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arXiv:2605.26610

End-to-End PDE-Based Quantum Algorithms for Multi-Asset Option Pricing under Local and Stochastic Volatility

Nikita Guseynov, Nana Liu, Chi Seng Pun, Tushar Vaidya

quant-phq-fin.CPSubmitted May 26, 2026• Announced May 27, 2026• 0 citations
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Abstract

Multi-asset option pricing under local- and stochastic-volatility models leads naturally to high-dimensional parabolic PDEs. We develop an end-to-end quantum PDE framework for European option pricing under local-volatility Black--Scholes and Heston models. The framework takes classical contract and model data as input and returns classical estimates of selected option values. We solve the pricing PDEs after finite-difference discretization on spatial grids. For $N=2^n$ grid points per spatial direction and $d$ assets, the end-to-end gate complexity for single-point recovery, counted in elementary CNOT gates and one-qubit Pauli-axis rotations, has leading grid-size dependence $\widetilde{O}(d^2 N^{2+d/2})$ for local-volatility Black--Scholes and $\widetilde{O}(d^2 N^{d+2})$ for Heston. Relative to grid-based finite-difference baselines, these scalings correspond to polynomial improvement factors $N^{d/2}$ and $N^d$, respectively. These estimates translate to Clifford+T resources via standard compilation. We complement the complexity analysis with numerical benchmarks against standard classical methods. In the Heston setting, the framework recovers option prices across strikes together with the associated implied-volatility smile/skew. Overall, this work provides a complete end-to-end quantum pricing pipeline with explicit resource accounting and theoretical performance guarantees.

