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, 57(28), 285302. ISSN 1751-8113 [Article]

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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

Data Access Statement:

No new data were created or analysed in this study.

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: