On American VIX options under the generalized 3/2 and 1/2 models

TL;DR

This paper extends the 3/2 and 1/2 models for VIX volatility, deriving new pricing formulas and optimal exercise boundaries for American VIX options using advanced mathematical techniques.

Contribution

It introduces generalized 3/2 and 1/2 volatility models, providing new pricing formulas and characterizations for American VIX options with free-boundary and integral equation methods.

Findings

Derived early exercise premium representations for American VIX options.

Established existence and uniqueness of optimal exercise boundaries.

Formulated integral equations characterizing the boundaries.

Abstract

In this paper, we extend the 3/2-model for VIX studied by Goard and Mazur (2013) and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options and, for the latter, we obtain an early exercise premium representation using a free-boundary approach and local time-space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We also consider a model mixing these two classes and formulate the corresponding optimal stopping problem in terms of the observed factor process. The price of an American VIX call is then represented by an early exercise premium formula. We show the existence of a pair of optimal exercise boundaries for the factor process and characterize them as the unique solution to a system of integral equations.