Paper: arXiv 2603.24154

Authors: Daniel Bloch

Abstract

This paper introduces a transformative framework for managing path-dependent financial risk by shifting from traditional distribution-centric models to a geometry-based approach. We propose the SigSwap as a new regulatory instrument that allows market participants to decompose complex risk into terminal price law and the underlying texture of the price path. By utilising the mathematical properties of the path-signature, we demonstrate how previously unmodellable risks, such as lead-lag dynamics and flash-crash spiralling, can be converted into transparent and linear risk factors. Central to this framework is the introduction of Signature Expected Shortfall, a risk metric designed to capture toxic path geometries that traditional methods often overlook. We also present a proactive monitoring system based on the Temporal Exposure Profile, which utilises anticipatory learning to detect potential liquidity traps and geometric decoupling before they manifest as realised volatility. The proposed methodology offers a rigorous alignment with global regulatory mandates, specifically the Fundamental Review of the Trading Book (FRTB), by providing a consistent bridge between physical stress-testing and risk-neutral hedging. Finally, we show that this algebraic approach significantly reduces computational complexity, enabling real-time, high-frequency risk reporting and capital optimisation for the modern financial ecosystem.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a highly novel mathematical framework based on path-signatures to address complex, path-dependent financial risks like lead-lag dynamics and liquidity traps. While the theoretical formulation and mathematical sophistication are exceptionally strong, empirical backtesting and detailed replication data appear limited in this initial conceptual presentation.

Research Flowchart

  flowchart TD
    A[Research Goal: Model & Regulate Path-Dependent Financial Risk] --> B[Input Data: High-Frequency Price Paths & Geometry]
    B --> C[Methodology: Path-Signature Decomposition & SigSwap Instrument]
    C --> D[Computational Engine: Linear Algebra & Anticipatory Learning]
    D --> E[Metrics: Signature Expected Shortfall & Temporal Exposure Profile]
    E --> F[Outcomes: Real-Time FRTB Compliance & Flash-Crash Risk Hedging]