Change of Measure in the Heston Model given a violated Feller Condition

TL;DR

This paper investigates the existence of equivalent martingale measures in the Heston model without assuming the Feller condition, extending the theory to cases with finite Laplace transform lifetime and true martingale conditions.

Contribution

It derives new sufficient conditions for measure change and martingale properties in the Heston model when the Feller condition is violated, filling a key gap in the literature.

Findings

Derived conditions for measure change without Feller condition

Extended analysis to finite lifetime of Laplace transform

Established criteria for stock price being a true martingale

Abstract

When dealing with Heston's stochastic volatility model, the change of measure from the subjective measure P to the objective measure Q is usually investigated under the assumption that the Feller condition is satisfied. This paper closes this gap in the literature by deriving sufficient conditions for the existence of an equivalent (local) martingale measure in the Heston model when the Feller condition is violated. We also supplement the existing literature by the case of a finite lifetime of the Laplace transform of the integrated volatility process. Moreover, we deduce conditions for the stock price process in the Heston model being a true martingale, regardless if the Feller condition is satisfied or not.