendeix a seguir una distribuci\u00f3 normal, independentment de la distribuci\u00f3 original de les variables.<\/p><p>Aix\u00f2 implica que, si els preus dels actius estan influ\u00efts per molts factors petits i independents (informaci\u00f3 de mercat, decisions d&#8217;inversors, soroll aleatori, etc.), els <strong>retorns<\/strong> (variacions percentuals del preu) poden aproximar-se a una distribuci\u00f3 normal en el curt termini.<\/p><h5>2. <strong>Facilitats matem\u00e0tiques i modelatge<\/strong><\/h5><p>La distribuci\u00f3 normal t\u00e9 propietats matem\u00e0tiques que faciliten els c\u00e0lculs estad\u00edstics i la presa de decisions financeres:<\/p><ul><li>Est\u00e0 completament definida per <strong>dues<\/strong> variables: la mitjana (<strong><span class=\"katex\">\u03bc<\/span><\/strong>) i la desviaci\u00f3 est\u00e0ndard (<strong><span class=\"katex\">\u03c3<\/span><\/strong>).<\/li><li>Permet c\u00e0lculs senzills de probabilitats i riscos mitjan\u00e7ant la taula de la normal est\u00e0ndard.<\/li><li>Es poden construir models anal\u00edtics com el <strong>Value at Risk (VaR)<\/strong> o el model <strong>Black-Scholes<\/strong> per a opcions.<\/li><\/ul><h5>3. <strong>Regla del 68-95-99.7% i an\u00e0lisi de risc<\/strong><\/h5><p>Quan es modelen els retorns financers amb una distribuci\u00f3 normal, es poden establir regles generals per quantificar riscos:<\/p><ul><li><strong>68%<\/strong> dels valors estan dins d&#8217;una desviaci\u00f3 est\u00e0ndard de la mitjana (volatilitat baixa).<\/li><li><strong>95%<\/strong> dins de dues desviacions (esdeveniments m\u00e9s extrems, per\u00f2 relativament comuns).<\/li><li><strong>99.7%<\/strong> dins de tres desviacions (esdeveniments rars, com crisis financeres greus).<\/li><\/ul><p>Aquesta propietat \u00e9s clau en la gesti\u00f3 de riscos i en la predicci\u00f3 de probabilitats de p\u00e8rdues severes.<\/p><h5>4. <strong>El Moviment Brownia Geom\u00e8tric i els models financers<\/strong><\/h5><p>Molts models financers assumeixen que els preus segueixen un <strong>moviment brownia geom\u00e8tric (GBM)<\/strong>:<\/p><p><span class=\"katex\">$$dS_t = \\mu S_t dt + \\sigma S_t dW_t$$<\/span><\/p><p>Aquest model assumeix que els increments percentuals dels preus segueixen una distribuci\u00f3 normal, la qual cosa permet derivar solucions elegants per al preu de derivats financers.<\/p><h5>5. <strong>Conceptes clau en l\u2019estudi de la distribuci\u00f3 normal als mercats<\/strong><\/h5><h5><strong>5.1 Esperan\u00e7a Matem\u00e0tica (<span class=\"katex\">\u03bc<\/span>) i Vari\u00e0ncia (<span class=\"katex\">\u03c3<sup>2<\/sup><\/span>)<\/strong><\/h5><ul><li><strong>Esperan\u00e7a Matem\u00e0tica (<span class=\"katex\">\u03bc<\/span>)<\/strong>: Representa la mitjana dels retorns d\u2019un actiu financer. Indica la tend\u00e8ncia central de la distribuci\u00f3 dels preus o retorns.<\/li><li><strong>Vari\u00e0ncia (<span class=\"katex\">\u03c3<sup>2<\/sup><\/span>)<\/strong>: Mesura la dispersi\u00f3 dels retorns al voltant de la mitjana, indicant la volatilitat de l\u2019actiu.<\/li><\/ul><h5><strong>5.2 Desviaci\u00f3 Est\u00e0ndard (<span class=\"katex\">\u03c3<\/span>)<\/strong><\/h5><ul><li>La desviaci\u00f3 est\u00e0ndard \u00e9s la <strong>arrel quadrada de la vari\u00e0ncia<\/strong> i s\u2019interpreta directament com la volatilitat.<\/li><li>En models financers, sovint es treballa amb la <strong>volatilitat anualitzada<\/strong>, que \u00e9s la desviaci\u00f3 est\u00e0ndard dels retorns diaris multiplicada per <span class=\"katex\">\\(\\sqrt{252}\\)<\/span>\u00a0(per dies de borsa en un any).<\/li><\/ul><h5><strong>5.3 Distribuci\u00f3 Normal Est\u00e0ndard (N(0,1))<\/strong><\/h5><ul><li>La versi\u00f3 est\u00e0ndard de la normal t\u00e9 mitjana <strong>zero<\/strong> i desviaci\u00f3 est\u00e0ndard <strong>u<\/strong>. \u00c9s clau per treballar amb estad\u00edstics tipus <strong>Z-score<\/strong>: <span class=\"katex\">$$Z = \\frac{X &#8211; \\mu}{\\sigma}$$<\/span> Aix\u00f2 permet transformar qualsevol variable normal en una normal est\u00e0ndard per fer c\u00e0lculs de probabilitats.