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arXiv:2208.02364

Quantum Encoding and Analysis on Continuous Time Stochastic Process with Financial Applications

Xi‐Ning Zhuang, Zhao‐Yun Chen, Cheng Xue, Yu-Chun Wu, Guo‐Ping Guo

Published Aug 3, 2022• 2 citations
arXivHTMLPDF

Abstract

The continuous time stochastic process is a mainstream mathematical instrument modeling the random world with a wide range of applications involving finance, statistics, physics, and time series analysis, while the simulation and analysis of the continuous time stochastic process is a challenging problem for classical computers. In this work, a general framework is established to prepare the path of a continuous time stochastic process in a quantum computer efficiently. The storage and computation resource is exponentially reduced on the key parameter of holding time, as the qubit number and the circuit depth are both optimized via our compressed state preparation method. The desired information, including the path-dependent and history-sensitive information that is essential for financial problems, can be extracted efficiently from the compressed sampling path, and admits a further quadratic speed-up. Moreover, this extraction method is more sensitive to those discontinuous jumps capturing extreme market events. Two applications of option pricing in Merton jump diffusion model and ruin probability computing in the collective risk model are given.

DOI
10.48550/arxiv.2208.02364

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The continuous time stochastic process is a mainstream mathematical instrument modeling the random world with a wide range of applications involving finance, statistics, physics, and time series analysis, while the simulation and analysis of the continuous time stochastic process is a challenging problem for classical computers. In this work, a general framework is established to prepare the path of a continuous time stochastic process in a quantum computer efficiently. The storage and computation resource is exponentially reduced on the key parameter of holding time, as the qubit number and the circuit depth are both optimized via our compressed state preparation method. The desired information, including the path-dependent and history-sensitive information that is essential for financial problems, can be extracted efficiently from the compressed sampling path, and admits a further quadratic speed-up. Moreover, this extraction method is more sensitive to those discontinuous jumps capturing extreme market events. Two applications of option pricing in Merton jump diffusion model and ruin probability computing in the collective risk model are given.
Modeling stochastic phenomena in continuous time is an essential yet challenging problem. Analytic solutions are often unavailable, and numerical methods can be prohibitively time-consuming and computationally expensive. To address this issue, we propose an algorithmic framework tailored for quantum continuous time stochastic processes. This framework consists of two key procedures: data preparation and information extraction. The data preparation procedure is specifically designed to encode and compress information, resulting in a significant reduction in both space and time complexities. This reduction is exponential with respect to a crucial feature parameter of the stochastic process. Additionally, it can serve as a submodule for other quantum algorithms, mitigating the common data input bottleneck. The information extraction procedure is designed to decode and process compressed information with quadratic acceleration, extending the quantum-enhanced Monte Carlo method. The framework demonstrates versatility and flexibility, finding applications in statistics, physics, time series analysis and finance. Illustrative examples include option pricing in the Merton Jump Diffusion Model and ruin probability computing in the Collective Risk Model, showcasing the framework’s ability to capture extreme market events and incorporate history-dependent information. Overall, this quantum algorithmic framework provides a powerful tool for accurate analysis and enhanced understanding of stochastic phenomena.
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 Discrete stochastic processes (DSP) are instrumental for modeling the dynamics of probabilistic systems and have a wide spectrum of applications in science and engineering. DSPs are usually analyzed via Monte-Carlo methods since the number of realizations increases exponentially with the number of time steps, and importance sampling is often required to reduce the variance. We propose a quantum algorithm for calculating the characteristic function of a DSP, which completely defines its probability distribution, using the number of quantum circuit elements that grows only linearly with the number of time steps. The quantum algorithm reduces the Monte-Carlo sampling to a Bernoulli trial while taking all stochastic trajectories into account. This approach guarantees the optimal variance without the need for importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. The Fourier approximation can be used to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented. Proof-of-principle experiments are performed using the IBM quantum cloud platform.
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.
Stochastic processes play a fundamental role in physics, mathematics, engineering and finance. One potential application of quantum computation is to better approximate properties of stochastic processes. For example, quantum algorithms for Monte Carlo estimation combine a quantum simulation of a stochastic process with amplitude estimation to improve mean estimation. In this work we study quantum algorithms for simulating stochastic processes which are compatible with Monte Carlo methods. We introduce a new ``analog'' quantum representation of stochastic processes, in which the value of the process at time t is stored in the amplitude of the quantum state, enabling an exponentially efficient encoding of process trajectories. We show that this representation allows for highly efficient quantum algorithms for simulating certain stochastic processes, using spectral properties of these processes combined with the quantum Fourier transform. In particular, we show that we can simulate TTT timesteps of fractional Brownian motion using a quantum circuit with gate complexity polylog(T)\text{polylog}(T)polylog(T), which coherently prepares the superposition over Brownian paths. We then show this can be combined with quantum mean estimation to create end to end algorithms for estimating certain time averages over processes in time O(polylog(T)ε−c)O(\text{polylog}(T)ε^{-c})O(polylog(T)ε−c) where $3/2
Quantum computers are expected to surpass the computational capabilities of classical computers during this decade and have transformative impact on numerous industry sectors, particularly finance. In fact, finance is estimated to be the first industry sector to benefit from quantum computing, not only in the medium and long terms, but even in the short term. This survey paper presents a comprehensive summary of the state of the art of quantum computing for financial applications, with particular emphasis on stochastic modeling, optimization, and machine learning, describing how these solutions, adapted to work on a quantum computer, can potentially help to solve financial problems, such as derivative pricing, risk modeling, portfolio optimization, natural language processing, and fraud detection, more efficiently and accurately. We also discuss the feasibility of these algorithms on near-term quantum computers with various hardware implementations and demonstrate how they relate to a wide range of use cases in finance. We hope this article will not only serve as a reference for academic researchers and industry practitioners but also inspire new ideas for future research.