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

Titos Matsakos; Stuart Nield

doi:10.22331/q-2024-04-04-1306

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

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

Published:	2024-04-04, volume 8, page 1306
Eprint:	arXiv:2303.09682v2
Doi:	https://doi.org/10.22331/q-2024-04-04-1306
Citation:	Quantum 8, 1306 (2024).

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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.

Featured image: Quantum states of a 6-qubit circuit that implements a Quantum Monte Carlo simulation for a stochastic 2-period equity model together with Quantum Amplitude Estimation to measure the probability of the equity price to go up twice. With the help of rotation gates, each of the two risk factor qubits (green) encode the probability of the equity price to go up at the first and second periods, respectively. Taken together, the two qubits represent the probability (green bar chart) of the equity price to be down (left bar), remain unchanged (middle), or be up (right) at the end of the second period. The risk measure qubit state (red) is prepared with a Toffoli gate that encodes the probability (red bar chart) of two upward price moves (right bar) in the angle $\theta$. The 3 output qubits (blue/orange with superscripts 0, 1, and 2) go through a Quantum Fourier Transform gate (second column), a phase kickback process to imprint a multiple of $\theta$ onto their phases $\phi$ (third column), and an Inverse Quantum Fourier Transform gate (fourth column) that leverages quantum interference allowing to directly read off the value of $\theta$, see blue/orange bar chart.

Popular summary

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.

► BibTeX data

@article{Matsakos2024quantummontecarlo,
  doi = {10.22331/q-2024-04-04-1306},
  url = {https://doi.org/10.22331/q-2024-04-04-1306},
  title = {Quantum {M}onte {C}arlo simulations for financial risk analytics: scenario generation for equity, rate, and credit risk factors},
  author = {Matsakos, Titos and Nield, Stuart},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1306},
  month = apr,
  year = {2024}
}

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