Pricing Lookback Options on a Quantum Computer

TL;DR

This paper introduces a quantum algorithm for pricing discretely monitored lookback options by reformulating the problem as a Schrödinger-type equation and applying variational quantum imaginary time evolution, addressing jump conditions.

Contribution

It presents two quantum-compatible formulations for handling jump conditions in path-dependent options, enabling quantum pricing of complex derivatives.

Findings

Quantum algorithms can handle jump conditions in path-dependent options.

The proposed methods are benchmarked against Monte Carlo simulations.

Trade-offs between resource requirements and accuracy are analyzed.

Abstract

We develop a quantum algorithm to price discretely monitored lookback options in the Black-Scholes framework using imaginary time evolution. By rewriting the pricing PDE as a Schrodinger-type equation, the problem becomes the imaginary time evolution of a quantum state under a non-Hermitian Hamiltonian. This evolution is approximated with the Variational Quantum imaginary time evolution (VarQITE) method, which replaces the exact non-unitary dynamics with a parameterized, hardware-efficient quantum circuit. A central challenge arises from jump conditions caused by the discrete updating of the running maximum. This feature is not present in standard quantum treatments of European or Asian options. To address this, we propose two quantum-compatible formulations: (i) a sequential approach that models jumps via dedicated jump Hamiltonians applied at monitoring dates, and (ii) a simultaneous multi-function evolution that removes explicit jumps at the expense of an increased number of dimensions. We compare both approaches in terms of qubit resources, circuit complexity and numerical accuracy, and benchmark them against Monte Carlo simulations. Our results show that discretely monitored, path-dependent options with jump conditions can be handled within a variational quantum framework, paving the way toward the quantum pricing of more complex derivatives with non-smooth dynamics.