This thesis studies option pricing under an inverse-stable time change from both classical and quantum computational perspectives. The operational-time pricing expectation is approximated by truncating and discretizing the inverse clock and, in the direct formulation, the latent Gaussian variables. Direct and conditional finite-state representations are developed, together with Quantum Amplitude Estimation (QAE) estimators and error decompositions that separate deterministic approximation, state-preparation, function-evaluation, and finite-query errors. The same expectation structure is used to estimate Galerkin load-vector entries and reconstruct a fixed-maturity European call-price curve on a bounded log-price interval. In the first numerical example, Iterative Quantum Amplitude Estimation (IQAE) is implemented on a noiseless Qiskit Aer simulator and compared with a classical finite-state Galerkin benchmark. A second example considers a two-asset arithmetic-basket call and compares classical Monte Carlo with the median of five independent ideal canonical QAE estimates for the same finite-state target. The numerical results illustrate the improved oracle-query accuracy scaling available from amplitude estimation while also showing that state preparation, reversible payo! evaluation, deterministic discretization error, and gate-level implementation costs remain essential. Accordingly, the ex- periments demonstrate estimator-level query scaling rather than an end-to-end practical quantum speedup.
This thesis studies option pricing under an inverse-stable time change from both classical and quantum computational perspectives. The operational-time pricing expectation is approximated by truncating and discretizing the inverse clock and, in the direct formulation, the latent Gaussian variables. Direct and conditional finite-state representations are developed, together with Quantum Amplitude Estimation (QAE) estimators and error decompositions that separate deterministic approximation, state-preparation, function-evaluation, and finite-query errors. The same expectation structure is used to estimate Galerkin load-vector entries and reconstruct a fixed-maturity European call-price curve on a bounded log-price interval. In the first numerical example, Iterative Quantum Amplitude Estimation (IQAE) is implemented on a noiseless Qiskit Aer simulator and compared with a classical finite-state Galerkin benchmark. A second example considers a two-asset arithmetic-basket call and compares classical Monte Carlo with the median of five independent ideal canonical QAE estimates for the same finite-state target. The numerical results illustrate the improved oracle-query accuracy scaling available from amplitude estimation while also showing that state preparation, reversible payo! evaluation, deterministic discretization error, and gate-level implementation costs remain essential. Accordingly, the ex- periments demonstrate estimator-level query scaling rather than an end-to-end practical quantum speedup.
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 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.
This work introduces an end-to-end framework for multi-asset option pricing that combines market-consistent risk-neutral density recovery with quantum-accelerated numerical integration. We first calibrate arbitrage-free marginal distributions from European option quotes using the Normal Inverse Gaussian (NIG) model, leveraging its analytical tractability and ability to capture skewness and fat tails. Marginals are coupled via a Gaussian copula to construct joint distributions. To address the computational bottleneck of the high-dimensional integration required to solve the option pricing formula, we employ Quantum Accelerated Monte Carlo (QAMC) techniques based on Quantum Amplitude Estimation (QAE), achieving quadratic convergence improvements over classical Monte Carlo (CMC) methods. Theoretical results establish accuracy bounds and query complexity for both marginal density estimation (via cosine-series expansions) and multidimensional pricing. Empirical tests on liquid equity entities (Credit Agricole, AXA, Michelin) confirm high calibration accuracy and demonstrate that QAMC requires 10-100 times fewer queries than classical methods for comparable precision. This study provides a practical route to integrate arbitrage-aware modelling with quantum computing, highlighting implications for scalability and future extensions to complex derivatives.
Quantum Amplitude Estimation (QAE) is an attractive financial derivative pricing algorithm with the potential for a quadratic speedup over classical Monte Carlo approaches. Yet, quantum advantage will only be found if not just fault-tolerant hardware but also an accurate representation of the financial problem on a quantum circuit is achieved. This paper offers a reproducible, diagnostic examination of the default Qiskit Finance application to European option pricing. We present a critical diagnostic marker: the classical discretized price, which is the classically computed expected payoff on the quantum model's actual grid. Our convergence analysis of low-qubit models (4, 5, and 6 uncertainty qubits) shows that for this particular configuration, representation error, which arises due to imprecise trigonometric encoding of the option payoff function, is the prevalent source of error, much larger compared to the discretization error. In a 6-qubit model, the representation error was 0.17699 USD, which is an order of magnitude larger than the discretization error of -0.01200 USD. We illustrate how a different Piecewise Linear (PWL) encoding can successfully remove this representation error, but at the expense of more circuit resources. Our results, for the vanilla, low-qubit Qiskit Finance circuits we used, indicate that the creation of high-fidelity, resource-conscious problem encodings is a pressing and essential problem on the road to applicable quantum advantage in finance.
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.