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arXiv:2303.09682 · Quantum

Quantum Monte Carlo simulations for financial risk analytics: scenario generation for equity, rate, and credit risk factors

Titos Matsakos, Stuart Nield

Published Apr 4, 2024• 22 citations
arXivHTMLPDF

Abstract

Monte Carlo (MC) simulations are widely used in financial risk management, from estimating value-at-risk (VaR) to pricing over-the-counter derivatives. However, they come at a significant computational cost due to the number of scenarios required for convergence. If a probability distribution is available, Quantum Amplitude Estimation (QAE) algorithms can provide a quadratic speed-up in measuring its properties as compared to their classical counterparts. Recent studies have explored the calculation of common risk measures and the optimisation of QAE algorithms by initialising the input quantum states with pre-computed probability distributions. If such distributions are not available in closed form, however, they need to be generated numerically, and the associated computational cost may limit the quantum advantage. In this paper, we bypass this challenge by incorporating scenario generation – i.e. simulation of the risk factor evolution over time to generate probability distributions – into the quantum computation; we refer to this process as Quantum MC (QMC) simulations. Specifically, we assemble quantum circuits that implement stochastic models for equity (geometric Brownian motion), interest rate (mean-reversion models), and credit (structural, reduced-form, and rating migration credit models) risk factors. We then integrate these models with QAE to provide end-to-end examples for both market and credit risk use cases.

