On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

TL;DR

This paper analyzes various martingale measures, including Esscher transforms and minimal entropy measures, for Barndorff-Nielsen and Shephard stochastic volatility models with jumps, highlighting differences in models with and without leverage.

Contribution

It provides a comprehensive comparison of multiple martingale measures for these stochastic volatility models, including explicit results and examples.

Findings

Different measures in models with leverage and jumps

Coincidence of measures in models without leverage

Explicit parametric examples illustrating the results

Abstract

We compute and discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of Ornstein-Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show, that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous return process, several measures coincide, some simplifications can be made and the results are more explicit. We illustrate our results with parametric examples used in the literature.