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.
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.
The accurate valuation of financial derivatives plays a pivotal role in the finance industry. Although closed formulas for pricing are available for certain models and option types, exemplified by the European Call and Put options in the Black-Scholes Model, the use of either more complex models or more sophisticated options precludes the existence of such formulas, thereby requiring alternative approaches. The Monte Carlo simulation, an alternative approach effective in nearly all scenarios, has already been challenged by quantum computing techniques that leverage Amplitude Estimation. Despite its theoretical promise, this approach currently faces limitations due to the constraints of hardware in the Noisy Intermediate-Scale Quantum (NISQ) era. In this study, we introduce and analyze a quantum algorithm for pricing European call options across a broad spectrum of asset models. This method transforms a classical approach, which utilizes the Fast Fourier Transform (FFT), into a quantum algorithm, leveraging the efficiency of the Quantum Fourier Transform (QFT). Furthermore, we compare this novel algorithm with existing quantum algorithms for option pricing.
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.
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.
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.
Options are financial instruments whose payoffs are determined by the value of an underlying asset. These underlying assets are typically stocks, interest rates, or exchange rates, but can also include commodities such as energy or agricultural products. Options serve diverse purposes, including hedging against price volatility and speculating on market movements. Additionally, they play a critical role in assessing certain positions on the balance sheets of insurance companies, particularly in the context of solvency capital requirement calculations. As an option’s payoff lies in the future and is stochastic in nature, finding a fair price today is a non-trivial task. The focus of this work is finding quantum algorithms for calculating these prices and compare them to algorithms designed for classical computers. This work follows two different directions on how to improve on classical methods for option pricing. The first direction is improving on Monte-Carlo (MC) methods and the second one is adapting Fourier based pricing algorithms. The Amplitude Estimation (AE) algorithm promises a quadratic speedup compared to classical MC. This fact makes this algorithm an interesting candidate for improvements by quantum algorithms. It was shown in the literature, that simple options with very crude discretization of the underlying stochastic models, can be implemented on a quantum device and evaluated with AE. One big challenge is the efficient implementation of the option’s payoff profile. To solve this problem colleagues and I have introduced a multi-objective genetic algorithm to automatically find well performing and efficient circuits to implement the payoff of non-path-dependent European options on one or more underlyings. Another strategy for pricing options is based on the fact, that in many models the characteristic function of the underlying is numerically more tractable than its distribution function. There are methods that make use of this fact and the performance of the Fast Fourier Transform (FFT) to efficiently calculate prices for European call options for many strikes in one go. The quantum version of the FFT, the Quantum Fourier Transform (QFT), is in some sense, even faster than its classical counterpart. We have adapted the existing classical method to make use of QFT, to benefit from its performance. Finally, the different strategies are benchmarked and compared regarding their accuracy, their potential for scaling and their adaptability to different styles of derivatives and models. The result of this comparison are requirements on the quantum hardware for the methods to be viable as well as use cases where each method has its benefits over the others.
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.