Pricing Lookback Options on a Quantum Computer

Paper: arXiv 2604.00389

Authors: Florence Paquette, Tania Belabbas, Emmanuel Hamel, Anne MacKay

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a novel application of Variational Quantum Imaginary Time Evolution (VarQITE) to handle non-smooth jump conditions in discretely monitored path-dependent options. It demonstrates strong mathematical rigor in mapping non-Hermitian PDE dynamics to quantum circuits and validates the results against Monte Carlo simulations.

Research Flowchart

  flowchart TD
    A[Goal: Price Discretely Monitored Lookback Options on Quantum Hardware] --> B[Map Black-Scholes PDE to Non-Hermitian Schrödinger Equation]
    B --> C[Inputs: Option Parameters & Hardware-Efficient Ansatz]
    C --> D{Handle Discrete Jumps at Monitoring Dates}
    D -->|Approach 1| E[Sequential Evolution via Dedicated Jump Hamiltonians]
    D -->|Approach 2| F[Simultaneous Multi-Function Evolution via Higher Dimensions]
    E & F --> G[Execute Variational Quantum Imaginary Time Evolution - VarQITE]
    G --> H[Outcomes: Validated Accuracy vs Monte Carlo & Resource/Complexity Trade-offs]

Cited References (30)

1. Quantum Computation and Quantum Information (2010)
2. Quantum Algorithms: A Survey of Applications and End-to-end Complexities (2025) — link
3. Variational Ansatz‑Based Quantum Simulation of Imaginary Time Evolution (2019) — link
4. Theory of Variational Quantum Simulation (2019) — link
5. Quantum circuits for isometries (2016) — link
6. Tensorized Pauli Decomposition Algorithm (2024) — link
7. Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates (2004) — link
8. Quantum Computing for Finance: Overview and Prospects (2019) — link

2 of 30 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.