$\alpha$-Hypergeometric Uncertain Volatility Models and their Connection to 2BSDEs

TL;DR

This paper introduces an $ ext{alpha}$-hypergeometric uncertain volatility model, linking it to 2BSDEs, and derives a limit model for worst-case option pricing using asymptotic analysis and deep learning simulations.

Contribution

It establishes a novel connection between $ ext{alpha}$-hypergeometric UV models, G-HJB equations, and 2BSDEs, providing a new approach to worst-case scenario analysis in option pricing.

Findings

Derived a limit model for worst-case option prices with slowly varying bounds.

Connected nonlinear PDEs with 2BSDEs for UV models.

Validated results through deep learning-based numerical simulations.

Abstract

In this article we propose a $\alpha$-hypergeometric model with uncertain volatility (UV) where we derive a worst-case scenario for option pricing. The approach is based on the connexion between a certain class of nonlinear partial differential equations of HJB-type (G-HJB equations), that govern the nonlinear expectation of the UV model and that provide an alternative to the difficult model calibration problem of UV models, and second-order backward stochastic differential equations (2BSDEs). Using asymptotic analysis for the G-HJB equation and the equivalent 2BSDE representation, we derive a limit model that provides an accurate description of the worst-case price scenario in cases when the bounds of the UV model are slowly varying. The analytical results are tested by numerical simulations using a deep learning based approximation of the underlying 2BSDE.