A Logistic Nonlinear Black-Scholes-Merton Partial Differential Equation: European Option

Abstract

Nonlinear Black-Scholes equations 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 may have an impact on the stock price, the volatility, the drift and the option price itself. Most modern models are 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 Verhulst’s Logistic equation. The objective of this study was to derive a Logistic Nonlinear Black Scholes Merton Partial Differential equation by incorporating the Logistic geometric Brownian motion. The methodology involves, analysis of the geometric Brownian motion, review of logistic models, 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 analysed 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 nonlinear pattern.