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.
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.
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.
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.
Fourier-based methods are among the most widely used techniques for pricing European options when the characteristic function of the underlying asset process is available. Their applicability to increasingly fine discretizations, however, is limited by the rapidly growing memory requirements of classical Fourier transforms, which become a computational bottleneck for large-scale pricing problems.
In this work, we overcome this limitation by reformulating the Carr-Madan pricing framework using tensor networks. Specifically, we employ the Superfast Fourier Transform (SFFT), a compressed Tensor Train representation of the Quantum Fourier Transform (QFT), and apply it directly to tensorized option pricing without ever explicitly constructing exponentially large vectors or Fourier operators. This formulation also enables a direct comparison between the classical tensor network algorithm and its quantum counterpart through QFT-based option pricing on quantum simulators and quantum hardware. Numerical experiments for European call options demonstrate that the proposed SFFT method maintains pricing accuracy while substantially reducing memory requirements and achieving subexponential computational scaling compared with conventional FFT-based pricing. The accompanying quantum simulations and hardware executions enable a direct comparison between the classical tensor network formulation and its QFT-based quantum counterpart, showing that both approaches avoid the exponential scaling of conventional Fourier implementations and provide complementary perspectives on large-scale option pricing.
Together, these results establish a unified framework connecting classical Fourier pricing, tensor network algorithms, and quantum computing approaches, demonstrating how tensorized Fourier methods can provide scalable alternatives for high-dimensional financial computations.
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.
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.
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.