Quantum mechanical approach to pricing multi-asset path-dependent options

This paper introduces a novel quantum-mechanical approach to pricing different multi-asset options. By leveraging the foundational principles of quantum mechanics, including Hermitian unitary operators, momentum-space wave functions, superposition, and the solution methodology for multiple particles in a box, we extend and refine the mathematical framework for multi-asset option pricing. In the formalism of the Hilbert space, the multi-asset Black–Scholes–Merton partial differential equation is equivalently represented as a multi-particle Schrödinger differential equation, with its time evolution governed by a specific Hamiltonian operator. Our study reveals that the multi-asset Black–Scholes–Merton Hamiltonian operator is pseudo-Hermitian and serves as a generalized form of the Hamiltonian operator already derived for single-asset options. We show that the time-independent Schrödinger equation offers a systematic method for solving a variety of multi-asset options, including path-dependent instruments. We demonstrate the utility of this quantum-inspired methodology by pricing general European multi-asset options and double knock-out rainbow options involving multiple correlated underlying assets, each characterized by distinct limit prices. This work highlights the potential of quantum mechanics to enrich the underpinnings of financial option pricing and provides new pathways for modeling complex correlated multi-asset financial instruments.