🤖 AI Summary
This work provides a geometric interpretation of the Heston stochastic volatility model, uncovering the intrinsic mechanism underlying its affine structure. By constructing an augmented local Lie groupoid formulation and introducing, for the first time, group quantization techniques into this framework, the study unifies the pricing operator in coordinate space with the Riccati equations in momentum space. Leveraging the Mellin transform together with geometric representation theory, the approach not only reproduces the classical characteristic function and Riccati solutions but also demonstrates that both arise as dual perspectives of a single geometric construction. This insight endows the Heston pricing formula with a clear and coherent geometric meaning.
📝 Abstract
We construct a lifted local Lie groupoid formulation of the Heston stochastic-volatility model and use it to give a geometric interpretation of its affine-transform structure. The construction extends the Group Quantization framework previoulsy applied to quadratic financial diffusion models.
The purpose of this paper is not to propose a new Heston pricing formula. The contribution is geometric: the Heston pricing operator in coordinate space and the Riccati equations in momentum space arise from two representations of the same lifted local groupoid construction. The usual characteristic-function and Riccati formulas are recovered.