Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process

• Author(s): Maina Erick ; Andanje Mulambula ; Brian D. Oduor
• Paper ID: 1709112
• Page: 753-760
• Published Date: 17-06-2025
• Published In: Iconic Research And Engineering Journals
• Publisher: IRE Journals
• e-ISSN: 2456-8880
• Volume/Issue: Volume 8 Issue 12 June-2025

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 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 Itô'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.

Keywords

Black-Scholes equation, logistic Brownian motion, jump diffusion, dividend yield, option pricing, stochastic processes

Citations

IRE Journals:
Maina Erick , Andanje Mulambula , Brian D. Oduor "Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process" Iconic Research And Engineering Journals Volume 8 Issue 12 2025 Page 753-760

IEEE:
Maina Erick , Andanje Mulambula , Brian D. Oduor "Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process" Iconic Research And Engineering Journals, 8(12)