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

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 Π0T takes the linear form Π0T(X^T)=P0Ttr(q^X^T) for any claim X^T , where P0T 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.