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

Titos Matsakos and Stuart Nield

Financial Risk Analytics, Credit & Risk Solutions, Market Intelligence, S&P Global, 25 Ropemaker St, London, EC2Y 9LY, UK

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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.

Monte Carlo simulations are widely used in financial risk management — from estimating value-at-risk (VaR) to pricing over-the-counter derivatives — but come at a significant computational cost. Previous studies have shown that quantum algorithms can provide a quadratic speed-up when starting from pre-computed probability distributions. When such distributions are not available, however, the associated cost to generate them may limit the quantum advantage. In this paper, we bypass this challenge by incorporating risk factor evolution to generate probability distributions within the quantum computation; for this, we use the term Quantum Monte Carlo simulations. In particular, we assemble quantum circuits that implement stochastic models for equity, interest rate, and credit risk classes, and provide end-to-end examples for both market and credit risk use cases.

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[1] Javier Gonzalez-Conde, Ángel Rodríguez-Rozas, Enrique Solano, and Mikel Sanz, "Efficient Hamiltonian simulation for solving option price dynamics", Physical Review Research 5 4, 043220 (2023).

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