STUDY OF TWO-DIMENSIONAL QUANTUM BLACK–SCHOLES EQUATION VIA THE NATURAL TRANSFORM DECOMPOSITION METHOD (NTDM)

Abstract

This study presents an analytical–numerical framework for solving the two-dimensional fractional Quantum Black–Scholes equation using the Natural Transform Decomposition Method (NTDM). The conventional Black–Scholes model, when transformed into its Schrödinger-type equivalent, captures the stochastic dynamics of financial markets in a quantum-mechanical sense. To incorporate market irregularities and memory effects, fractional-order temporal derivatives are introduced. The Natural Transform, an integral operator that unifies Laplace and Sumudu transforms, is combined with the Adomian Decomposition Method to form NTDM, enabling fast-convergent series solutions. Analytical and numerical results for the European-style two-asset put option demonstrate the method’s efficiency and stability. MATLAB simulations verify the real and imaginary components of the solution and corresponding probability-density distributions. The proposed NTDM framework provides a generalizable tool for solving fractional partial differential equations in quantum finance and related applied-mathematics domains.

OBJECTIVE: This study aims to get the probability of future market stocks in the form of random variables and to estimate the solution of the Schrodinger fractional modified partial differential equation in complex state form for two stocks. A random variable is what is utilised to determine a security's price. The cost and price of the option is equivalent to the solution, and the state of the Schrodinger equation which is a complex state function, according to a probabilistic interpretation.