Paper: arXiv 2305.10239

Authors: Lane P. Hughston, Leandro Sánchez-Betancourt

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 $Π_{0T}$ takes the form $Π_{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 $|ξ\rangle \in \mathcal H$ one has $\langle \bar ξ| \hat p | ξ\rangle = 0$ if and only if $\langle \bar ξ| \hat q | ξ\rangle = 0$. 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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents an intriguing and highly novel theoretical framework extending asset pricing to quantum measurement outcomes using advanced Hilbert space operator algebra and quantum mechanics theorems. However, it is purely theoretical with no empirical data or backtesting, resulting in low empirical rigor despite the high mathematical sophistication.

Research Flowchart

  flowchart TD
    A[Research Goal: Value a financial claim based on quantum measurement outcome at time T] --> B[Inputs: Known physical density matrix ρ, observable X_T, discount factor P_0T]
    B --> C[Axiomatic Derivation: Establish pricing function Π_0T]
    C --> D[Key Finding 1: Unique pricing state q equivalent to ρ on null spaces]
    D --> E[Pricing Formula: Π_0T = P_0T * tr_q_X_T]
    E --> F[Contract & Portfolio Optimization: Solve payout structure & analyze Kochen-Specker implications]
    F --> G[Extension: Generalize to multi-period quantum financial contracts]