A New Perspective for Financial Option Pricing with New Dynamic Solutions of the Black-Scholes Equation

Abstract

As a solution to the traditional limitations of financial option pricing models, this paper presents a new model that aims to more accurately reflect the evolving dynamics of financial markets. In this context, this study addresses the Black-Scholes option pricing model. The model places a particular emphasis on time-varying variables, deviating from the assumption of constant volatility and risk-free interest rate inherent in the traditional Black-Scholes model. Our objective is to visually present the outcomes derived from the analytical solution of the dynamic Black-Scholes equation, achieved through the utilization of the generalized exponential rational function method. The fact that this method is used for the first time to generate analytical solutions of the Black-Scholes equation with variable coefficients stands out as a remarkable innovation in the academic literature. For the analytical solution with financial implications, both mathematical and financial analyses were performed and compared with solutions in the literature. Different values of a physical parameter affecting the changes in the stock price per unit time under the initial condition reflecting the characteristics of the current state of the market, its relationship with the price of the underlying asset and the characteristics of the option are simulated. The study aims to enrich the literature by offering a nuanced perspective on financial market dynamics, one that intricately recognizes the changing nature of fundamental parameters over time. The findings not only deepen our theoretical understanding but also hold potential practical implications, contributing to the development of more accurate and responsive financial models for investors, risk managers, and policymakers.