The article provides a literature review and analysis of quantum computing methods in the context of option pricing, comparing them against a strict classical benchmark and recommending a viable way forward to quantum computing implementation. We assess the capabilities and the weaknesses of Windows-choice models closed-form legacy formulae of vanillas, lattice/PDE true hockeygoal models of early exercise, and Monte Carlo models of flexibleness and point out where classical run attracted and precision restraints are shown to apply in the new generation(s): complexity of variables, in this path dependence, in a portfolio composing. We proceed to describe quantum algorithms that are based on quantum Monte Carlo which is centered on amplitude-estimation-based algorithms to cut down on contact with sampling; avoiding the scale of classical Monte Carlo and are particularly appealing to complex derivatives; we describe circuit components to prepare states, encode payoffs, and approximate expectations. The study establishes research priorities that achieve scale implies: depth frequently used to design loaders in, and variants of payoff circuits, workflow standardizations that exchange the exchange of error-resilient quantum modules and technological risk engines in the NISQ age. In general, the paper offers a single source of information on classical-quantum option pricing, clean insertion architecture to deploy hybrids and a roadmap between demonstrations and reliable production.
The article provides a literature review and analysis of quantum computing methods in the context of option pricing, comparing them against a strict classical benchmark and recommending a viable way forward to quantum computing implementation. We assess the capabilities and the weaknesses of Windows-choice models closed-form legacy formulae of vanillas, lattice/PDE true hockeygoal models of early exercise, and Monte Carlo models of flexibleness and point out where classical run attracted and precision restraints are shown to apply in the new generation(s): complexity of variables, in this path dependence, in a portfolio composing. We proceed to describe quantum algorithms that are based on quantum Monte Carlo which is centered on amplitude-estimation-based algorithms to cut down on contact with sampling; avoiding the scale of classical Monte Carlo and are particularly appealing to complex derivatives; we describe circuit components to prepare states, encode payoffs, and approximate expectations. The study establishes research priorities that achieve scale implies: depth frequently used to design loaders in, and variants of payoff circuits, workflow standardizations that exchange the exchange of error-resilient quantum modules and technological risk engines in the NISQ age. In general, the paper offers a single source of information on classical-quantum option pricing, clean insertion architecture to deploy hybrids and a roadmap between demonstrations and reliable production.
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.
We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods. The options that we cover include vanilla options, multi-asset options and path-dependent options such as barrier options. We put an emphasis on the implementation of the quantum circuits required to build the input states and operators needed by amplitude estimation to price the different option types. Additionally, we show simulation results to highlight how the circuits that we implement price the different option contracts. Finally, we examine the performance of option pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. We employ a simple, yet effective, error mitigation scheme that allows us to significantly reduce the errors arising from noisy two-qubit gates.
Quantum computing has emerged as a transformative computational paradigm, offering novel capabilities to tackle complex problems in finance—most notably in derivative pricing, risk management, and portfolio optimization. This review synthesizes recent advances in quantum algorithms and hybrid quantum–classical frameworks as applied to financial services, examining how technologies such as quantum-annealing, quantum amplitude estimation, and variational quantum circuits can improve efficiency, accuracy, and scalability over classical methods. The study provides a systematic assessment of key use-cases: (1) derivative pricing and greeks computation via quantum Monte Carlo and amplitude estimation; (2) risk management enhancements, including Value-at-Risk (VaR) and conditional VaR through quantum gradient estimation; and (3) portfolio optimization via QUBO (quadratic unconstrained binary optimization) mapping, the Quantum Approximate Optimization Algorithm (QAOA) and quantum annealing approaches. In addition to reviewing technological potential, the paper addresses critical barriers—hardware noise, coherence limitations, data integration, regulatory considerations—and outlines future research directions, including quantum-resilient cryptography in finance and industry coupling. By bringing together algorithmic, financial and regulatory perspectives, this review seeks to guide academics and practitioners toward the next frontier in financial computing.
Financial modeling and risk analysis involve computationally intensive tasks -- Monte Carlo simulation for derivative pricing, portfolio optimisation over high-dimensional asset spaces, and Value-at-Risk estimation from complex multivariate distributions -- where quantum computing offers potential polynomial-to-quadratic speedups. This paper proposes the Quantum Financial Computation Framework (QFCF), a systematic evaluation of quantum algorithms for four core financial computing tasks: option pricing (quantum amplitude estimation for Monte Carlo), portfolio optimisation (QAOA and VQE on quadratic unconstrained binary optimisation formulations), credit risk assessment (quantum Bayesian inference), and Value-at-Risk estimation (quantum arithmetic circuits). QFCF is evaluated on 16 financial benchmark problems spanning European and Asian option pricing, 50-500 asset portfolio optimisation, and credit portfolio loss distributions on near-term (NISQ) and fault-tolerant quantum hardware models. Key results: quantum amplitude estimation achieves quadratic speedup over classical Monte Carlo for option pricing with circuit depth scaling as O(1/ε) versus classical O(1/ε2); QAOA achieves portfolio optimisation solution quality within 4.8% of classical branch-and-bound for 50-asset problems; quantum Bayesian credit scoring reduces inference time by 38x for 200-variable credit networks. The framework introduces the Quantum Financial Utility Index (QFUI) and quantitative quantum advantage thresholds for each financial task.
Quantum computing is revolutionizing computational methods in finance by enhancing efficiency and accuracy in financial modeling and risk management. This review explores the impact of quantum computing in finance, focusing on derivative pricing, risk management, and portfolio optimization. It analyzes the benefits, limitations, and future implications of integrating quantum technologies with classical financial systems. A comprehensive review of quantum Monte Carlo methods, quantum algorithms, and their financial applications was conducted. Theoretical foundations, including Chebyshev’s inequality and Grover’s search method, were evaluated for their role in enhancing sampling efficiency and predictive accuracy. Empirical studies and use cases demonstrate the practical implications of quantum computing in pricing financial derivatives and managing risk through Value at Risk and Conditional Value at Risk assessments. The critical findings reveal that Quantum Monte Carlo algorithms provide substantial efficiency gains, reducing sample size requirements by up to fourfold compared to classical methods. Techniques employed in prior studies highlight the effectiveness of Quantum Amplitude Estimation for derivative pricing and risk analysis. However, challenges persist in scalability, data quality, integration with classical systems, and compliance with regulatory standards. This research contributes to the current body of knowledge by providing a comprehensive analysis of quantum algorithms’ practical applications in derivative pricing, risk management, and portfolio optimization, demonstrating the efficiency gains and limitations of quantum approaches. Future research should explore hybrid quantum–classical frameworks, quantum applications in blockchain, and enhanced quantum cryptography to ensure secure and efficient financial transactions.
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.