Robust graph theoretical approach for the Black–Scholes equation regulating European call options

Financial systems are the building blocks of modern economies, enabling investment and generating wealth and prosperity. The randomness of the financial markets makes it challenging to determine the options’ values. The Black–Scholes equation is a conventional algorithm created by the renowned figures Fischer Black, Myron Samuel Scholes, and Robert Cox Merton for European-style options that have become an indispensable tool for analyzing the characteristics of the financial market in this context. This paper presents a novel numerical strategy, the chromatic polynomial collocation method, from the algebraic graph theory background for the generic Black–Scholes equation governing the European call options. An active matrix of integration of chromatic polynomials of the complete graph approximates the solution of the Black–Scholes equation. The proposed method converts the Black–Scholes equation into a system of algebraic equations with the standard collocation points. The Newton–Raphson method is the resultant system of equations to obtain the offered method’s solution to the Black–Scholes equation. We validate the efficiency and robustness of our approach with a set of numerical experiments and comparative error analyses. The tabular findings and the graphical plots demonstrate the reliability of the proposed numerical scheme compared to modern numerical methods. The current hybrid, semianalytical methods rely on controlling parameters, conventional orthogonal polynomial approximations, extensive iterative processes, and transformation techniques to attain the desired convergence rate. However, the sparse operational matrices and the real coefficients of chromatic polynomials of the present method exhibit a remarkable convergence rate and offer a potential likelihood for fixing other concerns in option pricing, including path-dependent issues in European- and American-style options.