Variational formulation of American option prices in the Heston Model

TL;DR

This paper provides an analytical framework for pricing American options within Heston models using variational inequalities, addressing existence, uniqueness, and characterization of solutions through advanced mathematical techniques.

Contribution

It extends variational inequality methods to American option pricing in Heston models, incorporating semi-group and affine process techniques for a comprehensive analysis.

Findings

Established existence and uniqueness of weak solutions.

Characterized the option price as an optimal stopping problem.

Utilized semi-group and affine properties for analysis.

Abstract

We give an analytical characterization of the price function of an American option in Heston-type models. Our approach is based on variational inequalities and extends recent results of Daskalopoulos and Feehan (2011). We study the existence and uniqueness of a weak solution of the associated degenerate parabolic obstacle problem. Then, we use suitable estimates on the joint distribution of the log-price process and the volatility process in order to characterize the analytical weak solution as the solution to the optimal stopping problem. We also rely on semi-group techniques and on the affine property of the model.