American option of stochastic volatility model with negative Fichera function on degenerate boundary

TL;DR

This paper analyzes American put options under a stochastic volatility model with degenerate boundaries, establishing existence, uniqueness, and properties of the solution using PDE methods and variational inequalities.

Contribution

It introduces a novel framework for American options with stochastic volatility involving degenerate boundaries and proves key properties of the solution.

Findings

Existence of a strong solution to the variational inequality.

Uniqueness of the solution despite degeneracy.

Characterization of the free boundary manifold.

Abstract

In this paper we study a general framework of American put option with stochastic volatility whose value function is associated with a 2-dimensional parabolic variational inequality with degenerate boundaries. We apply PDE methods to analyze the existences of the strong solution and the properties of the 2-dimensional manifold for the free boundary. Thanks to the regularity result on the solution of the underlying PDE, we can also provide the uniqueness of the solution by the argument of the verification theorem together with the generalized Ito's formula even though the solution may not be second order differentiable in the space variable across the free boundary.