Quadratic Volatility from the P\"oschl-Teller Potential and Hyperbolic Geometry

TL;DR

This paper links a financial model with quadratic volatility to hyperbolic geometry and quantum mechanics, providing a new analytical approach to derivative pricing and market anomalies.

Contribution

It establishes a formal equivalence between the generalized Black-Scholes equation with QNV and the Schrödinger equation for a P"oschl-Teller potential, revealing geometric insights.

Findings

Derived spectral properties of the financial Hamiltonian.

Constructed the pricing kernel from eigenfunctions.

Provided a geometric interpretation of market anomalies.

Abstract

This investigation establishes a formal equivalence between the generalized Black-Scholes equation under a Quadratic Normal Volatility (QNV) specification and the stationary Schr\"odinger equation for a hyperbolic P\"oschl-Teller potential. A sequence of canonical transformations maps the financial pricing operator to a quantum Hamiltonian, revealing the volatility smile as a direct manifestation of diffusion on a hyperbolic manifold whose geometry is classified by the discriminant of the QNV polynomial. We perform a complete spectral analysis of the financial Hamiltonian, deriving its discrete and continuous spectra and constructing the pricing kernel from the resulting eigenfunctions, which are given by classical special functions. This analytical framework, grounded in a gauge-theoretic perspective, furnishes a non-trivial benchmark for derivative pricing and provides a fundamental geometric interpretation of market anomalies. Future research trajectories toward integrable systems and formal field-theoretic analogies are identified.