Comments
49 pages, 10 figures, 10 tables

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Multi-asset option pricing under local- and stochastic-volatility models leads naturally to high-dimensional parabolic PDEs. We develop an end-to-end quantum PDE framework for European option pricing under local-volatility Black--Scholes and Heston models. The framework takes classical contract and model data as input and returns classical estimates of selected option values. We solve the pricing PDEs after finite-difference discretization on spatial grids. For N=2nN=2^nN=2n grid points per spatial direction and ddd assets, the end-to-end gate complexity for single-point recovery, counted in elementary CNOT gates and one-qubit Pauli-axis rotations, has leading grid-size dependence O~(d2N2+d/2)\widetilde{O}(d^2 N^{2+d/2})O(d2N2+d/2) for local-volatility Black--Scholes and O~(d2Nd+2)\widetilde{O}(d^2 N^{d+2})O(d2Nd+2) for Heston. Relative to grid-based finite-difference baselines, these scalings correspond to polynomial improvement factors Nd/2N^{d/2}Nd/2 and NdN^dNd, respectively. These estimates translate to Clifford+T resources via standard compilation. We complement the complexity analysis with numerical benchmarks against standard classical methods. In the Heston setting, the framework recovers option prices across strikes together with the associated implied-volatility smile/skew. Overall, this work provides a complete end-to-end quantum pricing pipeline with explicit resource accounting and theoretical performance guarantees.
The solution of option-pricing problems may turn out to be computationally demanding due to non-linear and path-dependent payoffs, the high dimensionality arising from multiple underlying assets, and sophisticated models of price dynamics. In this context, quantum computing has been proposed as a means to address these challenges efficiently. Prevailing approaches either simulate the stochastic differential equations governing the forward dynamics of underlying asset prices or directly solve the backward pricing partial differential equation. Here, we present an end-to-end quantum algorithmic framework that solves the Kolmogorov forward (Fokker-Planck) partial differential equation for local-volatility models by mapping it to a Hamiltonian-simulation problem via the Schrödingerisation technique. The algorithm specifies how to prepare the initial quantum state, perform Hamiltonian simulation, and how to efficiently recover the option price via a swap test. In particular, the efficiency of the final solution recovery is an important advantage of solving the forward versus the backward partial differential equation. Thus, our end-to-end framework offers a potential route toward quantum advantage for challenging option-pricing tasks. In particular, we obtain a polynomial advantage in grid size for the discretization of a single dimension. Nevertheless, the true power of our methodology lies in pricing high-dimensional systems, such as baskets of options, because the quantum framework admits an exponential speedup with respect to dimension, overcoming the classical curse of dimensionality.
In a global derivatives market with notional values in the hundreds of trillions of dollars, the accuracy and efficiency of pricing models are of fundamental importance, with direct implications for risk management, capital allocation, and regulatory compliance. In this work, we employ the Black-Scholes-Merton (BSM) framework not as an end in itself, but as a controlled benchmark environment in which to rigorously assess the capabilities of quantum machine learning methods. We propose a fully quantum approach to option pricing based on Quantum Neural Networks (QNNs), and, to the best of our knowledge, present one of the first implementations of such a methodology on currently available quantum hardware. Specifically, we investigate whether QNNs, by exploiting the geometric structure of Hilbert space, can effectively approximate option pricing functions. Our implementation utilizes a compact 2-qubit QNN architecture evaluated across multiple state-of-the-art quantum processors, including IBM Fez, IQM Garnet, IonQ Forte, and Rigetti Ankaa-3. This cross-platform study reveals distinct hardware-dependent performance characteristics while demonstrating that accurate pricing approximations can be achieved consistently across different devices despite the constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The results provide empirical evidence that QNN-based approaches constitute a viable framework for derivative pricing. While the analysis is conducted within the BSM setting, the broader significance lies in the potential extension of these methods to more realistic and computationally demanding models, including local volatility, stochastic volatility, and interest rate frameworks commonly used in practice.
ABSTRACT Quantum computing has attracted increasing attention as a potential tool for addressing computational challenges in quantitative finance. In particular, quantum amplitude estimation provides a theoretical quadratic speedup for expectation estimation compared with classical Monte Carlo simulation. However, most existing studies primarily emphasize algorithmic acceleration while retaining classical stochastic models for asset price dynamics. In this paper, we propose a unified quantum framework for European option pricing by formulating asset price evolution as a quantum continuous–time stochastic process (QCTSP) and integrating it with quantum amplitude estimation. By encoding both the stochastic dynamics of asset prices and the option payoff structure into a coherent quantum computational representation, the proposed approach enables option valuation to be performed entirely within a quantum probabilistic framework. Numerical simulations under the Black–Scholes setting are conducted to verify the feasibility and consistency of the method. Although the experiments are limited to a benchmark model, the proposed framework provides a systematic foundation for extending quantum option pricing to higher–dimensional and more complex financial models.
This work introduces an end-to-end framework for multi-asset option pricing that combines market-consistent risk-neutral density recovery with quantum-accelerated numerical integration. We first calibrate arbitrage-free marginal distributions from European option quotes using the Normal Inverse Gaussian (NIG) model, leveraging its analytical tractability and ability to capture skewness and fat tails. Marginals are coupled via a Gaussian copula to construct joint distributions. To address the computational bottleneck of the high-dimensional integration required to solve the option pricing formula, we employ Quantum Accelerated Monte Carlo (QAMC) techniques based on Quantum Amplitude Estimation (QAE), achieving quadratic convergence improvements over classical Monte Carlo (CMC) methods. Theoretical results establish accuracy bounds and query complexity for both marginal density estimation (via cosine-series expansions) and multidimensional pricing. Empirical tests on liquid equity entities (Credit Agricole, AXA, Michelin) confirm high calibration accuracy and demonstrate that QAMC requires 10-100 times fewer queries than classical methods for comparable precision. This study provides a practical route to integrate arbitrage-aware modelling with quantum computing, highlighting implications for scalability and future extensions to complex derivatives.
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 payoff of the option contract and no additional boundary conditions. Here, we present a digital quantum algorithm to solve the Black-Scholes equation on a quantum computer by mapping it to the Schr\"odinger 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 nine 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%.
This paper explores advancements in quantum algorithms for derivative pricing of exotics, a computational pipeline of fundamental importance in quantitative finance. For such cases, the classical Monte Carlo integration procedure provides the state-of-the-art provable, asymptotic performance: polynomial in problem dimension and quadratic in inverse-precision. While quantum algorithms are known to offer quadratic speedups over classical Monte Carlo methods, end-to-end speedups have been proven only in the simplified setting over the Black-Scholes geometric Brownian motion (GBM) model. This paper extends existing frameworks to demonstrate novel quadratic speedups for more practical models, such as the Cox-Ingersoll-Ross (CIR) model and a variant of Heston's stochastic volatility model, utilizing a characteristic of the underlying SDEs which we term fast-forwardability. Additionally, for general models that do not possess the fast-forwardable property, we introduce a quantum Milstein sampler, based on a novel quantum algorithm for sampling Lévy areas, which enables quantum multi-level Monte Carlo to achieve quadratic speedups for multi-dimensional stochastic processes exhibiting certain correlation types. We also present an improved analysis of numerical integration for derivative pricing, leading to substantial reductions in the resource requirements for pricing GBM and CIR models. Furthermore, we investigate the potential for additional reductions using arithmetic-free quantum procedures. Finally, we critique quantum partial differential equation (PDE) solvers as a method for derivative pricing based on amplitude estimation, identifying theoretical barriers that obstruct achieving a quantum speedup through this approach. Our findings significantly advance the understanding of quantum algorithms in derivative pricing, addressing key challenges and open questions in the field.