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 $\hat p$. 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, each such claim is represented by an observable $\hat X_T$ where the eigenvalues of $\hat X_T$ determine the amount paid if the corresponding outcome is obtained in the measurement. We prove, under reasonable axioms, that there exists a pricing state $\hat q$ which is equivalent to the physical state $\hat p$ on null spaces such that the pricing function $\Pi_{0T}$ takes the form $\Pi_{0T}(\hat X_T) = P_{0T}\,{\rm tr} ( \hat q \hat X_T) $ for any claim $\hat X_T$, where $P_{0T}$ is the one-period discount factor. By "equivalent" we mean that $\hat p$ and $\hat q$ share the same null space: thus, for any $|\xi \rangle \in \mathcal H$ one has $\langle \bar \xi | \hat p | \xi \rangle = 0$ if and only if $\langle \bar \xi | \hat q | \xi \rangle = 0$. 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 such a setting and we look at the problem of forming portfolios of such contracts. Finally, we consider multi-period contracts.