# Towards Quantum Advantage in Financial Market Risk using Quantum Gradient Algorithms

Nikitas Stamatopoulos1, Guglielmo Mazzola2, Stefan Woerner2, and William J. Zeng1

1Goldman, Sachs & Co., New York, NY
2IBM Quantum, IBM Research – Zurich

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### Abstract

We introduce a quantum algorithm to compute the market risk of financial derivatives. Previous work has shown that quantum amplitude estimation can accelerate derivative pricing quadratically in the target error and we extend this to a quadratic error scaling advantage in market risk computation. We show that employing quantum gradient estimation algorithms can deliver a further quadratic advantage in the number of the associated market sensitivities, usually called $greeks$. By numerically simulating the quantum gradient estimation algorithms on financial derivatives of practical interest, we demonstrate that not only can we successfully estimate the greeks in the examples studied, but that the resource requirements can be significantly lower in practice than what is expected by theoretical complexity bounds. This additional advantage in the computation of financial market risk lowers the estimated logical clock rate required for financial quantum advantage from Chakrabarti et al. [Quantum 5, 463 (2021)] by a factor of ~7, from 50MHz to 7MHz, even for a modest number of greeks by industry standards (four). Moreover, we show that if we have access to enough resources, the quantum algorithm can be parallelized across 60 QPUs, in which case the logical clock rate of each device required to achieve the same overall runtime as the serial execution would be ~100kHz. Throughout this work, we summarize and compare several different combinations of quantum and classical approaches that could be used for computing the market risk of financial derivatives.

Recently, quantum algorithms have been proposed to accelerate the pricing and risk analysis of financial derivatives. These algorithms use quantum amplitude estimation to achieve quadratic advantage compared to the classical Monte Carlo methods that are used in practice for most computationally expensive pricing. Given a desired error $\epsilon$, the quantum advantage stems from the runtime of a classical Monte Carlo simulation scaling as $O(1/\epsilon^2)$ while the quantum algorithms scale as $O(1/\epsilon)$.
A related and important financial application is the computation of the sensitivity of derivative prices to model and market parameters. This amounts to computing gradients of the derivative price with respect to input parameters. A primary business use of calculating these gradients is to enable hedging of the market risk that arises from exposure to derivative contracts. Hedging this risk is of critical importance to financial firms. Gradients of financial derivatives are typically called greeks, as these quantities are commonly labeled using Greek alphabet letters.
In this work, we examine the efficacy of quantum gradient algorithms in the estimation of greeks in a quantum setting. We introduce a method combining gradient algorithms and Maximum Likelihood Estimation (MLE) to estimate the greeks of a path-dependent basket option and show that quantum advantage for calculating risk may be achievable with quantum computers whose clock rates are 7 times slower than that required for pricing itself, indicating another possible avenue for quantum advantage in finance.

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### Cited by

[1] Steven Herbert, "Quantum computing for data-centric engineering and science", Data-Centric Engineering 3, e36 (2022).

[2] Mark-Oliver Wolf, Roman Horsky, and Jonas Koppe, "A Quantum Algorithm for Pricing Asian Options on Valuation Trees", Risks 10 12, 221 (2022).

[3] A. K. Fedorov, N. Gisin, S. M. Beloussov, and A. I. Lvovsky, "Quantum computing at the quantum advantage threshold: a down-to-business review", arXiv:2203.17181.

[4] Peter D. Johnson, Alexander A. Kunitsa, Jérôme F. Gonthier, Maxwell D. Radin, Corneliu Buda, Eric J. Doskocil, Clena M. Abuan, and Jhonathan Romero, "Reducing the cost of energy estimation in the variational quantum eigensolver algorithm with robust amplitude estimation", arXiv:2203.07275.

[5] Gabriele Agliardi, Michele Grossi, Mathieu Pellen, and Enrico Prati, "Quantum integration of elementary particle processes", Physics Letters B 832, 137228 (2022).

[6] João F. Doriguello, Alessandro Luongo, Jinge Bao, Patrick Rebentrost, and Miklos Santha, "Quantum algorithm for stochastic optimal stopping problems with applications in finance", arXiv:2111.15332.

[7] Hao Tang, Wenxun Wu, and Xian-Min Jin, "Quantum Computation for Pricing Caps using the LIBOR Market Model", arXiv:2207.01558.

The above citations are from Crossref's cited-by service (last updated successfully 2022-12-08 06:03:25) and SAO/NASA ADS (last updated successfully 2022-12-08 06:03:26). The list may be incomplete as not all publishers provide suitable and complete citation data.