Financial Relativity: An Information-Geometric Interpretation of Asset Pricing

Paper: arXiv 2604.03961

Authors: Li Lin

Abstract

Classical asset pricing relies on the risk-neutral measure $Q$ for valuation, yet its economic interpretation is typically anchored in a physical measure $P$. This creates an inherent asymmetry: pricing is governed by $Q$, while meaning resides in $P$, making it difficult to provide a unified account of asset pricing within a single conceptual framework. This paper proposes an alternative perspective based on information geometry, termed Financial Relativity. Its central principle is the relativity of probabilistic reference frames: $P$ and $Q$ have no intrinsic hierarchy, but instead represent geometric structures induced by different informational constraints. Terminal structural information shapes probability geometry, which in turn governs how information is expressed in prices. Within this framework, the risk-neutral measure is reinterpreted as a posterior probability geometry. Asset prices are characterized as geometric projections of terminal payoffs onto information subspaces, and their dynamics reflect the progressive manifestation of structural information under evolving geometry. We develop both discrete and continuous financial field equations to describe the formation of probability geometry and derive geodesic price dynamics in which volatility arises endogenously from posterior uncertainty. The framework provides a unified explanation for price fluctuations, event-driven behavior, and risk premia, and yields testable implications, including structural links between volatility and posterior variance and measures of price informational efficiency. By integrating structural information, probability measures, and price dynamics within a single geometric framework, the paper offers a coherent, extensible, and empirically tractable reinterpretation of asset pricing.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a highly novel theoretical framework using information geometry and field equations to unify P and Q measures, backed by dense mathematics. However, it lacks empirical backtesting and real-data validation, remaining largely a theoretical contribution.

Research Flowchart

  flowchart TD
    A[<b>Research Goal</b><br/>Unify physical P and risk-neutral Q measures via Information Geometry] --> B[<b>Core Methodology</b><br/>Treat P & Q as symmetric probabilistic reference frames]
    
    C[<b>Inputs & Constraints</b><br/>Terminal structural info & evolving market filtration] --> D[<b>Computational Engine</b><br/>Derive Discrete & Continuous Financial Field Equations]
    B --> D
    
    D --> E[<b>Dynamic Process</b><br/>Compute posterior probability geometry & geodesic price projections]
    
    E --> F[<b>Key Findings & Outcomes</b><br/>• Endogenous volatility from posterior uncertainty<br/>• Unified explanation for risk premia & event shocks<br/>• Testable measures of price informational efficiency]

Cited References (35)

1. The Mathematics of Arbitrage (2006)
2. Gravitation (1973)
3. The Pricing of Options and Corporate Liabilities (1973) — link
4. Theory of Rational Option Pricing (1973)
5. Martingales and Arbitrage in Multiperiod Securities Markets (1979) — link
6. Martingales and Stochastic Integrals in the Theory of Continuous Trading (1981) — link
7. A Theory of the Term Structure of Interest Rates (1985) — link
8. Asset Prices in an Exchange Economy (1978)

0 of 35 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.