Valuation of a financial claim contingent on the outcome of a quantum measurement

Abstract

Abstract 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 ^ T where the eigenvalues of X ^ T 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 0 T takes the linear form Π 0 T ( X ^ T ) = P 0 T tr ( q ^ X ^ T ) for any claim X ^ T , where P 0 T is the one-period discount factor. By ‘equivalent’ we mean that p ^ and q ^ share the same null space: that is, 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.