Logistic Black-Scholes-Merton Partial Differential Equation: A Case of Stochastic Volatility

Abstract

Real world systems have been created using differential equations, this has made it possible to predict future trends and behaviour. Specifically, stochastic differential equations have been fundamental in describing and understanding random phenomena. So far the Black-Scholes-Merton partial differential equation used in deriving the famous Black-Scholes-Merton model has been one of the greatest breakthroughs in finance as far as prediction of asset prices in the stock market is concerned. In this model we use the Logistic Brownian motion as opposed to the usual Brownian motion and we also consider volatility to be stochastic. In this stud y we have incorporated the stochastic nature of volatility and derived a Logistic Black-Scholes-Merton partial differential equation with stochastic volatility. This has been done by analyzing the Logistic Brownian motion and the Brownian motion, using the Ito process, Ito’s lemma, stochastic volatility model and reviewing the derivation of the Black-Scholes-Merton partial differential equation. The formulated Differential equation may enhance reliable decision making based on more rational prediction of asset prices.