Abstract Quantum computers have the potential to provide quadratic speedup for Monte Carlo methods currently used in various classical applications. In this work, we examine the advantage of quantum computers for financial option pricing with the Monte Carlo method. Systematic and statistical errors are handled in a joint framework, and a relationship to quantum gate error is established. New metrics are introduced for the assessment of quantum advantage based on sample count and optimized error handling. We implement and analyze a Fourier series based approach and demonstrate its benefit over the more traditional rescaling method in function approximation. Our numerical calculations reveal the unpredictable nature of systematic errors, making consistent quantum advantage difficult with current quantum hardware. Our results indicate that very low noise levels, a two-qubit gate error rate below 10−6, are necessary for the quantum method to outperform the classical one, but a low number of logical qubits (ca. 20) may be sufficient to see quantum advantage already.
Abstract Quantum computers have the potential to provide quadratic speedup for Monte Carlo methods currently used in various classical applications. In this work, we examine the advantage of quantum computers for financial option pricing with the Monte Carlo method. Systematic and statistical errors are handled in a joint framework, and a relationship to quantum gate error is established. New metrics are introduced for the assessment of quantum advantage based on sample count and optimized error handling. We implement and analyze a Fourier series based approach and demonstrate its benefit over the more traditional rescaling method in function approximation. Our numerical calculations reveal the unpredictable nature of systematic errors, making consistent quantum advantage difficult with current quantum hardware. Our results indicate that very low noise levels, a two-qubit gate error rate below 10−6, are necessary for the quantum method to outperform the classical one, but a low number of logical qubits (ca. 20) may be sufficient to see quantum advantage already.
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.
The ongoing progress in quantum technologies has fueled a sustained exploration of their potential applications across various domains. One particularly promising field is quantitative finance, where a central challenge is the pricing of financial derivatives-traditionally addressed through Monte Carlo integration techniques. In this work, we introduce two hybrid classical-quantum methods to address the option pricing problem. These approaches rely on reconstructing Fourier series representations of statistical distributions from the outputs of Quantum Machine Learning (QML) models based on Parametrized Quantum Circuits (PQCs). We analyze the impact of data size and PQC dimensionality on performance. Quantum Accelerated Monte Carlo (QAMC) is employed as a benchmark to quantitatively assess the proposed models in terms of computational cost and accuracy in the extraction of Fourier coefficients. Through the numerical experiments, we show that the proposed methods achieve remarkable accuracy, becoming a competitive quantum alternative for derivatives valuation.
The financial sector is anticipated to be one of the first industries to benefit from the increased computational power of quantum computers, in areas such as portfolio optimisation and risk management to financial derivative pricing. Financial mathematics, and derivative pricing in particular, are not areas quantum physicists are traditionally trained in despite the fact that they often have the raw technical skills needed to understand such topics. On the other hand, most quantum algorithms have largely focused on qubits, which are comprised of two discrete states, as the information carriers. However, discrete higher-dimensional qudits, in addition to possibly possessing increased noise robustness and allowing for novel error correction protocols in certain hardware implementations, also have logarithmically greater information storage and processing capacity. In the current NISQ era of quantum computing, a wide array of hardware paradigms are still being studied and any potential advantage a platform offers is worth exploring. Here we introduce the basic concepts behind financial derivatives for the unfamiliar enthusiast as well as outline in great detail the quantum algorithm routines needed to price a European option, the simplest derivative. This is done within the context of a quantum computer comprised of qudits and employing the natural higher-dimensional analogue of a qubit-based pricing algorithm with its various subroutines. From these pieces, one should relatively easily be able to tailor the scheme to more complex, realistic financial derivatives. Finally, the entire stack is numerically simulated with the results demonstrating how the qudit-based scheme's payoff quickly approaches that of both a similarly-resourced classical computer as well as the true payoff, within error, for a modest increase in qudit dimension.
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.
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.