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

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@article{1709112,
      author = {Maina Erick, Andanje Mulambula, Brian D. Oduor},
      title = {Deriving the Black-Scholes Differential Equation Using Dividend Yielding Logistic Brownian Motion with Jump Diffusion Process},
      journal = {Iconic Research And Engineering Journals},
      year = {2025},
      volume = {8},
      number = {12},
      pages = {753-760},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1709112.pdf},
      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 It?'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},
      month = {June},
  }