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Admissible Information Structures and the Non-Existence of Global Martingale Pricing

Alejandro Rodriguez Dominguez

TL;DR

This paper treats the information structure itself as endogenous, constrained by admissibility and time-ordering, and asks whether a single admissible filtration can price all traded assets as martingales. It proves that for any finite asset group a pricing-sufficient filtration exists in canonical minimal form generated by the assets, and that local pricing remains stable under restriction and aggregation when a common pricing measure exists. However, globally with three independent unspanned finite-variation drivers the authors show there is no admissible filtration and equivalent martingale measure that price all assets jointly; this sharp obstruction is equivalent to the failure of admissible dynamic completeness. Numerical diagnostics based on discrete-time Doob–Meyer decompositions illustrate how admissible structures suppress predictable components, while inadmissible enlargements introduce systematic predictability. The results delineate a clear boundary for global martingale pricing in arbitrage theory and connect the obstruction to fundamental completeness properties under endogenous information constraints.

Abstract

No-arbitrage asset pricing characterizes valuation through the existence of equivalent martingale measures relative to a filtration and a class of admissible trading strategies. In practice, pricing is performed across multiple asset classes driven by economic variables that are only partially spanned by traded instruments, raising a structural question: does there exist a single admissible information structure under which all traded assets can be jointly priced as martingales?. We treat the filtration as an endogenous object constrained by admissibility and time-ordering, rather than as an exogenous primitive. For any finite collection of assets, whenever martingale pricing is feasible under some admissible filtration, it is already feasible under a canonical minimal filtration generated by the asset prices themselves; these pricing-sufficient filtrations are unique up to null sets and stable under restriction and aggregation when a common pricing measure exists. Our main result shows that this local compatibility does not extend globally: with three independent unspanned finite-variation drivers, there need not exist any admissible filtration and equivalent measure under which all assets are jointly martingales. The obstruction is sharp (absent with one driver and compatible pairwise with two) and equivalent to failure of admissible dynamic completeness. We complement the theory with numerical diagnostics based on discrete-time Doob--Meyer decompositions, illustrating how admissible information structures suppress predictable components, while inadmissible filtrations generate systematic predictability.

Admissible Information Structures and the Non-Existence of Global Martingale Pricing

TL;DR

This paper treats the information structure itself as endogenous, constrained by admissibility and time-ordering, and asks whether a single admissible filtration can price all traded assets as martingales. It proves that for any finite asset group a pricing-sufficient filtration exists in canonical minimal form generated by the assets, and that local pricing remains stable under restriction and aggregation when a common pricing measure exists. However, globally with three independent unspanned finite-variation drivers the authors show there is no admissible filtration and equivalent martingale measure that price all assets jointly; this sharp obstruction is equivalent to the failure of admissible dynamic completeness. Numerical diagnostics based on discrete-time Doob–Meyer decompositions illustrate how admissible structures suppress predictable components, while inadmissible enlargements introduce systematic predictability. The results delineate a clear boundary for global martingale pricing in arbitrage theory and connect the obstruction to fundamental completeness properties under endogenous information constraints.

Abstract

No-arbitrage asset pricing characterizes valuation through the existence of equivalent martingale measures relative to a filtration and a class of admissible trading strategies. In practice, pricing is performed across multiple asset classes driven by economic variables that are only partially spanned by traded instruments, raising a structural question: does there exist a single admissible information structure under which all traded assets can be jointly priced as martingales?. We treat the filtration as an endogenous object constrained by admissibility and time-ordering, rather than as an exogenous primitive. For any finite collection of assets, whenever martingale pricing is feasible under some admissible filtration, it is already feasible under a canonical minimal filtration generated by the asset prices themselves; these pricing-sufficient filtrations are unique up to null sets and stable under restriction and aggregation when a common pricing measure exists. Our main result shows that this local compatibility does not extend globally: with three independent unspanned finite-variation drivers, there need not exist any admissible filtration and equivalent measure under which all assets are jointly martingales. The obstruction is sharp (absent with one driver and compatible pairwise with two) and equivalent to failure of admissible dynamic completeness. We complement the theory with numerical diagnostics based on discrete-time Doob--Meyer decompositions, illustrating how admissible information structures suppress predictable components, while inadmissible filtrations generate systematic predictability.
Paper Structure (47 sections, 25 theorems, 19 equations, 2 figures, 2 tables)

This paper contains 47 sections, 25 theorems, 19 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Main Results
  4. Information Structures, Admissibility, and Martingale Pricing
  5. Local Existence and Stability of Pricing-Sufficient Filtrations
  6. Time-Scale Separation, Unspanned Drivers, and Admissibility
  7. A Minimal Obstruction to Global Martingale Pricing
  8. Dynamic Completeness and Global Martingale Pricing
  9. Numerical Illustrations
  10. Conclusions and Future Work
  11. Limitations and scope
  12. Existence and Minimality of Pricing-Sufficient Filtrations
  13. Construction of the minimal pricing-sufficient filtration
  14. Sufficiency and minimality
  15. Uniqueness
  16. ...and 32 more sections

Key Result

Proposition 3.3

Let $\mathbb{H}$ be an admissible filtration and let $\mathbb{Q}\sim\mathbb{P}$ be such that $S^A$ is a local $(\mathbb{Q},\mathbb{H})$-martingale. Then $S^A$ is a local $(\mathbb{Q},\mathbb{F}^{S^A,\mathbb{H}})$-martingale.

Figures (2)

  • Figure 1: Predictable finite-variation paths $A_t$ for asset $S_1$ under different filtrations.
  • Figure 2: Distribution of estimated conditional mean increments $\hat{m}_t$ for asset $S_1$.

Theorems & Definitions (61)

  • Remark 3.1: Asset groups and martingale pricing
  • Remark 3.2: Role of admissibility
  • Proposition 3.3: Canonical local reduction of information
  • Proof 1
  • Definition 3.4: Locally pricing-feasible filtration
  • Proposition 3.5: Stability under restriction
  • Proof 2
  • Proposition 3.6: Pairwise compatibility under a common pricing measure
  • Proof 3
  • Definition 3.7: Unspanned finite-variation drivers
  • ...and 51 more