Quantum computational finance: martingale asset pricing for incomplete markets

• URL: http://arxiv.org/abs/2209.08867v1
• Date: Mon, 19 Sep 2022 09:22:01 GMT
• Title: Quantum computational finance: martingale asset pricing for incomplete markets
• Authors: Patrick Rebentrost, Alessandro Luongo, Samuel Bosch, Seth Lloyd
• Abstract summary: We show that a variety of quantum techniques can be applied to the pricing problem in finance. We discuss three different methods that are distinct from previous works.
• Score: 69.73491758935712
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A derivative is a financial security whose value is a function of underlying traded assets and market outcomes. Pricing a financial derivative involves setting up a market model, finding a martingale (``fair game") probability measure for the model from the given asset prices, and using that probability measure to price the derivative. When the number of underlying assets and/or the number of market outcomes in the model is large, pricing can be computationally demanding. We show that a variety of quantum techniques can be applied to the pricing problem in finance, with a particular focus on incomplete markets. We discuss three different methods that are distinct from previous works: they do not use the quantum algorithms for Monte Carlo estimation and they extract the martingale measure from market variables akin to bootstrapping, a common practice among financial institutions. The first two methods are based on a formulation of the pricing problem into a linear program and are using respectively the quantum zero-sum game algorithm and the quantum simplex algorithm as subroutines. For the last algorithm, we formalize a new market assumption milder than market completeness for which quantum linear systems solvers can be applied with the associated potential for large speedups. As a prototype use case, we conduct numerical experiments in the framework of the Black-Scholes-Merton model.

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