Time-changed \levy processes and option pricing: a critical comment

TL;DR

This paper critically examines Carr and Wu's framework for option pricing, revealing that their proposed models for time changes do not satisfy the necessary measurability assumptions, challenging its general applicability.

Contribution

The paper identifies a fundamental flaw in Carr and Wu's framework by analyzing the measurability of their proposed time changes, questioning its validity.

Findings

All proposed models for time changes lack the required measurability.

The framework's assumption of the stopping time property is invalid for these models.

This challenges the universality of Carr and Wu's approach in option pricing.

Abstract

Carr and Wu (2004), henceforth CW, developed a framework that encompasses almost all of the continuous-time models proposed in the option pricing literature. Their framework hinges on the stopping time property of the time changes. By analyzing the measurability of the time changes with respect to the underlying filtration, we show that all models CW proposed for the time changes fail to satisfy this assumption.