Paper: arXiv 2609.28463

Authors: Levin David Schwab

Abstract

In a finite discrete-time market, trading decisions may be predictable with respect to a filtration that does not adapt asset prices. The first fundamental theorem then characterizes absence of arbitrage by measures under which the optional projection of discounted prices is a martingale. We examine the corresponding completeness question. For a fixed projection, every claim measurable with respect to its terminal price history is attainable precisely when all equivalent martingale measures of that fixed process agree on the claim sigma-field. We give the finite-dimensional proof, retaining the distinction between the trading filtration and the claim sigma-field. If the projection is common to all optional martingale measures of the original prices, projected completeness implies uniqueness of their restrictions, but the converse fails even in a three-state model with a unique optional martingale measure. For the binomial model under its product martingale measure, a delay of $k$ periods yields a complete projected market with effective horizon $(T-k)^+$. An explicit replication construction and finite examples distinguish this completeness from replication at the original prices.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly theoretical and mathematically dense exploration of market completeness under restricted information, offering novel insights into optional projections and martingale measures. While strong in mathematical rigor and originality, it lacks empirical validation or backtesting, focusing purely on theoretical constructs and finite-state examples. The writing is clear for its intended expert audience, and the methodology is sufficiently detailed for replication of the theoretical results.

Research Flowchart

  flowchart TD
    A[Research Goal: Market Completeness under Restricted Information] --> B{Methodology: Characterize Arbitrage and Completeness};
    B --> C[Inputs: Finite Discrete-Time Market, Trading Filtration, Asset Prices];
    C --> D{Computational Process: Optional Projection of Discounted Prices};
    D --> E{Analysis: Agreement of Equivalent Martingale Measures on Claim Sigma-field};
    E --> F[Key Finding 1: Claim Attainability if Martingale Measures Agree on Claim Sigma-field];
    F --> G[Key Finding 2: Projected Completeness Implies Uniqueness of Restrictions (but not converse)];
    G --> H[Key Finding 3: Delayed Binomial Model yields Complete Projected Market with Replication];