DOI
10.22331/q-2024-04-04-1306

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Monte Carlo (MC) simulations are widely used in financial risk management, from estimating value-at-risk (VaR) to pricing over-the-counter derivatives. However, they come at a significant computational cost due to the number of scenarios required for convergence. If a probability distribution is available, Quantum Amplitude Estimation (QAE) algorithms can provide a quadratic speed-up in measuring its properties as compared to their classical counterparts. Recent studies have explored the calculation of common risk measures and the optimisation of QAE algorithms by initialising the input quantum states with pre-computed probability distributions. If such distributions are not available in closed form, however, they need to be generated numerically, and the associated computational cost may limit the quantum advantage. In this paper, we bypass this challenge by incorporating scenario generation – i.e. simulation of the risk factor evolution over time to generate probability distributions – into the quantum computation; we refer to this process as Quantum MC (QMC) simulations. Specifically, we assemble quantum circuits that implement stochastic models for equity (geometric Brownian motion), interest rate (mean-reversion models), and credit (structural, reduced-form, and rating migration credit models) risk factors. We then integrate these models with QAE to provide end-to-end examples for both market and credit risk use cases.
The continuous growth of financial markets dictates the need for its participants to seek new approaches to financial analysis to gain competitive advantages, including through the use of new approaches in the field of computing. Quantum computing can be used as a tool for obtaining these advantages over competitors. In particular, Monte Carlo modeling, although widely used in financial risk management, requires significant computing resources due to the large number of scenarios required to obtain an accurate result. To optimize this approach, quantum amplitude estimation algorithms are used, which accelerate this process if pre-calculated probability distributions are used to initialize input quantum states. However, in the absence of these distributions in existing approaches to this topic, they are generated numerically using classical computing, which completely negates the advantage of the quantum approach. This article proposes a solution to this problem by using quantum computing, including for the generation of probability distributions. The article discusses the creation of quantum circuits for modeling the evolution of risk factors over time for capital flows, interest rates, and credit risks, and presents the combination of these models with quantum amplitude estimation algorithms as an example of using the obtained algorithms for credit risk management. In conclusion, the article analyzes the possibility of using the obtained circuits in financial analysis.
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.
Monte Carlo simulations underpin computational quantitative finance, where they are used for derivative pricing, risk quantification and scenario analysis whenever closed-form solutions are unavailable. Their main limitation is the slow O(N^-1/2) convergence of classical sampling. Quantum amplitude estimation promises a quadratic improvement in query complexity over classical Monte Carlo, reducing the sample requirement from O(eps^-2) to O(eps^-1) under idealised conditions. This thesis measures the extent to which the quadratic improvement in query complexity survives a concrete risk task under gate noise when compared to classical Monte Carlo. A four-obligor correlated portfolio, whose loss distribution can be computed exactly, is encoded into quantum circuits, and its value at risk and expected shortfall are estimated by iterative amplitude estimation under exactly simulated depolarising noise on every two-qubit gate. Six configurations of 11 to 15 qubits vary the portfolio, the factor grid and the arithmetic construction. At matched loss-model evaluation budgets and without noise, amplitude estimation reduces the RMSE of the tail probability against classical Monte Carlo by a factor of 13.6 at the largest budget tested. The error budget is dominated by the circuit encoding, not the estimator. As a result, the value at risk at the 95% level picks the wrong loss level in none of the noiseless repetitions, while the encoding understates expected shortfall by 0.177 loss units. Under noise, the signal decays with every amplification round. The number of rounds that still help falls from 64 at an error rate of 3x10^-6 to none at 3x10^-2. This work concludes that amplification benefits only while the product of gate count and error rate stays small against one. Because every obligor added enlarges the circuit, the error rate at which amplification remains useful falls as the portfolio grows, and advantage starts depending on circuit compression instead of estimator design.
The article discusses the significant impact of Quantum-Inspired Computing (QIC) on financial risk modeling, surpassing traditional High-Performance Computing (HPC) and AI systems. In volatile financial markets, conventional methods often fail to provide required speed and precision. QIC utilizes enhanced classical architectures with quantum-inspired techniques like quantum annealing and hybrid algorithms, improving computational efficiency. QIC’s notable applications include Monte Carlo simulations for risk metrics like Value at Risk (VaR), where techniques such as Quantum Amplitude Estimation (QAE) and Quantum Monte Carlo (QMC) offer speed advantages. The article concludes that portfolio optimization is also enhanced through advanced techniques, leading to more advance strategies improving credit risk assessment and derivative pricing, utilizing Quantum Signal Processing (QSP) to streamline complexity. With major firms like IBM and Google Cloud adopting QIC in financial models, the technology demonstrates its practical applicability.
Abstract We present a quantum algorithm that analyzes risk more efficiently than Monte Carlo simulations traditionally used on classical computers. We employ quantum amplitude estimation to price securities and evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer. Additionally, we show how to implement this algorithm and how to trade-off the convergence rate of the algorithm and the circuit depth. The shortest possible circuit depth—growing polynomially in the number of qubits representing the uncertainty—leads to a convergence rate of O ( M −2/3 ), where M is the number of samples. This is already faster than classical Monte Carlo simulations which converge at a rate of O ( M −1/2 ). If we allow the circuit depth to grow faster, but still polynomially, the convergence rate quickly approaches the optimum of O ( M −1 ). Thus, for slowly increasing circuit depths our algorithm provides a near quadratic speed-up compared to Monte Carlo methods. We demonstrate our algorithm using two toy models. In the first model we use real hardware, such as the IBM Q Experience, to price a Treasury-bill (T-bill) faced by a possible interest rate increase. In the second model, we simulate our algorithm to illustrate how a quantum computer can determine financial risk for a two-asset portfolio made up of government debt with different maturity dates. Both models confirm the improved convergence rate over Monte Carlo methods. Using simulations, we also evaluate the impact of cross-talk and energy relaxation errors.
The analysis of credit risk is crucial for the efficient operation of financial institutions. Quantum Amplitude Estimation (QAE) offers the potential for a quadratic speedup over classical methods used to estimate metrics such as Value at Risk (VaR) and Conditional Value at Risk (CVaR). However, numerous limitations remain in efficiently scaling the implementation of quantum circuits that solve these estimation problems. One of the main challenges is the use of costly and restrictive arithmetic that must be implemented within the quantum circuit. In this paper, we propose using Quantum Singular Value Transformation (QSVT) to significantly reduce the cost of implementing the state preparation operator, which underlies QAE for credit risk analysis. We also present an end-to-end code implementation and the results of a simulation study to validate the proposed approach and demonstrate its benefits.