The Derivation of a Logistic Nonlinear Black Scholes Merton Partial Differential Equation: European Option

Abstract

\onlinear Black-Scholes equations have been increasingly attracting interest over the last twenty years. This is because they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor's preferences, which ruay have an impact on the stock price, the volatility, the drift and the option price itself. Recent models have been developed to take into account the feedback effect of a fund hedging strategy Or of the transaction costs of large traders tv[ost of these models cue represented by nonlinear variations of the well known Black- Scholes Equation.On the other hand, asset security prices may naturally not shoot up indefinitely (exponentially) leading to the use of Verhlusts Logistic equation. The objective of this study was to derive a Logistic Nonlinear Black Scholes f\. lertou Partial Differential equation by considering transaction costs (which \\ere oVBrlooked in the derivation of the classical Black Scholes model) and incorporating the Logistic geometric Brownian motion.The methodology involved, analysis of the geometric Brownian motion, review of logistic models, Ito's process and lemma, stochastic volatility models and the derivation of the linear and nonlinear Black-Scholes-Merton partial differential equation. Illiquid markets have also been analyzed alongside stochastic differential equations. The result of this study may enhance reliable decision making based on a rational prediction of the future asset prices given that in reality the stock market may depict a non linear pattern.