[Paper Review] Efficient Hamiltonian Simulation for Solving Option Price Dynamics

Javier Gonzalez-Conde, Ángel Rodríguez‐Rozas+2 more|arXiv (Cornell University)|Jan 11, 2021

Quantum Computing Algorithms and Architecture7 citations

TL;DR

This paper presents a digital quantum algorithm for solving the Black-Scholes partial differential equation in option pricing by mapping it to a Schrödinger equation via Hamiltonian simulation. Using a single ancillary qubit to embed the non-Hermitian Hamiltonian into a larger Hilbert space and applying periodic boundary conditions to stabilize the simulation, the method achieves classical-level accuracy with 9 qubits and a post-selection success probability exceeding 60% on a fault-tolerant quantum computer.

ABSTRACT

Pricing financial derivatives, in particular European-style options at different time-maturities and strikes, means a relevant problem in finance. The dynamics describing the price of vanilla options when constant volatilities and interest rates are assumed, is governed by the Black-Scholes model, a linear parabolic partial differential equation with terminal value given by the pay-off of the option contract and no additional boundary conditions. Here, we present a digital quantum algorithm to solve Black-Scholes equation on a quantum computer by mapping it to the Schrödinger equation. The non-Hermitian nature of the resulting Hamiltonian is solved by embedding its propagator into an enlarged Hilbert space by using only one additional ancillary qubit. Moreover, due to the choice of periodic boundary conditions, given by the definition of the discretized momentum operator, we duplicate the initial condition, which substantially improves the stability and performance of the protocol. The algorithm shows a feasible approach for using efficient Hamiltonian simulation techniques as Quantum Signal Processing to solve the price dynamics of financial derivatives on a digital quantum computer. Our approach differs from those based on Monte Carlo integration, exclusively focused on sampling the solution assuming the dynamics is known. We report expected accuracy levels comparable to classical numerical algorithms by using 9 qubits to simulate its dynamics on a fault-tolerant quantum computer, and an expected success probability of the post-selection procedure due to the embedding protocol above 60%.

Motivation & Objective

• To develop a quantum algorithm for solving the Black-Scholes PDE in financial option pricing using digital quantum computation.
• To address the challenge of non-Hermitian Hamiltonians arising from the Black-Scholes equation through unitary embedding in an enlarged Hilbert space.
• To improve numerical stability and performance by leveraging periodic boundary conditions in the discretized momentum operator.
• To demonstrate feasibility and accuracy comparable to classical numerical methods using a minimal qubit count (9 qubits) on a fault-tolerant quantum computer.
• To lay the foundation for extending the method to more complex PDEs, such as those with time- or price-dependent volatility or coupled options.

Proposed method

• Map the Black-Scholes PDE to a Schrödinger equation by reinterpreting the time evolution as unitary dynamics under a non-Hermitian Hamiltonian.
• Embed the non-Hermitian Hamiltonian into a larger Hilbert space using a single ancillary qubit to enable unitary time evolution.
• Apply periodic boundary conditions to the discretized momentum operator, which allows duplication of the initial state and enhances simulation stability.
• Use quantum signal processing techniques for efficient Hamiltonian simulation to evolve the quantum state over time.
• Implement post-selection to project the final state onto the desired computational subspace, where the ancillary qubit is measured as |0⟩.
• Normalize the final state using a factor Λ derived from the normalization condition to extract the option price.

Experimental results

Research questions

• RQ1 Can the Black-Scholes PDE be efficiently simulated on a digital quantum computer using Hamiltonian simulation techniques despite its non-Hermitian nature?
• RQ2 How can the non-Hermitian Hamiltonian arising from the Black-Scholes equation be embedded into a unitary evolution framework with minimal qubit overhead?
• RQ3 What impact do periodic boundary conditions have on the stability and accuracy of the quantum simulation of financial PDEs?
• RQ4 To what extent can this quantum algorithm achieve accuracy comparable to classical numerical solvers using only 9 qubits?
• RQ5 Can this approach be generalized to solve more complex PDEs, such as those with stochastic or spatially dependent volatility?

Key findings

• The algorithm achieves numerical precision comparable to classical numerical methods when simulating the Black-Scholes equation on a fault-tolerant quantum computer.
• The use of a single ancillary qubit enables effective embedding of the non-Hermitian Hamiltonian into a unitary evolution, allowing stable simulation.
• The post-selection success probability for measuring the ancillary qubit in the |0⟩ state exceeds 60%, indicating high efficiency of the protocol.
• The duplication of the initial condition due to periodic boundary conditions significantly improves the stability and performance of the simulation.
• The method is extendable to time- and price-dependent volatility models, where no analytical solution exists, offering a meaningful numerical approach.
• The framework can be adapted to simulate other financial derivatives, including American and Asian options, by modifying the boundary conditions and payoff functions.