A combination of dividend and Jump diffusion process on Heston model in deriving Black Scholes equation

Abstract

The reality that exists in a stock market situation is that assets do pay dividend to owners of assets or derivative securities. Dupire, Derman and Kani built an option pricing process with a dividend yielding diffusion process but lacked the jump diffusion component. Jump diffusion process that mostly been captured in modern stock market do exhibit discontinuous behavior on pricing of assets. Introduction of Black Scholes concept equation that assumes volatility is constant led to several studies that have proposed models that address the shortcomings of Black – Scholes model. Heston’s models stands out amongst most volatility models because the process of volatility is greater than zero and exhibit mean reversion process and that is what is observed from the stock market. One of the shortcomings of Heston’s model is that it doesn’t incorporate the dividend yield jump diffusion process. Black Scholes equation revolves around Geometric Brownian motion and its extensions. We therefore incorporate dividend yield and jump diffusion process on Heston’s model and use it to formulate a new Black Scholes equation using the knowledge of partial differential equations.