Valuation of a Financial Claim Contingent on the Outcome of a Quantum Measurement

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Hughston, L P and Sánchez-Betancourt, L. 2024. Valuation of a Financial Claim Contingent on the Outcome of a Quantum Measurement. Journal of Physics A: Mathematical and Theoretical, pp. 1-30. ISSN 1751-8113 [Article] (In Press)

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Hughston+et+al_2024_J._Phys._A__Math._Theor._10.1088_1751-8121_ad4cab.pdf - Accepted Version
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Official URL: https://doi.org/10.1088/1751-8121/ad4cab

Abstract or Description

We consider a rational agent who at time 0 enters into a ﬁnancial contract for which the payout is determined by a quantum measurement at some time T > 0. The state of the quantum system is given in the Heisenberg representation by a known density matrix p. How much will the agent be willing to pay at time 0 to enter into such a contract? In the case of a ﬁnite dimensional Hilbert space H, each such claim is represented by an observable X where the eigenvalues of X determine the amount paid if the corresponding outcome is obtained in the measurement. We prove, under reasonable axioms, that there exists a pricing state q which is equivalent to the physical state p such that the pricing function Π takes the linear form Π (X) = P_0T tr(q X) for any claim X, where P_0T is the one-period discount factor. By "equivalent" we mean that p and q share the same null space: thus, for any |ξ〉∈ H one has p|ξ〉 = 0 if and only if q|ξ〉 = 0. We introduce a class of optimization problems and solve for the optimal contract payout structure for a claim based on a given measurement. Then we consider the implications of the Kochen-Specker theorem in this setting and we look at the problem of forming portfolios of such contracts. Finally, we consider multi-period contracts.

Item Type:

Article

Identification Number (DOI):

https://doi.org/10.1088/1751-8121/ad4cab

Additional Information:

This is the Accepted Manuscript version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record will be available online at DOI 10.1088/1751-8121/ad4cab

Keywords:

Quantum mechanics, quantum measurement, contingent claims, discount bonds, absence of arbitrage, rate of return, density matrices, Gleason’s theorem, Kochen-Specker theorem

Departments, Centres and Research Units:

Computing

Dates: