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 ${\Pi }_{0T}({\hat{X}}_T)=P_{0T}\mathrm{tr}(\hat{q}{\hat{X}}_T)$ 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 $|\xi ⟩\in \mathcal{H}$ one has $⟨\bar{\xi }|\hat{p}|\xi ⟩=0$ if and only if $⟨\bar{\xi }|\hat{q}|\xi ⟩=0$. 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.

27 pages, 1 figure