ABSTRACT In this paper we provide a quantum Monte Carlo algorithm to solve multidimensional Black–Scholes PDEs with correlation for option pricing. The payoff function of the option is of general form and is only required to be continuous and piecewise affine, which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension of the PDE and the reciprocal of the prescribed accuracy . Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed‐up compared to classical Monte Carlo methods. Furthermore, we provide numerical simulations in two dimensions using our developed package within the Qiskit framework tailored to price continuous piecewise affine options with respect to the Black–Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.
ABSTRACT In this paper we provide a quantum Monte Carlo algorithm to solve multidimensional Black–Scholes PDEs with correlation for option pricing. The payoff function of the option is of general form and is only required to be continuous and piecewise affine, which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension of the PDE and the reciprocal of the prescribed accuracy . Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed‐up compared to classical Monte Carlo methods. Furthermore, we provide numerical simulations in two dimensions using our developed package within the Qiskit framework tailored to price continuous piecewise affine options with respect to the Black–Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.
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.
As one of the major methods for the simulation of option pricing, Monte Carlo method assumes random fluctuations in the distribution of asset prices. Under certain uncertainties process, different evolution paths could be simulated so as to finally yield the expectation value of the asset price, which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations. In order to solve the above computational problem, quantum Monte Carlo (QMC) has been established and applied in the relevant systems such as European call options. In this work, both MC and QM methods are adopted to simulate European call options. Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer, the amplitude estimation (AE) algorithm is found to play a secondary role in accelerating the pricing of European options. More importantly, the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.
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.
Option pricing is a fundamental problem in quantitative finance, where the accuracy and efficiency of valuation methods directly affect investment and risk management decisions. While the Black-Scholes model provides a closed-form solution for European options, numerical approaches such as Monte Carlo simulation are widely used when analytical methods are intractable. However, Monte Carlo suffers from slow convergence with complexity O(1/ε2).Quantum Amplitude Estimation (QAE), an algorithm in quantum computing, promises a quadratic speedup with complexity O(1/ε). This paper implements QAE, specifically Iterative Amplitude Estimation (IAE), for European call option pricing and compares its performance against Monte Carlo simulation and the Black-Scholes model. Experiments are conducted on both Qiskit simulators and IBM Quantum hardware (ibm_brisbane).Results show that QAE achieves estimates close to Monte Carlo and analytical benchmarks, with mean payoff deviations of less than 0.01. While QAE produces wider confidence intervals due to noise and discretization, the findings demonstrate its potential scalability advantage for future quantum finance applications.
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=2n 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 O(d2N2+d/2) for local-volatility Black--Scholes and O(d2Nd+2) for Heston. Relative to grid-based finite-difference baselines, these scalings correspond to polynomial improvement factors Nd/2 and Nd, 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.
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%.