Paper: arXiv 2212.13815

Authors: Jinge Bao, Patrick Rebentrost

Abstract

Quantum computers have the potential to provide an advantage for financial pricing problems by the use of quantum estimation. In a broader context, it is reasonable to ask about situations where the market and the assets traded on the market themselves have quantum properties. In this work, we consider a financial setting where instead of by classical probabilities the market is described by a pure quantum state or, more generally, a quantum density operator. This setting naturally leads to a new asset class, which we call quantum assets. Under the assumption that such assets have a price and can be traded, we develop an extended definition of arbitrage to quantify gains without the corresponding risk. Our main result is a quantum version of the first fundamental theorem of asset pricing. If and only if there is no arbitrage, there exists a risk-free density operator under which all assets are martingales. This density operator is used for the pricing of quantum derivatives. To prove the theorem, we study the density operator version of the Radon-Nikodym measure change. We provide examples to illustrate the theory.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly original theoretical extension of the First Fundamental Theorem of Asset Pricing to quantum state spaces using sophisticated functional analysis and operator theory. However, it is purely theoretical and lacks empirical validation, real-world financial data, or concrete market implementations.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Asset Pricing to Quantum Markets] --> B[Input: Market as Quantum Density Operator & Quantum Assets]
    B --> C[Define Quantum Arbitrage & Risk-Free Conditions]
    C --> D[Study Quantum Radon-Nikodym Density Operator Transformation]
    D --> E[Outcome: First Fundamental Theorem of Quantum Asset Pricing]
    E --> F[Applications: Pricing Quantum Derivatives & Illustrative Examples]