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# When Finance and Quantum Rub Shoulders

Financial markets are an integral part of the modern economy and our society. Several approaches exist to estimate the fluctuating value of market prices. Those that work well are generally highly complex. Quantum computing offers new tools that could ultimately change this.

Since 2022, a research project involving the UdeS École de gestion, IQ’s AlgoLab and Quebec’s Autorité des marchés financiers (AMF) has been exploring the potential of quantum computing tools that could, among other things, reduce calculation time for metrics useful to risk management in insurance products linked to financial markets, or for assessing the price of certain financial products such as options.

**Options and price estimates**

Options are contracts that give investors the right to buy or sell an underlying asset at a predetermined fixed price within a given timeframe. Options play a crucial role in financial markets, enabling investors to minimize some of their risk, speculate on price movements or manage portfolios more efficiently.

The complexity level of estimate calculation increases significantly when considering an option with several underlying assets, and their fluctuations over time. Financial institutions and academia are increasingly looking to quantum computing to improve this type of computation and potentially speed up stock market trading.

“*Generally speaking*, *prices are a mathematical expectation, which is essentially an integral of a function of the probability distribution. There are quantum algorithms that can theoretically calculate this quickly*,” explains Anne MacKay, Associate Professor at UdeS Department of Mathematics, and the driving force behind this research project along with her colleague, Professor Alain Bélanger. She is particularly interested in the paradigm shift in finance offered by certain quantum computing methods.

**Physics and finance, an unlikely pairing**

While a union between the two fields may seem unlikely, finance and physics have been rubbing shoulders for several decades. “*We often talk about the Black-Scholes model*, says Prof MacKay, *they were the first to use advanced mathematics to solve this type of problem in the 1970s, when they wrote the price dynamics of derivatives in the form of the heat equation, which is a tool of physics. Since then, models have multiplied and become highly sophisticated to better represent market prices. On the other hand, mathematically it’s becoming increasingly difficult to calculate with a [conventional] computer*.”

There are several classic approaches to determine the price of an option. Using random sampling, Monte Carlo simulations are used to model complex, multivariate systems (such as financial derivatives). “*Monte Carlo simulations are slow and not necessarily very efficient*, continues Prof MacKay. *But their big advantage is that they’re flexible, simple and versatile. That’s why they’re so widespread in finance. I believe there are other approaches, particularly with quantum algorithms*.”

**Two quantum tools**

Alexandre Foley, a quantum computer developer with the AlgoLab, also sees this project as an opportunity to apply theoretical methods in a practical way: “*There’s a good match between this type of prediction problem and the quantum computer’s ability to produce probability distributions. It’s possible to translate problems into a quantum mechanics form. Once that’s done, we can solve these in theory with a quantum computer.*”

Thanks to Alexandre’s work, the project implemented two methods for price estimation in a quantum computer. One estimates a weighted average by measuring the probability of a single qubit. The second method focuses on the preparation of a probability distribution in a quantum computer. The approach takes advantage of tensor networks to build an intermediate representation that can easily be converted into a quantum circuit. For the moment, both approaches work for a single variable.

**Starting today to have a better grasp of tomorrow**

For the Autorité des marchés financiers (AMF), this research project is an opportunity to keep abreast of breakthroughs that could have an impact on the world of finance: “*Quantum computers will most likely generate changes in the financial sector, particularly in terms of computer security, calculation algorithms used in risk management and the development of artificial intelligence models*, points out Emmanuel Hamel, data scientist at the AMF. *The Autorité’s participation in this project provides an opportunity to deepen its understanding of quantum computers and better assess their impact, both for financial sector players and for consumers of financial products and services.*

The first part of the project applied a few methods from the literature, in particular the two computational tools developed by the AlgoLab. The project team will now focus on the temporal aspect of state evolution including specific time intervals. In the near future, technical advances could also enable additional variables to be integrated into the quantum circuit, which would make the estimation tool more capable of solving practical problems in finance.

*“One of our aspirations is to put these tools together in an easy-to-use format for people who aren’t quantum computing specialists*, explains Alexandre. *We want it to be useful, and for that we need people to be able to describe their problem, and for the software to be able to translate it in a format that can be processed by a quantum computer to provide a result that they understand without making mistakes*.”

Quantum computing could also be applied in a hybrid manner for specific aspects of a problem in combination with classical approaches: “*In general, that’s what we’re trying to do, which is to use the quantum computer for the part where it’s going to be most effective. It’s a new tool, a bit like machine learning was ten years ago. It’s good to be involved with it right from the start, because it’s going to grow*,” adds Prof. MacKay.

For her, the scientific value of the project is obvious: “*I think that with the help of the AlgoLab programming team, we’ll be able to contribute to improving algorithms and the literature towards the application of quantum computing to finance. For me, contributing to research and the advancement of knowledge is more than enough, and that’s what I want to do*.”