Fundamental Theorem of Discrete Asset Pricing

Paper: arXiv 2610.21442

Authors: Masaaki Fukasawa

Abstract

Motivated by discrete price quotes in continuous time, we seek a fundamental theorem based on no arbitrage rather than a condition on limiting trading opportunities. We consider a vector of discounted prices taking values in a locally finite subset of $R^d$ in continuous time. For a locally bounded càdlàg price, we prove that no arbitrage, no free lunch with vanishing risk, no free lunch, and the existence of an equivalent local martingale measure are equivalent. An elementary finite-dimensional reduction of portfolios yields a common pathwise bound on admissible wealth. This bound reduces closedness to convex averaging in a weighted Hilbert space, after which the Kreps-Yan theorem applies. Trading gains are finite Stieltjes sums, and the proof uses no Itô integration.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly theoretical and mathematically dense contribution to the fundamental theorem of asset pricing under discrete price quotes. Its novelty lies in extending the FTAP to a setting without Itô integration, but it lacks any empirical validation or data-driven analysis. The focus is purely on rigorous mathematical proof.

Research Flowchart

  flowchart TD
    A[Research Goal: Fundamental Theorem of Discrete Asset Pricing in Continuous Time] --> B{Key Concepts: Discrete Price Quotes, No Arbitrage};
    B --> C{Methodology: Pathwise Analysis, Finite-Dimensional Reduction, Stieltjes Sums};
    C --> D[Computational Process: Convex Averaging in Weighted Hilbert Space, Kreps-Yan Theorem Application];
    D --> E[Key Findings: Equivalence of No Arbitrage, No Free Lunch, and EMM];
    E --> F[Outcome: Foundation for Discrete Asset Pricing without Itô Calculus];