<\/li><\/ul><h5><strong>5.4 Valor en Risc (VaR &#8211; Value at Risk)<\/strong><\/h5><ul><li>Basat en la distribuci\u00f3 normal, el VaR estima la <strong>p\u00e8rdua m\u00e0xima esperada<\/strong> en un horitz\u00f3 de temps donat per a un nivell de confian\u00e7a determinat.<\/li><li>Exemple: un <strong>VaR al 5% diari<\/strong> indica la p\u00e8rdua que, amb un 95% de probabilitat, no es superar\u00e0 en un dia de trading.<\/li><\/ul><h5><strong>5.5 Test de Normalitat<\/strong><\/h5><ul><li>Els retorns financers <strong>no sempre segueixen una normal perfecta<\/strong> perqu\u00e8 poden tenir:<ul><li><strong>Asimetria (skewness)<\/strong>: Si la distribuci\u00f3 t\u00e9 m\u00e9s probabilitat a la dreta o esquerra.<\/li><li><strong>Curtosi (kurtosis)<\/strong>: Si t\u00e9 cues m\u00e9s pesades (exemple: m\u00e9s esdeveniments extrems).<\/li><\/ul><\/li><li>Per testar si els retorns segueixen una normal es poden fer:<ul><li><strong>Test de Jarque-Bera<\/strong>.<\/li><li><strong>Test de Shapiro-Wilk<\/strong>.<\/li><li><strong>QQ-Plot per veure les desviacions respecte a la normal<\/strong>.<\/li><\/ul><\/li><\/ul><h5>6. <strong>Limitacions de la normalitat en els mercats<\/strong><\/h5><p>Encara que la distribuci\u00f3 normal \u00e9s una primera aproximaci\u00f3 \u00fatil, no sempre captura correctament la realitat dels mercats:<\/p><ul><li>Els mercats sovint tenen <strong>cues m\u00e9s gruixudes<\/strong> del que prediu la normal (m\u00e9s esdeveniments extrems).<\/li><li>La <strong>volatilitat no \u00e9s constant<\/strong>, fet que viola una de les hip\u00f2tesis de la normalitat.<\/li><li>Hi ha <strong>correlacions din\u00e0miques<\/strong> i efectes de depend\u00e8ncia temporal entre els moviments de preus.<\/li><\/ul><p>Per aquests motius, es fan servir models m\u00e9s sofisticats, com les <strong>distribucions de L\u00e9vy<\/strong>, la <strong>teoria del valor extrem (EVT)<\/strong> i els <strong>models heteroced\u00e0stics (GARCH)<\/strong>.\u00a0<\/p><h5><strong>Conclusi\u00f3<\/strong><\/h5><p>L&#8217;atractiu de la distribuci\u00f3 normal en l&#8217;estudi dels mercats financers es deu a la seva simplicitat matem\u00e0tica i al suport te\u00f2ric del Teorema Central del L\u00edmit. Tot i que no \u00e9s un model perfecte, continua sent una eina essencial per entendre la volatilitat i el risc. Per capturar millor els fen\u00f2mens extrems i les irregularitats del mercat, sovint s&#8217;utilitzen models m\u00e9s complexos.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6c57e8c elementor-widget elementor-widget-wp-widget-custom_html\" data-id=\"6c57e8c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"wp-widget-custom_html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"textwidget custom-html-widget\">\r\n<\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>L&#8217;an\u00e0lisi quantitativa dels mercats financers sovint assumeix que els preus o els rendiments dels actius segueixen una distribuci\u00f3 normal. Aquesta aproximaci\u00f3 no \u00e9s perfecta, per\u00f2 \u00e9s \u00fatil en molts contextos. A continuaci\u00f3 expliquem per qu\u00e8 la distribuci\u00f3 normal s&#8217;ha convertit en un dels models b\u00e0sics per entendre els moviments del mercat. 1. El teorema central [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":25701,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[16],"tags":[],"class_list":["post-25574","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-didactic"],"_links":{"self":[{"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/posts\/25574","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/comments?post=25574"}],"version-history":[{"count":0,"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/posts\/25574\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/media\/25701"}],"wp:attachment":[{"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/media?parent=25574"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/categories?post=25574"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qooant.com\/ca\/wp-json\/wp\/v2\/tags?post=25574"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}