Formulating black Scholes equation using a jump diffusion Heston’s model

Abstract

In modern financial mathematics, accurate values are obtained by taking into account a considerable number of more realistic assumptions in logistic Black Scholes equation. The aspects considered here are cost of transactions in trading, perfect illiquid markets and risks that occur from non – protected portfolio or large investments that have a lot of impact on price of the assets, volatility, the percentage drift and the life of the portfolio itself. In modern world of finance, Jump diffusion process is used to assess the behavior of non – continuous asset when pricing of options. Since the introduction of Black – Scholes concept model that assumes volatility is constant; several studies 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 the zero and follows mean reversion and this is what is observed in the market world. One of the shortcomings of Heston’s model is that it doesn’t incorporate the aspect jump diffusion process. The Black – Scholes partial differential equation that has been studied so far revolves around Geometric Brownian motion and its extensions. We therefore incorporate jump diffusion process on Heston’s model and use it to formulate a new Black – Scholes equation using the knowledge of partial differential equations.