Affine Pricing Models from Group Quantization and Holonomy
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.