Paper: arXiv 2609.28863

Authors: Santiago Garcia

Abstract

The analytic tractability of affine pricing models is usually expressed through two complementary formulations: a coordinate-space pricing operator and an exponential-affine transform representation governed by generalized Riccati equations. We develop \emph{Affine Holonomy Group Quantization} (AHGQ) as a geometric framework in which these two formulations arise from the same underlying structure. The construction separates the affine pricing symbol into a homogeneous quadratic sector and a complementary affine sector. The first generates a finite-dimensional symplectic transport and a centrally extended Lie group, while the second is represented by a multiplicative holonomy carried by a thin-path groupoid. Their combination determines an affine Poincaré–Cartan form. Its characteristic dynamics reduce in momentum variables to the generalized Riccati system and its scalar amplitude, whereas the coordinate representation recovers the standard affine pricing operator. Representative Gaussian and square-root models illustrate the construction. The contribution is structural: AHGQ gives a common geometric origin to the coordinate and transform representations of continuous-path, time-homogeneous affine pricing models.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly theoretical and mathematically dense framework for affine pricing models. While offering significant novelty in its geometric approach, it lacks empirical validation or practical implementation details, focusing purely on structural contribution. The clarity is reasonable given the complexity, but reproducibility is limited without code or data.

Research Flowchart

  flowchart TD
    A[Research Goal: Unified Geometric Framework for Affine Pricing Models] --> B{Key Methodology: Affine Holonomy Group Quantization (AHGQ)};
    B --> C[Core AHGQ Construction: Homogeneous Quadratic Sector & Complementary Affine Sector];
    C --> D{Derivations: Symplectic Transport, Centrally Extended Lie Group, Multiplicative Holonomy, Affine Poincaré-Cartan Form};
    D --> E[Outcome 1: Generalized Riccati System (Momentum Representation) & Standard Affine Pricing Operator (Coordinate Representation)];
    E --> F[Outcome 2: Common Geometric Origin for Coordinate & Transform Representations of Affine Pricing Models];