{"id":5262,"date":"2025-06-30T08:31:00","date_gmt":"2025-06-30T08:31:00","guid":{"rendered":"https:\/\/fs.kibu.ac.ke\/?p=5262"},"modified":"2025-12-26T22:49:48","modified_gmt":"2025-12-26T22:49:48","slug":"deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process","status":"publish","type":"post","link":"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/","title":{"rendered":"Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"5262\" class=\"elementor elementor-5262\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-749aed40 elementor-section-full_width elementor-section-height-min-height elementor-section-items-stretch elementor-section-height-default\" data-id=\"749aed40\" data-element_type=\"section\" id=\"ourgoal\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5e67bfbe\" data-id=\"5e67bfbe\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1845ac51 elementor-widget elementor-widget-theme-post-title elementor-page-title elementor-widget-heading\" data-id=\"1845ac51\" data-element_type=\"widget\" data-widget_type=\"theme-post-title.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process<\/h1>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-35d68c5f elementor-section-height-min-height elementor-section-boxed elementor-section-height-default elementor-section-items-middle\" data-id=\"35d68c5f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-131eb975\" data-id=\"131eb975\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-1029197e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1029197e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-6b6c431a elementor-invisible\" data-id=\"6b6c431a\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7e5216fb elementor-widget__width-auto elementor-widget elementor-widget-heading\" data-id=\"7e5216fb\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">2025<\/h2>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-3a1cdc6a elementor-invisible\" data-id=\"3a1cdc6a\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1ef75fdf elementor-widget elementor-widget-heading\" data-id=\"1ef75fdf\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">KIBU Authors<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9c60e62 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"9c60e62\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>Andanje Mulambula<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-7bb784d1 elementor-invisible\" data-id=\"7bb784d1\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-52e5821d elementor-tablet-align-left elementor-invisible elementor-widget elementor-widget-button\" data-id=\"52e5821d\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInLeft&quot;}\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-shrink\" href=\"https:\/\/www.irejournals.com\/paper-details\/1709112\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">VIEW ON PUBLISHER SITE<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-1ead7568 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1ead7568\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-34b66cdc elementor-invisible\" data-id=\"34b66cdc\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-36f55c3c elementor-widget elementor-widget-heading\" data-id=\"36f55c3c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Abstract<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9e72738 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"9e72738\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>This paper presents a novel approach to deriving the Black-Scholes differential equation by incorporating dividend yielding logistic Brownian motion with jump diffusion processes. Traditional Black-Scholes models assume constant volatility and neglect dividend payments and price discontinuities, which are prevalent in real financial markets. We extend the logistic Brownian motion framework to include both continuous dividend yields and jump diffusion components, creating a more realistic model for asset price dynamics. Using It\u00f4&#8217;s lemma and stochastic calculus, we derive the modified Black-Scholes partial differential equation that captures market complexities including price jumps and dividend distributions. The derived model demonstrates enhanced capability in describing asset price behavior under volatile market conditions, particularly during periods of economic uncertainty. Our theoretical framework provides a foundation for more accurate option pricing and risk management strategies in modern financial markets.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>2025 KIBU Authors Andanje Mulambula VIEW ON PUBLISHER SITE Abstract This paper presents a novel approach to deriving the Black-Scholes differential equation by incorporating dividend yielding logistic Brownian motion with jump diffusion processes. Traditional Black-Scholes models assume constant volatility and neglect dividend payments and price discontinuities, which are prevalent in real financial markets. We extend [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_header_footer","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5262","post","type-post","status-publish","format-standard","hentry","category-research-list"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process - Faculty of Science<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process - Faculty of Science\" \/>\n<meta property=\"og:description\" content=\"2025 KIBU Authors Andanje Mulambula VIEW ON PUBLISHER SITE Abstract This paper presents a novel approach to deriving the Black-Scholes differential equation by incorporating dividend yielding logistic Brownian motion with jump diffusion processes. Traditional Black-Scholes models assume constant volatility and neglect dividend payments and price discontinuities, which are prevalent in real financial markets. We extend [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/\" \/>\n<meta property=\"og:site_name\" content=\"Faculty of Science\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-30T08:31:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-26T22:49:48+00:00\" \/>\n<meta name=\"author\" content=\"kibabii\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"kibabii\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/\"},\"author\":{\"name\":\"kibabii\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/fc9b7f13eeb65cf0be520197c35b17d6\"},\"headline\":\"Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process\",\"datePublished\":\"2025-06-30T08:31:00+00:00\",\"dateModified\":\"2025-12-26T22:49:48+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/\"},\"wordCount\":158,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#organization\"},\"articleSection\":[\"Research List\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/\",\"url\":\"https:\/\/kibu.ac.ke\/fs\/deriving-the-black-scholes-differential-equation-using-dividend-yielding-logistic-brownian-motion-with-jump-diffusion-process\/\",\"name\":\"Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process - 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Traditional Black-Scholes models assume constant volatility and neglect dividend payments and price discontinuities, which are prevalent in real financial markets. 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