Quantum algorithm for copula-based risk aggregation using orthogonal series density estimation
Abstract: Quantum Monte Carlo integration (QMCI) provides a quadratic speed-up over its classical counterpart, and its applications have been investigated in various fields, including finance. This paper considers its application to risk aggregation, one of the most important numerical tasks in financial risk management. Risk aggregation combines several risk variables and quantifies the total amount of risk, taking into account the correlation among them. For this task, there exists a useful tool called copula, with which the joint distribution can be generated from marginal distributions with a flexible correlation structure. Classically, the copula-based method utilizes sampling of risk variables. However, this procedure is not directly applicable to the quantum setting, where sampled values are not stored as classical data, and thus no efficient quantum algorithm is known. In this paper, we introduce a quantum algorithm for copula-based risk aggregation that is compatible with QMCI. In our algorithm, we first estimate each marginal distribution as a series of orthogonal functions, where the coefficients can be calculated with QMCI. Then, by plugging the marginal distributions into the copula and obtaining the joint distribution, we estimate risk measures using QMCI again. With this algorithm, nearly quadratic quantum speed-up can be obtained for sufficiently smooth marginal distributions.
- Ashley Montanaro. Quantum speedup of monte carlo methods. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 471, No. 2181, p. 20150301, September 2015.
- Quantum computing for finance. Nature Reviews Physics, Vol. 5, No. 8, pp. 450–465, 2023.
- Correlation and dependence in risk management: properties and pitfalls. Risk management: value at risk and beyond, Vol. 1, pp. 176–223, 2002.
- Roger B Nelsen. An Introduction to Copulas. Springer, New York, NY, USA, second edition, 2006.
- Sam Efromovich. Orthogonal series density estimation. WIREs Computational Statistics, Vol. 2, No. 4, pp. 467–476, 2010.
- G. Maruyama. Continuous markov processes and stochastic equations. Rend. Circ. Mat. Palermo, Vol. 4, pp. 48–90, 1955.
- Georg Mainik. Risk aggregation with empirical margins: Latin hypercubes, empirical copulas, and convergence of sum distributions. Journal of Multivariate Analysis, Vol. 141, pp. 197–216, 2015.
- Hermite series estimates of a probability density and its derivatives. Journal of Multivariate Analysis, Vol. 15, No. 2, pp. 174–182, 1984.
- Gabor Szegö. Orthogonal polynomials. American Mathematical Soc., Providence, 1939.
- Quantum amplitude amplification and estimation, 2002.
- Creating superpositions that correspond to efficiently integrable probability distributions. 2002.
- Black-box quantum state preparation without arithmetic. Phys. Rev. Lett., Vol. 122, p. 020502, Jan 2019.
- Preparing arbitrary continuous functions in quantum registers with logarithmic complexity, 2022.
- Efficient quantum circuits for accurate state preparation of smooth, differentiable functions, 2020.
- Quantum state preparation without coherent arithmetic. 2022.
- Linear-depth quantum circuits for loading fourier approximations of arbitrary functions. Quantum Science and Technology, Vol. 9, No. 1, p. 015002, October 2023.
- Efficient state preparation for quantum amplitude estimation. Physical Review Applied, Vol. 15, No. 3, March 2021.
- Quantum pricing with a smile: implementation of local volatility model on quantum computer. EPJ Quantum Technology, Vol. 9, No. 1, p. 7, 2022.
- Quantum risk analysis. npj Quantum Information, Vol. 5, No. 1, feb 2019.
- Option pricing using quantum computers. Quantum, Vol. 4, p. 291, July 2020.
- Optimizing quantum circuits for arithmetic. arXiv preprint arXiv:1805.12445, 2018.
- Edgard Muñoz Coreas and Himanshu Thapliyal. Everything you always wanted to know about quantum circuits. Wiley Encyclopedia of Electrical and Electronics Engineering, pp. 1–17, 2022.
- Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science, Vol. 43, pp. 169–188, 1986.
- Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York, 1968.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.