Simulating the non-Hermitian dynamics of financial option pricing with quantum computers

TL;DR

This paper extends quantum imaginary time evolution (QITE) to simulate non-Hermitian Hamiltonians, enabling quantum-based solutions to PDEs like the Black-Scholes equation for financial option pricing.

Contribution

The authors generalize QITE to handle arbitrary non-Hermitian Hamiltonians, broadening its applicability to real-world PDEs such as the Black-Scholes equation.

Findings

Successfully simulated European option pricing using generalized QITE.

Demonstrated feasibility of quantum algorithms for non-Hermitian PDEs.

Extended QITE to non-Hermitian systems for practical financial modeling.

Abstract

The Schrodinger equation describes how quantum states evolve according to the Hamiltonian of the system. For physical systems, we have it that the Hamiltonian must be a Hermitian operator to ensure unitary dynamics. For anti-Hermitian Hamiltonians, the Schrodinger equation instead models the evolution of quantum states in imaginary time. This process of imaginary time evolution has been used successfully to calculate the ground state of a quantum system. Although imaginary time evolution is non-unitary, the normalised dynamics of this evolution can be simulated on a quantum computer using the quantum imaginary time evolution (QITE) algorithm. In this paper, we broaden the scope of QITE by removing its restriction to anti-Hermitian Hamiltonians, which allows us to solve any partial differential equation (PDE) that is equivalent to the Schrodinger equation with an arbitrary, non-Hermitian Hamiltonian. An example of such a PDE is the famous Black-Scholes equation that models the price of financial derivatives. We will demonstrate how our generalised QITE methodology offers a feasible approach for real-world applications by using it to price various European option contracts modelled according to the Black-Scholes equation.