Derivation of Black-Scholes-Merton Logistic Brownian Motion Differential Equation with Jump Diffusion Process

Abstract

Black- Scholes formed the foundation of option pricing. However, some of the assumptions like constant volatility and interest among others are practically impossible to implement hence other option pricing models have been explored to help come up with a much reliable way of predicting the price trends of options. Black-scholes assumed that the daily logarithmic returns of individual stocks are normally distributed. This is not true in practical sense especially in short term intervals because stock prices are able to reproduce the leptokurtic feature and to some extent the ``volatility smile". To address the above problem the Jump-Diffusion Model and the Kou Double-Exponential Jump-Diffusion Model were presented. But still they have not fully addressed the issue of reliable prediction because the observed implied volatility surface is skewed and tends to flatten out for longer maturities; The two models abilities to produce accurate results are reduced. This paper ventures into a research that will involve Black-Scholes-Merton logistic-type option pricing with jump diffusion. The knowledge of logistic Brownian motion will be used to develop a logistic Brownian motion with jump diffusion model for price process.