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

By Friedrich Hubalek and Carlo Sgarra
Thiele Research Reports
No. 14, August 2007
We compute resp. 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.
Published in Stochastic Processes and their Applications Volume 119, Issue 7 (July 2009) Pages 2137-2157.
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This publication also serves as Research Reports no. 501