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Finance

While several industries stand to benefit from quantum computing, the financial services industry has historically been an early adopter of quantum technology by investing in research and development efforts in the area of quantum finance. Finance has the distinct feature that more powerful and more accurate simulations can lead to direct competitive advantage, in a way that is harder to identify in other industries. In this application area, researchers strive to find quantum speedups for use cases of interest to financial services. A number of use cases have been proposed as candidates for quantum solutions, such as:

  • Derivative pricing (such as options [1], and collateralized debt obligations (CDO) [2]). Derivatives are financial instruments that are built upon an underlying asset (or assets) that can depend on the value of the asset in potentially complicated ways. In the derivative pricing problem, one needs to determine a fair price of the financial instrument, which typically depends on an expected value of the underlying assets at some later date. A similar and related problem is known as computing the Greeks [3]. The Greeks of a financial derivative are quantities that determine the sensitivity of the derivative to various parameters in the problem. For example, the Greeks of an option are given by the derivative of the value of the option with respect to some parameter, e.g., \(\Delta:=\partial V/\partial X\), where \(V\) is the value of the option and \(X\) is the price of the underlying asset.
  • Credit valuation adjustments (CVA) [4]. CVA is the problem of determining the fair price of a derivative, portfolio, or other financial instrument that is extended to a purchaser on credit, and that takes into account the purchaser's (potentially poor) credit rating, and the risk of default. CVA is typically given by the difference between the risk-free portfolio and the value of the portfolio taking into account the possibility of default.
  • Value at risk (VaR) [5]. Many forms of risk analysis can be considered, with VaR being a common example. VaR measures the total value a financial instrument (such as a portfolio) might lose over a predefined time interval within a fixed confidence interval. For example, the VaR of a portfolio might indicate that, with 95% probability, the portfolio will not lose more than \(\$Y\). A similar technique works as well for the related Credit Value at Risk (CVaR) problem.
  • Portfolio optimization [6]. The goal of portfolio optimization is to determine the optimal allocation of funds into a universe of investable assets such that the resulting portfolio maximizes returns and minimizes risk, while also respecting other constraints.

While there are are many more use cases and several approaches for generating quantum speedups, broadly speaking, many uses cases stem from one of two paths to quantum improvements: quantum enhancements to Monte Carlo methods (for simulating stochastic processes), and constrained optimization. In the first case, the approach generally involves encoding a relevant, problem-specific function into a quantum state, and then using quantum amplitude estimation to sample from the distribution quadratically fewer times than classical Monte Carlo methods [7]. In the second case, a financial use case is reduced to a constrained optimization problem, and a quantum algorithm for optimization is used to solve the problem.

Among the use cases studied in these two areas, option pricing and portfolio optimization often serve as archetypal examples of Monte Carlo and constrained optimization problems, respectively, and their associated quantum algorithms have the most follow-up work. Moreover, these two classes of problems comprise a considerable fraction of the classical compute used in the financial services industry. For these reasons, we will focus on these two use cases in this section, though the approaches, caveats, and complexities can (usually) be readily carried over to other relevant use cases.

In addition to the use cases described above, other areas of interest to the financial services industry include post-quantum cryptography, quantum-secure networking and quantum key distribution, etc. However, many of these topics or their proposed quantum implementations are outside the scope of this document. Quantum machine learning is yet another popular use case within quantum finance, but oftentimes these results are quantum approaches to standard machine learning problems, which are then applied to a financial application. As such, we will also not study machine learning in this finance-specific section, and we instead refer interested readers to any of the excellent review articles on quantum finance (e.g., [8, 9]) for more details.

Bibliography

  1. Nikitas Stamatopoulos, Daniel J Egger, Yue Sun, Christa Zoufal, Raban Iten, Ning Shen, and Stefan Woerner. Option pricing using quantum computers. Quantum, 4:291, 2020. arXiv: https://arxiv.org/abs/1905.02666. doi:10.22331/q-2020-07-06-291.

  2. Hao Tang, Anurag Pal, Tian-Yu Wang, Lu-Feng Qiao, Jun Gao, and Xian-Min Jin. Quantum computation for pricing the collateralized debt obligations. Quantum Engineering, 3(4):e84, 2021. arXiv: https://arxiv.org/abs/2008.04110. doi:10.1002/que2.84.

  3. Nikitas Stamatopoulos, Guglielmo Mazzola, Stefan Woerner, and William J Zeng. Towards quantum advantage in financial market risk using quantum gradient algorithms. Quantum, 6:770, 2022. arXiv: https://arxiv.org/abs/2111.12509. doi:10.22331/q-2022-07-20-770.

  4. Jeong Yu Han and Patrick Rebentrost. Quantum advantage for multi-option portfolio pricing and valuation adjustments. arXiv: https://arxiv.org/abs/2203.04924, 2022.

  5. Stefan Woerner and Daniel J Egger. Quantum risk analysis. npj Quantum Information, 5(1):15, 2019. arXiv: https://arxiv.org/abs/1806.06893. doi:10.1038/s41534-019-0130-6.

  6. Patrick Rebentrost and Seth Lloyd. Quantum computational finance: quantum algorithm for portfolio optimization. arXiv: https://arxiv.org/abs/1811.03975, 2018.

  7. Ashley Montanaro. Quantum speedup of monte carlo methods. Proceedings of the Royal Society A, 2015. arXiv: https://arxiv.org/abs/1504.06987. doi:10.1098/rspa.2015.0301.

  8. Dylan Herman, Cody Googin, Xiaoyuan Liu, Yue Sun, Alexey Galda, Ilya Safro, Marco Pistoia, and Yuri Alexeev. Quantum computing for finance. Nature Reviews Physics, 2023. arXiv: https://arxiv.org/abs/2201.02773. URL: https://doi.org/10.1038/s42254-023-00603-1, doi:10.1038/s42254-023-00603-1.

  9. Adam Bouland, Wim van Dam, Hamed Joorati, Iordanis Kerenidis, and Anupam Prakash. Prospects and challenges of quantum finance. arXiv: https://arxiv.org/abs/2011.06492, 2020.